Embed Size (px)
Citation preview
ED 215 859
AUTHORTITLEINSTITUTION
SPONS AGENCYPUB DATECONTRACTNOTEAVAILABLE FROM
EDRS PRICEDESCRIPTORS
DOCUMENT RESUME
SE 036 395
Higgins, Jon L.; Kirschner, VickyCalculators, Computers, and Classrooms.ERIC Clearinghouse for Science, Mathematics, andEnvironmental Education, Columbus, Ohio.National Inst. of Education (ED), Washington, DC.Dec 81400-78-0004196p.Information Reference Center (ERIC/IRC), The OhioState Univ., 1200 Chambers Rd., 3rd Floor, Columbus,OH 43212 ($6.00).
MF.01/PC08 Plus Postage.*Calculators; Computer Assisted Instruction;*Computer Literacy; Elementary School Mathematics;*Elementary Secondary Education; *MathematicsCurriculum; Mathematics Education; MathematicsInstruction; *Microcomputers; *Problem Solving;Secondary School Mathematics; Simulation
ABSTRACTSuggestions for using four-function calculators, ,
programmable calculators, and microcomputers are considered in thiscollection of 36 articles. The first section contains articlesconsidering general implications for mathematics curricula implied bythe freedom calculators offer students from routine computation,enabling them to focus on results and relationships, and is balancedby Section Two, exploring inappropriate ways calculators can be used.Freedom from thinking about routine calculations provides freedom forthinking about problem solving is the theme of Section Three.Articles in Section Four include some specific lesson ideas for usingcalculators in the classroom. Section Five focuses on programmablecalculators. Section Six contains articles which consider ways in
which microcomputers can be introduced into schools, addressingphysical, economic, and political issues. Section Seven exploresimplications of the computer on mathematics curricula, consideringboth new:topics and new approaches to old topics (such as computerassisted instruction). Computer literacy is the theme of SectionEight, suggesting that although all students need to know aboutcomputers, "what" they need to know is debatable. The ability tosimulate real-world events (computer simulations) is considered inthe final section, suggesting that this ability opens new areas for
mathematical exploration. (Author/JN)
**********************u************************************************Reproductions supplied by EDRS are the best that can be made
from the original document.************************t**********************************************
0
CALCULATORS, COMPUTERS, AND CLASSROOMS
U.S. DEPARTMENT OF EDUCATIONNATIONAL INSTITUTE OF EDUCATION
EDUCATIONAL RESOURCES INFORMATIONCENTER (ERIC)
This document has been reproduced asreceived from the person or organizationoriginating it
r,- Minor changes have been made to improvereproduction quality
' Points of view or opinions stated in this documerit do not necessarily represent official NIE
Positron or policy
"PERMISSION TO REPRODUCE THISMATERIAL HAS BEEN GRANTED BY
er4 (A),owe
TO THE EDUCATIONAL RESOURCESINFORMATION CENTER (ERIC)."
0
ERIC
BY
JON L. HIGGINS!;.
AID
VICKY KIRSCHNER
CALCULATORS, COMPUTERS,
AND
CLASSROOMS
CLEARINGHOUSE FOR SCIENCE, MATHEMATICS,AND ENVIRONMENTAL EDUCATIONTHE OHIO STATE UNIVERSITY
1200 CHAMBERS ROAD -: THIRD FLOORCOLUMBUS, OHIO 43212
DECEMBER, 1981
Thii publication was prepared with funding from the NationalInstitute of Education, U.S. Department of Education undercontract no. 400-78-0004. The opinions expressed in thisreport do not necessarily reItect the positions or policies ofNIE or U:S. Department of Education.
Introduction
CALCULATORS, COMPUTERS, AND CLASROOMS
Table of Contents
Section I: Calculators and Mathematics Curriculum
_Wheatley, Grayson. Calculators in .he Classroom: A Proposal
for Curricular Change. AERA Paper, 1979. ERIC:
ED 175 631 3
Girling, Michael. Towards a Definition of Basic Numeracy.
Mathematics Teaching 81: 4-5; Deceiaber 1977 11
vii
1
Section II: Calculator Cautions a3
Reys, Robert E. Calculators in the Elementary Classroom: How
Can We Go Wrong! Arithmetic Teacher 28: 38-40; November
1980
Johnson, David C. Calculators: Abuses and Uses. Mathematics
Teaching 85: 50-56; December 1978
Kessner, Arthur and Slesnick, Twila. Myths About Calculators
in the Schools. Calculators/Computers 2: 78-81;
September/OctOber 1978 27
Section III: Calculators and Problem-Solving 31
Rogers, Jean B. Using Calculators for Prdblem Solving..
Calculators/Computers 2: 19-21; March 1918 33
Shields, Joseph J. Mini-Calculators and Problem Solving.School Science and Mathematics 80: 211-217; March 1980 37
15
19
Aidala, Gregory. Calculators: Their Use in the Classroom.
School Science and Mathematics 78: 307-311; April 1978 45
Section IV: Usine_Caldulators:' Suggestions from Surveys,
Research and Cl4ssrooth Trials
Vannatta, Glen, D. and Hutton, Lucreda A. A Case for the
Calculator. Arithmetic Teacher 27: 30-31; May 1980
Reys, Robert E.; Bestgen, Barbara J.; Rybolt, James F.; andWyatt, J. Wendell. s Hand Calculators: What's Happening
in Schools Today? Arithmetic Teacher 27: 38-43;
February, 1980
51
53
55
Etlinger, Leonard E. and Ogletree, Earl J. Calculators in
the,,,Elementary School: A Survey of How and Why. Chicago:
Chicago, State University, (1980). ERIC: ED 191 741
Lowerre, George V.; Scandura, Alice M.; Scandura, Joseph M.;
and Veneski, Jacqueline. Using Electronic Calculators with
Third and Fourth Graders: A Feasibility Stub. School
Science and Mathematics 78: 461-464; October 1978
Zakariya, Norma; McClung, Margo; and Winner, Alice-Ann. The
Calculator' in the Classroom. Arithmetic Teacher 27:'
12-16; March 1980"*
Meyer, Phillis I. When You Use a Calculator You Have to
Think! Arithmetic Teacher 27: 18-21; January 1980
Lichtenberg, Betty K. Calculators for Xids Who Can't
Calculate. School Science and Mathematics 81: 97-102;
February 1981
Davies, Brian. Using Calculators with Juniors. Mathematics
Teaching 93: 12-15; December 1980
Wheatley, Charlotte L. Calculator Use in the Middle Grades.
School Science and Mathematics 80: 620-624; November
1980
61
69
73
79
83
89
93
Woodward, Ernest and Hamel, Thomas. Calculator'Lessons
Involving Population, Inflation, and Energy. Mathematics
Teacher 72: 450-457; September 197999
Section V: Programmable Calculators107
Edinger; Leonard E.; Krull, Sarah; Sachs, Jerry; and Stolarz,
Theodore J. The Calculator in the Classroom: Revolution
or Revelation? Chicago: Chicago State University, (1980).
ERIC: ED 191 740109
Stolarz, Theodore J. The Programmable Calculator in the
Classroom. Chicago: Chicago State University, (1980)
ERIC: ED 193 026
Wavrik, John J. The Case for Programmable Calculators
Calculators/Computers 2: 63; April 1978
Maor, Eli. DA Summer Course with the TI 57 Programmable
Calculator. Mathematics Teacher 73: 99-106; February
1950
Section VI: Computers for Schools
Bell, Fred. Classroom Computers: Beyond the 3 R's. Creative
Computinj 5: 68-70; September 1979
117
125
127
135
137
O
0
ti
Robbins, Bill and Taylor, Ross. Getting Started in a Junior
High School: A Case Study. Mathematics Teacher 74:
605-608; November 1981 141
Gesshel-Green, Herb. Getting Started in a High School: A
Case Study. Mathematics Teacher 74: 610-612; November
1981145
Section VII: Computers and the Mathematics Curriculum 149
Wiebe, James H. BASIC Programming for Gifted Elementary
Students. Arithmetic Teacher 28: 42-44; March 1981 151
Hatfield, Larry. L. A Case and Techniques for Computers: Using
Computers in Middle School Mathematics. Arithmetic Teacher
26: 53 -55; February 1979 155
Norris, Donald O. Let's Put Computers into the Mathematics
Curriculum: Mathematics Teacher 74: 24-26; January 1981 159
Hutcheson, James W. Computer-Assisted Instruction is Not
Always Drill. Mathematics Teacher 73: 689-691, 715;
December 1980 163
Section VIII: Computer Literacy 167
Johnson, ,David C.; Anderson, Ronald E.; Hansen, Thomas P.; and
Klassen, Daniel L. Computer Literacy--What Is It?
Mathematics Teacher 73: 91-96; February 1980 169
Luehrmann, Arthur. Computer Literacy--What Should It Be?
Mathematics Teacher 74: 682-686; December 1981 175
Anderson, Ronald E.; Klassen, Daniel L; and Johnson, David C.In Defense of a Corehensive View of Computer Literacy- -A Reply to Luehrmann. Mathematics Teacher 74: 687-690;
December 1981 181
Lawson, Harold W., Jr. Explaining Computer Related Concepts &
Terminology. Creative Computing 7: 92-102; October 1981 185
Cupertino Union School District. Cupertino School District
Develops Computer Literacy Curriculum. Computing Teacher
9: 27-34; September 1981 193
k'N)
6
Section IX: Computer Simulations in Mathematics 201
Flake, Janice L. _Computer Simulations in Mathematics
Education. Contemporary Education. 47: 44-48; February
1975
Lappan, Glenda and Winter,- M. J. Probability Simulation in
Middle School. Mathematics, eacher 73: 446-449;
September 1980
r
7
203
209
CALCULATORS, COMPUTERS, AND CLASSROOMS
Introduction
The developers of electronic chip technology may have dreamed about
revolutionizing the industrial world, but it is doubtful that they realized
0
the extent to which this technology would revolutionize the educational
world.. Nowhere does that revolution have more potential for impact than,in
mathematics classrooms from kindergarten to college. Paradoxically, these
silicon-based chips free us from routine thinking and stretch our minds to
new thinking patterns at the same time. This collection 'of readings explores
that paradox and points the way to the mathematics curriculum-of the c.uture.
The book begins by considering simple hand-held calculators, proceeds
to investigate programmable calculators, and concludes with an examination
of the educational implication of microcomputers. The distinction between
these three is often fuzzy and hard to discern. The reader should not be5
overly concerned about forming fixed categories. Microcomputers do basic
four-function arithmetic just as hand-held calculators do. But both pro-
grammible,calculators and microcomputers do more than calculators. Calcula-
tors add, subtract, multiply, and divide. They may raise to powers, take
roots, find logarithms, and calculate trigonometric functions. To these
feats, the programmable calcuL-;or adds one rore: it can test an equality
or inequality to determine if is true or false. This is the basic step
for logical prnramMing and decision-making. To this extent, the program-
mable calculator is essentially a computer. But a microcomputer usually can
do one additional thing: it can operate with a programming language, using
numbers, letters, and/or symbols as commands as well as elements to operate
4' If there is a logical progression between these machines, there is
0
probably a logical progression (in a similar mannep) in the way the machines
may be usedsin classrooms. The Calculator can free students from routine
computation and enable them to focus upon results and relationships. The '
programmable calculatol-adds the aspect of logical testing and sequencing0
to the realm of study. And finally, the microcomputer expands our capabili-41
ties to give instructions and to study the instruction-giving process. Iron-
ically, this is a return to the idea of algorithm -- but algorithm on a
thinking, analytical level rather than on a routine, mechanical level. Thus,
the silicon chip both frees us from thinking and stretches our thinking re-
O
quirements at the same time.
The first section of this hook considers general implications for the,
mathematics curriculum implied by this calculator freedom. It is balanced
by Section Two exploring inappropriate ways in which c.21culaors can be
used. Freedom from thinking about routine calculations provides freedom
for thinking about problem-solving, the theme of Section Three. Section Four
concludes the consideration of calculators by including some specific lesson
ideas for using calculators in classrooms.
Section Five tontiders programmable calculators. Just as the program-
mable calculator is a bridge between calculators and computdrs, this-section,
is a bridge between the initial and final sections of the book.
Section Six considers ways in which computers can be introduced into
schools. As such, it is concerned with physical, economic, and even polit-
ical considerations. The following section (Section Seven) explores impli-
cations of the computer for the mathematics curriculum,oand considers not
only new topics, but new approaches to old topics as well. Computer literacy
is the theme of Section Eight. It is generally agreed that all students need
to know about computers; it is not generally agreed just what they. need
A
know in order to function in tomorrow's society. This section explores
questions of "need to know" and "nice to know." The book concludes with a
hint of the possible range of the use of, computers: the ability to simulate
real-world events: This ability to mimic change-and the change-process
opens new areas for mathematical exploration and makes them accessible to
all classrooms.
This book is a glimpse at the future. Lie.S quick glance, it may be
incomplete and perhaps even distorted in parts. But we think it is an ex-
citing glimpse, and we hope you will-share that excitement with ds.
= Jon L. Higgins
Vicky Kirschner
PEditors
2
ere
10
--P
. Section I
CALCULATORS
and
MATHEMATICS
CURRICULUM
a
tz.
CALCULATORS IN THE CLASSROOM:
A PROPOSAL '?OR CURRICULAR CHANGE*
GRAYSON H. WHEATLEY
PURDUE UNIVERSITY
O
Q
c
*Paper presented as part .of a Symposium "The EffeCts of Calculator
Availability on School Mathematics Curripulum at the Annual Meeting of the
American Educational Research Association, San'Francisco, April 9, 1979.
4
a
4
0
4
CALCULATORS IN THE - CLASSROOM:
A PROPOSAL FOR CURRICULAR CHANGE*
Historical Perspective
A careful study of the history of mathematics educatior will reveal that
computation has always' been the focus of the elementary school mathematics
curriculum: In the 18th century, children were taught ciphering. Rote
computation was taught without any attempt to develop an understanding of
the process. During the 19th century a few persons, like Warren Coburn,
called for attention to meaning but the curriculum remained computational.
During the 1930s there was a movement toward social utility and developing
meaning in mathematics. Then in the period from 1958 to 1971 there was an
emphasis on teaching the structure of mathematics. Viewed from the
perspective of today, there was one unfortOate aspect of theso called
"modern mathematics" movement. Much attention was given to rationalizing
Algorithms.. Each of the complex,algorithms (e.g.,-the division algorithm)
was taught in great detail using a subtracting approach so that students
would understand why the algorithm worked. But computational algorithms
are nothing more than a set of rules for efficient paper and pencil
answer-finding.
While it is important for students to understand the mathematics they are
learning, time spent teaching children why algorithms work does little to
help pupils understand mathematics and its applications. Vestiges of this
unfortunate curriculum trend remain in. current textbooks. Since 1971 there
hasbeen a well-defined swing toward the "Basics." In elementary school
classrooms, this has been interpreted as more emphasis and time teaching
basic facts and computation with even less time for cdticepts and
applications.
Throughout the eras described above, the curriculum has remained
d'auputational. There is great contrast between the recommendations of
leading mathematics educators and practices in the classroom.
At the same time this country was experiencing a Back-to-Basics movement in
the 70s, aerances,in electronic technology were reshaping the mathematical
needs di! society. Today the availability of inexpensive calculators has
eliminated the need for complex computational algorithms. Before
calculators (B.C.) it was necessary to be proficient in computations to
apply mathematics. That hds now changed. Cash registers show the amount of
change and nearly every home and office has a calculator which can perform
complex computations rapidly. It is quite ironic that as the calculator
availability was reducing the need for computation, schools were increasing
their emphasis ,on paper and pencil computation proficiency.
Current Status of the Elementary School Mathematics Curriculum0
A study of'currently available elementary school mathematics texts suggests
that the curriculum is computationallyoriented. A typical fifth grade
text devotes 139 pages (37%) to computation. However, tdaaters do not
teach all the chapters. For a number of reasons decisions are made to omit
certain chapters. Most classes'cannot finish the text in the time alloted
for mathematics. In practice,'the chapters omitted are usually
noncomputational. Chapteri covering geometry, measurement, probability.,
.13
statistics, and problem-solving are first targets for omission. The
highest priority is given to teaching computation. In fact, while certain
topics are omitted, daily work is supplemented with additional
computational worksheets. In many classes, computation completely
dominates the year's work. My experience and that of my colleaguesobserving in schools almost daily for an entire school year suggests chat
computation is the major emphasis in nearly every classroom. Calls for
accountability and minimal competency testing have increased the time spent
teaching computation since parents and school boards want to be sure that
children know their "Basics." The newer texts reflect this trend.
The Proposal
1. Shift from a computation-based curriculum to a conceptually oriented
curriculum utilizing the calculator as an instructional tool.
2. Eliminate the teaching of complex computations in the elementary
school.
Rationale
1. We cannot afford the cost in time of teaching .complex computations.
Through surveying teachers, observing, and analyzing textbooks, I nave
concluded that of the first nine school years in studying mathematics
more than two years are devoted to teaching the division algotithm.
When the level of proficiency attained by the pupils is considered,
along with this information, I conclude that we can no longer afford to
teach complex long division. An analysis of other computationalprocedures will reveal that the cost benefit ratio is far too high. The
time saved by dropping this topic could be devoted to applications and
problem-solving where the focus would be on when to divide, not how to
divide.
2. Complex computations by paper and pencil are no longer necess :ry.
3. A computationally oriented curriculum inhibits the development of
problem solving. A heavy emphasis on algorithmic thinking tends to
encourage the application of rule-seeking behavior, even when it is not
appropriate. Successful problem - solvers need to try a variety of
approaches, apply heuristics, and recognize that more than rule
identification is required. With a calculator to perform computations,
pupils are freed to focus on the problem rather than_ being distracted
by switching to algorithmic thinking to compute.
4. The NAEP data suggest that 13-year-olds are not proficient in complex
compu*ations, even with all the time devoted. An examination of the
National Assessment data (Carpenter et al., 1978) reveals that less
than two-thirds of the 13-year-olds could perform a long division
problem (three digit by two digit). Only 45% of the 13-year-olds
could solve a single-step word problem requiring single digit
division.'
14
5. Other topics are of more importance. In order to use mathematics
meaningfully in today's society an understanding of certain othertopics is important. More attention should be given to estimation,interpreting data, geometry, measurement, and the use of decimals and
percents. Time will be available for these important topics only ifcomputation is deemphasized.
Recommendations for Curricular Mange
Specifically, I am proposing that the following complex computations beexcised from the curriculum.
I. Division with two or more digits in the divisor (e.g., 37.596).
This copic consumes more time than any other single computational topicin the elementary school curriculum. On the other hand, the singledigit divisor algorithm is relatively easy to teach and could be
retained. It is critically important that pupils learn the meaning of
division. This can certainly be accomplished with single digit divisor
computation. While I am recommending that calculators be used toperform complex divisions, attention should be given to interpretingquotients and estimating results. It is essential that children
attach meaning to their calculator results. Calculator activities have
been published which focus on this objective (Vervoot and Mason, 1977,and Reys, et al., 1979a&b).
462
2. Multiplication by two or more digits (e.g., x 89 ). Children should
gain proficiency-with,problems such as
326 39000
x 8 and x 4 .
3. Addition of fractional numbers with unlike denominators. This topic
has proven to be one the most difficult and thus timeconsuming topicsin the curriculum. The difficlty may result from the cognitivedemands of the task. Addition of fractional numbers requires formalthought and most elementary school pupils have not reached this
cognitive level. This skill can probably be taught more effidiently if
begun in the junior high school.
4. Complex computations with decimals. Such tasks as 3.45 x .865 or
456.78 4. 6.7 are more appropriately performed on a calculator.
What Should We Teach?
I will not attempt to describe a new curriculum in detail but instead makecertain clarifying points related to the proposal.
I. Insist on mastery of basic facts.
It is,important that children memorize the addition and multiplication
1 5.
facts. As Professor Schoen has shown, the calculator can be quite
useful in teaching the basic facts.
2. Stress the concepts of addition, subtraction, multiplication, and
division. It will be easier to do this if less emphasia is placed on
computation.
3. Stres estimation and mental arithmetic.
With calculators used for complex computations there is much importance
in children being able to estimate the answer, and to judge whether the
result makes sense. 'Along with this, 'more time teaching mental
arithmetic would be quite useful.
4. Emphasize applications.
Mathematics is studied in large part because of its utility, yet the
curriculum has not included much attention to applications for two
reasons. We have reasoned that pupils need to learn to compute before
they can apply mathematics and time has not been available because of
the emphasis on computation. This must change.
5. Emphasize decimals.
6. Be sure that such topics as measurement, statistics, probability, and
problem-solving are taught.
These topics have appeared in elementary school texts but have been
given low priority. They should be considered as basic.
7. Include computer literacy.
Since computers play such a central role in today's society, children
should begin to learn about them at an early age.
The NCSM list of 10 basics (1977) should become the foundation for
curriculum development. Number five.on the NCSM list is "appropriate
computational skill." This paper suggests one interpretation of this
statement.
The Role of the Calculator
Although this proposal seems to stress the calculator as a substitute for
paper and pencil computation, the-calculator is actually a highly versatile
instructional aid. A calculator can be USed in concept development quite
effectively since many examples can be explored quickly. Considef the -
concept of prime number. A number can be eested,for primeness quickly by
dividing on a calculator. Pupils must-know what divisors to try but the
focus can remain on prime since the child does not have to take the time to
perform the paper and pencil division. The concept of decimals can be
introduced quite early and developed through calculator activities. More
realistic numbers can be used since the calculator-handles "grubby" numbers
r. 16
just as well as "nice" numbers. Many motivational calculator games develop
import nt concepts such as place value, estimation, integers, and
functional rules.
Summary
The curriculum changes I am proposing may seem radical but the mathematics
needs of society have changed in the last few years with the advent of
electronic technology and will change even more in the near future. A
compqtationally oriented curriculum is archaic and will not prepare our
students for the 21st century. We must shift the emphasis to applications
and topics that now are crowded out by the time spent on computation. Only
by excising complex computations can time be made available for,
applications, problem-solving, and other import4pt topics. Such changes,
have far reaching implications, from teacher education to textbook
preparation.
This document may be obtained from EDRS as ED 175 631.
t
References
Carpenter, T., T. Coburn, R. Reys, J. Wilson. Results from the first
mathematics assessment of the National Assessment of Educational
PProorss. Reston, VA: National Council of Teachers of Mathematics,-...:.z....--___
1978.
National Council of Supervisors of Mathematics, "Position on Basic Skills."
25, 18-22, 1977.
Reys, R., B. Bestgen, T. Coburn: H. Schoen, R. Shumway, C. Wheatley,
G. Wheatley, A. White. Keystrokes: Addition and Subtraction..
Palo-Alto: Creative Publications, Inc., 1979a.
Reys, R. B. Bestgen, l'. Coburn, H. ,Schoen, R. Shumway, C. Wheatley,
G. Wheatley, A. White. Keystrokes: Multiplication and Division.
Palo Alto: Creative Publications, Inc., 1979b.
Vervoort, G., D. Mason. Calculator Activities for the Classroom. Copp
Clark Publishing, 517 Wellington Street West, Toronto, 1977.
,
Towards a Definition of Basic, NumeracyMICHAEL GIRLING
At a time when a cheap calculator can bebought for the price of two good cabbages, weneed to redefine our aims for numeracy.
I suggest the following definition.
Basic numeracy is the ability to use afour-function electronic calculator sensibly
If this definition is accepted- it is obviouslynecessary to re-examine our objectives in theteaching of calculation in mathematics, parti-cularly where it is related to the paper and pencilalgorithms which form the core of this work inprimary schools.
The definition needs tightening up by expandingwhat I mean by "sensibly". This seems to me to bethe crucial question, and I will try to define"sensibly" generally and by giving specific exam-ples. I will then consider the place ofalgorithms inthe scheme of our approach to mathematicalcalculation.
1 We need to be able to check that we or the calcu-lator have not made a mistake and giv awrong result. This checking can take severaforms and, depending on the seriousness of oconcern in getting the correct. er (to e
necessary accuracy), we-may uscetne ' oreof them.a)-1-Does-the-answer make sense? (323.63 miles
is a long walk.)b) Repeat the calculation in a different order,
using, .if possible, a different cperation, atleast three times.
c) Very rough approximation. (Is the decimalpoint in the right place?)
d) Approximation to one figure accuracy (interms of salaries, say, £3,000 is very differentfrom £1,000 or £9,000).
e)
f)
g)
Use of pattern. (32 x87 should end in 4;3002 x 9007 probably won't though!)Work out intermediate results and use theprevious checks on them. (Mr. Hope-Jones,when he was a master at Eton, was presentedwith a rubber stamp by one of his sets whichsaid, "You have substituted for 1z too early".This stamp is now obsolete!)It is important to check that the input datais reasonable. (9,291 boxes of chocolates at3p each ?)
2 We need an understanding of the relative size ofnumbers. What numbers are appropriate todescribe the number of pages in a book or thenumber of words on a page, say. How long is amillion seconds; how short is a millionth of asecond? This skill is not easily acquired. Wewould make a start if every calculation we askedfor, except for those needed to investigatepattern and structure in the number system,were related to a realistic problem. We shouldnever require a calculation for which "Does theanswer make sense?" is an irrelevant check.
I3 We need to be able to perform mental calcu-
lations for speed, for convenience and so as to beable to hold our own in the commercial andindustrial world. The standard of mental skillwe achieve will vary, and depend on our abilityand interest. The minimum should probablycontain:a) the ability to give and receive change by
"counting on";b) multiplying and dividing by 10;c) addition facts to 20 (quick recall);d) doubling and halving (varying degrees of
difficulty);e) multiplication facts to 10 x 10 (abolish table
squares!).
Reprinted by permission from Mathematics Teaching 81: 4r5; December 1977.,
19
The will vary mainly in the ability tochain these processes together and in the speedwith which they are cperfornied. Speed shouldalways be considered less important than accuracy.New strategies need to be devised to encourage andpromote mental work, perhaps a new campaign:"Dont show your working" !!
AlgorithmsI am not going to suggest that pencil and paper
algorithms should not be taught, but that theyshould only be taught as part of the armoury oftechniques that we have to help in an under-standing or number and not because they are useful.
This is not just a slight difference in emphasis but,I believe,-has quite-dramatic consequences:-a) 'The concentration is now unequivocally on
understanding the process involved.b) There is a very clear advantage in studying as
many different algorithms as can be manu-factured; for example all possible methods ofsubtraction should be used!
c) The most refined methods of long division, forinstance, which may be the least illuminating,
, need not be taught at least not to everyone.d) There is no need for anyone to be stopped in
their progress in mathematiclethrough beingunable to perform the useless algorithms wenow require.
It is perhaps worth adding that the time thatmight be saved by cutting out the practice of tech-niques could be well used by more attention toa) increasing rental facility;li) early introduction and manipulation of scien-
tific notation;0 developing techniques of approximation;d) serious investigation of pattern in number.
I, (Mr. Girling wishes it to_be_known that his views areunderstood to be controversial, and are not necessarilyshared by his colleagues. The implications need to bediscussed not just by teachers but by employers, parentsand teachers in tertiary education.)
20
.....:,
,
.
-
u
,
:
Section II
CALCULATOR ' i
c CAUTION&
(1)
,
,
o
..
.
1,
Calculators in theElementary Classroom:How Can We Go Wrong!
By Robert E. Reys
Hand-held calculators are widelyavailable. The 1977-78 National As-sessment of Educational Progress re-ports that 75 percent of 9-year-olds, 80percent of 13-year-olds, and 85 percentof 17-year-olds have access to at leastone'calculator. State and local surveysdocument that this statistic increasesyearly.
The current proliferation of calcu-lator-related activities for school pro-grams are evidence of the impact ofhand-held calculators. Dramaticchanges have been reflected in basalseries and supplementary publicationsthat are being designed to take advan-tage of the existence of calculators.Thus the role of calculators in elemen-tary school mathematics must be seri-ously considered and raises a pertinentquestion, namely, How. can we gowrong?
We can go wrong by
1. Banning" tfie-use.ofcalculators in mathemalics,classes
A tremendous amount of inertia is tobe overcome if calculators are to havean impact on elementary schoolmathematics programs. Several thingscontribute to this inertia.
(a) Few calculators exist in srbools.Despite the fact that most children
either own or have access to ahand-held calculator, only slightlyover 20 percent of Schools reportedhaving calculators (Wyatt et al.1979).
(b) Basal textbook series have yetto integrate calculators into -theirprograms as an instructional tool.Only 11 percent of the teachers re-ported that their mathematics text-books included activities writtenfor calculator use. Interestinglyenough, a majority of teachersthought their texts should includeactivities that use calculators (Reyset al. 1980). Furthermore, few sup-plementary materials exist thatprovide pragmatic use of calcu-lators with content typical oftoday's mathematics program.(c) Lack of familiarity with calcu-lators. Only 4 percent of ele-mentary teachers had attended acalculator_ workshop, yet morethan two-thirds of them wanted tolearn ways of using calculators in
- their mathematics classes. (Reys etal. 1986)._
This documents the need for a massivetransfusion designed to provide 'aware-_ness and effective instructional uses ofcalculators in elementary schools.
2. Forbidding the use ofcalculators on standardized
Rabat Reys is a professor of mathernatLeduea.-------glom at Atli! irerslitof-Milsouiiiit Columbia. More than 80 percent of the elemen-
22Reprinted by permission from-Arithmetic Teacher 28: 38-40;
meamduir 14EILL-
tary teachers said calculators should beavailable to children in school. (Reyset al.) This is a strong endorsement ofCalculator use, yet nearly all of theteachers cited standardized tests as theprincipal reason for not using calcu-lators. Teachers feel responsible forpreparing students for these ,tests andsince computation is a separate strandin almost every standardized test. em-phasis is placed on paper-and7pencilcomputation. The second mathematicsassessment by the National Assessmentof Educational Progress reports thatstudents using calculators do better ondirect computation, but the use of cal-culators has little effect on problemsolving. (Carpenter et al. 1980)
Suppose your school uses calculatorsthe next time standardized tests are ad-ministered. What would likely happen?There will be a jump in computationscores, with little change on other por-tions of the test. Foul! Foul! you cry.This isn't fairchildren aren't sup-posed to uie'calculators on tests. Why?The following problem is one selectedfrom the mathematics problem- solvingportion of a widely used standardizedtext and slightly modified:
In servicing a car the attendant used5 quarts of oil at 750 a quart and 15gallons of gasoline at 86e a gallon.What was thettotal cost for oil andgas?
If a child were allowed to use a calcu-lator to find the answer, would the cal-culator do the problem for the child?Does the calculator decide what keysto punch? Does the calculator interpretthe result? Calculators don't think foryou; they only do what they are told todo. Many employers recoenize this andask prospective employees to bring acalculator to use on job applicationtests.
Test developers are now preparingforms with which students can use cal-culators. If a few schools allowed stu-dents to use calculators on standard-
ized tests, it would have a tremendousimpact on test development and accel-erate the construction of assessmentsusing calculators. Calculator forms ofstandardized tests would-soon appear.Most importantly, when teachers knowthat calculators can be used in all
stages of the learning processfromintroductory development to evalua-tionthey can concentrate on problemsolving and higher level mathematicslearning.
3. Failing to recognize theintangibles associated with/calculator use.Dramatic increases in computationperformance are consistently associ-ated With calculator use. Less objec-tive. yet extremely strong evidencesuggests that pupils show more enthu-siasm toward and confidence in solvingproblems; greater motivation for learn-ing, which is accompanied by a morepositive attitude toward mathematics:greater persistence in solving problems,which results in more time on th'. task;recognition of different calculator solu-tions to the same problem: and in-creased awareness of weaknesses andlimitations of calculators, such as cal-culators perfoilning only what the op-erator keys in, overloading. stroking er-rors, or mechanical errors thatsometimes occur in the calculator.(Wheatley et al. 1979) All of these areextremely valuable educational goalsand should be considered in evaluatingthe impact of calculators in the class-room.
4. Creating another falsedichotomy
In recent years, mathematics educationhas experienced a number of false di-chotomies such as new mathematics'versus old mathematics. basic skillsvcrsus regular mathematics programs,skill versus concept development, anddiscovery versus expository lessons.
To consider calculator programs ver-sus noncalculator programs is to createanother false dichotomy in mathe-matics education. Calculators must beviewed as another tool within themathematics program and used assuch. There are many topics appropri-ate for calculator usage and othersequally inappropriate. These differ-ences must be recognized and treatedaccordingly in developing instructionalprograms. Thus, the question is notshould calculators be used, but when,where, and how can they be used mosteffectively.
1623
5. Using calculators tocheck paper-and-pencilcalculations.
Teachers ofteri cite checking paper-and-pencil calculations as the best useto be made of calculators by their stu-dents. Over 85 percent of the elemen-tary teachers indicated that calculatorsshould be used to check paper-and-pencil computation. (Reys et al. 1980)This is perhaps the most popular use ofcalculators by teachers in elementaryschools today. Yet asking pupils to dulengthy calculations and then askingthem to- check their work with a calcu2lator is of questionable educationalvalue. Providing an easily accessibleanswer key is a much more efficient useof a- child's time. Furthermore: chil-dren wonder why they should spend somuch time doing the paper-and-pencilcalculation if a calculator, is available.This reinforces the notion that use ofcalculators is cheating. a feeling thatshould be avoided under any circum-stances. If calculators are to be used tocheck work, it is Jr a more realistic toperform the calculation twice using thecalculator and to compare these an-swers.
6. Emphasizing thecalculator in developingpaper-and-pentilcomputation algorithms.
It has been suggested that paper-and-pencil computation algorithms be de-veloped with the calculator. For ex-ample, the calculator could be used tofind the partial products in figure 1. butwhy would you want to? This is a veryartificial use of a calculator. If youhave a calculator available, it is muchmore reasonable to do the computationwith a single operation.
There are. of course, calculator al-gorithms which are important and re-quire explicit deyelopment. For ex-ample,
1234567x 7007
presents an interesting problem thatcannot be solved directly with mostcalculators. Yet thoughtf.T.1 use of a cal:-culator and a modification of the typi-cal multiplication algorithm provides aspeedy and accurate solution. In fact.
the most appropriate algorithm formany computational problems will de-pend on the quantities and the internallogic of the calculator. Our teachingmust be designed to help students de-cide which algorithm or combinationof algorithms is needed to solve a par-ticular problem.
Fig. 1
4 X 46.3
20 x 463
463x 24
7. Using calculatorsonly after basic factshave been mastered.
Many exciting ways of using calcu-lators to help develop basic facts exist.Research evidence indicates that thedevelopment of basic facts is, in fact,enhanced in the primary grades,through calculator use. (Channel1980). To unequivocally ban calcu-lators or ignore their potential contri-bution in the development and masteryof basic facts is a serious error of judg-ment.
8. Using calculatorsor.: i after the concepthas been taught.
Calculators can be used to 'reinforceand expand many mathematics con-cepts that have been introduced. Noresearcu evidence exists\ to support theclaim that concepts must be developedprior to calculator use: (Suydam,1979). The notion of developing under-standing through examples, followedby explanation and discussion is acommon technique in mathematicsteaching. Why shouldn't a calculatorbe used to give pupils many examplesof concepts. such as negative numbers,decimals, or exponents before, or at
.least in conjunction with formal in-struction on these topics? This questiondeserves careful discussion by teachersand attention by researchers. In themeantime, the use of calculatorsshould be used in all learning stages
and not arbitrarily restricted to con-cepts that have already been taught.
9. Assume that concretematerials or manipulatives areless important in the learningprocess.
With the increased computational fa-cility among yot: ig pupils, it is tempt-ing to bypass models (such as counters.an abacus and multi-base blocks) indeveloping place value concepts and/or computational algorithms. Omissionof, or even less attention to such mod-els would be dangerous and increasethe likelihood of mechanical ratherthan meaningful learning. The chal-lenge is to plan concrete models whichparallel and complement the calculatorexperience.
10, Assuming that theprocess of formulatinglarge numbers isaccelerated.
Calculators certainly provide many op-portunities for experiences with smalland large numbers. Facility of dis-playing or reading numbers may oefalsely interpreted as understanding.Research has shown that understand-ing the formulation of numbers is adelicate cognitive process and that it isdeveloped over a long period of time.There is no research evidence to sug-gest that this period of concept devel-opment will be shortened or that anystages will be bypassed through calcu-lator experiences.
Where to from Here?Much research related to the use ofhand-held calculators in elementaryschools has been complattd. Withoutdoubt the most important result fromthis research has been its consistentfindings that "almost all of the studiescomparing achievement of groups us-ing or not usinl calculators favor thecalculator 'groups or reflect, no signifi-cant differences." (Suydam 1979). Thissuggests that schools that have calcu-lators have everything to gain in usingthem as an instructional tool in theirmathematics program. At this stage,the most clearly needed research is lon-gitudinalat least several years in
17.24
lengthstudies of students ...The, havehad sustained school experience withcalculators. In the meantime, we can-not ignore *calCulators and simply hideour heads in the sand.
As with any new technological de-velopment, there are many things thatwill change. As a resule`there are manyplaces to gosome right, others wrong.The only clear wrong decision at thistime would be for teachers and schoolsto ignore or ban hand calculators, thusnot considering them in future devel-opment of their mathematics pro-grams. It is clear that hand calculatorshold great potential as an instructionaltool for the elementary school. The ed-ucational value has been proclaimedby many, including the NationalCouncil of Teachers of Mathematicswhich "encourages the use of calcu-lators in the classroom as instructionaids and instructional tools."
ReferencesCarpenter, Thomas P Mary Kay Corbin, Henry
Kepner. Mary Montgomery Lindquist, andRobert E. Reys. "Results and Implications ofthe Second NAEP Mathemaiics Assessment:Elementary School." Arithmetic Teacher 27(April 1980):10 -12, 44-47.
Channel, Dwayne E. "The Use of Hand Calcu-lators m the Learning of Basic MultiplicationFacts." Columbus, Ohio: Calculator Informa-tion Center. May 1980.
-A Position Statement on Calculators in theClassroom. Reston. Va.: The National Councilof Teachers of Mathematics. September 1978.
Reys. Robert E., Barbara J. Bestgen, James F.Rybolt, and J. Wendell Wyatt. "Hand Calcu-latorsWhat's Happening in SchoolsToday?" Arithmetic Teacher 27 (February1980):3& -43.
Suydam, Marilyn N. "The Use of Calculators inPre-College Education: A State of the Art Re-view." Columbus. Ohio: Calculator Informa-tion Center. May 1979.
Wheatley, Grayson A., Richard J. Shumway,Terrence G. Coburn, Robert E. Reys, HaroldL. Schoen, Charlotte L. Wheatley, and ArthurL. White. "Calculators in ElementarySchools." Arithmetic Teacher 27 (September1979):18-21.
Wyatt. J. Wendell, James F. Rybolt, Robert E.Reys. and Barbara J. Bestgen. "The Status ofHand -Held Calculator Use in SchooL" PhiDelta Kappan 61 (November 1979):217-18.
Author's note: This article is based on a pre-sentation at a symposium entitled "The Ef-fects of Calculator Availability on SchoolMathematics Cumculum," conducted at theApr31 .1v79 Annual Meeting of the AmericanEducational Research Association in San
;"'' Francisco. California.
Calculators: buses and LIRAS
A look at much of the calculator materialcurrntly available for use in classrooms suggeststhat we are in the midst of a fast growing. techno-ogical 'movement, one which has not allowed usmuch time to consider how best to implement thistechnology. Calculator books and magazines arein abundance, and rmuch of what appears in these_supplementary materials represents merely play ":activity, or worse forces the use of the machinewith little attention to the goals of school mathe-matics. It is possifile to classify the most commonabuses (abuses in terms of the activities used as apart of school mathematics, not necessarily interms of their potential interest to individuals)into four general categories.I. Calculations, often with awkwa'rd numbers,
for no apparent purpose other than to require the useof the calculator. Some of the proposed calcu-lations might be interesting or motivating to somebecause of the chance to use the calculator, but afull page of `birnumber' calculations looks formi-dable whether or not the exercises are to be donewith a calculator. One can raise the questionregarding this type of abuse: Will such activitycreate different attitudes towards mathematics?Pupils will still make mistakes pressing the buttons,and when practising in such situations may well beinclined to say "So What".Jt is possible to usesome of the big number calculations which areembedded in sd-called applications to developsome tense of large numbers, but because of thecrude measurements and approxiMations whichmust be used, most of these activities are moreappropriately estimation exercises rather thancalculator exercises. As an example, take theactivity about the length of the line if all cars in theU.S. were lined up bumper to bumper, whichappeared in the article Calculations You .WouldNever Make Witicut a Minicalculator [1]; one mightsuggest that the article should have been titledCalculations You Would Niver Make.-2: Gaines and puzzles with no apparentmathematical objectives. This 'is not meant tosuggest that games and puzzles are not a valuableteaching aid (in fact, the November 1976 Arith-mak Teacher describes a number of such activitieswhich fit very nicely into a school mathematicscurriculum, and a great number of the calculatorgames have a logical reasoning component animportant goal in school mathemittics), butrather it is intended as a caution against an indis-criminate use. So the listing of tbis as a category
DAVID C. JOHNSONChelsea College, London
of-abuses is not intended- to suggest that learningshould not be fun, but rather that such fun shouldbe used to advance our cause. This is particularlyimportant since an often heard teacher complaintis that the curriculun. is already so full that theyare hard pressed to "cover the e;cisting material".3,. Misdeal button pushing. One of the mostfrequent abuses, and a good example for thiscategory, is 'making words by turning your calcu-lator upside down'often a fun activity, but onebest left for home or pleasure. It represents just onemore attempt to coerce children into tiding somebasic operations with numbers, utilising a calcu-
,-lator.The button pushing category also includes an
even worse abuse; children are asked to use thecalculator to perform some activity forwhich theylack some basic prerequisite understanding. Thisis not an argument for limited keyboard display, asI do not feel there is any problem if a child justwants to see what happens when certain buttonsare pushed. Rather, the concern is about requiringresponses or procedures for which the child lacksthe necessary background for 'understanding. Agood example of this abuse appears in a recentU.S. calculator publication intended for use withprimary age children. Early in the first book thechildren (aged 5 to 6) are asked to use theircalculator to count the ones in a display
1111111111
11by pressing 1 + each time they count. Andthis is before _any consideration has been givento the concept of addition or the `+' symbol. Onecan reasonably ask what the author expected toachieve with such an activityand whether thereare not better ways of doing this.4. Checking answers. I expect to find con-siderable disagreement with listing this categoryat an abuse. However, why use the calculator tocheck when it is usually the best device for per-forming the calculations in the first place? 'Whynot merely give pupils an answer sheet for checkinganswers; do we need a relatively expensive pieceof equipment for this task ? The Editor makes thisvery point in the December 1977 .1/T [2].) Forthose who welcome the motivational aspect of thecalculator, this may well become `old-hat' whenthe curriculuin is finally revised to take full ad--vantage of this device in teaching and learning.It seems that with today's technology the more
Reprinted by permission from Mathematics Teaching 85: '50-56; December 1978;
25
eit
,
.. .,,,
, .
critical checking skill is the ability to estimate andassess Ale reasonai*ness of answers' (More about
ne hates to begin an, hrtic e by listing.all thethis ter). 1 . . .
i" bnegative points, pz.trticularlr when, the calculatoroffers so much'that,iS good. However, some Caution the number of possible bridgehands:is called 'for before we go too far, We do not want
to leave a result in notcttional form, e.g.:the number of posiible poker hands:
sscs - 52!5 452 -5) ! ;
to be in a position where a critic questions our s: Cu 52.
of this device, demanding to know-svhfiliei it is ,13!(52-13)!
not just another expensive' educational gadget,and we are unable to supply,acceptable answers.While it may be true that .Many of the currenteducational materials tend to lack any firm philo-sophical or pedagogical basis other than.an empa-sis on the motivational aspect of 'calculator ,Use,it is also true that the majority,of these materialsdo contain a few, sometimes- Very few, gocciactivitiesor at least the basis for some good ones-. .when reworked. a ,
While working with teachers in Minneapolis!St. Paul, I have found it helpful first to establisha scheme for categorising the different tfres ofcalculator activities which at the same dine relatesthese activities to the school curriculum. Thescheme has been particularly useful for assessing,the potential contribution of published materialsand identifying areas which need development...,(For an alternative list of categories, see [3].) Theplac'ement of a particular calculator lesson in oneor more of the categories is based on the purposeof the activity. The categories are not intendedto be distinct or toepresent disjoint sets, but theymerely 'provide a structure for thinking about-calculator activities: one should not get hung upOn whether a particular activity belongs, say, underpatterns or 44/oration. A consideration of thesecategories should also enable a teacher to providesome, balance in types of use. The category new!renewed content is included to provide an opport-unity to consider curricular changes or, the re-newed emphasis which might be expected inschool mathematics because of the availability of
.calculators and computers.The examples below come from a variety of
areas and are only intended to be illustrative ofthe types of activities in each category.
CalculationsThis is really the most obvioui category, and it
is usually ecasier to describe it last after showingexamples for each of the others. It is often the casethat mathematical ideas and procedures can bebest illustrated with small, easily handled, num-bershence it may not be necessary or even des-irable to change this just to make use of a calcu-lator. It is important that we identify situations.where the contribution is:rear ratherthan forced.
One_example-, for which the calculator's role isprimarily for doing the arithmetic. is in workingwith the formula for the number of combinationsof n things taken r at a time. Ve are often forced
This may be pretty unsatisfying and pupils mayhave littlefeel for the relative sizecof these numbers.Do you? Guess which of the above you think is
-larger (,and how much larger) and then checkwith your calculator. The calculator does notremove the need.to be ablelo work with factorialsand to be able to represent the expansion inreduced form, as 52 would involve a lot of buttonPushing and may overflow ihe'machine. However,the calculator does remove the computationaldrudgery involved in multiplying many numbers.The answ.- s to- the two exercises' are 2.598,960
and 635,013,559,600 respectively, and the secondis considerably larger. Does this surprise you?)
Another example involve? the solution of trig-onometric equations. If the equation has theunknown in the denominator, e.g.,
tan 63°=22/x,solving for x requires division by a four or fiveplace decimal, usually takth from a table or witha shrewd 'manoeuvre, multiplication by a four orfive place decimal). Since the emphasis is reallyon setting up the equation and using the appro-priate trigonometrical function, the calculationsinvariably end up getting in the way of the straight-forward ideas. Examples from trigonometry reallyrequire little discussion as it is readily apparentthat for such things as building a table for sketchinga reasonably accurate 9icture of the graph of thesine or,cosine functions, and for applying the sineor cosine law, the calculator can be an invaluableaid. The availability of the calculator also suggeststhat table-reading activities generally included incertain mathematics courses can be considerablyreduced.
An entry under calculations has the characteristic= that its primary purpose pis to remove comput-
ational drudgery and the actual input or outputis not intended to demonstrate or reinforce con-cepts or have a strong application or problem-solving emphasis. (In the latter cases the activityfalls into one of the other categories.) The needfor including this category will become moreapparent whcn thc examples fore the others havebeen presented. It will also become, apparent thatthe categories- are in fact somewhat hicrachial,as it is generally the case that the calcuip.tionalaspect of calculator use will generally be a sec-ondary purpose in activities listed elsewhere.Remember that the key element in determiningwhere a particular activity or lesson tits in thescheme is in terms of the primary purpose of the lesson.
20
26e.
e o
is
PatternsThis may well be considered a part of renewed
content or a type of exploration. However, since it isconsidered to be an important mathematicalactivity in its own right and a useful skill forsolving certain types of problems and for learningnew mathematics. it is listed as a separate item.In this way it does not get slighted or lost. Manyof the articles and booklets which promote the useof calculators make a big point of emphasising:thisactivity; however. if one looks carefully at thesituations which are described, one finds thattypically they tend to emphasise the generation ofeasily observed patterns, as opposed to generatingoutput which then requires some study and testingto find a pattern. Thus, one might wish to dividepattern activities into pattern generation and patternsearch. Bot,h have a role, but unfortunately the firstis often left without taking full advantage of thesetting. Consider the following. quite common,example.
Complete the following.37 x 1 x3=37 x2 x3=37 x 3 x 3=37x4x3=
Is there a pattern? Use the pattern to complete the f.'s:
of chart. (And the children are asked to find theresults up to 37 x %x 3.)
While this might be interesting, the situation asgiven does not go anywhere. The pattern is obviousand one merely writes down the triples. Childrenat an early level catch on very iuickly and com-plete the items without much tought. When thepatterns are of such an 9bvious nature, one needsto ask how the situation can be used to promotemathematical thinking. One natural extension isto ask the children to try to figure out why it worksIn the example .the key is to notice that 37 x 3=111. (One can extend this same example to moreof a pattern ;search activity by asking the child totry to figure out the pattern up to 37 x 60 x 3. In thiscase they will have to try to figure out how thepattern seems to change in each decade. and "twhit point. A pupil worksheet and accompan..:ngteacher notes for the extenaed'calculator activitywas produced by a team of Minneapolis teachersand included in a set of,materials called Calculator
Cookery. [4] )If the pattern lessons are to be ;Irimarily pattern
starch, then the pupil should be involved in asearching and testing situation, that is he shouldbe required t6 study the various' examples, tohypothesise a relationship, to test or verify therelationship. to modify on the baiis of new inform-ation, et(. Esamples of this type of activity aboundin the literature, but are probably not as prevalantas the first' type. Here are three examples which
-illustrate the type of problem-solving thinkingwhich must be employed to find the pattern.
I. Use the calculator to find the recurring cycle of digitsfor 1/7, 2/7, 3/7, . 6/7 (and 7/7). What do youobserve? Try the set of fractions with denominator13. Again, what cl. you observe? The decimalexpansions for some fractions like the 7ths are cyclicin the sense that the repeating digits appear in thesame sequence, but with a different initial digit. Theset of fractions with denominator 19 are also cyclic.
Knowing this use your calculator to find the decimalexpansion for 1119.
(For a discussion of methods sec several MTarticles culminating in [5].)
2. Generalise: 13=13+23=Is+23+33.
13+23+33+ ... +n3=Use your calculator to test your generalisation for
10.
3. The question to'be investigated is: If a is assignedan integer value such that a is odd and greater than1, is it always possible to determine integers b aact cso that a2-1-b2=c2? Use the following examples.1. 32+b2=c: 1. 52+ b2=c22. 32+42=52 2. 52+122=1333. 9+43 =53 3. 25+122=1334. 9+16=25 4. 25+144=169Try these.I. 72+b2=c2 1. 92+b3=c32. 2.3. 3.4. 4.When you find the number write out ai the examples.1. 112-1-42.c2 1. 132-1-b2=c2Do you see a pattern? If so test your conjecture withthese.
372+b2=c2 992+62=c2 1512-14*=--c3
In each of these examples, the patterns are not soobvious. While the role, of the calculator is stillprimarily one of pattern generation, the major purposeof the activity is the search itself. (Even in' thedecimal expansion for 1 /19 the task is not trivialas the search involves looking at set's of digits, not
' just the single digit. Try it; the cycle has 18 digits.)One further point should be made. There are
also many pattern search type of activities whichlead to the cliscovery_or generalisation of a 'main-stream' concept or important and/or usefulmathematical relationship. These will generallybe listed under the category of exploration if theyhave as a primary purpose that of the learning ofthe specific piece of mathematics and the patternsearch activity is of secondary importance. This isillustrated in the first example of the next section.
ExplorationThis is probably the most fruitful and wealthiest
21
27
0
0
area for calculltor activities, particularly in termsof the usual mathematics curriculum. The mainfeature of activities which are to be placed in thiscategory is that tfie pupil uses the calculator togenerate output with the put-pose that the outputwill demonstrate a concept or relationship, or thatthe actual generation of the output will se. e tohelp reinforee a concept Which has been taughtpreviouily, or that the output will assist in solvinga mathematical problem. Thus the category reallyhas three somewhat distinct types of examples.
The example of exploration for concept..demonstration is of decimal multiplication.The traditional approach is to show how themultiplication is done by a number of examplesin which the decimal fraction is first converted tothe standard fraction representation with denom-inators of ten, hundred, etc. The examples are usedto justify the rule regarding the placement of thedecimal point (in terms of the number of decimalplaces in each of the factors). Thus, the usualpedagogical sequence is first to justify the newrule and then actually to implement in practicewith decimals. For some reason, some pupilsoverlook the justification stage and tend to con-centrate on memorising the rule, leading to
errors later, on when practice with decimalsinvolves mixed operations, An alternative app-roach using the calculator is given below. Thepupils are first asked to look for a, pattern whenthe calculator is used to do some calculations thepupils have noc yet learned how to do (discover arule), theit justify the result with a few, well-chosen examples, and then practise. The emphasis
'is on finding the pattern and then asking why itworks (very much along the same lines as thatrecommended as the extension to the commonpattern generation activities described in ,theprevious section). Also, in this example, studentsare asked to concentrate immediately on what isto be learned, rather than first look at a numberof seemingly less related examples and wonderingwhere the lesson is going.
Use your calculator to fine the products:x 0.2
0.8 x 0.63.2 x0.82.2 x 64
0.02 x 0.34241 x 1.220.72 x 0.6
0.026 x 0.003
(Approximately 15 -25 items, none with a zero as atrailing digit in one of the factors, or a 5 as atrailing digit when the other factor has an evennumber as the trailing digit.)
What do you observe about the placement o .1 -the demol
point in the answers?(Now the class can discuss any observations and
22
come to the generalisation. The teacher now askswhy this is so, and the justification should .bereadily apparent from a few examples withfractions which illustrate that tenths times tenthsgives hundredths, tenths times hundredths givesthousandths, etc.)
Now try the following with your calculator:2.44 x0.35126 x 0453.60 x 0.40
(5-10 items which involve zero or 5 as a trailingdigit)
Does your rule still work? What's wrong?(This is used to help reinforce the idea that the rulestill holds as the calculator suppresses the trailingzeros in the result. Also this reinforces the idea thattwo-tenths is twenty-hundredths.)
When a particular piece of mathematical contenthas already been introduced and the calculatoractivity is planned to provide an opportunity topractise or apply what has been learned, this canbe thought of as exploration for concept-rein-forcement. An example of this type of use can betaken from elementary algebra and involves theevaluation of a^ for various non-negative values ofa. taking n to have integer values from say I to 10or 1 to 20. This can be given after the pupils havebeen introduced to the idea of what is meant byan exponent, and used to reinforce the idea thatthe exponent says how many times a particularnumber is to be used as a factor. If pupils are toldto consider a=0, 0<a<1, a =l, and a>l, thensome important relationships are also reinforced(or demonstrated if the ideas are new); in parti-cular the fact that for 0<a <1, a^ gets very smallquite quickly, even when a is close to 1, say 0.9.This reinforces (or calls .to one's attention) the factthat when one multiplies two proper fractions, theresult is always less than the smallest fraction.The calculator use in this lesson illuitrates thedynamic nature of the resultone of the realadvantages of using calculators. Of course, thesame example can be used to illustrate how quicklythe result grows for numbers greater Than 1. (Com-
., pare, say, a=4 with a=8; the successive terms forthe second are not just twice the first.) Thus theactivity can be used to provide an interesting firstexposure to the exponential function.
Another example of exploration for concept-reinforcement is the calculator_ game Wipeout(described in the Arithmetic Teacher, November1976, p. 516). This is included here to illustratethat there are some calculator games which havethe potential for a -very real contribution to thelearning of mathematics. The game involvesentering a given number, with all digits different,and then asking the pupils to use subtraction toremove a specified digit (i.e. replace it with a zero)without changing-any other digit in the number.For example, if the game is used to reinforce
28-
concepts of place value, one might start withsomething like 834.562)9 and ask pupils First toremoi, c the 2, then the 4, and so on. (This rcin-forccs place value, since to remove the 2 requiresthe subtraction of -002. However, note that thisshould not represent the only place value practiceas it could become a rote activity of the form:count the places, use zeros and subtract.) Asindicated previously, thgre are many games of thistype, which can be both motivating and alsocontribute to the learning of mathematics. How-ever, major consideration should be given to thelatter point in deciding whether or not to includesuch an activity.
Exploration for problem - solving is gener-ally concerned with questions of the form "whathappens if ", and examples for this categoryabound in situations which involve formulas.(Formulas are not the only examples, but theseeasily illustrate the ideas.) One which is bothmotivating and interesting is to consider theformula (function) which relates temperature indegrees Celsius to degrees Fahrenheit,
F.--.,-1C+32.One can ask the question: When, if ever, are thetwo readings the same? After trying a number ofvalues for C. and looking for a trend or pattern,which is the way one would expect pupils At anearly level to proceed with the relationship as given,the pupil should finally arrive at the result,-403.(Of course at later levels one would expect pupilsto recognise that this problem is readily solvedwithout the calculator by solving an equation.)
Other interesting explorations for problem-solving can be described for such topics as the
pythagorcan theorem: what happens with differentshapes on the sides and hypotenuse; area andvolume formulas: what happens if a certain dimen-sion is halved. doubled. tripled. etc.; the classical1 penny for the first day and double your salaryeach day for a month; and so on.
Additional explorations can be found in numbertheory activities as well as many of the classicalproblems dealing with series and sequences.Examples from this single category could wellmake up an entire article.
ApplicationsConsumerSince the applications area is so important if we
are to produce some level of basic -numeracy(ability to use mathematical ideas and processes),it is useful to separate the applications into twodifferent sub-categories. Consumer applicationsare those applications which have immediateimplications for the individual in everyday living,e.g., comparative shopping; personal consumptionof water, gas, and electricity, constructing ormaking something; etc. Social applications aremore oriented to decision-making activities withimplications for benefiting society as a whole:population growth; conservation of res-mrces(water, fuel, minerals, etc.); health and healthcare (e.g., smoking): inflation; and so on.
It is not diwlult to find ideas for consumerapplicationsthe real problem is in pr rentingthese in a realistic fashion and in emphasising therole of mathematics in decision-making. A niceexample of an activity (problem) which is statedmuch along the lines as it might occur in real lifeis in the booklet by Dolan [6), p. 38.
PICNIC HA MS
As chairman of the food committee for your club picnic,
it is your job to purchase the ham for the sandwiches.the committee plans to make two sandwiches for eachperson and figures 2 oz. of meat per sandwich. Assume
that the bone in the shank ham is 20% of the totalweight, and that there are 35 people in the club.Which ham or hams would you buy to make the sand-wiches for the least cost?
23
29
Another, example is to use selected formulas tocalculate the cost of running certain electricalappliances (or for lighting). The exercises can beextended actually to figuring monthly bills, usingthe graduated scale of pricing which is used by theelectricity boards, and finding out how one mightsave by conserving. (What does it cost to leave aheater on all night instead of for only one or twohours in the morning? What would be gained bychanging all non-reading lights in your home to25-watt bulbs?)
ApplicationsSocialWhile it would be nice to include two or three
examples in this category, one problem is that theygenerally take some space to develop. Hence I willjust describe briefly some of the characteristics ofsocial application lessons/ As indicated previously,one key feature of most of such lessons is that thepupil should use the mathematics of a given unit orcourse to assist in decision-making or to comparealternative solutions. Some lessons may not lendthemselves to actual decision-making, but ratherenable the pupil to investigate some real worldphenomena of an, impersonal -nature--e.g. developequations to predict future times for track andfield events, look' at the manufacturing of someproduct, etc. However, it is problems of thedecision-making type which are sorely lackingin most of the usual textbook applications; thepupils are asked tb use their mathematics in whatappears at first glance to represent an application,but it goes nowhere or has no apparent purpose.We need to motivate pupils to want to use mathe-matics and to recognise the value of using mathe-matics to assist in making decisions. For example,why should one support. or vote against. 'ban thecan' legislation (legislation concerning tih,posablebottles-or cans)? What about the use- of certainpesticides? One other-feature-of social applicationsis that each usually involves, either a big problem(for which one needs a reasonable length of timefor study) or consists of a number of ;elated
24
exercises, instead of the typical single exercisewhich culminates in a somewhat open-endedquestion of a 'why' or `what' nature. Lessons couldbe based on such topics as wildlife management,camera lenses, predicting, birth rates. sound. etc.(Lessons on these four topics appear in [7j.) Themain emphasis is on applying mathematical ideasto real world situations. They involve either someaspect of decision-making or the study of some realworld phenomena, and any mathematical resultsneed to be interpreted back to the real orld.Thisis quite different from such questions as "How higha pile would we have if all the McDonald ham-burgers sold to date were stacked one on top ofanother ? "; at best this is an activity for looking atlarge numbers (number awareness, exploration)and at worst it is merely a button pushing activitywhich is probably a waste of time.
New/Renewed ContentAs indicated earlier, this category is included
since it is related to today's technology:--and whilesome of the topics do not necessarily involve theactual use of the cakurator on any regular orintegral basis, they'd° in fact relate to the effectiveuse of this technology. The key topics are esti-mation, errors, algorithms and iteration, andmathematical modelling. (For a discussion of someof these ideas see Michael Girling's Towards aDrfinition,of Basic Numeracy (4) Each of thesetopics easily warrants an article and it will only bepossible to touch briefly on some important ideashere.
Estimation is probably one of the easiest to con-sider, since it is already a part of today's curri-culum. Current practices in many textbooks tendto confuse estimation with rounding of and treatthem as if they arc the same thing. Even worse,pupils are typically told what to round off to, sothey treat this as another mathematical procedureto be implemented when instructed to do so. Thisresults in a failure to identify the procedure as anaid in checking resultsand the pupils typically
30
ask what they are supposed to do. With the avail-ability of calculators, the skill of estimation is evenmore important, but we no longer need the pre-cision in our estimates which was often demandedfor checking hand calculation. In fact, to check acalculator result one is primarily concerned withorders of magnitude and hence the only real needis the ability to do single digit arithmetic and workwith powers of 10. To estimate the product of, say;567 x 6324 (for the purpose of checking a calcu-lator answer) we need only think of this as 500 x6000that is, we need only consider the leadingdigit (rather than round), a much easier procedure;the product then becomes 5 x 100 x 6 x 1000, or5 x 6 x 100,000 or 3,000,000. Pupils will' soonnotice when applying this relatively straight-forward algorithm that the resulting estimates areoften unsatisfying. This then motivates the needfor rounding off. (On the other hand, even if apupil is unable to round, the leading digit algo-rithm is still a useful procedure.) After learning toround off, the pupil may still wish to sharpen thisskill, and use some number relationship notionsand common sense to obtain better estimates. Forexample, 65 x 14: using the initial algorithm weestimate 600; using rounding of to the leadingdigit we estimate 700; using some number senseand thinking about the problem we see we canconsider 65 x 10, or 650, plus 60 x 4, 240, or a sumof about 900 (and the correct result is 910). Noticethat with this threS stage approach, the pupil is
able to estimate at least the lowest level, and theskill is developed as one feels the need.
The inclusion of the topic of errors should pro-bably begin in the primary school. In general thistopic can be subdivided into two main categories,one dealing with the concepts of error analysisthe mathematics of errorsand the second relatedto ideas of checking answers through estimationand on the basis of the reasonableness of the resultsin terms of physical reality. These concepts are notnew, and in the past textbooks have attempted toprovide some instruction on the topics. However,it seems that pupils today do not have much of afeel for errors and it would be worthwhile takingInother look at what might be done. At the veryleast we need to include more exercises of the typewhere an answer is also given. and the student is
asked to determine whether or not the result is
reasonable (and why or why not).The mathematical notions of relative and absolute
errors should probably be introduced .after pupilshave some facility with fractions. Pupils shouldhave an idea of the importance of these ideas andhow actually to calculate them numerically Thatis, the pupil should be able to tell you tnat droppingthe 7 in 0.0067 is a large..relativeerror,..while. .
dropping the 7 in 0.6007 is small..(However, wedo need to be careful about "large" and "small"as the context of the problem may demand acertain precision.) For the two settings the absolute
error, in terms of droppii the 7, is the same.Notions of errors of measurement, precision andaccuracy also fit under this topic.
The concc; of algorithm and the use ofiterative procedures are basic in today'scomputer world. More needs to be done with theseideas throughout secondary school mathematics.Pupils need to be given the opportunity to designtheir own algorithms or to modify existingprocedures to do new tasks. In order to developsome appreciation of what an algorithm is, theyshould be provided with situations which requirethe following of a new algorithm. One interesting
. and useful algorithm bisection, for finding orapproximating zeros of polynomial functions; this isdefinitely within the grasp ofpupils at an early pointin secondary school mathematics. The procedure isa natural one foecalculator or computer use. (Sincethis is another of those topics which could be thefocus of an entire article, the reader is directed totwo interesting papers, [9] and [14)
The final topic is that of mathematicalmodelling. This involves notions of translation,but even more important, the development ofequations if you will, which can be used to explainknown phenomena as well as predict. Theavailability of calculators and computers will enablea student to work with and modify models; suchactivity has been hampered in the past because ofthe excessive calculations usually encountered. TheNCTM 1979 Yearbook, Applications in SchoolMathematics, treats this topic in some depth (see mychapter).
References[I] B. Litwiller & D. Ducan: Calculations rou Would .Never
Make without a .lJinicalculator (The Mathematics Teacher,Nov. 1977)D. S. Fielker: Editorial (MT No. 81)D. S. Fielker: Using Electronic Calculators (MT No. 76)C. Humphreys (project leader) ct al: Calculator Cookery(Minneapolis Public Schools, 1977)J. Ounsted: Pocket Calculators and Recurring Decimals(MT No.J2)D. Dolan: Let's Calculate (Lakeshore Curriculum Mat-erials, San Leandro, Calif., 1976)Corcoran. Devlin, Gaughan, Johnson & Wisner: AlgebraOne & Algebra Two (Scott Foresman, 1977)M. Girling: Towards a Definition of Basic .1ti'urnerary (MTNo. 81)
,,International Federation for Information Processing,Working Group 3.1: Ana4sed and Algorithms (from series,Computer Education for Teachers in Secondary Schools,
British Computer Society, 1978)DO] Calculators Have Come (Mathematical Association, 1976)
See especially the paper by Ethel.
[2](3)[4]
(5)
[6]
(7)
[8]
(9)
ABOUT CALC.ULATORfIll THE... ./C.HOOLf
by Arthur Kernerand Twila fle.fnick
Arthur Kessner is the Mathematics Specialist and CoDirector for theElementary Mathematics Concepts with Calculators Project lEMC2),at the -Lawrence Hall of Science, University of California, Berkeley.
Twila Slesnick is on the EMC2 staff, finishing her Ph. D. in education.
Every now and then a glimpse is caught by peopleoutside the field, of how much more there is to mathe-matics than school arithmetic it is art and science andit is creative, as well as logical and practical. And so itis also true that there is much more to arithmetic thancomputation. Since most people's math education wasnearly all computation, most people are unaware of thisbroader definition. and the general public continues to'demand that children spend most of their time learn-ing the same complicated computational routines thatthey themselves were forced to learn during their elementary school years. In fact, this is now a waste of expen-sive teaching resources. From now on, repetitaie drilland practice with comptitation techniques will be asworthless as they are demeaning. It should be clearthat all that time on computation denies teachers andchildren a chance to participate in other areas of mathe-matics which are less mechanistic, more creative andfar more practical.
In 1972, electronics companies introduced thefour-function hand-held calculator at retail pricesaround S200. By 1976, the price of a calculator, nowa more durable model, had dropped to about S10. Itbecame an invaluable tool in business and at home. Itsportability and responsiveness, as well as the ease withwhich it can execute computations, made its officialor unofficial appearance in school inevitable.
Reaction to school use of calculatorswas emo-tional and divided. People favored total prohibition;most who could countenance it at all favored control-led and restricted use from fourth grade on, with lit-tle if any use in kindergarten through third grade.Very few people advocated unrestricted use of the,machines. All in all, the calculator is having a hardtime making its way into the elementary classroom.
The reasoning of the prohibitionists seems to us tobe confused, short-sighted, or uninformed. Our guess isthat, not having used calculat irs themselves to learn arithmetic, people feel anxious about allowing their childrento use them because it seems to invalidate their own hard
work in learning arithmetic. It is our intention in this pa-per to dispel anxieties. Perpetuated and unsupportedfears become myths. The following are some of themost common myths about using calculators withchildren. -
1. CHILDREN WILL BE HANDICAPPED BY THEIRDEPENDENCE ON CALCULATORS.
a. Should we give children crutches when they don'tneed them?
The calculator has been called a crutch primarilybecause it enables people to do computations that theyare perfectly capable of doing or leaf ning how to dowithout a calculator. We, however,have no objectionto this use of calculators because valuable time will besaved. This will help ,nift major teaching-prioritiasaway from-me chanistiCripetitivicalculation. Today,most children leave elementary school with at least oneof two severe mathematical disabilities caused by thecurrent emphasis on computational facility. Calculatorscan be a legitimate remedy for either of these.
The most common handicap is the inability to solvesimple real world problems, such as finding averages orselecting necessary quantities of paint. (1) The reasonsfor this handicap are not very complicated. Childrenspend 75% of the tirst seven years of elementary mathe-matics instruction learning and relearning the complicated,step-by-step procedures (called algorithms) necessary todo an addition, subtraction, multiplication, or divisioncomputation quickly and accurately. Then they spend .the tiny remainder of their math time trying to learn amyriactof other things, including measurement, decimals,estimation, geometry, logic, statistics, real world appli-cations, mathematical games,-other-base numeration systems, and set theory. The traditional high priority ofcomputation results in a population which is able to doa computation when specifically told which operationsto perform on which numbers_ Yet these people lackother mathematical competencies such as logical reason.ing and the ability to apply that reasoning to the solvingof problems. (2) In today's perspective, this deficiency is
(I) CaspArtter, T. and T.G. Coburn and R.E. Revs and J.W. (2) Carpenter, T. and T.G. Coburn and R.E. Revs and J.W.Mimi. Results from the First NattoPal Asseurnent of Wilson. Results ;tom the Forst National Assessment of
4, Educational Progress. (NCTM, 1978). Educational Progress. (NCTM, 1978)..ts
Reprinted by permission from Calculators/Computers 2: 78-81; September/October 1978.
27
:3?
a serious handicap. A ten dollar machine can performmost ordinary computation, bUt society needs peoplewho are able to clearly define a problem and a methodof solving it.
The second handicap is the inability to do an arith-metic computation even when the numbers and opera-tions are specified. For example, some children cannotsuccessfully compute 346 x 28, 1869 4- 72, 4273 986or 112.32 + 89. These children are doubly disabled if, asis often the case, they also suffer from the first handicap.Calculators offer these children a powerful remedial aidor crutch. At least, if they are taught how to manipulatethe calculator they will be able to compute.
IfftIlertwx*,
4INCItie -1: sek
!Ar' fLs)
-41%.411111,
For children, a calculator becomes a vehicle for communicatingabout numbers. Here, two 2nd graders from Denton, Texas,share a number discovery. (The boy on the left speaks littleEnglish; he's from Mexico and nas been in the US. only a fewweeks.)
b. What if a child needs to do a computation anddoesn't have a calculator?
Consider what happens when you need to do an ex-act computation and you have no pencil and paper. Ei-ther you borrow them, or you save the computation untilyou can get your own pencil and paper. Calculators arenow so inexpensive and ubiquitous that this analogy withpencil and paper is reasonable.
Certainly all businesses which need to do any calcu-lations have calculators available. Even street vendors usethem. It is just not efficient to do otherwise. On the otherhand, if exactness is not required, then an estimate is need-ed, For this reason, we believe 'Iat the teaching of estima-tion in all the grades needs to b given a high pr:ority.
c. What will a child do when the batteries die?
New types of calculators .1se far less energy. As a re-Suit, calculators are available that have a lifetime of 10,000hours. At five hours per day of continuous use, seven daysa week, that amounts to more than five years' time. More-over, there are now slim, portable calculators with alarmclocks and stopwatches attached that are never turned off.The batteries are simply replaced on a regular scheduleabout once a year. This we feel is little Maintenance forsuch a useful tool.
2. CALCULATORS SHOULDN'T BE USED BEFOREFOURTH-GRADE.
a. Young children's minds could atrophy if they use cal-culators instead of thinking.
First, using a calculator does not diminish thinking. In3rder to enter data into the calculator correctly, students mustthink carefully about the related problem. 1 his has a highercognitive demand than does manipulating paper and pen-cil algorithms by rote.
Second, this statement assumes that the brain is amuscle that needs to be continuously exercised. Conse-quently, it is thought that doing complicated algorithmsis good for the mind, just as lifting weights develops thebiceps or running exercises the heart. However, the brainis not a muscle it thinks whether we are conscious ofthinking or not (for example, when we are dreaming).We realize that practicing computation can result in _
faster and more accurate computig;_but- there4s-virtUallf-no enhancement- of-other mathematical abilities as a di-rect result of computational proficiency.
Further, it is commonly believed that whatever ishard to learn must be good and valuable. and what iseasy to learn probably isn't valuable. We do not believethis. It is true that people value those things in whichthey have invested a great deal of time and energy. Butif the same things can be accomplished more effciently,those long hours of work could be used on somethingelse. Now that we have calculators to do the routineaspects of mathematics, it Will be more important thanever to invest time and energy teaching creative thinking,
b. There are no good learning activities with calcula-tors for young children.
s statement is usually based on the miscor: p-ton that a child has to, or wants to, use a calculato asan adult does as a fast and accurate computation ma-chine. If you hand a calculator to an adult who has ne-ver used one before, after a very few minutes of routinemanipulation that adult is bored. The feedback the adultreceives is nothing new. This is not the case with chil-dren. For a child, a calculator is a responsive toy thatalmost says, "Play with me". Because of this our devel-
opment group, EMC2, has devoted four years to research-ing ways to use calculators with children of all ages. Inkindergarten and first grade, before many children havemastered the fine motor skills necessary to write num-bers with ease, a calculator makes it easy for them tocommunicate with and about numbers. This is impor-tant. Numbers become things with which they can doenjoyable things. The calculator is a writing instrument.for them and can help make *numbers more concrete andless abstract. At this early age, children enjoy count-ing with a calculator by I's, 2's, 3's .... Counting, then,is a succession of "one mores", "two mores", "threemores", and so on. Thus, beginning even in kindergarten,counting can be taught as a foundation for addition andmultiplication, rather than as a rote chant. In grades twoand three, the calculator can help teach the similaritybetween one-digit and multi-digit multiplication. The pa-per and pencil ways of doing 15 x 8 and 15 x 24 are clit-fecent, whereas the calculator methods are the same. In
is 33
these grades, the connections between subtraction and di-vision can be elucidated with a calculator. The relation-ship between an operation and its inverse can also be ef-fectively demonstrated. With practice, problems invol-ving large numbers may be tackled. Up to now this hasbeen unrealistic. Finally. time can be devoted to solvingproblems finding diverse solutions to one problem typeas well as diverse problems that are solved by one solu-tion type. In other words, various problem solving tech-niques and problem types can be examined if less timeneeds to be spent on pure computation.
Some educators argue that children have to- under-stand the concepts of addition, subtraction, multiplicationor division before they can be allowed to use a calculator.Here again is the misunderstanding of what it means to"use a calculator". We have noted several ways in whichthe-calculator itself can help teach understanding of con-cepts. The calculator is not just used to apply concepts;--it is used to understand the concepts. Understanding is
--especially important now that it is no longer necessaryto focus solely on carrying out the operations.
Calculators might even allow teachers to look care-fully at their children's private use of arithmetic and atthe informal, intuitive strategies that children have in-vented or found outside of school. Recent in-depth in-terviews with children indicate that they make use of arich repertoire of mathematical techniques to solve prac-tical out-of-school math problems. (3) For example,counting on fingers to solve addition problems is one suchtechnique which is, unfortunately, discouraged by mostteachers. Few, if any, of these techniques were learnedformally in school. These informal partial understandingsneed to be given validity by teachers and_parentsrif-arith.----
metic-is-tcrbe-reallytrilful to children. While children douse their incomplete understanding to do school math,it is rare that children.rnake out-ofschool use of the com-putation techniques they spend 75% of their in-schoolmath time mastering. If calculators were to be adoptedby teachers and parents as necessary learningidevices,children would have a tool they could use easily, bothin and out of school. It would connect inschool mathwith their out-of-school math. This could increase therelevance and decrease the drudgery of mathematicsfor children.
3. WITH CALCULATORS AVAILABLE, DECIMALSWILL BE EASY TO TEACH IN THE PRIMARYGRADES.
A calculator can display to any curious child num-bers like 0.333 and 1.5. Thus, children will probablyask teachers questions about such numbers and perhapsbe more motivated to learn about them.'
Also because operations with decimals resembleoperations with whole numbers, they probably will be easierto teach them operations with fractions.
However, the part-to-whole relationship is easier tovisualize with fractional notation than with decimal nota-ton: 1/3 represents one part of a'whole which has been di-vided into three equal parts. The decimal equivalent, 0.333,is not easily interpreted as the same thing.
Further, increased computational facility will not neces-sarily enhance understanding of decimals. Ordering of deci-mals, for,example, takes a good deal more understanding
of place value than most primary-age children presentlyhave. (Which is larger: 0.9 or 0.888?) Even with calcu-lators, teaching decimals in the primary grades will requirenew materials and careful thought.
4. WITH CALCULATORS AROUND, THERE IS NOMORE REASON TO TEACH FRACTIONS:
Simple fractions, which become a part of our every-day vocabulary very early, are used quite often in oursociety and probably will continue to be used. "Half ofthe pie", "a quarter of an hour", or "a third of the money"are expressions which are not likely to disapoear with in-creased use of decimals. Hence, even in the elementarygrades, we will need to maintain the skills of recognizingwriting and ordering simple fractions. --
algebra, when one is solving equations, computingwith fractions is as common as computing with integers.Operations like addition subtraction, multiplication anddivision with fractions could be performed more easily andmore accurately on a calculator. However, students willstill need to know when to operate on fractions, whichoperations are to be used, and how to manipulate thecalculator to perform the operations.
It is true that most calculators display nonintegers(like 2-3/4 and 1/6) as decimals. However, there arecalculators that display and compute with fractions ineither decimal or numeratorand-denominator form. Iffractional notation is really important to the generalpublic, all calculators could be manufactured with thiscapacity.
5. CHECKING PROBLEMS YOU HAVE DONE BYHAND AND DOING THE COMPONENT PARTSOF A LONG PAPER AND PENCIL ALGORITHMARE VALUABLE USES OF THE CALCULATOR.
Using calculators to check hand computationsteaches the child little if anything about maiheMatics.It does over emphasize the importance of the one rightanswer. But it reveals no additional information aboutthe operation, especially since the calculator does notdemonstrate to the user the component parts of its in-terne; algorithm and the role that each plays. Thisrepetitive process will only let students know if theircomputation is accurate. If the intent is to increase ac-curacy, then problems should be done on the calculatorat the start and hand computation should be eliminatedaltogether.
Similarly, using the calculator to perform the com-ponent steps of a paper and pencil algorithm generally haslittle instructional value and is certainly not efficientiorcomputation. It is somewhat like using parts of a goodwatch to make a sundial. Those who encourage childrento use the calculator in this way with the division algorithniargue that understanding of the division process will be en-hanced. However, simply ising the calculator for the inter-mittent multiplications and subtractions which occur in thelong division algorithm Will not necessarily illustrate theprocess of partitioning a set of objects, nor will it reveal theplace value Witem of numeration that helps explain why thelong division algorithm works.
6. THE STANDARD COMPUTATIONAL ALGORITHMSARE BASIC AND IMPORTANT AND SHOULD ALWAYSBE TAUGHT.
(3) Ginsburg, Herbert; "The Psychology of Arithmetic Think-ing' . The Journal of children's Mathematical Behavior,
-Urine 1977. prh1410.29,
34
For hundreds of years around the time of Christ, pea.
pie used the Roman numeral system of naming numbers.Anybody who needed to multiply whole numbers greater
than one hundred had to translate the problem from theRoman notation onto an abacus-like device, do the com-putation on the abacus, and carefully translate the resultback to the Roman system. This method was ingeniousbut time-consuming to use and difficult to learn.
When the Arabic numeral system (our present base-10
place-value system) became known, the computational algo-rithms became simpler to learn and use. Indeed, as we men-
tioned previously, most children can use our com_putational-- -algoritms reasonably-well by-the-tirrifThey leave eighth grade.,
131-ktfii primary purpose of these algorithms has always been
to get fast and accurate answers to quantitative questions.The calculator is a better algorithm in this sense than are anyof the paper and pencil algorithms we now teach, just as theplace value numeral system was better than the Roman system.The calculator is faster and more accurate and it requires less
effort to learn. Some of the algorithms like the long division
or long multiplication algorithms are more difficult for youngchildren to learn than the actual concepts of multiplicationor division. Consequently, computational facility comesonly after.a number of years of learning and relearning. But,most important, there is no indication that facility with analgorithm enhances understanding of the related concept.Listen to a child or to yourself as you go through a multipli-cation procedure to do 365x 748: "Eight fives are forty,put down a zero, carry a four; eight sixes are forty.eight andthe four makes fifty-two, put down the two, carry the five,"eto.,_et.c.retc,-This-is-simpty-a Tetifiliiiii of a previouslymemorized algorithm to do the multiplication. In fact, thepurpose of memorizing it so carefully is so we do not haveto reason our way through the algorithm each time we useit. That's why algorithms are useful. We drill them intoourselves (and our children) precisely because we want to beable to complete the computation quickly and accuratelywith as little conscious effort as possible. However, mostadults could not explain why and how the procedures work.Thus, if they forget one step in the procedure, they mayhave sufficient understanding to reconstruct for themselves
the procedure needed. ,
We must conclude then that what is basic and im-portant is understanding of the problem and possibleways of solving it, rather than understanding of thespecific algorithm. If computation is necessary for deter-mining a solution, then a calculator can and should beused.
Cot elusion
We have stated some of the common fears that pre-vent school use of calculators and we have mentioned anumber of positive wayszalculators can be used in theprimary grades to teach new concepts rather than simply
to reinforce those already learned.
No longer does arithmetic have to be synonymouswith paper and pencil computation. In fact, all hand com-putation probably will be replaced by calculator compu-tatkin soon - if not in one year, then in fifteen years.The implications of these facts are that teachers need toaccept the reality of the calculator and to change theiremphasis and focus. Since it is known that computationalability is not closely related to problem-solving ability, (4)teachers have little to fear from using the calculator forall computation, and good reason to look forward todevoting more time totieveloping other mathematics`skills and abilities. .
(4) Beale, Edward; "Some Lemon Learned by SMSG". TheMathematics Teacher, March 1973, pp. 207.214.
, ^)
.41
Section III
e
CALCULATORS
aril
PROBLEM-SOLVING
c
31 36
It
.
..,
Using Calculators forProblem *Solving
by JEAN B. ROVIERSJean B. Rogers is 'a member of the Executive Council of the Oregon Council for Computer Education
and is active in cr,mputer and calculator circles in Oregon. Editor
This unit is designed to present the user with information on how calculators can be usedin mathematical problem solving methods. The two techniques invo\ved are TRIAL AND COR-RECTION and MAKE A TABLE, both easily andsommonly- used.
After completing this unit, the user:a) will be able to apply two specified probkm_sol-ving-met ho-dctoippropriate problems.
is'usefLil when these methods are applied to mathemati-cal problems.
The unit presupposes an ability to understand what a square root is, and the ability to cal-culate simple interest. A basic four-function calcula,tor is the only tool required.
It is important to emphasize that the methods presented in this material are general prob-lem solving methods, and not restricted to the specific types of examples given.
CALCULATORS AND PROBLEM SOLVINGA calculator is a tool that can increase the problem solving pO'wer of the user. There are
many techniques that are helpful when we approach a problem, be it mathematical or non-mathe-matical, and knowing these general methods increases the ease ,with which we solve the problem.A calculator can make some of these techniques even better when we are working on a probleminvolving numbers.
TRIAL AND CORRECTIONOne of the most commonly used problem solving techniques is called "trial and error",
but to take a more positive approach, let's call it TRIAL AND CORRECTION. A calculator -
makes it so much simpler to do the "trials" that this becomes a very efficient way to work outa problem.
For example, let's look at finding the square root of a number. We know that we canfind a square root by using an algorithm (a step-by-step procedure that will determine the result),if we can remember it. For square root it looks like this:
However, with a calculator to increaseour power, we could take a different approach:TRIAL AND CORRECTION. To do this weare going to use a machine with just the four ba-sic functions on it. If yours has a square rootkey, just ignoretit for now,
A way to find the square root of 3136: (V3-1ZChoose a number, multiply it by itself, and if the result is 3136, that number is 3136.
15W2r1 1
12525 125
TRIAL NUMBER506055
** 56
TRIAL TIMES TRIAL2500 Too little3600 Too big3025 Too little, but close3136
After each trial, 1 used the information yielded by'my multiplication result, either toolittle or too big, to correct my old trial so that the next trial is a better one. Thus we have the
TRIAL AND CORRECTION method.
Reprinted by permission from Calculators/Computers 2: 19-21; March 1978.
33
tise,the TRIAL AND CORRECTION method to find: the square root of 729 .
the square root of 42.25
N.r7Y9-
TRIAL NUMBER TRIAL TIMES TRIAL
MAKE A TABLEAnother problem solving technique is to MAKE A TABLE, and this is also a place where a
calculator can be helpful. Having a table is very beneficial when values are changing while a process
is being carried out, or when there are many values that need to be available for comparison. Calcu-
lating interest on a loan or a savings account is a good time to use this method.Suppose I make a credit purchase for $70 and pay it off at $10 a month. Since I must pay
Ph% interest on the remaining balance each month, only the remainder of my $10 payment is cre-
dited to my account.
TRIAL NUMBER TRIAL TIMES TRIAL
Month 1 Month 2 Month 3 Month 4 Month 5 Month 6 Month 7 Month 8
Balance $70.00 $61.05 $51.97 $42.75 $33.39 $23.89 '$14.25 $4.46
Interest 1.05 .92 .78 .64 .50 .36 .21 .07
Creditedto account
8.95 9.08 9.22 9.36 9.50 9.64 9.79)
****
**** Last payment is balance'plus interest.
From this table we see that it will take 8 payments to pay the $70 debt. By summing the
entries in the Interest row of the table, we find the the total interest cost will be $4.53.
Make a similar table for payments of,$20 per month.
$2.48.
t
Month 1 Month 2 Month 3 Month 4
Balance $70.00
interest
reditedlaccount
From your table you should find that 4 payments are required with an interest cost of
34 38
Make two more tables, finding similar data for a $70 debt if theinterest rate is only 1% while the payments are again $10 per month and320 per month.
ft
Now make a summary table that shows the interest cost and number of payments for thefour different circumstances we worked out above. This final table will give you an easy way tocompare the data on the four plans.
Total Interest Cost. Number of Payments
$10 payments at 11/2%
$20 payments at 11/2%,
$10 payments at 1 %
$20 payments at 1 %
CONCLUSIONS
' We have seen-hoW a calculator can be used in two problem solving techniques,TRIAL AND CORRECTION and MAKE-A TABLE-.'Remember that the examples welooked at are only a few mkt of the many different types of problems to which thesemethods can be applied, and that there are many occasions when these techniques canhelp people increase their problem solving power. '
..,:i.,00.....,
Apire..21. 'At:.,,4..!): -1:4, C tor .- ....!. .... c., -.
1' A1'1:0d -I /1-:;:** r'4\ .°
ti 4 '.::`.'. ' ..
tt:::: .\,, ..4.. e ..-
Xrt4. e "V \ tivik:' (....- f:- .."4ti-:., .0% 'IN,' . "r\ - ,.... - .
. -Vi:,,-., qk i. ... .
.1.t:f;;;PK%-:,.414.
gci .!"1" `'tits
C1(1 C 04',4%, X
. -/..n. 0 el 1-. ' '-q
- "... "Ci: .' 1 . .t. 4: .4;,. "
640.4; 1:7 ..14:t 1:: r /*-
4ZM.- v:' (..fr.A: Naksti,,..2:kliZz', ;01'ref.rr,
:AY"' .o.";*;:f- A.-
4.00.4,- t
.144 Pt.,t`i-so.e. '11 6- IP lib k.f
1.d 0,"-
zo t,ct:
,0 'II 0 t phk.
%. AA'- 1".5-%it e A11Y
% %
.s?Pit
35:
9
Mini-Calculatots And Problem Solving
Joseph J. ShieldsDeportment of Mathematics
Missouri Southern State CollegeJoplin, Missouri 64801
During the past five years, one of the most widely discussed topics atany meeting where math educators gather has been the impact of theminicaltulator on the classroom environment: "Do children become de-pendent on calculators?" "How can calculators be used constructive-.ly?" "Are calculators durable enough for classroom use?"
University educators were not allowed their leisure at coming to gripswith the problem. Manufacturers were producing hand calculators with-
, in the price range of many families in a short twa or three years alter thequestiori came to national attention. Students brought thern`to class andteachers wanted direction immediately. Articles, pro-and con, appearedoften. State, Regional, and National meetings were filled with sessions,workshops, debates, and research concerning the use of these tiny mar-vels. The Board of Directors of the National Council cof Teachers OfMathematics adopted a pi:sition which read in part:
, Mathematics teachers should recognize the potential contribution of this calculator asa valuable instructional aid. In the classroom the mini-calculator should be used inimaginative ways to reinforce ,learning and to motivate the learner as he becomes pro-ficient in mat/le-snacks. (NCTM, Newsletter. December, 1974)
The prodatnatich. Ikhich appeared less than' five years after the issuebecame'zciininent, gave a great deal of support to the protagonists ofthis device.
Today, most mathematics educators have accepted thii position(some, less eagerly than others), either because of.continuing researchsupporting the use of hand-held calculators, or because rejection wouldbe tantamount to burying .ones head in the sand. Several sttidiei (Rud-nick and Krulik, 1976; Sehnur and Lang, 1976) have demonstrated thatstudents who use mini- calculators do no worse, and in some cases imrprove significantly, in overall mathematical achievement than youngsterswho do not us* calculators. School districts'in IYew York, California,Indiana, and other states across the Nation are putting calcnlators in anumber of experimental classes. Preliminary results seem to indicate thatstudents take an interest in using minicalculators and ,that pencil andOaper computational skill test scores are as high for students who regid-
, arly use calculators as those without access to these electronic instru-ments.
Such widespread acceptance is by no means a fact of life among,` parents. Rudnick and Krulik (1976) indicate that the majority of parents
O
dv.
Reprinted by permission from School Science and Mathematics
80: 211 -217; March 1980.
. 374 0
O
0
whose children participated in their study Shad sincere reservations con-cerning the use of haal-held calculators in the classroom. They reportedthat parents were overwhelmingly certain that children would becomedependent on calculators with continued use.
In the last five years, authors and speakers have continued to suggestUses of the calculator in "imaginative and enjoyable" ways. Many of
these uses seem to 'focus on the motivational value of the calculator,stressing games-or computations necessitating their use. Somewhat less issaid about their 'functional use (as a tool). A consensus among thesevocal leaders of 'the movement is that the word problems'in the texts be-come 'almost trivial when calculators are used in place of pencil andpatter algorithms and echo theneed for more meaningful problem SON-
° ing-experiences. Indeed, widespread use of the hand-held calculator mayshrink the pencil and paper algorithm curriculum, but it will certainly ex-pand the curriculum in the direction of real world problem solving.
Consider the model for human problem solving based ,on,an informa-tional processing theory -which views problem solvingas sequencial proc-ess. The solver defines the problem; decides on a plan of attack, carriesout. this plan and chooses the 'optimal solution. Ther^ is evidence(Shapiro, 1973; Shields, 1976), to suggest that children clt. ..e problems-using a discernible pattern, and that solutions to comprehensive prob-lemsbecome more Yealistie and complete as age and problem solviii2 ex-perience increases. Hence; it seems reasonable to begin comprehensiveproblem solving practice in. the early=grades and continue such effortsthroughout the school experience. It is within the context of such practicethat hand-held calculator may have its most valued impact. Studentsusing calculators as their adult problem solving counterparts use them, asa tool.
Consider the problem solving process in greater detail. During the firststep, the solver the question being asked and translates theproblem into terms that make sense to him. Step two, development of aplan of attack, often involves the use of one or more generalized problemsolving strategies. These "rules of thumb" include working backwards,using contradiction, identifying subgoals or looking at a related prob-lem.
After the solver has sonic plan, he proceeds'with this plan to reach oneor more possible solutions. This thirdstep-hi:the-solving process involvesthe use Pf a number of heuristics which are taught in the elementaryschools. For example, the solvennay make a list or table, guess and testvarious hypotheset, use a graph or draw Nures and set up equations. Itis this step of the solving process in which the mini- calculator will aidstudents_in_reaching more accurate and realistic solutions. In addition,during the fourth step in thepRicess, selectiqg the optimal solution, the
38
41
calculator can be used effectively to check the solution against giveninformation.
For rample, in one sixth grade class, a discussion concerning thesafety of an elevator in a nearby department store prompted the teacherto challenge the class to determine the number of persons who couldsafely use the elevator, which had a posted weight limit of 1600 pounds.After some additional discussic,n, it was determined that the solutiondepended on the "average" user of the elevator. Two days later, groupsof students posted themselves outside the elevator to ask users how muchthey weighed: After several days and many weights, the average weightwas determined with the aid of calculators. The sample of over one
---.7_,hundred respondents was handled easily using calculators. An added' borii-nt vas,that the youfisters were able to average after each group had
completed its quesiionffig,--and found that after the first fifty weights, theaverage weight fluxuated very With additional data. (This is an
S,excellent example of the Law of Large Numbers, that is, tends to
stabilize as n becomes large, which;an alert teacher could exploit to teachsome basic probability). After it was determined that ten and a half"average" people could use the elevator, the discussion began to focuson the accuracy of this solution. Some students noted that some indi-viduals questioned did not respond; other students wondered about peo-ple who lied about their weight. The consensus was that some safety mar-gin should be allowed. Finally, the number of people, (men, women, andchildren), was limited to ten, and further discussion was postponed for afewclays.
When they elevator problem was again raised; a young man suggcstedthat they, do some investigation, concerning weight variations for thepopulation using the elevator. Certainly ten businessmen could exceedthe 1600 pound weight limitation, whereas, this limit probably wouldnever be reached by ten, sixth graders: For this part Of the problem thestudents used a table of heights and weights, rather than sampling indi-viduals at the elevator. Using the range of weights indicated for the adultmale, female, and eleven year old boys,,and _girls, the students con-structed :Confidence intervals for .ihe safe number of elevator users. TheirconclusiOns are shown in Figir 1.
.
Bois, 11 years old 117.251
0,;:s. 11 yearSol(. 117,241
Ad,it; females (10,13)Muir isles 17,91 ,
FIGURE 1Confidence intervals for safe number of elevator users":
3942
The students could not use this additional information, except to agreethat a "surefire" safe limit could be set at seventhe minimum allow-able riders for.the four populations studied.
This final solution to the problem gave rise to some additional ques-tions: "Since we have a safety margin in our limit, did the elevator manu-facturer also include a margin of safety iipthe 1600 pound limit?" Maybethe elevator won't hold (accommodate) seven very large men." Dis-cussion continued until the bell signaled the end of class. Interest in theproblem diminished and the challenge did not come up for general dis-cussion again. Perhaps the students recognized that they had achievedonly a partial solution, but one they could live with.
This example provides several instances where the calculator instnedaccuracy and, more importantly, prevented the solving prc,:ess from be-coming "bogged down" in laborious algorithms. Often, young childrenwill spend a great deal of time on pencil and paper algorithms, and forgetthe purpose of their calculations.
If our intent is to proVide experience in problem solving so that stu-dents gain insight into the process, then we should minimize the numberof instances where children can becoine sidetracked. In our example, thenumerical solution is not nearly as important as learning that any lawwhich guarantees the safety of everyone in a particular instance can be-come overly restrictive. This class was beginning to recognize this prob-lem while being ensconced in the solving process. This type of insight andinvolvement is necessary to provide for transfer to other similar problemsolving situations.
The kind and number of comprehensive problems which can be con-structivelyundertaken by children is limited only by your imagination.Some popular challenges which children enjoy are design problems.Design a playground, park, or bike trail. Children may elect to improvethe lunch room service or the crosswalk in front of the school. Any prob-lem which your students attempt to solve should embody the followingprinciples:
I. Itmot beof interest to youngsters so that they are willing to undertake the time con-suming problem solving task..
2. It must be a situation or question for which the students do not already possess asolution, or a direct method of obtaining the solution (e.g. soh c a readily availableequation).
.1. It must be solyable using previous learning.4. !f possible. the solution should become reality.
The first three conditions insure that youngsters work at the highestcognitive level creating something new i,) their experience. The fourthcondition is necessary so that children will accept future challenges. Onefourth grade class tried to solve the continuing logistics problem in theschool cafeteria and worked on the various aspects of the problem for
4043
about six weeks. The culmination of this work was a collection of pro-posals which they expected would "improve the service in the cafeteria".This list was taken to the principal who took a great deal of time to evalu-ate, suggest modifications, and present the proposals to the requiredadministrative personnel. As a result of their work, several changes,wereimplemented, and the students were anxious to attack other problemsthey found around the school.
The Unified Science and Mathematics for Elementary Schools(USMES) project has developed and tested several interdisciplinary unitscentered on long-ra ° investigations of real and practical problemstaken from the local/L.,mmunity environment. Some of the units involvechallenges in advertising, bicycle transportation, designing for humanproportions, consumer research and soft di-ink design. These units, eachembodied in a teacher resource book, have been classroom tested byteachers from all over the country during the past five years and haveproved popular with students and teachers alike. Any of these problemsor the comprehensive problems defined by you and your students willprovide hundred of computational task over the many weeks it takes toreach a satisfactory solution.
Consider the following comprehensive problem which children enjoy:Invent a new soft drink that would be popular and producedat a low cost-(USMES, 1973). Typically, an intermediate grade or middle school classwill spend two or three periods per week, for several weeks, reachingsome solution to this question. During that time interval students willconduct surveys on taste, carbonation; and price. They will test drinksfor individual preferences, and devise a rating system for soft drinks.They must analyze taste results for intermediate and final experimentaldrinks. In one particular class, the students used this data to find means,variations percentages and cost analysis. In addition, two youngsterstried to find an equation which predicted the cost-preference relation-ship. Although their efforts were fruitless, their 'teacher reported thatthey tried such a task because of their belief in the power of their calcu-lator. Therein lies one of the most striking arguments for using hand-held calculators in the context of comprehensive problem solving: thesolver learns the limitation of the computing device and begins to.under-stand the need to organize and direct the, solving process. Indeed, one ofth° most crucial ingredients in effective problem solving is the organiza-tion of, and inferences made from the given information.
How does the teacher plan her instructional sequence so that her classis in a position to take advantage of the symbiosis of problem solving andthe hand calculator? Before any interesting problem solving can occur,the student has to be conversant (with the calculator and the "fun time"atmbsphere prevalent When calculators are first introduced must dimin-
4144
4
ll
The student needs a little instruction and some time to practicepunching' the correct buttons. A successful way to provide adequate
. .preparation is through games and large number calculations.' Concur-rent With this practice, the tefther should supply some short puzzle pro-blems from which their pupils can learn the organization of data andgeneralized problem solving strategies.," This simultaneous practice maycontinue'as long as your students enjOy and profit from'the experience,but it sho\uld not keep you from getting into comprehensive problemsolving within a few weeks after the calculators are introduced into yourclassroom.
Calculator instruction Short puzzle andand practice problems and discussion
comprehensive problem solvingexpe.ence
FIGURE Seciu.: ;ce for introducing problem solving using calculators.
Once a real world challenge is introduced, allow your'students to findtheir own direction without letting them stray from their intended goal.Most teachers find that class discussion prior to data gathering, compu-tations; experimentations, etc., prevents the thrust of such work frombecoming divergent. Teachers should not discourage enthusiasm in a
, direction which is foreseen ,as an unproductive search. Failure in onedirection maybe as instructive as continued teacher engineered success.Periodic work on the challenge should continue until interest begins towane in the majority of your youngsters.
Summary
Our focus, in this article has been to show that mini- calculators andproblem solving, two of the most topical issues in mathematics educationtoday, complement each other perfectly. The natural combination ofreal world problem solving experience and the hand-held calculator pro-vides an answer to the recurring teacher question, "Now that we knowhow to push the buttons, what do we do?".
Comprehensive problem solving supplies many realistic computationalsituations where youngsters can use their hand-held calculators. Further-
t. Many examples of these kinds of acurnies can be found in the Anshmetic Teacher. November. 1976: Vol 23. No. 7.2. For example, see the Save ThutArni SktItSersts by Anna Harnacrek. hlidreu Pubbeations Co.. Troy. Michigan:
Irak
42 45
more, in the context of this problem solving experience, children begin toappreciate the strengths and limitations of the mini-calculator.
On the other hand, the calculator aids pupils in reaching accurate solu-tions to computational questions quickly. This speed and accuracy helps
'Ahem get through the forest of problem solving, without getting side-tracked while theyore working around a single tree. So waft no longer,start using those miniature marvels iu meaningful ways through investi-gations of long range comprehensive problems.
SELECTED BiBLIOGRAPHYa
HARNADEG, A., Basic Thinking Skill Series. Troy, Michigan: Midwest Publications Co.,1977.
NCTM, Newsletter. National Council of Teachers of Mathematics. Reston, Virginia,December. 1974.
RUMICK; J.A. and KRULIK, S., "The mini-Calculator: Friend or Foe?", The ArithmeticTeacher. Nov. 1976.
Scustat, J.O. and LANG, J.W., "Just Pushing Buttons or Learning? A case of Mini-Cakulators", The Arithmetic Teacher. Nov., 1976.
SHAPIRO, B.J.. The Notebook Problem: Reporr of Observations on Problem SolvingAciivity in USMES and Control Classrooms. Newton, Massachusetts: Education
. Development Center, 1973. , . -
SHIELDS, J.J., The Detection and Identification of Canprehensive Problem SolvingStrategies Used by Selected Fourth Grade Students. (Michigan State University, 1970DA1- 37A: 3481-3482; Dee., 1976.
USN1ES, USMES Guide. Education Development Center. 1973.USMES, Soft Drink design: Teacher Resource Book. Education Development Center,
1973.
,11
Calculators: TheiroUse In The Classroom
Gregory AidalaT !d Guilder Place
Ballston Lake. New York 12019
The use of pocket calculators within a classroom-setting has received a
great deal of attention in recent months. Briefly, the proponents of thesedevices argue that calculators can: 1. reduce the time spent on tediousand routine computation; 2. allow its users more time to focus onproblem solving techniques; and 3. provide increased motivation toward
the study of mathematics. . .
Opponents of the use of calculators in the classroom rear that these de-
vices may becoine a crutch and thus students Will become weak in theirability to perform basic arithmetic computation. This philosophy pro-Poses that students may become so dependent on their machines that
0 once removed, a student may not be able to independently execute basic
number skills.The solution to this controversy rests with the. investigation and pro-
motion of various experimental programs involving calculators. Anyweaknesses or imperfections in existing programs must be ironed out sothat those with justified criticism can accept the use of calculators as aworthwhile learning device in the classroom.
PURPOSE
In November, 1974 a proposal was approved by the Bethlehem CentralSchool District, thus inaugurating a program utilizing minicalculators in
mathematics class at the eighth grade level.In accordance with the National Council of Teachers of Mathematics
policy statement on the utilization of calculators in the classroom, ourprogram was intended to reinforce learning and to motivate the learnerin the study of mathematid. The function ofihe calculators were: 1) toprovide continued and increased motivation toward the subject matter;2)to furnish faster and more efficient ways of solving problems; 3) to al-
low problems of greater intricacy to be attempted; and 4) to contribute tofurther applications and-eXproration-ofrelated topics._
It was our belief that the use of a calculator was both a faster and moreefficient means of achieving solutions to mathematical problems. How-ever, the calculator was not meant to,be a substitute for skills which oth-envise could be performed manually. In order to operate such a device,
the student must possess a basic understanding of the mathematical con-
cepts of addition, subtraction, multiplication., and division. Therefore,the use of a calculator was not an end in itself, but a means to an end.The emphasis in the classroom.was to improve the problem solving skill
rather than their ability to compute.
Reprinted by permission from School Science and Mathematics
78: 307-311; April 1978.
f,4547
'REPARATION.
Our program° proposal called for the purchase of fifteen Bowmar-25calculators and adaptors. At the time of purchase in November 1974, thecost of each calculator and adaptor was approximately thirty-five dol-lars. However, present market prices of calculators have shown a sub-stantial reduction dyer the past.two years. One calculator and adaptorsimilar to the ones we have employed in our program can be procured forunder twenty dollars.
The physical requirements for calculators to be utilized in a classroomsetting were minimal. A locking two drawer,file cabinet was used to storeall equipment. The top drawer of the cabinet was divided into fifteennumbered areas corresponding to the specific number on each adaptor.Similarly, the bottom drawer was subdivided and labeled according tothe same number imprinted on each calculator.
In our program we opted for adaptors over the niore conventional re-charging devices which other calculator programs seemed to prefer. Wefelt that since we had fifteen calculators which were in operation duringfive different-class periods in a school da adaptors would be more ad-vantageous. Recharging devices could coigeivably lose their charge dur-ing the day through overuse or misuse. It was our belief that rechargingdevices would have been more valuable if the calculators were used by a,small group of students. In our case, adaptors provided a continuous andlonglasting performance to a large number of Students.
Our final barrier in preparing the classroom for implementing calcula-tor usage was solved with the assistance of our school electrician. Theclassroom had only two electrical .outlets but our electrician designedthree twenty foot extension cords with outlets every four feet. Thus, eachextension cord was planted between a set of two desks and each cord pro-_vided-five-extra outlets in the room from which a source of current couldbe drawn.
CLASSROOM OPERATION
In utilizing calculators in a classroom environment certain regulationswere necessary to insure the systematic operation of the program. In ourprogram, rules and regulations regarding the me of the calculators werespecifically outlined and strictly: enforced. Students were fully aware oftheir responsibility in taking care of their assigned calculator. This In-cluded careful handling of the calculator and adaptor at all times. shar-ing the calculator with a respective partner, and other applicable instruc-tions. We found that students rarely took advantage of the situationsince misuse of a calculator resulted in a loss of privileges for those indi-viduals.
In each mathematics class where calculators were employed. one re-sponsible student was selected to assist the teacher in the distribution and
46
collection of machines. The dual function of the student monitor was tocarefully distribute the appropriate calculator to individual classmates astheir number was called' by the teacher. Upon receiving each calculator atthe end of a class period, the student monitor returned each calculator toits proper resting place and checked to see that all machines were proper-ly turned off. The teacher was responsibleefor the distribution and collec-tion. of Each adaptor.
The student monitor was an energetic contributor to the efficient func-tioning and overall success of the calculator program. By having an able
-assistant,theteacher was in a _better position_to control the completion ofactivities for that classroom period.
Utilization of calculators in mathematics class began with instructionon how to operate the calculator. Initially, ten minutes of free time wasset aside for students to explore theicapabilities of the calculator.
As the popularity a calculators has continued, many:companies-haveproduced various models. Ir. our program we had to overcome the factthat a large, percentage- of students had accesslo calculators at home.More often than not, differences did exist between the calculators stu--dents had used previously and the Bomar-25 calculators. However, amodel calculator made of construction paper, (I V: feet by 3 feet) was em-ployed to explain all canabilities and limitations of the.Bowmar-25.
Although many stuc. .nts claimed to be proficient in the of a cal-culator; we found that at least two full class periods devcned strictly to'problems in addition, subtraction; multiplication, and division werenecessary to insure mastery of fundamentals. In particular, divisionproblems proved to be most difficult for students to solve.'For example,
.3 FE, 432 or 432 =.3 all represent identical division problems bu:.3
on a calculator many students often confused the order of imput. As aresult, a great deal of practice in addition, subtraction, multiplication,and especially division was provided before proceeding to specific
problem solving situations.
USES IN MATHEMATICS CLASS
As we approach the end of our fourth year of using calculators inmathematics classes, there does exist one method of operation superiorto all others. We have experimented with the use of calculators on a dailybasis for a short period of time, as well as the utilization of these deviceson a once a week basis. It seems that no matter how familiar students be-cometwith the use of calculators, new material could not be presented ina class period where the calculators were in operation. The level of ma-turity of eighth grade students combined with the distracting powers ofthe calculator presented too great a hindrance to allow normal learningactivities to take place. However, we did conclude that in order to profit
47
49
most from the use of calculators in a mathematics program, usage of thedevices should take place once a week.
Usually at the end of each week, following a fifteen minute quiz re-viewing the material for that week, the calculators were used to solveMany related problems. In this case, any explanation of a problem by themathematics teacher could generally be bypassed since in the minds ofmost students the material was very fresh. This technique prevented stu-dents from becoming too dependent on the calculator's, but at the sametime provided increased motivational support for the required curri-culum.
The eighth, grade curriculum at the Bethlehem Central Middle Schoolis composed-of many diversified topics epresenting various branches ofmathematics. Units of study coverea during the school year include
,probability, statistics, proportion, percent coordipate geometry, solidgecimetry, and algebra. Within each unit there exists several exercises in-volving the use of calculators which serve to reinforce learninoctivities .from previous class periods.
By way of example from the unit on solid geometry, the volume of arectangular prism and the volume of a cylinder were studied during oneweek of classes. In class on Friday students were tested via a short quizreviewing the material on volume covered during the previous four days.In the remaining twenty to tw-nty-five minutes of class, students wererequired to complete an exercise with the aid of calculators. The exerciseinvolved calculating the volume of two rectangular prisms and fourcylinders. Students were given the dimensions of each figure andfcrmulas were written on the board for reference. The substance of thatexercise is summarized below.
FIGURE LENGTH OF WIDTH OF HEIGHT *VOLUME=
BASE BASE LxWxH
-V.
I) Rectangular .
Prism 19.85 ft 11.3 ft 7.7 ft 1727.15 cis ft
2) RectangularPrism 42,3 cm 39.6cm 15.5 cm 25.963.74 cu cm
. RADIUS HEIGHT VOLUME = nll'h
3) Cylinder* 7.4 in 16.7 in 2871.50 cu in4) Cylinder 29.5 cm 52.8 cm 144.280.48 cu cm
5) Cylinder .54 m 1.6 m 1.46 cu m
6) Cylinder 72 ft 125.8 ft - 2.047.742.2 cu ft'Students lb at instructed to round off sassier% to the newest hundredth place.
t "*.3.I4
The preceding exercise satisfies the, objectives stated earlier in thisarticle concerning the function of calculators in a classroom-setting. Theuse of calculators to assist in problems on volume illustrates that moreintricate and exact dimensions of geometric figures which ordinarily havenot been part of the curriculum can be included as required classwork.
One of the goals of our progrim was to include exercises involving the
48 50
use of calculators which reinforced recent classroom activities. In achiev-ing thii goal, the facilitation of student interest in mathematics has alsobeen increased. The implementation of a faster and more efficientmethod of problem solving' through the utilization of calculators sub-stantiates still another advantage of our program.
SUMMARY
The use of calculators in the classroom' adds a distinctive feature toany mathematics program. Based on four years of experience, it is thefeeling of the author that calculators have a definite function in a class-room setting. At the eighth grade level calculators; if available, should bean integral part of the mathematics program but only as a supplementalaid to learning:
The following-list ,of guidelines is meant to assist those educators in theprocess of initiating ayrogram involving the use of calculators.
1) The purchase of all calculators should include a one year ssarraitty to replace or re-pair any malfunctioning toichine.
2) Distinct and permanent identification is necessary for all calculators and adaptors.3) The authors highly recommend the use of electrical adapters as opposed to any type
of recharging device. Adaptors %sill provide uninterrupted and longer lasting sersiee
in the utilization of calculators.4) A locking cabinet must be prolided to enhance the easy distribution, collection, and
protection of all calculators and adaptor;.5) Designated calculators should be assigned to students so that a particular machine is
utilized by the same pair of students on a continuous basis.6) Rules and regulations insolving the use of calculators must be clearly stated and en
forced so that students will exercise care in the operation of each calculator.7) A trustssorthy student should assist the teacher in the distribution and collection of
calculators during a class period.8) At least two full class periods of instruction should be pro% ided to all students %k-a-
vis methods of operating a calculator.'9) Although educators should be urged to explore all asenues of incorporating-calcula-
tor Usage into daily lessons, we highly recommend that the utilization of calculatorsnot exceed one'experience pei week. The no% elts of calculators in a classroom en-vironment can easily be eroded by oseruse that more importantly basic computa-tional skills might esent ually become weaker.
Enrichment problems dealing with such topics as probability, statis-tics, proportion, and geometric figures illustrate how easily calculatorscan become a valuable tool in the study of mathematics. As an instru-ment of great Motivational prowess, the incorporation of calculators intoa mathematics program is'rapidly and deservedly gaining prominence asan essential resource. ^
49 51st
Section IV
USING CaeULATORS:
Suggestions froM Surveys, Researchand Classroom Trials
1
51.
During the 1976-77 school year, 560hand-held calculators were introducedinto the mathematics instruction ofthirty-eight intermediate -level class.;rooms in six Title I schools in the In-dianapolis Public School System.Approximately 1000 students were in-volved and all teachers of fourth, fifth,and sixthgrades in the six schools par-ticipated in the project. The projectiris continued during the 1977-78school year, but during the sect d yearteacher Participation was entirely vol-untary.
The purpose of this project was toinvestigate the possibility of improvingproblem-solving performance and in-creasing student interest in mathe-matics by incorporating calculators inthe mathematics curriculum.
Classroom Materials andMethods
To enhance the use of calculators:, avariety of written materials wasprovided to project teachers. A pub-lication titled "How to Use,Your Mini-Calculator" was prepared and distrib-uted for orientation and familiarizationofteachers and their students. Also, a .
guide correlating calculator use withthe adopted mathematics textbookswas given to each teacher. The guideproceeded page by page through thetextbooks and gave directions and ad-vice on the many ways of using thecalculators as an integral part of in-struction. The primary useSothe cal-culators included reinforcement of
Glen Vannatta is supervisor of mathematics,grades kindergarten thraugh twelve, in the In-ilianapolis (Indiana) Public Schools. LucredaHutton is an assistant professor of mathematicalMentes at Indiana UniversityPurdue University
Induiniipohs. The authors were codsrecto ofterpropct reported In this ankle.
A CASE FOR THE CALCULATOR
By Glen D. Vannatta and Lucreda A. Hutton
computational skills, textbook problemsolving, supplementary practice withlarge numbers, and extended out-of-text problem solving.
In addition to the guide, a correlatedseries. entitled "Problem of the Week."was developed for calculator solution.These problems concerned appli-cations that were relevant and of cur-rent interest to intermediate-grade pu-pils. For example, dtiring the very coldwinter period, one problem concernedearning money by shoveling snow.The problems were graded in difficulty
Aso that fourth, fifth, and sixth graders'could work on the same general topicbut at different levels of complexity.Each "Problem of the Week ' was puton transparencies so that teacherscould easily present the problem to anentire class for discussion and calcu-lator solution.
Teachers had considerable latitudein the manner of their.use of the calcu-lators. Pupils worked in pairs for mostcalculator periods.,One pupil wouldwork a problem while the other ob-served and niade a written record ofthe work. Then an exchange was made.In some activities the pupils played acompetitive game. For example, pupilA would enter a multiplication factsuch as 7 x 9. Pupil B would have tosay aloud the product and then pushthe = key to reveal the correct answer.A point was scored for pupil B if he orshe was correct, or a point for pupil A,if not. Pupils alternated in giving prob-lems until a goal score was reached. Inthis way skill in basic facts was devel-oped or reinforced.
Teacher cooperation in the first yearof the project was highly variable. Al-thoughhere was no open refusal toparticipate, it was evident that a fewteachers did not support the premise of`using calculators in mathematics class-rooms to improve problem-solving
Reprinted by permis3ion,from Arithmetic Teacher 27: 30-31;
53
57
abilities. Others were very ositive andcreative in fulfilling their obligations.'One teacher developed a series of les-ons on nutrition that involved prob-lems about calories, vitamins, size ofservings, and soon. The integratidn ofsubject matter was excellent, and thecalculators were used to arrive at con-clusions that were interesting andineaningfuL
Equipment
The calculators chosen for the projectwere basic four-function machineswith memory, percent, square root,and sign-change keys. The keys werelarge and color.coded. Wooden boxescapable of holding one classroom set ofcalculators in compartments weremade by the industrial arts depart-ment. Each calculator was coded toin,dicate school, teacher, and student.Varying routines of checking out andchecking in the calculators from the. .
boxes were developed by the teachers.The identification numbers and pupilmonitors proved very effective. Thematter of security was less serious than'anticipated. Perhaps we were lucky;but only two calculators were missingat the end of the second year, and thesewere lost in a school break-in, not as aresult of careless classroom manage-ment. The calculator boxes were usu-ally stored in a lockable cabinet or in acloset when not in use. Only one calcu-lator was damaged due to the extent ofmalfunction by pupil abuse. Aboutfour percent of the calculators devel-oped malfunctions due to defectivemanufacture and were replaced underwarranty.
Another concern centered on thepower source for the calculators. Thecalculators used were powered by fourAA disposable batteries. Battery con-sumption was less than anticipated. An
May 1980.
,.estimate of battery cost is between $1and $2 per, per calculator for mod-erate or normal usage. As a backup.AC adapters were included in the pur-chase price of the calculators but werenot distributed or used. The specter ofa tingle of extension cords, the hazardof electrical outlets in antique class-looms, and security.problems limitedthe usefulness of adapters in our judg-ment.
Project Results
After two years-of operation ofthedianapolis PalicSchools calculatorproject, we reached several con-elisions, including the following: .
I. Management, Security, and powersource are minor concerns that canbe easily handled.
2. Most pupils are interested in andmotivated by calculators.
3. Most pupils can learn to use calcu-lators quickly, carefully, and accu-rately.
4. During the 1976=77 school year thesixth-grade students registered a sig-nificant increase in problem-solvingperformance.
S. During the 1977-78 school year cal-culator classes showed achievementwell above normal expectations inboth computation and problem ap-
. plications on the CaliforniaAchievement Test.
.6. Use oicalculators in intermediate-level classrooms does not seem toresult in a decline in computationperformance. Although during the1976-77 school year the fourth-grade students registered a signifi-cantly smaller gain in computationperformance than did the controlgroup, the 1977-78 results show noindication'of a decrease. At the endof the two year period, improvedperformance was noted for mostclasses,' r
The California Achievement Testwas administered to the calculator
:classes in September 1977 and again inMay 1978. For this period a gradeequivalent score difference of 0.8 could'be expected in combined computationAnd concepts/applications. The out-COme observed was that 86 percent of
the calculator classes scored meangrade equivalent gains greater than 0.8.Actually, in 1977-78, computationgains exceeded those for concepts/ap-plications. No control group data wereavailable during the second year of theprogram. The results were similar foreach of the three grades, fourth, fifth,and sixth.
There are many factors that affectedstudent performance. It is not our in-tention to claim that the impressivegains were solely the result of calcu-lator use. Other factors were present. asin any typical school setting. Duringthe first year the research design in-cluded a control sample. The Metro-politan Achievement Test was admin-istered for evaluation of computatiOnand problem solving performance. Thesecond year did not involve a researchdesign with a control sample and theCalifornia Achievement Test was ad-ministered for evaluation of computa-tion and concepts/applications. It is afact, however, that the calculators wereavailable and were incorporated intothe mathematics instruction cf theseclassrooms and the students did showvery good gains on the CaliforniaAchievement Test iiibtests on compu-tation and applications.
The teacher factor remains the mostimportant aspect of effective instruc-tionwith or without calculators. Theteachers need and want curriculumsupport from,the administration. In
fur opinion, a successful calculatorprogram must include effective teach-ing materials correlated with the ongo-ing mathematics program as well ascalculators..
If this study has value for elemen-tary teacheri, it is the positive in-dication that they can experiment withcalculators in-the classroom withoutfear of endangering computationalskills, and thata thoughtful program ofinstruction integrating calculators withtext material can be presented. Theonly danger, it seems to us. is to placecalculators into a classroom and leavethe outcome to chance.
Editor's note: As this issue goes to press. theauthors report that the calculator project in theIndianapolis Public Schools continues alive andwell. Three other Title I schools have beenadded in an expansion of the project. Although
r5V 54
1,
no formal evaluation has been undertaken, feed-back from teachers and standardized testinggives evidence that the goals and approach areeffective. Another calculator project for high -
achieving pupils in grades four through six hasadopted some of thyame features and matenalsof this first project. The great importance of in-service training of teachers in instructional usesof calculators has also been demonstrated. 0
4
14
anCaleUlators:What's Happeningin $chools Today?
By Robert E. ReYs,-Barbara Bestgen,James F. Rybolt, and J. Wendell Wyatt
What's happening with calculatorsin school's today? Are they tieing used?If so, by wftat students? How do teach-ers feel about usini,calculators in themathematics program? Should calcu-lators be used on standardized tests?Shdttld use of calculators be integratedinto basakmathematics textbooks? Ac-curate answers to such questions areessential in assessinh ae current statusof calcg:ator use in schools today and
Barbara te.agen is elementary mathematics sr!.eialist for Parkway School District in St. Louts.Missouri. Robert Reys is professor of mathematic:education at the lni.iersity of Missouri--Columbia. James Rybolt is a ntathemtincs teacher at Es-condido Nigh School. Escondido. California.Wendell Wyatt as an assistant professor of educa-tion" University Higf: School. Southeast Misiouristate Uniyersit. Cape Girardeau. "Missouri.
more importantly, preparing for calcu-lator usagt in the mathematics curricu-
lum during the 1980s.During the 1970s many voices have
been heard debating the virtues anddangers of calculator usage. A fewhave cried out in fear that use of calcu-lators will result in pupils who cannotremember basic' facts or do traditionalpaper- and - pencil computation. Teach-ers, in particular. are concerned abouthow calculators will affect students'computational skills (Palmer). The"rot-the-mind theory" has not beensupported by research (Suydam). Al-though long-term effects of sustainedcalculator usage are not yet known.there iS ample.evidence that frequent,use of calculators in elementary:schools has no detrimental effect on
Reprinted by permission' from Arithmetic Teacher
55
achievement in mathematics. Somehave proclaimed that the rapid growthand sales of inexpensive calculatorsand their consequent widespread avail-ability to pupils demand that hemathematics curriculum be reexam-ined and that teachers use calculatorsas an instructional tool. Still others,through the NACOME Report (Gin-ference Board of The MathematicalSciences 1975) and official poliq state-ment by the National Council ofTeachers of Mathematics, have en.:couraged schools to formulate calcu-lator policies and to make these toolsavailable to pupils to be used crea-tive and innovative ways of learningmathematics.
;lea: for information about calculator usage have been made at the
27: 38-4 3; February 1980.
I
grassroots level by parents and teach-en as well as by research groupsforexample, the 'event Report of the Con-ference on Needed Research and Devel-opment on Hand-Held Calculators inSchool Mathematics. One of the recom-mendations from this conference wasto survey current instructional policiesto answer questio such as, How ex-tensively are calculators being used inclassrooms? It is toward that end thatthis trearch was conducted.
This Study
In the spring of 1979, A survey of class-room leaders in Missouri was con-ducted to Check the current attitude to-ward and the extent of hand calculatorusage. Rather than mailing question-
-A
mires, which typically has less than anacceptable rate of return, it was de-cided to involve fewer teachers and tointerview each of them individually inorder to obtain more complete and ac-curate information. Ten school districtswere randomly selected from the 1978-79 Missouri School Directory. In orderto insure the representation of a varietyof school districts, two urban, threesuburban, three small co.nmunity, andtwo rural school districts were selected.The following criteria were used toCharacterize these four different typesof school districts.
Urban:. located in a metropolitanarea with a population ex-ceeding 125 000.
Suburban: located in a municipality
56,
adjacent to an urban area orwithin ten miles of an urbanarea.
Small Community not suburbanand servicing a populationbetween'10 000 and 125 000.
Rural: not suburban and located ina town with a populationless than 10 000.
Once a school district was selected. asecond stage of random sampling wasdone to zhoose one elemertary, onejunior high, and one senior .o,h schoolwithili the district. The superintendentof each district was contacted andasked for permission to contact princi-,pals within the selected schools. Eve 7S,snptrintendent expressed keen interestin the survey and agreed to cooperate.
7.
o I
The third and final sampling stage in-volved the identification of classroomteachers. Every full -time mathematicsteacher in the junior or senior highschool was scheduled for an interview.Two elementary teachers at each gradelevel from grades I through 6 wererandomly chosen to be interviewed.
This procedure was used to scheduleinterviews with 202 teachers in the tendistricts. Eight teachers were absentwhen the interviews were conducted intheir schools. It was not feasible to re-schedule these interviews so the resultsof this survey are based on informationgathered from 194 teachers. Thirteenpercent of these teachers indicated theywere members of the National Councilof Teachers of Mathematics. Each in-
terview followed a similar set of ques-tions and required from 10 to 25 min-utes to complete. The questions used inthe interviews addressed key issues re-lated to calculator usage. The protocolswere developed with the help of localclassroom teachers, mathematics coor-dinators from several school districts.and several mathematics educators.
Prior to the interviews, the teacherswere asked to survey their classes todetermine bow many of the childrenhad calculators available at home.SixtY-eight percent of the students re-ported having access to at least one cal-culator. This percentage ,varied littlewhen looking at the data by gradelevel. These percentages are she. (lybelow those found idthe 1977-78 Na-
tional Assessment of Educatibital Prog-ress, which reported that over 75 per-cent of the nine-year-olds, 80 percentof the thirteen-year-olds, and 85 per-cent of the seventeen-year-olds had ac-cess to at least one calculator.
The opportunity to sit down withteachers individually, outside the class-room, allowed for gathering not onlyfactual information,`but also teacherinsights, questions. and concerns re-garding calculator usage in the schools.Highlighted here are several of thequestions used in the interview, alongwith a description of the teachers' an-swers ind comments.'
Does your school (school district)have a calcularor policy?
An.overwhelming majority of the 194teachers interviewed said no currentcalculator policy existed in their dis-trict. Seventeen teachers reported theirschool had a calctflator policy, but theexact nature of this policy was neverdetermined. The confusion among -
teachers' regarding this question appar-ently stemmed from what,they thoughtconstituted a policy, whether it be writ-ten or perhaps what their principal oranother teacher might have said tothem informally. Some Actual teachercomments illustrate this uncertainty:-
"I don:t know if we have a policybut I have heard we are not sup-posed to use them.""1 think our policy.prohibits calcu-lator use.""They can be used in the senior highbut our board policy'prohibits usfrom using them in the junior high.""Qur principal doesn't want calcu-lators used in mathematics classes."
After talking with the teachers, in-terviewers frequently contacted princi-pals, curriculum coordinators, or dis-trict superintendents in an effort totrack down specific policy.-In no casewere they able to locate a written pol-icy formulated by a Board of Educa-tion, superintendent, principal, ormathematics department..
The interviews revealed that someschools are beginning to prepare a pol-icy or identify some guidelines for cal-culator usage. If such a policy reflectsteacher input, it will be very flexible.Of the teachers interviewed, none felt a
school policy should unequivocallyban calculator usage and there werenone who felt it should be required.The overwhelming feeling was that cal-culators exist, that there are many ap-propriate places for using them at alllevels of the mathematics curriculum,and that the type and extent of this us-
_ age-should_be left up to the discretion ofMe individual classroom teacher.
This provides a background for thequestions that follow and the specifictgatistics that are reported in table 1.Although percents are reported in table1, color coding is also used to provide avisual interpretation of the results. Dif-ferent shades, from light (0-20'0) todark (81-100%), suggest some trendsand patterns. Statistics are reported forthe whole group and for each gradelevel. In some cases, totals for a givenquestion do not add to 100 percent.This is due to rounding and in part tosome teachers not answering certainquestions. A careful examination oftable I will provide insight into inter-preting the results of this survey.
Should calculators be available tochildren in school? If so, to which chil-dren? At what grade level?
Eighty-four percent of the teachers saidCalculators should be available to chil-dren in school. Many were careful toadd that supervision by the teachershould be an important element. TableI shows a breakdown as these teacherscharacterized which children and atwhat grade levet calculators should beused. There are certain trends in-dicated by these data. One interestingtrend is that. in general. teachers arelikely to say calculators should beused, but they shc,uld be uscd in highergrades. For example. often fourth-grade teachers would say calculatorsshould be available. "probably in theupper grades and high school. but notin my grade." This feeling that "it'sokay for everybody except my grade"may lit evidence of the teachers' lackof confidence in using calculators as aninstructiona! tool. Teachers were morefavorable toward using calculatorswith all children rather than with onlythe best or poorest students. At thehigher, grade levels, how4, it wascommon tot teachers to commentstout calculators being a time saving.
more efficient computational devicethan paper-and-pencil calculations forthe best students.
Teachers who had used calculatorswere asked ifit had changed their em-phasis on mathematics content. Manycommented that not only could theywork more problems if a calculatorwas- available, but also they-actually
*covered more topics. ThPv also re-ported dealing more witn concept de-velopment and less with computationduring their mathematics class. Fur-thermore, more than 80 percent ofthem reported observing attitudinalchangel in their students. Without ex- .ception these changes were positive'and were Tharatterized by teachercomments describing students as beingeager to attack problems. showinggreater confidence in ability to solvemathematics problems. and becomingmore excited about doing mathematics.
Several teachers reported an inter-esting psychological problem resultingfrom calculator use, namely, con-vincing children that calculators are le-gitimate tools to use in a mathematicsclass. Some children felt it was cheat-ing to use a calculator and expressedguilt feelings when they used them.I hese teachers said they worked hardto dispel' this notion and to nurture theidea that use of calculators in mathe-matics class was perfectly acceptable.
Can children bring their calculators toyour class and use them?
Forty-two percent of the teachers re-ported that children could bring calcu-lators to their class and use them.Many teachers, however, said thisquestion had never come up. Their pu-pils occasir,nally brought their calcu-lators to school to show their friends orfor play during free time, but they hadnever asked to use them during amathematics lesson. Primary, seniorhigh. and experienced teachers aremost likely to allow calculators to bebrought and used in their classroom.Whereas, table I reports far less inter-mediate and junior high teachers al-lowed children to bring and use calcu-lators. Why? One explanation is theheavy emphasis on computational al-gorithms that are introduced and prac-ticed and practiced and practiced ...during this time. Use of calculators
68 58
could greatly alter such a mathematicscurriculum!
Have you used cakuliztors in yourmath class? What for? Would you liketo?The most frequent use of calculatorswas reported at the senior high level.with little being done at the elementarylevel.teachers who had used calcu-lators reported using them most oftenwith problem solving, word problems.and basic facts. These teachers in-dicated they had found the calculatoruseful for developing certain mathe-matical concepts as well as for compu-tition.
Of those teachers who had not usedcnielilntfNrc. fnrtv-tn pPrcAnt thPiiwould like to. By far. the most desirewas expressed at the senior high level.Elementary teachers generally ex-pressed caution, saying they would usecalculators if they were convinced cal-culators were an appropriate instruc-tional tool and if they could receiveproper training. Several also indicateda concern for declining test scores ifchildren used calculators.
Is special training needed to ,use cal-culators effectively?Elementary and junior high teacherswere clear in expressing the need fortraining. In talking with some teachers.it became apparent that they under-stood training to mean learning to op-erate their particular calculator. Theyhad not thought of training in terms oflearning ways to use the calculator ef-fectively and incorporating it in theirdaily lessons as another instructionaltool. Only 9 percent of the entiregroupmostly senior high teachershad attended a calculator workshop ortraining session, although 67 percentsaid they would like to attend such asession. Palmer (1978). in a survey ofleadership pertonnel responsible formathematics instruction found thissame attitude: "Teachers felt that therewas a definite need for workshops tohelp them develop and improve theircompetence in th° use of calculators inmathematics instruction."
Teachers cited several reasons fortheir lack of use of calculators in theclassroom. Primarily. inservice trainingwas not available. They also men-tioned that they saw little reason to get
Table 1: Percentage of Teachers of Mathematics Who RespCnded "Yee to Selected
Questions from a 1979 Calculator Usage Survey. (N = 194)
Chrgenon,"---
Should calculators be availableto children in school?
Which children:
Legend
0- 20%
Eli61- 80%
41- 60% El 81-100%
AInter- Junior Saribr
Whole Primary modiste high high
25
8
What grade level:
primary'
Intermediate*
junior high
senior high..,
Can children bring their calculatorto your class and use them? 42
Have you used calculators in yourmathematics class? 35
Would you like to?
Is special training needed to usecalculators effectively?
21 26 28
Have you attended a calculatorworkshop? 2 6 13 17
Would you like to attend a(another) calculator workshop? : .---
. ..
7
a
Should children master the fouroperations of arithmetic'beforethey use calculators?
Does use of calculators causechildren to rose ability to computeor to remember basic facts?
Would you support calculator useon standardized tests measuringconcepts or applications?
Does your mathematics textbookhave activities written forcalculator usage?
Should your text have activitiesfor calculator usage?
37 49
1;T
%poi% want of towhoei who responded yea to this question and the procook2g
59 53
training in this area when no calcu-lators were available for use in theirclassrooms. Although about 35 percentof the teachers reported calculatorswere available for classroom use, fur- -
ther examination showed this typicallymeant a few calculators were availablein a mathematics lab or Title I mathe-matics classroom, or perhaps a set ofcalculators was available on a check-.out system. Rarely were there suf-ficient quantities to provide a calcu-lator for every student or even forevery pair of students in a classroom.Often it was found that teachers withina school where calculators were avail-able were unaware of their existence.In no instance were there sufficientquantities of calculators available toprovide for two or,more teachers whowanted to use them at the same time.
When asked what sr :cific informa-tion or material should be included ina calculator training session the teach-ers identified the following as high pri%orities: sharing of specific instructionalactivities and lessons that use calcu-lators; discussions with teachers whohad used calculators and who wouldshare their perceptions and experi-ences; learning of effective ways tocommunicate with parents regardingthe role and value of calculators in amathematics program; and reporting ofrecent research related to calculator us-age.
Should children master the four oper-ations of arithmetic before they usecal-
culators?
Eighty percent of the teachers felt chil-dren should master this material priorto using calculators. Several indicatedthat if this was not done, the calculatorwould be a crutch and children wouldhave little incentive to learn to com-pute. Other teachers, however, saidsome of these skills could be developedwith the calculator. One teacher re-marked that if students could not com-pute, they probably would not be ableto use a calculator effectively either.
Teachers generally agreed that withslow students or students at the seniorhigh level who had never learned tocompute, a calculator should be usedbecause these students would probablynever learn to compute otherwise. Thiscomment is in the same spirit as a rec-
commendation found in the Overviewand Analysis of the School MatheMaticsCurriculum, K -12 report. althoughnone of the teachers interviewed men-tioned this NACOME Report. None ofthe junior high teachers reported thatcalculators should be made 'available
10-theirpoorer-students.-whichis in di-rect conflict with the NACOME rec-ommendation "that beginning no laterthan the end of the eighth grade. a cal-culator should be available for eachmathematics student during eachmathematics class. Each studentshould btpermitted to.use the calcu-lator during all of his or her mathemat-ical work including tests."
Does use of a calculator cause chil-dren to lose the ability to compute or toremember basic facts?
Few teachers said they could answerthis from first hand experience. but.43percent felt that using a calculatorwould cause students' ability to com-pute to decline. Twenty-two percentsaid they were not sure but would like'to know. The ambivalency of the're-sponses to this question reveals theneed for dissemination of recent re-search on calculator usage. One of the.common findings among such studiesis that children with calculators avail-able during mathematics classes do aswell on traditional paper-and-pencilcomputation tests as classes who havenever used calculatOrs. (Wheatley, et al1979)
If a calculator is made available toeach student, would you suppoit calcu-lator use on standardized tests measur-ing concepts or applications?
When this question was posed. teach-ers were reminded that most standard-ized tests have a computation subtestalong with portions measuring appli-cations, concepts, or problem solving:and furthermore, their answers shouldbe directed toward the parts of the testnot dealing exclusively with computa-tion. Over 40 percent of all teacherssaid yes to this question and supportedtheir position with statements such asthe following:
*This would provide a better mea-sure of what they know.""Calculators are not going to give
$
,
them any answers,:they will do onlythey are told to do." e`'
"Calculators are a part of the realworld and Should be, used,"1--,e,,,"Calculators area toollike a pen-ciland should be available.""Calculators allow more time to con -t entrate on the process."
On the other hand. teachers who feltcalculators should not be used on thesetests commented as follows:
o
"How will we know what theyknow?""Standardized tests should reflectwhat students know without a calcu-12tne."
"Using calculators is not fair. stu-dents must do their own thinking.""I want students to have the skills Ileumed.""It's like digital watches, kids won'tknow how to tell time any more."
Does your mathematics textbookhave activities written for calculatorusage? --
A review of current textbooks revealsfew activities explicitly written for cal-culator.usage. The interviews confirmthis: only 11 percent of the teacherssaid their textbooks had such activitiesand most of those were senior highteachers. Over half of theIeachers saidthe textbooks should include instruc-tional activities proyiding for calcu-lator usage. Several teachers felt that itshould be their responsibility .o adapttextbook activities for calculator use,but the overwhelming majority in-dicated that problems taking advan-tage of the calculator's capability aswell as instructional suggestions shouldbe included in their mathematics text-books. Some teachers also stated thatsuch activities should be an integralpart of the content being learned andnot simply tangential problems, chal-lenges, or artificial uses for the calcu-lator.
SummaryThis statewide survey of calculator us-age in Missouri schools provides muchfood for thought. Here are four of thejuciest morsels to chew on:
I. Sixty-eight percent of students,
60 60
grades I through 12. have at least onecalculator available at home..
2. Eighty-five percent of all teacherssaid calculators should be available tostudents in school.
3. Almost two-thirds of the teacherssaid that special training was needed touse calculators effectively.
4. Over half of the teachers would.like to have calculator activities In-tegrated within their regular mathe-matics textbook.
There are many other interestingfindings but these four document thatteachers perceive the need for calcu-lators within the school mathematicsprogram as well as their own need forinservice training with them. Further-more, a sizeable portion would like cal-culator activities included within theirregular mathematics program.. Takencollectively, these morsels as well asthe overall results from this calculatorusage survey suggest,that things are k'changing; but ,much more leadership.direction, and training is sorely neededif calculators are to be used to, theirfullest potential.
References
Conference-Board of the Mathematical Sciences,National Advisory Committee on Mathemati-cal Education (NACOME). Overview andAnalysis of School Mathematics. Grades K -12.Washington. D.C.: The Board. 1975.
Missouri State Department of Education. /978-79 Missouri School Directory. Jefferson City,Missouri, 1979.
National Institute of Education/National Sci-ence Foundation' (NIE/NSF.) Report on theConference on Needed Researchbril Develop-ment on Hand-Held Calculators in SchoolMathematics. Washington, D.C.: 1978.
Palmer, Henry B. A. "Minicalculators in theClassroomWhat Do Teachers Think?"Arithmetic Teacher 25 (April 1978):27 -28.
Suydam. Marilyn N. "The Use of Calculators inPre-College Education: A State of the Art Re-view." Columbus. Ohio: Calculator InforMa-tion Center, May 1979.
Wheatley, Grayson H.. Richard J. Shumway,Terrznce G. Coburn. Robert E. Reys. HaroldL. Schoen. Charlotte L. Wheatley. and Arthur1.. White. "Calculators in ElementarySchools." Arithmetic Teacher 27 (SeptemberI979):18-21.
Wyatt.). Wendell. James F. Rybolt. Robert E.Reys, Barbara 1, Bestgen. ''Status of Hand-Held Calculator Use in School. Phi Delta Kap-pan (November 1979). 0
iN
CALCULATORS IN THE ELEMENTARY SCHOOL:
----- A SURVEY OF HOW AND WHY
by
Leonard E. Etlinger and Earl J. Ogletree
The availability and decreasing cost-of hand-held calculators has put
pressure on educators Lid carefully consider the issues related to classroom
utilization of these devices. During a series of National ScienceFoundatibn-funded activities* in Chicago and St. Louis from 1978-1980,
inforv'ation was CollectedY4rom more than 400 participants with regards to
how and why calculators shOUld be used in the classroom. The objectives of
the sessions were information spread and exchange, and facilitation of
decision-making: the participants included teachers, administrators, and
parents.
Opinions on the issues appeared to have a common consideration;71..e.,
implementation of hand-held calculator programs should be based on sound
policy, well-defined procedures, and the caution due any innovative
activities in education. 'The participants agreed that teachers need to be
aware of how, when, with whom, and under what conditions calculators can be
used in the classroom. They felt that calculators should be usd to
reinforce basic facts, discover new concepts, explore number patterns, play
instructional games, solve applicable problems, and aid in".a wide variety
of math, science, and other subject-matter learning The only significant
caveat developed by the participants was that
Calculators, like any educationalinnovations--whether ideas, mater-ials, or devices--can be misusedin the classroom. With calculatorssuchrmisuse could be highly detri-mental to the students' arithmetical
development.
61 61
Findings
The participants first responded to a variety of questions in discussion,
debate, or panel response formats and later formed working groups for the
development of recommendations. Responses to selected questions and issues
are presented next.
*This material was prepared with the support of the National Science
Foundation Grant Number SER 77-20715. Any opinions, findings, concluSions,or recommendations expreSsed herein are those of the authors and, do notnecessarily reflect the views of the National Science Foundation. In the
operation of this and other projects, Chicago State Universit. has not andwill nct discriminate against any person on the grounds of sex, age, creed,color, or national origin.
6262
CALCULATORS IN THE ELEMENTARY SCHOOL:A SURVEY OF HOW AND WHY
Can Children Learn Mathematics From Using Calculators In School?
Responses were almost entirely positive. Some typical responses were:
Children can explore the ideas behind arithmetic,concepts of mathematicd, numerical patterns, verylarge and very small numbers, etc.
Calculators can help with basic facts, multiplica-tion and division, order of magnitude, intuitiveconcepts, an0 motivation in math.
Children will learn more because it will be more.interesting and fun.
Kids can work with very large numbers and experi-ment with a variety of processes such as the squareand root of numbers.
Calculators can reinforbe place value and numbersequence. Approximation techniques can bedeveloped.
Should Students Be Permitted To Use A Calculator All Of The Time in School?
Probably Not! Children need to learn facts and'
basid skills without : alculators.
Math activities should be planned specificallyfor calculator or noncalculator usage. Attentionshould be given to the objectives of the activities.
Children in junior and senior high school shouldprobably be allowed to use calculators in most oftheir school work and tests.
Very young children (kindergarten and-first grade)should,only use calculators in a few carefullydesigned instances.
Can Calculators Facilitate Creative Thinking In Students?
(.4
Most responses were affirmative. It was felt that calculators couldenhance creativity
There are many fine examples of activities where childrenuse their own creativity together with the calculator innumerical exploration.
Pencil and paper calculation, particularly When notmastered properly, restricts creativity.
Students are not limited to "nice" or "easy" numbers.
Decimals, nonintegral quotients, large numbers, andnegative numbers can More easily be used in problems
which are to be solved by calculators.
Some students will attempt difficult problems which
they would not think of trying without calculators.
The gifted or advanced student will be challenged toexplore sophisticated problems and concepts.
Should Children Use Calculators Before They Have Masteied The Basic
Facts?
Children must, eventually, be able to perform basic
arithmetic without calculators.
Calculators are motivational; this motivation may result
in kids learning basic skills.
Children should still learn,basic skills without calcula
tors, for the most part, and use calculators for other
types of math learning. This other math learning could,
in some instances, take place before basic skills are
mastered.
Possibly, older kids, who have still not mastered basic
skills can benefit by moving on in math and science
with the aid of the calculator.
Should Calculators Be,Used On Tests?
Clearly not on tests of basic arithmetic skills.
They probably should be allowed on most higher level
tests.
Tests should reflect real life needs. When appropriate,
test rules should be changed to allow the use of
calculators. National and regional tests, includingstandardized achievement tests, college admissions
tests, and vocational aptitude tests, should be
reviewed n light of this changing technological
aspect of society.
The project participants also generally agreed on the following points:
Calculator activities must be carefullydesigned if genuine learning is to take
place.
Teacher training will be critical toproper school utilization of calculators.
64
64
- Many math actiVities, particularly in the
prictiry grades, must still be done without
calculators.
- The greatest threat posed by calculatorsis-that of misuse in the classroom. Poor
teachers might use calculators as an excuseto avoid real teaching and learning.
- Continued exploration, research, curriculumdevelopthent, and policy formation Is needed.'
Recommendations
In a variety of working droups the project Participants developed the
following recothmendations and statements concerning school utilization of
calculators.
1., Primary Grades (K -3).
A.', Recommend usage of calculators to:
1. Develop an understanding of the number system
(a) provide additional practice with number sequencing and
number recognition2
(b) develop place-value concepts
(c) provide additional drill with basic facts
(d) become more familiar with basic operations performed on
numbers and number patterns
(e) to aid in introducing the concept of fractions
(f) to help the student become proficient in estimation
(g) reinforce the base ten system metric, money)
2. To concentrate onTroblem-solving skills
3. To promote success in arithmetic for slow learners
4. To challenge the gifted student
2. Middle Grades (4 -i)
A. Reinforce and follow-up of ideas in grades (K-3)
B. Develop objectives for hand-held calculators in the curriculum at
each grade level
65
65
C. Each grade level and/or department should
.., 1. Integrate calculators into the curriculum by means of various
topics, such as
- (a) Science - interpreting data- measurement- metrics
(b) Math - money, estimation, averages, place value, story
problems,
(c) Economics - consumer education
(d) Social Studies - map skills.
2. Utilize calculator activities that correlate with the scope and
sequence of their math texts
3. Grades (7-12)
Uses - In review for preparation for high school placement tests
- To reinforce-fractions, decimals, and related patterns
- Review concepts rather thad tedious calculations
- Formula work - quick calculation (e.g., areas and volumes)
- To promote creativity in students to develop their own games
and problems
- Creating math lab activities to reinforce concepts
Solve statistical 'pioblems and their application
- Use to aid in the solution of real life problems
To check answers and formulae of algebraic problems
= Introduction of programmable calculators to calculate
commissions, mortgages,taxes, and interest.
The following recommendations were made regarding teacher training.
- Staff Development
Staff dev opment program should be sufficient to meet the objectives
establishe for each of the grade and/or department levels: Primary,
Intermedia e, Junior High,"Senioi' High.
0
Suggestions include:
1. In-service workshops
2. In-servicecredit o41: payment for courses
3.' Funds for resource people, professionals, video-taped lessons
4. Released time to observe programs functioning
Facilities (beyond those available through an individual school district)
for providing in-service training should be:
(a) University and college, extension programs
(b) University and college short,toUrses
(c) Facilities such as a learning center or teachers tenter.The teacher centers should provide facilities for staff
development in the use'of hand-held calculators.
5. Schools should train their own teachers to use calculatorsproficiently and have one "very proficient," or troubleshboter
teacher, to helpout with teacher problems; could be another
professional, or a resource, to help with teaching problems.
6. Faculty must be supportive so there Is no one undermining the
program. School personnel (and parents) must be convinced.
7. Some guidelines should be established on use of calculators.
In addition to these'sUggestions additional research should be done to
discover the "best" we of calculators in teaching and learning. T
The participants in the workshop and survey.were asked to list materials
needed to implement and develop a Calculator program in their school.
Materials
'- Supply enough-calculators for largest class in school(It is probably important that this set consists of
one standard model)
- ,Booklet with suggested outline format including con-
ceptual and skill activities appropriate to grade.level
- Individualized packet for student enrichment
Teacher packets that can be used in classroom situ-
ations
67 67
Identification of the use of the calculator in subject
matters other than math and science;..setting up pac-
,kets to be used
ro Identification of the model of calculator used
Identification of the updated resource materials
available
Learning Center activity books
This document may be obtained from EDRS as ED 191 741.
6868
Using Electronic Calculators WithThird And Fourth Graders:
AFeasibilityStudy*,
George F. Lowerre, Alice M. ScanduraJoseph M. Scandura, and Jacqueline Veneski
MERGE Research Institute1249 Greentree Lane -
Narberth. Pennsylvania 19072
During the summer of 1974, a sequence of mini-experiments was con-ducted by the authors,'exploring ways to use electronic calculators withchildren ages 5-7. The results showed that the children responded en-thusiastically to the calculator, and that the students who used themshowed considerable gairs in mathematics achievement.
As a sequel to this sequence of mini - experiments, a small experimentwas run using the hand held calculator with children ages 7-9. The exper-iment was designed to help answer these questions for grades 3 and 4: (1)which standard mathematics topics can be taught most effectively usingthe hand held calculator? (2) What implications does the hand held cal-culator have for problem solving situations? (3) What new mathematicaltopics can be successfully introduced because of the availability of theelectronic calculator? For comparative purposes, a standardized mathe-matics achievement test was administered both before and after the in-struction.
Five third and fourth grade children participated in the study during aten week period in the fall of 1974. Three of the children attended regu-larly and were given the Metropolitan Achievement Test, both beforeand after the instruction. The other two attended irregularly, and werenot' tested. Instruction took place during 32 class meetings during the 10week period, with each meeting lasting 30 to 60 minutes.
Among the topics which were introduced were the following:Operation of the calculator.** '
Using large numbers on the calculator. The children discovered thatsome large nuMbers, sucu as 180 billion, cannot be shdwn on the calcula-
.,
tor.
Writing numbers using only the 1, 0, + and = keysNegative numbers (introduced using temperature and a 'number line
with space's to the left of 0); addition of signed numbers:Internal logic of the hand held calculator; discussed visual appearance
and unseen memory of the calculator as a problem is entered.Estimating: getting "ball park" answers for addition and subtraction
problems, then using the calculator tc check the estimate.
This mearch as supponed Inas instruments. Inc. orkna under contract wars the MERGE Instinne.
Texas Instrantenu Model Tf 2300+as used throughout 0:restudy.
Reprinted by permission from School Science and Mathematics
78: 461-464; October 1978.
69
Number tricks (take a number with 3 different digits, reverse the.digits, subtract the smaller from the larger, the sum of the difference andthe difference with the digits reversed is 1089)
AreaDecimals (introduction to tenths and hundreths using a square divided
into 100 parts; addition, subtraction, and multiplication of decimals)Decimal-fraction equivalenciesInequalities with decimalsUnit pricingFlow charts for adding and multiplying using the hand held calcUlatot..Prime and composite numbers; prime factorization of numbers up to
SO.Throughout the ten week petiod, a great deal of time was spent review-
ing and practicing the arithmetic operations with whole numbers anddecimals.
For the three children who attended regularly, the results on theMetropolitan Achievement Test (reported by glade level) were asfollows:
Before _.litstruction
: AfterInstruction Gain
Computation 4.3 6.2 + I yr. 9 mo.A Concepts 7.3 7.7 + 4 mo.
(age 9) Problem Solving 7.0 7.0Total 5.6 6.9 + I yr. 3 mo.Computation 3.8 5.5 + I yr. 7 mo.
B Concepts 3.8 5.6 + 1 yr. ,tt mo.-(age 7) Problem Solving 17 4.9 + (.yr. 2 mo.
Total 3.6 5.6 + ryr.C.mputation 14 2.9 + 5 mo.
C Concepts 3.8 4.9 + I yr. I mo.(age 9) Problem Solving 3.4 4.4 + I yr.
Total 3.0 3.7 + 7 mo.
While these large gains in mathematical ability are quite impressive,several limitations must be mentioned. Small group, essentially indi-vidualized instruction, is not typical, of the classroom teaching-learningenvironment. In a one-to-one situation, a skilled and careful teacher canimmediately correct any misconception, and adjust work level to studentneeds. For example, one child was given a quick review of subtraction ofwhole numbers when he experienced difficulty in subtracting decimals.Also, the Metropolitan Achievement Test relies heavily on accuracy inaddition and multiplication tables, and much time was given to practiveon tables, albeit with the calculator.
However, these facts are not enough to entirely discount the very real
70 70
gains that were shown. Several of the children were in the upper range ofthe scale origir ally, and the retest did not necessarily represent a realpower test for them. Gains in areas such as understanding of negativenumbers, comprehension of flow charts, real world problem solvingskills, and actual ability to compute with the hand held calculator werenot measured at all. Nor will they be measured in other standardized testsgenerally in use.
SUMMARY
As a result of this study, we would like to offer these tentative answersto the three questions asked about third and fourth grade mathematics.
Question 1: Which standard mathematical topics can be taught mosteffectively using the hand held calculator?
a) The hand held calculator is especially useful for testing and practic-ing place value skills with whole numbers and decimals. Quick, non-tiring practice (as contrasted with written practice) can be given.
b) Negative-numbers hre discovered through just "playing" with thehand held calculator. Formulas such as A <B A B = can be
-tried with-many-numbers so the child can See the pattern.c) Decimal and metric measures,. including area and volume, can eas-
ily be taught.d) Prime and composite numbers can be more easily identified, be-
cause of the capability of quickly and accurately checking factors.e) The calculatdr is especially good for going from fractions to deci-
mals, but difficult the other way.f) In general, a guided discovery method can be used because the
children can quickly try many examples and detect patterns and -algo-rithms.
Question 2: What implications does the calculator hold ;or problemsolving situations?
a) The students are able to obtain much more practice with "realworld" verbal problems (which are closer to real life than typical verbalproblems are). The children do not have to write the equation, which canbe tiring, nor do they have to remember the numbers and operationsmentally.
b) One surprising benefit of using the hand held calculator in a verbalproblem solving situation was a test of how quickly different childrenprocess oral information. A teacher can readily detect children with weaklistening skills and weak short-term memories by taking note of childrenwho continually need the problems repeated. In fact, practice with thistype of problem and increasingly longer and more difficult examples
, seemed to help the children gain speed in listenihg skills.c) The children enjoyed and were highly motivated to try fairly com-
plex real world problems involving several operations. Teachers used thecalculator to explore large number, problems that occurred in other sub-ject areas such as social studies and language arts. Students made uptheir own problems and tried solutions, often discovering a need foroperations or algorithms which hadn't yet been taught.
d) Because the computation is done easily, the emphasis in teaching
can be placed on problem definition, delineation of relevant and irrele-vant information, operations involved, formula and equation writing,pattern recognition, and algnrithm formulation.
Question 3: What new mathematics topics can be successfully intro-duced because of the availability of the electronic calculator?
a) In general, from the brief experience in this study (grades 3-4) anddiscussion with teachers, the standard mathematics curriculum can be ex;
panded in computation to include use of numbers of greater magnitude.A shift of emphasis occurs, so that estimating skills, the use of negativenumbers, and decimals occur at a much earlier time than normally taught
the standard mathemadcs curriculum. In other areas, such as pre-corn-puter_skills, t he_use_of_fl zw charts_ and discoveryofJ_!ciebugging'_'_tech- _
niques become an intrinsic parfof the curriculum.b) The teacher has an opportunity to spend more time on concrete
representations of concepts since she can check instantly for studentunderstanding by having the children show answers on the calculator.Patterns can be more easily detected and explored. The children are nottied to a writing surface when they explore mathematics.
c) The teachers expressed a desire for problems using large numbers.As one teacher said, "a book of 'fantastic' large number word problemsis-needed. Things kids can relate to,-such as large number attendance fi-
,gures as parades, fairs, or numbers of hamburgers or pizzas eaten at a
restaurant."Finally, we should comment on one concern 'expressed to us by
teachers: How will students who have used the calculator do on stand-ardized tests? Our strong, but tentative results should help in this regard.
In this study we incorporated many activities using the calculator to"drill" the children in addition combinations and times tables. Theseactivities could easily be incorporated into any calculator curriculum.
72 72
The Calculator in it
the ClassroomBy Norma Zakariya, Margo McClung, and Alice Ann Winner
The hand-held calculator was intro-duced into the mathematics Curriculumof the United Nations InternationalSchool at the upper-elementary levelduring the 1977-78 school year. Thenew progra:n was inspired by a sum-mer. workshop at Teachers' College.Columbia University. Calculator in-struction begari on an experimentallevel in a fifth-grade class in November
Parma Zakariya has been teaching third grade atthe United Nations International School in NewYork since 1975. Iler previous experience includesteaching third grade in Scotland. fifth ETA, inVenezuela. and grades three hr ^h seven m Ma-
laysia Margo McClung has been teaching fourthgrade at VMS since 1975,11er previous experi-enceincludes teaching grades two through six rn
the Internatiowl School in Manilla. Philippines., and third grade in Columbia. Alice-Ann ginner
has been teaching fifth grade at VMS since 1976.She previously :ought fifth grade at the AmericanInternational School is Henna, Austria.
1977. This was followed by instructionin a fourth-grade class, and later in athird-grade class.
The goal at the fifth-grade level wasenrichment of the existing curriculumand instruction was organized to dem-
onstrate-the-use-of the calculator as alearning tool for exploration of con-cepts too difficultio<execute without it.The main direction of the instructionat the fourth-grade level was extensionand expansion of the existing curricu-tun! , especially in problem solving.\Lessons were developed, also. to dem-onstrate the use of the calculator inreal-life situations.
The experiment was not begun inthe third grade until February 1978.This was not by design, but happenedto be the point in the school year whenthe experiment had already shown en-couraging results in the twohighergrades. It did prove, however, to be theoptimum time to begin: the children
Reprinted by permission from Arithmetic Teacher 27: 12-16; March 1980.
73
73
had reached a maturation level in theirskill development that enabled them toprofit from calculator instruction. Theinstructional emphasis was varied: re-inforcement of basic concepts, devel-opment of number sense, problemsolving, and a greater appreciation of amathematical system.
The parents were informed of theprogram and the rationale behind it ei-ther at the open-house at the beginningof the school year. or by letter prior tothe start of instruction. The solid sup-port of the administration and the en-thusiasm of the children greatly con-tributed to the success of theexperiment.
Three examples of lessons taught inthe various classes are described here.These particular lessons are repro-duced in this article to illustrate thediversity of the program and not be-
% cause they are necessarily the best ex-amples of the many lessons prepared.
74
Third Grade
Subtraction with regrouping
Third grade children often have diffi-culty in understanding the concept ofplace value. Subtraction with regroup-ing is, therefore, a process which theyfrequently find hard to master. Thelesson in figure 1 was designed to helpthem overcome their problems in thesetwo important areas.
The steps were as follows:
1. The child decidei if regrouping is -necessary.
2. If so, the child regroups withoutthe calculator.
3. The child then enters the ele-ments of the regrouped number on thecalculator to check that the regroupednumt,er does equal the original num-ber.
4. The child proceeds to subtract.
We found that those children whohad been unable to cope with the con-
, cept prior to instruction with the calcu-lator were able to do so upon com-pletion of these exercises. Evenchildren who had been successful be-fore the calculator was used found thislesson fascinating and developed aclearer understanding of the operation.We felt that the lesson not only facili-tated the teaching of the subtraction-with-regrouping concept, but also in-creased the children's basic numbersense.
Fourth Grade
Estimation
Fourth-grade children frequently havedifficulty understanding the concept ofestimation and the value of this proc-ess. The lesson in figure 2 was designedto demonstrate that estimation is a us-able, everyday process. It was plannedto(I) broaden the understanding of esti-
mation;,(2) give an added dimension to the use
estimation;(3) show how estimation may be ap-
plied to everyday life.
The children were asked to bring agrocery advertisement from a newspa-per or a flyer from a grocery store.
The children found that the calcu-lator freed their minds to concentrateon developinia method of spendingthe money within certain limits. Inproblem D, the children faced a newsituation and found the calculator in-valuable in finding the cost per ounceand finding the better buy. The exer-cise increased the pupils' understand-ing of grocery shopping and the use ofShe calculator in an everyday situation.They thought about the problem to besolved and not just the arithmetic.
Fifth Grade
Number theory: Quadratics
In this introduction to quadratic equa-tions, the goals were to
(1) extend mathematics knowledge byexploring new mathematical ideas;
Fig. I
Regrouping
Copy the examples and write the missing numeral.Use your calculator to check your work.
For example:
988 8 18 0
Find the sum of 800, 180 (18 tens), and 8 ones. The sum 98988 will appear, which is the number you regrouped.
1 . h t o 2. h t o 3. h t o 4. h t" 0
7 8 5 6 4 3 4 0 2 3 9 7? 18 5 ? 14 3 ? 10 2 2 7
5. h t 0 6. h t o 7. h t 0 8. h t 02 5 9 0 7 5 3 2 3 1
2 4 ? 10 7 5 2
Subtraction (Regrouping When Necessary)
Do the following subtraction examples
E 260 2. ) 60 3. 87 4. 89:-130 70 36 43
5. 106 6. -144 7. 129 8. 14312 63 82 757
9. 474 10. 146 11. 243 12. 243187 107 137 162
Now,use your calculator to check your answers. If you made an error, do you know whatyou did incorrectly? Did you
(1) Regroup incorrectly?
. (2) Subtract incorrectly?
Try not to make the some error again.
(2) explore alternate methods of prob-lem solving;introduce, at a basic level, an alge-braic concept as a transitional ex-perience preparatory to formal in-struction in the middle school.
(3)
The students were told that the exer-cises would help them extend theirknowledge of mathematics by explor-ing new ideas. They were given the ex-amples in figure 3 with the informationthat the numbers that were needed tomake each sentence true formed a typeof pattern. "They were challenged tofind the pattern, and asked if there wasmore than one pattern.
75
Conclusions
Questions have been raised in the re-cent literature concerning the best useof the calculator in the elementaryclassroom. Should it be used as a toolof the existing curriculum or shouldthe curriculum be Changed to imple-ment the calculator? Both uses were at-tempted at U.N.I.S. The third-gradelesson on regrouping demonstrateshow the calculator can be used effec-tively to promote better understandingof a textbook lesson involving a con-cept that many children find difficult.The fourth-grade lesson on estimationgoes one step beyond this approach.A
Fog. 2
A. list the costs of six items on your grocery list in the column morked "real." Round this
figiire to the nearest ten (or hundred) and list the rounded figure in the column
marked "estimate." Then find the real total of your grocery list and the estimated,
total.Real , Estimate
1.
2.
3.
4.S.
6.
Total
What was the difference between the real and estimated totals?
B. If you gave the grocer $30, would you receive change?
C. You are in the grocery with 510. From your grocery list, estimate the cost of eight
items that you could buy with your $10 and hove the least amount of money left
over.item - Estimate
1.
2.3.
4.
5.
6.7.
8. Total
D. You wish to buy an item that is sold in two sizes:
Size 1
Size 2
240 g @ $1.09
480g @ $1.79
Without using your colculotor, which one is the better buy per gram?
With the calculator, how much does a grom cost in each size?
Size 1 Size 2
Which size would be the better buy?ings per gram be?
How much would your SONN
textbook topic was selected, but thelesson was made more challenging.The calculator relieved the children ofthe burdensome calculations that moredifficult new material involves. Thefifth-grade lesson illustrates how thecalculator makes it possible to gobeyond the curriculum, studying atopic not normally included in a fifth-grade textbock.
New dimension
Concepts once thought too difficult fora particular age group were exploredand understood with the help of thisnew tool. The children seemed to de-velope a much better number sense.Problems that the vast majority of thechildren were unable to solve withoutthe calculator became manageable for
all. It was possible to concentrate onanalyzing how to solve the problemoncethe calculator relieved the chil-dren of the cumbersome computationinvolved in arriving at the solution. Inshort, the calculator was not simplyused within the-limitations of the text-book. Its use compelled the teacher torethink traditional ideas and added awhole new dimension to the teachingof mathematics.
Increased understandingMany children elected to work in pairsrather than individually. Invariablythis promoted discussion and increasedtheir uuderstanding of the ideas andprocesses in their specific assignments.which, in turn, meant an increasedlevel of interest and success. The work
Fig. 3For each number sentence there is a .
number that will make the sentence true.fincro number for each sentence. Do you .
see a pattern in the numbers?
1 (--._ X ___) (5 X _____) + 6
2. (........... X _____) (8 X _____) + 15 = 0
3 (...:_. X _......) (15 x ____.) + 50 = 0
4. (.___..... x ___) (13 x ____) + 22 = 0
S. ( x ........) (102 x _____) + 200 = 0
6. (..._ X _____) (37 X ......_) + 70 = 0
7. (.._ X _____) (108 X _____.) + 800 = 0
8 (.____ x ._) (7 X _____) + 10 = 0
9. (..... X _) (28 X _____) + 75 = 0
10. (_____ X ._._) (16 X _) + 55 = 0
From A Teacher's Notebook' Mathematics CoOrtight1975 National Association of Independent Schools.
Used by permission.
7676
,-I
with calculators also established thatthose children with a good visual mem-ory found it easier to memorize com-putation facts with the continued useof the calculator.
Exploration and discovery
Some of the calculator lessons viere de-signed so that the children were com-pelled to explore and discover. In alesson on number theory, for example,they were first asked to try certain ex-amples. with the calculator. They thenhad to solve similar examples. applyingwhat they had discovered. without us-ing the calculator.
Expansion
Problems which the vast majority ofthe class had been unable to solvewithout the calculator now becamemanageable for all. It was possible toconcentrate on analyzing how to solvethe problem once the calculator re-lieved the children of the cumbersomecomputation involved in arriving at thesolution.
Motivation
Without exception the children foundthe calculator added excitement tomath lessons. The fact that the weekly
77
7 7
calculator lesson was a class activity,rather than the customary group or in-dividual instruction. provided an inter-esting change. Children with little priorsuccess in math were elated to find thatthey could also attain success. Themore advanced pupils found the mate-rial challenging. They were encour-aged to assist any classmate having diffi-culty. The children discovered that thecalculator did not lend itself to everytype of problem. but was invaluable inthe more complex mathematical opera-tions at every grade level. The generalconsensus was that the calculator is"fun." 0
Mnn Youn rrcasir am EmweTriBy Phillis I. Meyer
C i - t
-
.
.... - ,,- . .% - -=''''' i ;, .,...
: - -,-;.- --- ...-- 5-4. -* '.....04r. /. ....:. ,.."' . 1. ; . 1. A .
-. e Tv 1 ,,, ..;,..
41fet:/.2-(
Reprinted by permission from Arithmetic Teacher
79
78
.: " 1 : 1 le, . '?:1
- - '.1. . .-r.-
-1",. ..C -1, ..4. r ..
2 . 11 % , "s
y
e. et A- r c s
27: 18-21; January, 1980.
We had just finished a semester ofusing simple handheld calculators inour fourth-grade mathematics class. Ihad given every student a sheet of pa-per and asked each to write what, forthem, was the hardest thing about us-ing the calculator. I got a variety of an-swerspushing t I wrong key and for-getting to "clear" it, turning thecalculator off before a problem wasfinished, making a mistake and havingto start all over again, and most reveal-ing of all, "When you use a calculatoryou havesto-think:,
Several years ago wethad securedseveral old office calculators, the noisy"clunkety-clunk" variety of the fortiesand fifties. The kids had had a greattime adding and subtracting. But mul-
tiplying and dividing were more diffi-
cult since you had to wait for all those
Phyllis Meyer teaches mathematics in the fouith
rout fifth grades at the Watt Elementary School
in Denver. Colorado. She was involved in estab-
fishing the first public school mathematics labora-
tory at an elementary school in Denver. She has
alto had expertence in teaching computer pro-
yawn* in ice elementary school.
"clunkety-clunks." The paper-tapeadding machine was better, but stillmultiplication and division were nearlyimpossible for my fourth graders.
For several years the sixth-grademathematics students of our schoolhave operated a School Store that car-riesschool supplies such as pencils. pa-per, and so on, at a very low price.Profits were accumulating and we de-cided to purchase twenty-five simplefour-function, rechargeable, hand-heldCalculators. We had earned half of thefunds through the School Store and thegenerous PTA donated the other half.Having read many articles about theadvantages and disadvantages of hand-held calculators in the classroominParticular, the November 1976 issue ofthe Arithmetic Teacher, 'a special edi-tion on minicalculatorsI was readyto give it a try. My students were cer-tainly eager to have "instant answers"for their arithmetic problems.
I began by using a series of activitiesdesigned to help'students learn thefunction of each key. The + key waseasy, so was the x key; but the
tricky key sometimes gave you a ""in front of your answer. We exploredto and why this happened and discov-ered that zero was not the beginning ofthe number linethere were negativenumbers. We worked with negativenumbers for a few days, trying. in par-ticular, to understand when and whythey Might occur in everyday lifethethermometer, the family checkbook,and your allowance when you borrowon next week's allowance. We experi-mented with using the function keyswith negative numbers.
30 79
,A14.777"."."at,;01.Pu..
4-gai
it 01411
A %
4.-I 1
21t4.....aiiii
Some students discovered that push-ing a function key more than oncewould cause the calculator to keep re-peating the process with the numberlast entered. We used this with the xkey to talk about scientific notationand powers of numbers. The key
gave us some very long. strange look-ing answers, usually with a decimalpoint somewhere in the string of num-bers. Place value had to be rediscov-ered. The ones column was not alwaysthe column on the far right. Once thebasic understanding of place value was .developed, students could easily con-vect fractions to their decimal equiva-lents and add, subtract, multiply, and
divide them.Now that we knew what the function
keys would do, we could get started.We played some simple games, withthe children playing .with partners. We
took the School Store price sheet and"spent" all kinds of money. We boughtthings &Om catalogs and "playedstore," using the calculators. We dis-covered that some numbers looked likeletters when we turned the display up-
si e down. We worked problems thatga e us answers that would shell words
if v,, got them correct. Students wrotethci own problems to give "word" an-
eswerf. Some of the problems camequiteinvolved. with several t pes of
opera cons required to get the answer.All of here activities aoquaidted thestuden with the capabilitie4 of their
t?calculat rs. as well as with same of thehazards nvolved in,using them.
So far, all was a bed of roses. Youpunched ertain keys and glt certainanswers. The game sheets alt had "+","", "x", r "+" telling yoti what todo next. The buying activities were alittle harder but adding and sub-
-tfacting-sum of money was 'somethingthe children ere all familiar with, be-sides being lo of fun. I
Probably th most valuable part ofour project in sing calculators wasin-solving some q al" problenis. Now wehad to think. W talked about how wewould go about olving each problem.We used role pla ing on some of theharder ones. Did we want the answerto be a "big" number or a "small"
r. .
4 t-
a
41:s...- `,.
13.. 1111,
411
'
81 SO
number? We began with some simpleproblems, solving them Together as a
,Cass. After we had discussed the prob-lem, the students worked it individ-ually on their calculators. Then weasked ourselves, "Does our answerMake sense? Is it about the right size?"If not, then we tried again. We learnedto roundnumblrs and to estimate. Andwe discovered that we still needed toknow some basic arithmetic facts to dothis.'
We used our developing problem.solving skills in consumer mathematicssituations with a variety of materials.some even on the junior high level.Sometimes the solutions got very in-volved. Our simple calculators had nomemory and sometimes we had to useour paper and pencil to "save" num-bers we needed to use later.
I made up problems with outrageousnumbers to be computed. Students be-gan to create their own story problems.using a variety of operations requiredfor solution. Surprisingly. come of themost successful pioblem solving wasdone by the average students. The stu-dents who could "reason" through aproblem and know what should bedone were successful because theywere not burdened or hampered by alaci of computational ability. The cal-culator could do that for them. Thefrustrated students were the ones whohad always been told "what to do."Even though they could recite all themultiplication facts in two minutes.their reasoning abilities had not beendeveloped. Soon, though. with muchencourageinent, these students beganto experiment: Will I get a reasonableanswer if I do this? No? Then I'll trythis. They began to reason and tothink, but not always before theyacted. If they did have to "try again." it
was not a defeating. laborious prObess
because the calculator; would do thecalculations for them. They began tolearn how to "reason through" a prob-
km.A change in attitude was evident
very early in the project. Students whohad previbusly been of the "I hatemath" variety were now saying. "Dowe have to stop now?" We still hadsome basic drill and practice, but thedrudgery was gone when you couldcheck the answer with a calculator and
2 .
111W
4.; V .e:*04- ` .46:r:44,040.16",et:
*1/4 I
14-
Is 40,lb.r4
;-4 4,tik *
know right away if you were right orwrong.
As I look back over my lesson plansfor fourth-grade mathematics, I won-der what some stranger would sayupon seeing the results and outcomesof the fourth-grade mathematics les-sons. Students had had experiencesin
adding, subtracting; multiplying, anddividing with negative numbers;using powers "numbers in correctscientific notation;place value, including 8-place deci-mals;converting any fraction to its decimalequivalent;adding, subtracting, multiplying, anddividing with decimals;reasoning logically to solve com-plicated story problems using two ormore operations with whole num-9'bers, fractions, or a combination ofboth.
Without the calculator. these processeswould not have been possible for anaverage nine- or ten-year-old student.
The calculator is here to stay, atprices so low that nearly every childhas access to some form of simple cal-culator. Throbgh guided activities in
82,
the classroom, it can be a very valuableteaching tool in exploring a variety ofmathematical topics. We used the cal-culator in numerous ways and learnedmany things. The most valuable bene-fit to most students was'learning howto reason logically through a problemto reacka solution. We found thatwhen you use a calculator, you have tothink
ReitmanBeisse, Karen, and David Moursund. "Uses of
Calculators in the EleMentary School.- Ore-gon Computing Teacher 3 (February 197451.
Caravella. Joseph R. Minitaltulaioes in theClassroom. Washington, D.C.: National Edu-cation Association. 1977.
Huntet, William 1... Getting the Most Out of YourElectronic Calculator. Blue Ridge Summit.Penn.: Tab Books. 1974.
Immerged. George. **It's 1986 and Every Stu-. dent Has A Calculator." Instructor Magazine
LXXXV (April 1976)..Judd. Wallace P. "A New Case for the Calcu-
lator." Learning Maga:int 3 (March 1975).Rogers, James T. The Calculating Book. New
York: Random House: 1975,Rudolph. William B.. anti A. D. Claassen. The
Calculator Book. Boston. Houghton MifflinCo.. 1976,
Schlossberg. Edwin. and John !Rodman. ThePocket Calculator Game Book. New York:William Morrow and Co.. Inc.. 1975.,
Schmalz. Rosemary. -Calculators: WiWt Differ-ence Will They flake' Ariihmenc. Teacijer2tr(December 1978). 0
Calculators for Kids Who Can't Calculate*
Betty K. LichtenbergUniversity of South Florida
Tampa, Florida 33620
It is not uncommon these days to find classroom sets of calculatorsavailable for use in mathematics instruction. ** However, when mathe-matics teachers at the high school level are asked what they do with them,a common response is that they're used with the Trig class or other ad=vanced mathematics classes. Thesestudents find calculation tedious anda waste of time, and they certainly should be using calculators. (So do Iand so I do!) For the lower level mathematics students, though, someteachers believe that calculators have no legitimate or appropriate use. Iflessons for these students consist of worksheets devoted to computationpractice, then this is true. Calculators turn such "lessons" into trivialand illegitimate exercises. On the other hand, there are many instanceswhere calculators for these kids (who can't compute efficiently and accu-rately, anyway), turn the lessons into stimulating mental challenges. The
--following collection of possible uses of the calculator provides some ex-,
amples to try!
calculator CompetitionOne type of activity that can be used with almost any mathematics
topic you're teaching at:this level involves competitions with the calcu-lator. Students without calculators are encouraged to compute arith-metical expressions mentally more quickly than students who are us-ing the calculator. The choice of suitable expressions is crucial to thesuccess of this activity. For example, you wouldn't use 497 x 368- -The calculator would win handt down, but you can use 400 x 700.Students who have had a brief review of multiplication of multiples often and who know that 4 x 7 = 28 can determine that 400 x 700 is thesame as 28 x 100 x 100 or 280 000 nicker than another student who
is depressing 111 0 0 , , 0 0 , and Eon the cal-culator.
The psychological advantages of this type of activity include immedi-ate positive reinforcement. There is a high probability of success for thestudents and their confidence will be increased. The competition is non-threatening because they're playing against a machine and you'll rig it sothey'll win!
Paper Presented at the Annual Convention of School Science and Mathematics Association, Pittsburgh. Peansylva-
elk November. 1977.' ma papa is a pmentation of some ideas and activnies that can be used with students ut general mathematics dais-
,Int at the high school kvel. Munn 'modification will make them appropriate foe middle school or junior NO school au-
dos.
Reprinted by permission from School Science and Mathematics
81: 97-102; February 1981.
83 82
Management of this type of activity will depend upon he number ofcalculators you have available. With or two, or even one calculatoryou can pit the Calculator-Kid(s) against tilt! Thinkers. Cards, regularnotebook paper ort_transparencies with expre.-Ainas written on themshould be prepared before class.
Expressions such as these can be used to review place value ideas:
70001-400 + 20 + 61 4x 100+ 3 x 1 145800 1001 10
You may want the students conclude on the list card that 104 is allright the way it is (as ;ong as we're sure itis not 60 and is the same as 10 x10 x 10 x 10 x 10 Y 10 or 1,000,000).
Other expressions that involve computations with whole numbers are:140 x 60 1
4893 x 631 x 0
10 + 97642
91-9-7 185
11234 9991
5 x 2468
Using basic facts of multiplication and the associative and commu-tative properties of multiplication, students can quickly determinethat
40 W 10) 10) 24 xx ;4 x 10) x (6 x = (4 x 6) x (10 x or100 or 2400.
Recalling that for any whole number n, 0 x n = 0 makes this aquick mental computation.Remembering that for any non-zero whole number n. 0 n 0 en-
ables siudents w compute this with ease.Using the idea that 185 can be renamed as 3 + 182 helps translatethis into an easier computation:
997 + 185 =si=
=
997 + (3 + In)(997 + 3) + 1821000 + 1821182.
Using the principle that for any whole numbers a, b, and ca b (a + c) (b + c)
yields (1234 + 1) (999 + 1) or 1235 1000, and an easier com-
putation to ca.-ry out.Rethming2468 as 2 x 1234 gives
5 x 2468 = 3 x (2 x 1234)(5 x 2) x 1234
- 10 x 1234,a very easy computation.
Calculations Too Big for the Calculator
Using the most inexpensive calculators that can perform computations11 in addition, subtraction, multiplication and division but do not havescientific notation capabilites provides an opportunity to explore anothermethod for performing computation of large products. This methodmakes-use-of the distributive property of multiplication over addition.That is, for any whole numbers a, b, and c, a x (b + c) = (a x + (ax c).
To establish the need for this approach to a particular computationstudents should try to compute an expression such as 19 x 5,876,213with the calculator. In some way the calculator will indicate that this pro-
i
duct is too large to display. (Mine flashes on and off.) A quick estimationdiscussion will reveal that the product is a little less than 120 million andmore than 100 million. Then you're ready to perform the computationsome other way:
19 x 5.876,213 = 19 x (5,876,000 + 213) , renaming the larger number as aconvenient sum
2. (19 x 5,876,000) + (19 x 213) using the distributive property ofmultiplication over addition
It (19 x 5.876 x 1000) + factoring so that there is a power
(19 x 213) of ten in the first expression(Powers of ten are "convenient"in this method.)
Now using the calculator to compute 19 x 5,876 and 19 x 213 wehave
(19 x 5,876 x 1000) + (19 x 213) = (111,644 x 1000) + 4047-
= 111,644,003 + 4047= 111,648,047
.
Students can make up their own computations, estimate the mu' 3,carry them out, and then exchange with other students as a check on thperformance. They will be using some important mathematical ideas andthe calculator will be doing the "hard" part.
Square Roots, Cube Roots, etc.
The inexpensive four-function calculator can be used to find accurateapproximations for square roots, cube roots, fourth roots and so on. Theideas are the same. The realistic interpretation of roots becomes apparentwith simple questions such as:
%AK What numbe; rnliltiplied by itself gives 46?a What number used as a factor three times gives 58?
What number used as a factor four times gives 85?
Estimation that uses multiplication facts that are relatively simpleshould be an integral part of exercises such as these. For example
6' st 36andr is 49 so6< Nr4377
orr = 27and41 " 64 so 3 < 01---<4
At this point attention must be given to place value ideas related to or-der and detimal representation of rational numbers. Selecting numbersbetween 6 and 7, or 6.7 and 6.8 or 6.78 and 6.79 involves important skillsand concepts thatdeserve the review and extension that this type of activ-ity provides. The following procedure is an example of classroom tech-nique appropriate to determining a rational approximation forV1048:
,1.
85 84
Consider V1048 10' = 1000(without a calculator)
So 10 <'r8 < I ITry 10.1' with a calculator
10.2' with a calculator10.1 <Ma< 10.210.15' 1045.6783 in between10.16' 1048.77210.75 <Ma< 10.1610:158' 1048.1528 in between10.157' 1047.8432
So, correct to the nearest hundredth we can approximate the cube rootof 1048 by 10.16, which is probably close enough for practical purposes.In the meantime sound mathematical principles have been reinforced andthe computation is a breeze.
SoTry
SoTry
11' = 1331(with a calculator)
1030.301 (Not big enough)1061.208 (Too big)
Prime Factorization
Loren Henry in the November, 1977 issue of School Science andMathematics pointed out that calculators can be used efficiently to deter-mine prime factorization for large (previously unmanageable) numbers.Again, review or reteaching of the prerequ.isite concepts is necessary,particularly in general mathematics classes. The definition of a primenumber as being a whole number with.,exactly two !whole number factorswill determine the set d numbers from which to select trial divisors. Thisshould be followed by a discussion of the fact that any whole number canbe expressed as a product of prime numbers. Then students can partici-pate in a sequence such as the following:
To find the prime factorization of 1066, we begin by considering the first prime num-bs, 2. 1066 has 2 as a factor since the last digit of the numeral is "6".
1066 + 2 = 533, so 1066 = 2 x 533
Now we proceed to find prime factors of 533. Since '333" does notcad in either "0," "2," "4," "6" or "8", 533 does not have 2 as a fac-tor and a calculator isn't necessary to determine this.
Primes2 533 + 2 2 is not a factor (not necessary to compute)
.3 533 + 331. 177.66666 3 is not a factor533 + 5 5 is not a factor (not necessary to compute)
7 533 + 7'76.142857 7 is not a factor11 533 + 11'48.454545 11 is not a factor13 533-+ 13 = 41 13 is a factorSo 533, = 13 x 41. The other factor that was found is a prime number, as well, andwe're finished. That is 1066 = 2 x 13 x 41.
It is beneficial to point out to students that as the primes we divided bybecame larger the results of the computation became smaller. This can beused to determine where to stop in this procedure, and a quick glance at
868,5
the calculator reveals that to us. When the number that we've divided byis larger than the result of our division, we've passed the square root of ,the number. Any factors that could be detected now would have been de-tected earlier by smaller primes, just by looking at the calculator. Sup-pose we're considering 521.
521 + 3 173.666666521 + 74.428571521 + 1 l * 47.363636521 + 13 * 40.076923521 + 17* 30.647058521 + 19 k 27.421052521 + 23 * 22.652173 We can stop here because
23 > 22.652173.
This procedure works well with some numbers, those whose factorsaren't obvious. However, students should realize that the procedure isn'tappropriate for a number such as 3;000,000.
3,000,000 = 3 x 1,000,000 by inspection.1,000,000 has 10 as a factor six times.Each of the six 10's has 2 and 5 as factors.
So 3,000,000= 2 x2x 2 x 2 x2x2x 3 x5 x5x5x5x5x5= 2' x 3 x 5'
. . . without r. calculator at all!
Patterns
Many interesting number patterns can be discussed with generalmathematics students. If they are allowed to peform the calculationswith a calculator there is more time to be spent on forming generaliza-tions and testing hypotheses. The following are examples of some pat-terns for students to examine.
1. Products involving two 2-digit numerals where the numbers can be expressed ,s 10a+ b and 10a +,(10 b) where a and b are natural numbers less than 10. The tensdigits are the same and the sum of-the ones digits is If).For example, consider 76 x 74 = 5624.
The tens digit is 7 x (7 + 1)
76 x 74 = 56. 24
Sum is 10 6 x 4
Similarly, 48 x 42 = 2016 and 65 x 0 3. 4225.A little algebra reveals that
(10a + b) x (10a + i10 bj) = 100a(a +.1) + b(10 b)Thus65 x 65 .1 (100 x 6 x 7) + (5 x 5),
48 x 42 = (100 x 4 x 5) + (8 x 2), and76 x 74 al (100 x 7 x 8) + (6 x 4)
The pattern can be used without being verbalized and it can be easily discovered bystudents at this level. The calculator can be used to check the examples and to pro-tide counter-examples for faulty generalizations.
87"86
2. Patterns that encourage students to guess what comes next are fun to use in the class-room. Some students will be able to provide a reasonable mathematical analysis orjustification. Others may not know why a pattern works, but, nonetheless, will be
- amazed that it does.For example, consider the following patterns and piedict what comes next in each:
6 x 6 = 36 563- 45' = 1111
66 x 66 = 4356 5563- 4452= 111111
666 x 666 = 443556 55563- 4445' = 11111111
12345679 x 9 = 11111111112345679 - 18 = 22222222212345679 x 27 = 333333333
The preceding ideas are just a few examples of appropriate activitiesfor students who are often unmotivated and even unable to perform in asatisfactory way. The calculatcir provides them with the opportunity toexplore mathematical ideas . . . successfully. Let's use calculators withkids who really need them!
REFERENCES
ENGEL, C. WILLIAM, "Meet Mini-Calculator," Fostering Creativity Through Mathematics,Florida Council of Teachers of Mathematics, 1974.
HENRY, LOREN L., "An Invaluable Aid to Understanding N,Thematics: The Hand-heldCalculator," School Science and ,Sfathemattcs, November, 15 '. pp. 585-591.
LICHTENBERG, DONOYAN R., "It's' Greater Than You Think," The Mathematical Log,January., 1976. May, 1976. .-
Ltrwit.titt. BONNIE H. and DUNCAN, DAVID R., "Calculations You Would Never MakeWithout a Calculator," The Mathematics Teacher, November, 1977. pp. 654-656.
TROUTMAN, ANDRIA M. and I2r4ITENBERG, BETTY K., Mathematics: A Good Beginning,Brooks/Cole Publishing Company, Monterey, California, 1977.
-..
P
, 8887
is
Using Caiculators with Juniors
It all started with the class teacher saying shewould like to try something new! I had just beento a workshop on the use of pocket calculators inthe classroomso we had a deal; although neitherof us realised how stimulating it would turn out tobe.
The first stage was to obtain the calculators.The school only had two, so she asked thechildren. All the children who had them at homebrought them in well in advance. One or two evenhad them bought specially, and I was pleased tosee a dozen working calculators. The teacher hadeven eone to the trouble of getting sparebatteries--an indication to the children of herinterest and concern. So we had one between twochildrenHust enough for the whole class to beinvolved in some things, and enough for one eachin group work.
I thought we might start with a trick or two justto demonstrate the 'number crunching' power ofthe calculator.
Choose any three digit number, e.g. 365;repeat it, i.e. 365 365;divide by 13;divide by II;divide by 7;and see if the answer is the number you startedwith. (This also acted as a check to make sure theywere working properly.)
The children were interested and winked on theirown for some time, trying different numbers. Thisgave us the opportinity to talk to individualgroups.
Could it be done with 2 digit numbers (put 0 in themiddle)? single digit numbers? 4 digit numbers?(No!)
Next we started introducing games in groups ofabout four. Although we had copies ofinstructions for the games, by far the mosteffective method was to introduce the game to agroup by playing it with them. The first one weused,-and still one of the most popular, was simplya "guess the answer" game, played in groups ofabout four. One child makes up sum of any sort.e.g.
346+ 48 + 274or 36x 18or half 3178,or double 468, etc.
That child then writes it down on paper in the
BRIAN DAVIES1.L.E.A.
middle of the table and the -other children writedown their guesses of the answer on the same bitof paper., The child who invented the sum thenworks it out on the calculator and whoever was thenearest gets four points, the next three, and so on.It is then the turn of the next person in the group.
Only experience can tell you the best way to playthis eame, but we found it best if the questionswere not made too hard, and if the group stuck toone sort of operation at each session. Halving anddoubling big numbers went down particularlywell, giving an opportunity for talk about largenumbers.
Another popular gaine was "Zero's the limit",played by two children with one calculator.
First player enters a two digit number;second player punches minus, then a single digitnumber, then 4=each player punches minus, then single digitnumber, then 4= ';but each number must be vertically, or,horizontally, adjacent to the previous numberpressed.The '0' &anon may not be used.The player who makes a minus sign appear on thedisplay is the loser.
The sort of games where you aim for a 'goal'are very adaptable for use with a calculator. Withone group of four the goal was 29. We all sta:tedwith zero and could add any single digit number inturn. If you were the one who ended up making29, then you were the loser This game was verypopular and gradually the rules became morecomplicated, highlighting knowledge of tables!
If you landed on an even number you lost a life;if you landed on an odd number yo lost a life;if you landed on the three times tabies you lost alife;if landed on the four times table you lost alife;if you landed on the three or four times ;able youlost a life;
and so on, whoever lost the least number of livesbeing the winner.
Although puzzles and games were popular andgreatly increased the children's awareness ofnumber, we both felt that if calculators were tocatch on and hold a permanent place in theclassrooni they should be of benefit for more thanjust puzzles and games.Could they help with the children's normal
Reprinted by permission from Mathematics Teaching 93: 12-15; December 1980.
? 89
mathematicsdecimals, tables, place value, fourrules?Could they help with work from their mathsschemes?Could they be used to enable children to tacklereal situations which would not have been feasiblewithout calculators, i.e, could they enlarge and
enrich the curriculum bya) highlighting which arqhmetic operation
should be used in various circumstances,b) allowing the children to tackle situations
where the numbers and calculationsinvolved would otherwise have been out oftheir grasp,
c) throwing up answers which are not alwaysunderstood?
We decided to tackle the last first, wondering howmuch of the previous would be used. 'We need not
have wondered so much- since, once the childrensaw a need, they made great strides in their under-standing of such things as large numbers, placevalue, decimals and the four rules.
So, what situation.; could we see, within thechildren's understanding; which would benefitfrom the use of calculators? We sat and lookedaround and talked to the children. Alas. nothingcame up, until I walked past the gerbil asleep in hiscage and noticed for the first time, how fast hebreathed. So this became our starting point,although in retrospect I think there could havebeen any number of them_ This started us on aseries of question and problems, some of themquite eccentric, but none the less still holding a lotof interest for the children.
How many limes does the gerbil breathe in a day?How Many times does the gerbil breathe in a year?How many times do we breathe in a day/year/lifetime?How many times does our heart beat in a day/year/lifetime?flow many seconds in a day?
With the help of calculators these problems camewithin the children's grasp, although none found
them particularly easy.For example, when they tried "How many
seconds in a day?" it highlighted first of all thatnot all the children (fourth yearn knew how manyseconds in a minute, minutes in an hour, or hoursin a daythings that the teacher assumed theyknew! Then because nobody wrote down thisinformation, part of it was consistently omitted ittthe calculations and many did 60-x 24 = 1440
seconds in a day! Other children did not use themultiplication function and just wanted to keep
on adding 60. They, of course, lost count of howmany 60s they had added.
Yet andther child did 60 + 60= 120 seconds in
an hour, then 120x 24= 2880 seconds in a day.Another did 60+ 60= 120, then added 120 twelve
90
times on the calculator to get 1440, then added
another 1440 to get 2880 seconds in a day.Peter tried:
60 seconds in 1 minute120 seconds in 2 minutes180 seconds in 3 minutes240 seconds in 4 minutes300 seconds in 5 minutes600 seconds in 10 minutes
1,200 seconds in 20 minutes3,600 seconds in 60 minutes7,200 seconds in 2 hours
14,400 seconds in 4 hours28,800 seconds in 8 hours57,600 seconds in 16 hours86,400 seconds in 24 hours
These Sorts of problems highlighted three
particular kills:a) solving a big problem by breaking it down
into manageable, steps:b) selecting the appropriate operation on the
numbers;c) interpreting your answere.g. How many seconds in a day?
Does 60x24 give us seconds in a day,seconds in an hour, or hours in a day?
The sort of questions mentioned above seem tohold their own intrinsic interest although, ofcourse, they would not be suitable for all ages. Ihave come across other, perhaps even more weirdproblems that have been tackled.
How many hairs on your head?How many names in the telephone- directory?How many blades of grass in a field?How many wooden blocks in the hall floor?How many words in a book?How much does it cost to light the school each
year?If 45,000 attend a football match and there arenine sections, how many people in each section? Ifthere are 50 rows in a section, how many in a row?One policeman is needed for each 400 people, howmany policemen needed for each section? Howmany for the whole 'round? At £15.00 perpoliceman, how much will it 'cost for theafternoon? How much is that on each ticket sold?
This last story was tried because the teacher wasconcerned about how the children tackled divisionproblems, and in fact her concern was mostjustified, as illustrated by a single problem.
Share 48 sweets between I6, children.
Half of the class of 11-year-olds tried 48-i- 16 = 3
on the calculators, whilst the rest chose to do16+ 48=-1 0.3333 . . . . Why? Because they reallydid not understand the nature of division, orbecause they are used to this notation:
89
16 Pr.Whatever,the reason, the children seemed a lothappier about operating division after furtherproblems of this nature and stories like the above.(At one stage we actually had to get the Unifix outto prose who was right. 48 Unifix shared out to 16children gave them three sweets each. Half thecalculators showed not three but 0.3333 . . . .
Some children were so sure they had done theoperation the right way round that they accusedthe calculators of not working properly!)
Two other areas of mathematics crop up verysoon after children start using calculators. Theyvery soon come across decimal points on theirdisplays and, equally as soon, come across largenumbers. Because of their involvement with theirinvestigation they can be motivated and thereexists an ideal teaching opportuni:y. Children'sreactions to strings of decimals are interesting,varying from "The calculator has gone wrong" toignoring the decimal numbers altogether. In fact itseems that children and adults can operate happilywith decimals without a real understanding ofwhat they are.
When decimals come lip for the first time withDerek he asked what it was. I said,."It is a tenth.Do you know what that is?"
"No," he replied.. "Is it something like aquarter?"
Eventually he decided to ignore the decimalscompletely in his calculations, although when hepronounced his esentual answers he said, "Exceptit's a bit more than that really!" ,
This is an important aspect of the use ofcalculators in this sort of investigation. We foundthat children already knew mathematics we hadnot taught them, and also did not know thingsthat we thought they might or even should know.
One boy had wildly suggested, "How manyblades of grass on the field?" The teacher, neverone to miss a likely opportunity, said O.K. Thisboy seemed to appreciate straight away the needfor sampling or grouping, i.e. he realised he couldnot count all the grass and he thought quitequickly of using a small square to start with. Manychildren would not see tnis for themselves and it isa key stage in investigations of this sort. He madehis sample in a square centimetre and then did asecond, only to find he had different answers. Histeacher suggested trying a few more, lie did 5 andthen added them up and wanted to halve theanswer to get a good number. However, when hedid thii he got a number far higher than any of hisoriginal samples. His teacher. suggested dividingby five. He did this and came up with a decimal hecould norunderstand. So two sources of work hadarisenaverages and 'decimalswhich could befollowed up in more depth at a later stage.
He eventually decided to ianore the decimalfractionnot a bad deciSion considering that the
41
,-whole exercise was one of approximationandproceeded to a square metre. To his amazementthe teacher now found out that this boy, normally`eood at maths', did not know how manycentimetres there were in a metre, let alone howmany square centimetres in a square metresoout with the calculator and paper. Next camemeasuring the field. starting with a trial patch "tosee if it would really work". It did, so he measuredmost of the field but left out the awkwardbitsanother follow-Up for the teacher.Eventually came the first calculation involvingmultiplying by 10s, 100s. 1000s and he ran out ofspace on the display panel. So he and the teacherhad a long talk about what happens to numberswhen you multiply by 10, 100, or 1000 andes entually the child did the calculation without thezeros.
At the end of this, she knew further work wasneeded on decimals (w hen is it not needed!), largenumbers, averages and area, but at leait the childhad begun to see the need for such ideas. Further-more he is now starting to compare fields andlawns, under trees and in the sun, gaining some
: insight into what calculations are needed in realcircumstances.
A fterthe class had been working on these thingsfor about a month we wondered where tc go next.I tried to encourage thr teacher to try thecalculator within the mathematics scheme.However, she was still unhappy about thechildren's use of the four rules of number. Shewas still doubtful as to whether they could applythem in real situationsa problem highlighted inthe recent APU report on 11-year-clds. Couldchildren actually construct the appropriatemathematical model to help them solve realproblems?
We once again relied on the children for ideasand this time we asked them to think of a 'howmany' or 'how much' type question that wouldgive them large numbers in the answer, and thatinvolved the school or the people in if. This timethere was more respOnse and the childrensuggested problems that ranged from th quitesimple to the very complicated.. Some could becounted quite eazil; others needed extensive useof the calculators. All of them involved breakingthe problem up into manageable parts andoperating on these parts.
How much money did Alan, who bought a packetof crisps and a bottle of Tizer each day, spend in aweek?How much in four years in the Junior school?What could he have bought insteadamotorbike?How much milk gets drunk each day/year in theschool?How many cows would the school need to keep?How many exercise books in the school?
9()
One girl later branched out intoHow many pages?How many squares?How much did they weigh?How much did one square weigh?How many squares made a gram?
How many bricks in a wall?Later on a boy whose father is a bricklayer went
on to:How long to make the wall?How many journeys from the pile of bricks tothe wall for the hod-carrier?How many lorry loads? .
How many bricks in d cubic metre?How many doors in the school?Then later:
How many hinges?How many screws?
How much Jo the children in the school weigh?Do they weigh the same as a double decker bus?An elephant?
Does it give false confidence to some? Timothyplayed the guessing game by himself, with a chartlike this.
NUMBER OPERATION GUESS CALCULATORC -CHECK
43 +1+10+100+1000-110x1x10x100+50+150+250etc . . .
..
He was not guessing but simply working it out onthe calculator straight away. This seem -to bewhere the teacher has to make a judgement, as shehas to about many activities in the classroom. asto whether the calculator and/or the acti% icy issuitable, for that particular child. It might well bethat a group game, played with children of hisown ability, could stop him 'cheating' and stillincrease his confidence.
Thoughts after two months workHow exciting and interesting it has all been.How it has converted the teacher to an imesti-gational approach to mathematics.What a liberating force the calculatth havebeen, tnabling the children to tacklecalculations that vould have t len tedious.time-consuming and error-prone without them.How impOrtant is the teacher's attitude, giving
the children time, taking their ideas seriously,taking time for discussion and being interestedin the process as well as the answer.The importance of di -cussion about sampling,and grouping, writing down the calculationsbeforehand if. possible, so avoiding thesituation of
"How did you get that answer ?""Er, don't know, forgotten, er . , . think itwas . ."
i.e.,planning the investigation and breaking itinto manageable chunks.,How it cnabled the children to thinkof a coupleof problems and gave them opportunities tosolve them.How the less able children developed anenthusiasm, especially with the games.How, by watching the children use thecalculator, a teacher can check how sure a childis with the operational side of the wOrk.
How hesitatingly are the buttons pressed?Are errors made with x and ?
Are they fluent with the symbol?How this is leading onto other things in that thechildren are asking about the buttons which arenot being used (i.e. memory, r, Az, 00).
In fact it seems important to have calculatorspermanently available as an essential tool to beused as occasion arises in classroom discussionand work.
92 91
tzi
Calculator Use In The Middle Grades
Charlotte L. WheatleyDepartment of Education
Purdue UniversityWest Lafayette, IN 47907
ti
RATIONALE
The world is presently experiencing an explosion of electronic technol-
ogy which is threatening to render obsolete much of what we teach in
school mathematics. Electronic calculators, unknown in the early 70's,
are so prevalent'that they can be found in nearly every home or office.
Because of their low cost and widespread use in society, calculators have
the potential of reshaping the mathematics curriculum in our schools.
Schobl practices generally lag far behind societal changes and-the utiliza-
tion of calculators in schools is no exception. While on-the-job and in-
the-home use of calculators to perform complex computations is nearly
standard, schools have yet to incorporate Calculators into the mathe-
matics and science curricula. This must change.
When asked to identify problems in schools today, teachers quickly list
pupil apathy and lack of motivation as a major concern. Studies by
Wheatley ' and Wheatley et. al. show that the calculator is a highly
motivating instructional tool. Teachers report that pupils will tackle
problems_with the calculator they would never attempt otherwise. Activ-
ities presented later in this article provide examples of motivating mathe-
matical material made possible through calculator use.Often students are not successful in learning and applying concepts be-
cause the need to compute with paper and pencil may obscure the tar-
geted mathematical ideas. However, with calculators to perform burden-
some computations, the learner can focus on concepts, applications, and
problem solving. The advantage is particularly evident for slow learners
who are greatly aided in learning by a computational device. Likewise the
more capable students can focus on problem solving heuristics and ad-
vanced topics. Teadhers can present problems using- realistic data (large
numbers, decimals) when calculators are available to perform complex
computations. These factors allow emphasis to shift from learning com-
putational rules to applying mathematics in meaningful contexts. Thus
we see that calculators in the clissroom can be highly motivating and fa
I. Viheatley. C. **Calculator Use and Problem Soleil%) Performance." Journal for Remark in Mathematics Edon.
tbs. 1. pas.2. Marley. G.. Shumway. It. Coburn. T.. Reys. A.. Schoen. H.. Wheatley. C and Whne. A. 'Calcutalon m
EkattestearySehook."Anrionene Teacher, 27. (Septeraer 1979). pp. 1$41.
Reprinted by permission from School Science and Mathematics
80: 620-624; November 1980.
9392
1
I
litative of learning. As Phillips ' says we must, ". . . i udge a studentinto organizing things in his head, putting the pieces together to build aneerenlarging structured whole . . . ."
Described below are four activities which capitalize on the calculatoras an instructional aid. A brief description of each activity is presentedwith suggestions for the teacher. These activities may be adapted to othergrade levels or ability groups.
,
CALCULATOR ACTIVITIES AND SUGGESTIONSFOR IMPLEMENTATION IN THE CLASSROOM
Activity One: Estimating sums and addendsPurpose: To develop estimation skills and order relationsGrade Lthl: May be adapted to any gradeProcedure: 1. Present the problem using the diagram below.
440 470
137 + =If theball must land in the range shown, what dis-tance must the ball' be kicked? What number addedto 137 will put you between 450 and 470?
2. Write suggested answers on the board.3. Encourage pupils to explore all possibilities.4. List all answers.S. Ask, "How many answers are possible? Why?"Extension activity (Multiplication and division of whole
numbers) f9oa1. Use the representation shown.
Present problem 56 + = _Find a whole number which multiplied by i 750
56 puts you between 750 and 900..2. Write suggested answers on the board.3. Obtain agreement an the solution set.4. Ask, "Why are there only three numbers
in the solution set?"5. Supply students with similar prOblems to
solve.56
3. 111:15ps. 3. "Go Sack`lo the (*.manias? Where's That?" School Snencr end Mathematics. LXXX (February 19110),
13L
94 93
Activity Two:Purpose:
Grade Level:Procedure:
Activity Three:Purpose:
Grade Level:Procedure:
'Problem solvingTo demelop problem solving skills, to reinforce the useof metiic units, to illustrate the use of mathematics inhealth education:Grades 5 to 81. Present the problem.
Your heart usually pumps about 65 milliliters ofblood Per 'heart beat. If your heart beats 68 times aminute, \how much blood is pumped in 1 minute? In1 hour? In a 24-hour day? In 1 week?
2.* May supply hints below:a. Could\'ou make a chart?b. Try simplifying the problem,' e.g:, suppose the
heart beats 2 times a minute.c. Is your answer reasonable?
3. When Paul is jogging, his heart beats !45 times perminute. Howl much blood does his heart pump in 1minute? In 16\ minutes?
4: Let students discuss their solutions.5. Present a different problem such as:
Sarah's heart pUmped about 4.7 liters of blood in 1minute. How manyiimes was he heart beating?
6. May supply hints befow:a. How is this problem like the problems above?How is it different')b. Try making a, guess and checking it in the
problem.c. Is your answer d,etween 60 and 90 beats per min-
ute? ,
7. Have students find their own pulse.rate. Ask, "Howmuch blood does your heart pump per minute?"
ApplicationTo make comparisons usi g division, to reinforce theuse of metric units, and to develop the concept of rate.Grades 6 to 81. Present the problem. W ose vehicle is the most en,
ergy efficient?2. Discuss computation of kilometers per liter using the
formula below:Kilometers 4- Number of liters used = Kilometersper liter
43. Provide students with the chart below:
Distance.
Gas Used
.
KilometersPer Liters
. .
Mr. RemoMs. KesslerMr. Carter
161 kilometers435 kilometers289 kilometers
.
11 liters28 liters24 liters
ActivityFour:Purpose:
Grade Level:Procedure:
4. Ask, "How does your family car compare?"
5. Challenge students to find the kilometers per liter rat-
ings for the leading automobiles.Developing the concept of decimal
To develop the concept of .1, to compare decimals, to de-
velop decimal ordinal sense.Grades 5 to 81. Teacher states, "Let's teach the calculator to count.
Press 1 + = = = ...."(Most calculators have a constant addend and will
count but you should check the logic and keystroking
of calculators being used.)"Count to 100 by ones.""Now let's teach the calculator to count by tenths.
Press .1 = = = Count to 10. How many times did
you press = ?Does 3.6 come before or after 3.4?Does 5.1 come before or after 1.5?Which is smaller, 6.5 or 3.7?Where is 4.2 on the nutaberline?"
2.
0 1 '2 3 4 5 6 7 8
3. Have the pupils count by tenths. again.,Ask, "When
counting by tenths, what comes after .9?
2.0? 2.3? 2:9?"
4. Have pupils count back from 10 by tenths.
10 .1 = = ,
5. Ask, "What comes before 2.0? 3.1? _ .9? 10?"
6. The concept of hundredths can be developed in a sim-
ilar manner.Many excellent calculator activities can be found in the sources below:
96 95
t
IMMERUEL, G. Problem solving with calculator. University of Northern Iowa. Cedir Falls.Iowa: Price Laboratory. 1977.
IMIELZEEL, G. and OCNENGA, F.. Calculator activities for the classroom. Book I. PaloAlto, California: Creative Publications. Inc.. 1977.
1.14MERIEEL; G. and OCKENGA, E. Calculator activities for the classroom. Book 2. PaloAlto, California: Creative Publications. Inc.. 1977.
kw. W. P. Problem solving kit for use with a calculator. New York: Science Research As-sociates, Inc., 1977.
REYS, R., BF-570EN. B.. COBURN, T.. SCHOEN, H.. SHUmw AY. R.. WHEATLEY. C..WHEATLEY, G., and WHITE. A. Keyitrokes, addition and subtraction. Palo Alto. Cali-fornia: Creative Publications. 'Inc.. 1979.
REVS, R., SEMEN, B.. COBLhtN, T.. SCHOEN. H.. SHUNIwAY, R. WHEATLEY. C..WHEATLEY, G.. and WHITE, A. Keystrokes, mul Iplication and division. Pak Alto. Cali-fornia: Creative Publications. inc., 1979.
REYS, R., BESTGEN, B.. COBLN, T., t4.4Aacecci. R.. SCHOEN. H.. SHuml..y. R..WHEATLEY, C. WHEATLEY, G.. AND WHITE, A. Keystrokes, exploring new topics. PaloAlto. California: Creative Publications, Inc., 1980.
SUMMARY
Calculators' have the potential for reshaping our computationallyoriented curriculum. Its place in the classroom is presently unknown. Butit clearly has a place. The activities in this article suggest ways calculatorscan be effectively used to develop four types of mathematical thought.
6
r
97
CALCULATOR LESSONS INVOLVINGPOPULAT1ON, INFLATION,AND ENERGY
By ERNEST WOODWARDand THOMAS HAMEL
Austi. Poay State University71,137040 L,
One of the joys of studying mathematicscomes from the surprising results that areoften obtained. An example of such a situa-tion is the Rule of 72. This rule states that.if money is invested at r percent com-pounded annually, the amount will,doublein approximately `72/r years. On the otherhand, if the interest is compounded semi-annually, the doubling time is approxi-mately 70/r years; and if the interest iscompounded instantaneously, then thedoubling time is approximately 69.3/ryears. In the interesting article "The Ruleof 72," which appeared in the November1966 issue of the Mathematics Teacher,Brown argues that these results are appro-priate for "small" r. His argument, whichmakes use of logarithms, is not reproducedhere.
The purposes of this article are (1) tosbow how a hand-held calculator can beused to help students discover the Rule of72 and (2) to indicate how the Rule of 72can be used to investieate problems in-volving population, inflation, and energy
, reserves. The lessons described in this pa-per are lessons that each author has usedindividually v:ith both college and second-ary students.
Lesson 1The first lesson is introduced with the
story about the king and the inventor of thechess game. The king wishes to reward theinventor, who is also a mathematician, andasks how he can do this. The mathemati-cian-inventor asks for the wheat obtained
by putting one kernel of wheat on the firstsquare of the chessboard, two kernels onthe second square, four on the third square,eight on the fourth square, and so on forthe entire sixty-four squares of the board. ,
The observation is made that the totalamount of wheat on the chessboard is
+ 2 + 2' + 2' + + 26'
grains, and
1 -r 2 = 3 = 2' 1,
1 + 2 + 2' = 7 = 2' 1,
1 + 2 + 2' + 2' = 15 = 2. - 1,1+ 2 + 2' + 2' + 2' = 31 = 2' 1,
1+2+234-+2'
and thus,
1 + 2 + 22 + 2' + + 263 = 2" 1.
The statement is made that 2" kernels ofwheat is estimated to be approximately 500times the total present yearly world wheatproduction (Bartlett 1976).
The purpose of introducing this story isto emphasize that in the case of a continu-ous doubling circumstance, the numbersget very large very quickly. Some studentsare interested in determining how large anumber 2" is. For those students havingcalculators with an x2 function, the calcu-lation of
2" 1.8447 x 10"
is simple. Students are impressed by therapid change in the display from 216 io 2"to 2".
Next, the formula for compound interestwith inte.,:st compounded yearly is devel-oped. This formula is
Reprinted by permission from Mathematics Teacher 72: 450-457;September 1979.
99 97
p(I +100
where A. represents the total amount at theend of n years, p represents the principle,and r percent represents the rate. f his re-sult is derived in a typical textbook proce-dure.
Next, inflation is discussed. The point ismade that a constant annual inflation rateis an application of compound interest.Students are asked what they think the costof a new bicycle would be in the year 2146,assuming it now costs S100 and that therewill be a 6 percent annual inflation rate.(Most students predict less than $500,whereas the actual amount is over$1500 000.)
No attempt is made to find the solutionto the problem of the cost of the bicycle inthe standard way, that is. an evaluation of
.= 100(1.06)'bs. Such an evaluationwould involve the use of logarithms (whichsome students have not studied) or the useof the exponential function y` (which somecalculators do not have); but these ap-proaches would detract from a strategy thatis perhaps more important than the numer-ical result.
The preferred solution to the bicycleproblem, and other similar problems. ne-cessitates introducing the concept of dou-bling time, that is, the number of years itwill take a certain ,amount of money todouble in value when invested at a com-pound yearly interest rate Of r percent. Stu-dents are easily convinced that the originalamount of money is arbitrary; therefore S1is used. To find the doubling time, it is nec-essary to find n such that
Agi= 1(1 4- (1100 100
e-- 2.
Students then calculate n's for severalgiven possible r's. For example, if r = 6percent, the following values are obtainedwith the calculator showing that n = 12when r = 6 percent.
(1.06)' = 1.06(1.06)2= 1.12(1.06)2=1.19
lob
(1.06)" 4-- 1.90(1.06)12 -4-- 2.01
Thes 'ata are recorded in table 1. Circlednumerals are determined by the use of thecalculator and are integral approximationsfor n. Next, the students are requested toinvestigate this table. Some students willindicate that in each case the product of thepercentage and the doubling time is 72.This conjecture is checked by the class. Thequestion posed is How could we calculatethe doubling time. given the rate? Studentstypically respond that the doubling time is72/r.
TABLE I
Doubling Time in Yearsr (Approximately)
3
4689
12
18
24
Returning to the topic of inflation, stu-dents are asked to help complete table 2.The entries in the first two columns aregiven and all others are completed (circlednumerals), assuming a 6 percent annual in-flation rate. For this example, the doublingperiod is 12 years (72/6). The entries in thethird column are double the correspondingentries in the second column, since onedoubling period will have elapsed. The en-tries in the fourth and fifth columns are de-rived similarly. The date for the last col-umn, 2146, is 132 years (11 doublingperiods) after the date for the previous col-umn. Hence, the entries in the last columnare found by multiplying the correspond-ing entries in the previous column by 2".
The world population in 1975 was ap-proximately 4 000 000 000. or 4 x 10°. andthe annual population increase has beenfairly constant at about 2 percent (1976Census, p. 866). Table 3 is then completed,assuming the rate of increase in populationWill remain at 2 percent. Observe that the
98
TABLE 2Costs at a 6 Percent Annual Inflation Rate
Item 1978 Cost 1990 Cost 2002 Cost 2014 Cost 2146 Cost
Soft drink
Movie ticket
Bicycle
Compact automobile
Modest house
$4C70°
(549 15)
(51 638 40D
'16 000) 32 000) Q65 536 000)
120000) ($24000 491 52000)
doubling time is 36 years. As before, thecircled numerals are calculated by the_ class. Next, the observation is made thatthe total land area (including inland lakesand rivers) is approximately 1.36 x 10"
square meters (1976 Census, p. 867). As-suming a constant 2 percent populationgrowth, there would be approximately oneperson for each square meter in the year2515. A calculator is not needed to com-plete this table but it does make the com-
putation easier.A rather complete class development of
this lesson takes approximately one hour.At times, only portions of the previous twotables are completed in class, and the stu-dents are asked to complete these tables as
an out-of-class assignment. (In one in-stance this lesson,ivas used alone, withoutthe next lesson, when only one hour ofclass time was available.)
The first class is ended by presenting thestudents with the following problem:
Many years ago a fictitious , king had2000 barrels of wine in his cellar. Giventhat his court consumed 20 barrels of winelast year, and past increases in consump-tion would indicate a 12% annual increase,how long will the king's wine supply last?
The students are asked to guess at a solu-
tion and then challenged to solve the prob-
,:.
101
lem by whatever means they can devise.Most students guess many years more thanthe actual answer of approximately 22years. Sometimes students are able to solvethe problem correctly prior to the next class
TABLE 3World Population Assuming 2 Percent Growth Rate
Year
1975
2011
Predicted World Population
4 x 10'
2047 (1.6x 1010)
2083
2119
2155
2191
25/5
( 6.4 x 10)
(1.28 x 10)
(2.56 x 10)
meeting but only after considerable effort.The difficulty they have solving the prob-lem generates considerable interest in themethods that are developed in the secondlesson.
Lesson 2The second lesson is involved with the
investigation of problems concerning howlong energy reserves (coal and petroleum)would last, assuming a constant annualpercentage increase in usage. These prob-lems (e.g., the wine problem) are much
-more complex than the prolilem of deter-.
mining how much energy reserves wouldbe used in a given year (e.g.. the inflationand population problems). In order to solvethese more difficult problems easily, let xbe the initial amount of fuel used, x, theamount of fuel used in the ith year after theinitial year, and r = 6. Next, the data intable 4 are presented, Of particular interestis the fact that
and
24 12
E x, 2E.;0.13 #.1
34 24
Ex, 2 E x,i-25 I12 0.4
These data are determined by use of ahand-held calculator. For example, x6 =(1.06)6x. Many students recognize that the
numbers 1.06, 1.1236, 1.1910, and so on arethe same numbers that were obtained inthe completion of the doubling-time tableof the first lesson, using r = 6.
The observation is made that for 1 i12,
.X..624 7E1 2,4142 4x, .
This result is to be expected since the dou-bling period is 12 years when r = 6. Of par-ticular significance is the fact that 2:1,2_,x, isthe amount of fuel used in the first dou-bling period, that r,..4,3 x, is the amount 'of.fuel used in the second doubling period,and that r62,x, is the amount of fuel usedin the third doubling' period. Letting.
= y, the amount of fuel used in thesecond doubling period is approximately"2y, the amount of fuel used in the thirddoubling period is approximately- 4y, theamount of fuel used in the fourth doublingperiod is approximately 8y, and the amountof fuel used in the nth doubling period isapproximately 2-'y. These conclusionssuggest that it is important to develop ,atechnique for finding y without using therather tedious computation nresentedabove.
A point is made that r2, x, is the sum ofthe terms of a geometric progression andthat it is easier to approximate this sum byusing an appropriate arithmetic series. Ifa, = x, a,2 = 2x, and d = 1/11(x), then
TABLE 4
x, ot 1.06xx2 1.1236xX3 a 1.1910xx4 1.2625xx2 1.3382xxek 1.4185x
xls .1. 2.1329x .1. 2x,x,4 2.2609x .1. 2X2Xs,* 2.3966X .1. 2X3X1 .1. 2.5404A .4. 2x4x,21. 2.6928x *2xsXls .4. 2.8543x 2x6
x22 * 4.2919x .1. 2x,3 * 4x,x24 * 4.5494X .1. 2x14 4X2
X27 *4.8223x *1-x,s 4x3
x22*5.1117x *2x104"4x4x2, +. 5.4184x *2x,, +. 4x,xx,* 5.7435x .1. 2x,2* 4x6
x2 I.5036xx2 A. 1.5938x'
X19 .1. 3.0256x .1. 2x,x20 * 3.2071x 2xs
xs, .1. 6.0881x +. 2x12 4x7
x32 .1. 6.4534x +. 2x20 4x2
x2 * 1.6895x x21 * 3.3996x 2x9 xss * 6.8406x .1. 2;21* 4x9
x10* 1.7908x X22 4. 3.603$.a +. 2.x10 x24 A. 7.2510X A. 2X22 4X10
xis .1. 1.8933xx12 2.0121x
x23 * 3.8197x 4. 2x,,x24 *,4.0489x ± 2x,2
X23 .4. 7.6861x *2X23.b 4x11x36 * 8.1473x A. 2x24 4x12
x, 2. 17.8819x 35.9832x - 72.4042x3J
102
100
a1= 1:00xth= 1.0xa3=1.-i-gxa.= 1.0xas 1.1.6x
,a.= 1.713xa7=1.31-1x
a.= -1.63xa,=, LUZ
= 1 Ex1.Mx
a12 2.00x
These values are corn-.pared with the corre-sponding values of x,through X12.
44k= x + 2x 3= (12) = x 12 = 18x.
2 2
Since E2,x, = 17.8819x and Dita, = 18x,using E;;;,a, in place of r.2.,x, results in anerror of less than 1 percent,
.1181x17.8819x
The important generalization is made thaty -÷- 3/2(xm), where m is the length of thedoubling period.
Purists are probably shocked that anarithmetic series is used to approximate ageometric series on thehasis of a single ex-ample, but students are willing and able to
accept this obvious impreciseness for thesake of convenience. Also, estimates ofavailable energy reserves are questionable,at best, as a result of ecology constraints,prices of a given resource, and tech-nological advancements. As long as thecomputed results are accepted for therough estimates they are, the authors feel
justified in using reasonable Mathematical
approximations.
The lesson is continued byletting A be the amount of fuel available
(estimated reserves) andletting n be the number of doubling peri-
ods the fuel will last.
The contention is that
A y + 2y + 4y + 8y + + 2"-'ymiy(1+ 2 +4 + 8+--+2"-')- y(2"-+ 1).
103
e A2" 1 and
yr.-+1.
yIn 1977 the world petroleum reserves
were estimated at about 6.46 x 10" barrels,and about 2.17 x 10,0 barrels were con-sumed that year (Energy Statistics 1978).Jointly, table 5 is completed (circled nu-merals) using the information given in thefirst three columns.
The only column in table 5 that is diffi-cult to complete is the next-to-last columngiving values of n for which 2" = A/y + 1.Some students have studied logarithmspreviously and are able to quickly recallthat the solution is given by
log frt + I)yit =
log 2For those students unable or not wanting touse logarithms, the graph of f(n) = 2" (fig.1) is used. To make sure that the studentsrecall what the graph of f(n) = r lookslike, a few of the points are plotted beforeplacing a transparency of the graph on theoverhead projector. From the value ofA/y + 1 on the f(n) axis, students are di-rected to go horizontally to the graph of thefunction, then down vertically to find thevalue of n such that f(n)= A/y +1. For ex-ample. if A/y +1 = 1.55 then n ---a 0.63. Thevalues of t1 in the table are those found byusing logarithms; therefore, use of thegraph may result in values of n beingslightly different (but still within 0.1) fromthose listed in the table. Errors caused bythe impreciseness of the graph rarely havea significant effect on the value of mn.
The problem of coal resources is studiednext. In 1974 it was estimated that the U.S.coal reserves were about 4.34 x 10" tons(Coal Resources of the United States 1974),
and approximately 5.58, x 10 tons wereconsumed in 1975 (Monthly Energy Review
1978). Table 6 is completed by the studentsas a group.
Recent annual increases in usages haveusually been under 5 percent; however, it
appears this figure may increase to 10 per-
101
a
TABLE S
EstimatedPetroleumReserves
A
AnnualUse Last
Year
x
Rate ofIncrease
r
DoublingTime
.-mnr
AmountUsed in
FirstDoubling
Period
3P -I. (x) (n1)A +IY
No. ofDoublingPeriods
n
No. ofYrs. FuelWill Last
MR
6.46 x 10" 2.17 x 101° 0
6.46 x 10" 2.17 x 1010 2
6.46 x 10" 2.17x 1010 4
6.46 x 10" 2.17 x I0'° 6
6.46 x 10" 2.17 x 101° 9
6.46 X 10" 2.17 x I0'° 12
ft*
0 Q.I7 x 10"
9 (3:91 x 10)
0 Q.60 x 101)
® C .95 x 10")
f 0000
Where 2' maA + 1Y
Does-not apply
cent or above because of the shortage ofpetroleum.
The last few minutes of the second lessonare spent discussing some of the results ob-tained and the implications of those results.In particular,
I. if inflation increases at the present rate,'the money system will probably need tobe replaced by another one prior to theyear 2100;
2. famine, war, disease, and so on will notallow population to get to the level pro-jected 'for 2515; and
3. as reserves of fuel become depleted, thefuel becomes harder to get, thus drivingup prices and decreasing usage.
(The students were delightfully per-ceptive, and a couple of their comments
and out."One student who had solved thewine problem, after considerable effort,said of the procedure developed in class for
it,,)
104
solving such problems, "Boy, it's a lot eas-ier that way." In a discussion about how in-accurate the estimate of reserves might be,the students deduced that even if there istwice as much of a resource available as theamount estimated, the fuel will be depletedafter only one more doubling period. Afterconsidering this dilemma for a few seconds,another student said; "Changing the per-cent (rate of increase) is the only solution?'That statement sounds a lot like the mes-sage energy experts have been giving.)
The second lesson is closed with a hypo-thetical example Albert Bartlett used in atalk the authors heard him give. In his ex-ample, there is bacteria in a bottle and thisbacteria doubles in amount each minuteand the bottle is large enough to last 'onehour. If this situation originates at 11:00o'clock, then at
11:01 lots of room;11:02 still lots of room;
102
.,
80O
.
:.-
70
60
. An)
,,
50
40
30
20
10
1 2
11:59 still lots of room(only half full);
12:00 full.
3 4 5 6n
Fig.'I. Graph of f(n) - 2".
' completed in about two hours. Portions ofthe -tables could be completed in but-of-class assignments, thus cutting down onclass time. For the most part, students whowere capable of understanding compoundinterest seemed capable of understandingthe lessons. Students did react favorably tothe problems studied.
These lessons are recommended in thatthey
Bartlett said that right now in the inflation,population, and fuel situation, it is 11:59.
Evaluations and ConclusionsAs indicated earlier, the authors used
these lessons with a variety of students(both college students and high se; stu-dents) and found that the lessons were
I
105
1. concern interesting and vital problems,
103
a
o
TABLE 6
EstimatedCoal
Reserves
A
AnnualUse Last Rate of
Year IncreaseDoubling
Time72-r
AmountUsed in
FirstDoubling
Period
3y -2-(x)(m)
A- +
No. ofDoublingPeriods
n.
No. ofYrs. FuelWill Last
ntn
4.34 x 10" S.58 x 10* . 0
4.34 x 10" 5.58 x 10* 2
4.34 x 10" 5.58 x 10* 4
4.34 X 10" 5.58 x 105 6
4.34 x 101' 5.58 X 10* 9
4.34 X 10" 5.53 x lOs 12
° Where 2° -A + 1
6° Does not apply
IP
C)0 (.01 x 10")
(.00 x 1010)
(6.70 x to')
0 (5.02 x 1.0')
Oe 48
a
2. relate mathematics to evaluation of so-cial and scientific problems, and
3. give an example of a situation where useof a hand-held calculator makes the in-terpretation easier.
The authors suggest the possibility of in-vestigation of similar problems involvingthe availability and.use of other natural re-sources, such
coinaluminum and copper.
Teachers coal! give students data on esti-' mated resources,and current usage patterns
or ask students to find the information in. the library.
REFERENCES
Brown, Richard G. "The Rule of 72." MathematicsTeacher 60 (November 1966):638-39.
Bartlett. Albert. "TI Exponential Function. Part I."Physics Teacher 6, 1976):393-401.
Department of Enc. Monthly Energy Review (No-vernixr 1978).
Geological Survey Bulletin 1412. Coal Resources ofthe Milted Slates. January'., 1974. p. 33.
:06104
Institute of Gas Technology. Energy Statistics vol. I,no. 4. Fourth Quarter 1978. p. 45.
U.S. Bureau of the Census. Statistical Abstract of theUnited States: 1976, 97th ed. (Washington. D.C.1976): pp. 866-67.
RELATED READINGS
Dunn. Samuel L., and Lawrence W. Wright. "Modelsof the U.S. Economy." Mathematics Teacher 70(January I977):102-10.
Dunn. Samuel L., Ruth Chamberlain. Patricia Ashby.and Kenneth Christensen. "People. People.People." Mathematics Teacher 71 (April 1978):283-91.
Vest, Floyd. "Secondary School -Mathematics fri'mthe EPA Gas Mileage Guide." MathematicsTeacher 72 (January 1979):10-14.
Wagner, Clifford H. "Determining Fuel Consump-tion-an Exercise in Applied Mathematics."'Mathematics Teacher 72 (February 1979):134-36.
Weyland, Jack A.. and David W. Ballew. "A RelevantCalculus Problem: Estimation of U.S. Oil Re-serves." Mathematics Teacher 69 (FebruaryI976):125-28.
The authors are indebted to Mrs. Gene Moigan forher assistance in the preparation of this manuscript.
$
Section V
o._
PROGRAMMABLE
...
107105
1
0. T
*
c
THE 'CALCULATOR 'IN THE CLASSROOM:
REVOLUTION OR REVELATION?*
Leonard,E. EtlingerAssociate Professor of CurriCulum and Instruction
and Directorvof Teacher CorpsChicago State University
Sarah KrullDirector of Teacher Corps
St. Lonts University
----- Jerry SachsPast President
Northeastern Illinois University
tr
Theodore J. StolarzProfessor of Psychology
Chicago State University
O
O
*This material was prepared with the support of the National Science
Foundation Grant Number SER 78-1416. Any opinions, findings, conclusions
or recommendations expressed herein are those of the authors and do not
necessarily reflect the vi s of the National Science Foundation. In the
operation of this and other .projects, Chicago State University has not and
will not discriminate against any person' on the grounds of sex, age, creed,
color, Or national origin.
109
06
THE CALCULATOR,IN:THE CLASSROOM:REVOLUTION OR REVELATION?*
I was an elementary s-hool student in the late 1930's and attended theErnest Prussing Elementary School in Chicago's farms northwest side. It was
a rather new school then. It liad'an ekcelleni reputation for achievement'and, as I recaili-it was provided With-the best of everything that could beprovided during the great depression. We had one of the first "adjustmentteachers" in Chicago and I remember taking a lot of tests that ultimatelyresulted in my skipping a grade. But what I noticed about the adjustmentteacher (apart from the fact that she was an attractive' young woman) wasthat she used a stopwatch and a slide rule. It was the first time in mylife that I had seen such things:- I remember thetI had to build up myCourage to ask her about them and, she demonstrated them to me. I had nouse for a stoopwatch, but a ruler that could do arithmetic seemed like amiracle. I had to have one.
- Ted Stolarz
The microelectronic revolution is with us and the world will never be thesame as it was before it started. That megiC ruler that could ioarithmetic is now obsolete. The hand-held pocket calculator:car doeverything the slide-rule could and more. Calculators can do arithmetic,algebraic, trigonometric, and statistical s computations and do them faster
'than they can be done by the algorithmS' most students learn in school.',What is more, they operate with an amazing degree of accuracy.
Therere those who are disturbed by this and who feel that children should,not be allowed to use calculators because if they' d6 they will not learnthe basic operations of arithmetic. The danger is there, of course, butthe calculator; when used widely under the direction of good teachers cangenerate interest in learning the facts and skills of mathematics:1 ircethese skills are learned, the calculator,,especially the programmahle'one,can take a great deal of the drudgery out of computation.
To illustrate this point consider the "ruler that could do arithmetic."The slide rule, which has been available for more than 350 years, maKes anumber of calculations quite easy. And, until the recent advent of thehand -held calculator, it was always standard equipment for engineering andscience students. The same fear that today accompanies the idea' of usingcalculators in the classroom also was prevalent when the slide rule began"taking over" the calculating process.
But the slide rule did not destroy students' ability to do fundamentaloperations longhand. It did not destroy their ability to do computationwithout it because to use the instrument skillfully, one had to know theprocess. The slide rule simply gave an approximation to.. the answer withoutsome of the long computation.
Just as the slide rule did not destroy computational ability, the ,
calculator will not do so either if used Properly. Many facets of,mathematics cairbe enhanced and supported by the use,of calculators,especially the processes of estimation and approximation (for which theslide rule was often Used).
110
107
complicated 'sequence in which it does one part over and over a set numberof times or until a predetermined value has been raached and then goes on 'to do other things. It can store information in what are called memoriesand recall whatever is needed for use whenev4i-Tt is needed. This meansthat the programmable calculator, unlike the ordinary calculator, can dosophisticated operations without needing the operator to initiate eachstep.
_
MOWever, it does these things only if one can "teach" it to do so. One
must make the calculator "learn" the separate steps in a sequence. Everymathematical step is possible on a nonprogrammable calculator. But theadVantage of the programmable calculator is its ability to "learn" to domany operations with processes not possible on a nonprogrammablecalculator. It is important to note that if one does not undqrstand the
.'process he is trying to program, the programing is a hopeless task. The
programmer must identify and plan sequential steps for solving the problemas well as plan the order of arithmetic operations. Thus with theprogrammable calculator, even more than with an ordinary calculator, thethinking process of the user is primary if he is to build his own program:
Most of the applications of mathematics to real problems involve severaloperations. The programmable calculator permits Is to address manydifferent real problems without spending hours and hours in longhandcomputation. In addition, the use of,the programmable calculator helps todevelop skills in designing algorithms for the solution of problems which
o recur with different inputs. These skills wil be as essential as basicarithmetic in the 21st century.
e4iThe programmable calculator or computer can also enable a person to, se a 0
special program (prepared by experts in a certain field) to do a complexoperation he cannot do longhand. For example, there are statisticalpackages which can be very useful in providing results which are meaningfulto the user even though he does not have the statistical training to do the
computation or program thet.calculator.
The use of the programmable calculator in a creative way'(i.e., having theuser develop his own programs) is preferred over using the calculator orcomputer to run existing programs. Several examples of creative uses ofthe programmable calculator can be made. For instance, students can gainseveral geometric insights by using a programmaitle calculator. Sappose
o there are youngsters at the level of using basic geometric formulae forlengths, areas, and volumes.
Consider these problems for the student to explore:
1. How does the area of a circle of diameter d comparewith the area of a square of side d? Try severalcases and see if you can get a generalization. Look
112.
`,108
a
at the formulae and see if the generalization makes
sense.
d2/4 and As is (12
1r/4 < 1, therefore, Ac is a little more
than 3/4 of As
2. What happens_to the perimeter and area of a square,rectangle, circle if all basic dimensions are
doubled? Tripled? ,Do the same for volume formulae.
3. If we have a rectangular box ofof sheet metal, what dimensionsthe sheet metal a minimum?
4. You have a can in the shape ofsheet metal. If the volume isheight will make the amount of
fixed volume Made outwill make the area of
a cylinder made offixed, what radius andsheet metal a minimum?
These problems can be quickly explored with the use of a programmable
calculator. First, the student must know how to use the formulae required.
Next he must tell the calculator how to use the formulae; that -is, what
sequence of steps it must take to arrive at a solution. Once-this is
achieved, the student experiments with different inputs to arrive at the
final conclusion.
-The programmable calculator should be introduced as soon as the child is
faced with problems which occur frequently or which require more than a
single step. This will happen not only in the computation but also in the
application of mathematics to real problems such as interest, installment
buying, cost compirisons, etc.
The use of the programmable calculator will not only make computation less
'tiresome, it will also allow -the- child to learn simple programming at an
early date. Computers and their programs affect everyone in many ways. In
the near future it will be as natural for one to understand how to use
programming as it is now to understand spoken and written language. In the
secondary schoolsit will become essential to give many, if not most,
students some hands-on experience with microprocessors. The present
generation of professional men and women, business people, and skilled
workers has-a handicap because the computer, its uses, and its languages
were not a part of their early experience. Those who have had to learn
this later have not had the advantage of'an_early introduction. Even more
serious is the problem of those who must blindly use the results without
any understanding of how they came about or, in the case of management,
make decisions affecting computer operations with only a smattering of
knowledge about what they do and how they do it. This generation, those in
school now, must be given some background on this tool which they find .
everywhere in their
113
1 09
It is obvious that the calculator, especially the programmable calculator,
has much to offer mathematics students of any age. The following
recommendations are made to promote and encourage the use of the calculator
in the classroom.
It is recommended that:
1. Calculators be used in our schools.
2. School districts adopt a policy with guidelines inregard to the use of calculators.
3. Calculators be used as tools to reinforz..e thebasic skills, not as substitutes for teaching thebasics.
4. Calculators be used:
to facilitate the learning,of basic arithmeticskills at all levels,
to facilitate the use of comparison in problem
solving at all levels,
to facilitate the use of estimation in problem
solving at all levels, and
to facilitate problem solving in real lifesituations at all levels.
5. Emphasis in calculator programs should be to teachchildren critical thinking. .Calculators enhancecritical thinking by:
reinforcing the basic skills,
helping in the basic skills of reasoning,
reinforcing the thinking process,
reinforcing problem-solving ability,
promoting logical thinking,
encouraging creative usage,
providing stimulation and motivation,
helping to develop number sequencing concepts, and
aiding in discovering mathematical concepts.
114 1
A4
All of this has important implications for teacher education. It isessential that those who will teach future citizens be prepared tointroduce them to the tools which they will use and which will affect theirlives in so many ways. It is suggested that elementary school teachers berequired to learn how to use a calculator and be urged to learn how to usea programmable one. Secondary school teachers should understand how toemploy the programmable calculator as well as the microprocessor orminicomputer.
The responsibility for this training will have to be assumed by theuniversities. Although there will be specialists who can teach theadvanced material, most, if not all, teachers will have to have somebackground,knowledge. At the moment it looks as though the universitieswill have to be-sure that experience in the use of hand calculators,programmable and otherwise, plus experience in the use of microprocessorsand mainframe computers is available to all who desire it. Further, itwill be the responsibility of the universities to encourage all students toget at least minimal training with such equipment.
It is difficult to predict what the future will bring. The revolution inminiaturization, along with decreasing costs, will undoubtedly produce newwonders in the next few years. Whatever is made available in schools andcolleges should reflect what is available on the market and in general use.The equipment purchased at the present should be minimal and consistentwith good usage so that schools can take advantage of new developments. In
short, we are entering a new age. It is hoped that in education we willenter iL with enthusiasm and not be dragged into it reluctantly.
This document may be obtained from EDRS as ED 191 740.
115
THE PROGRAMMABLE CALCULATOR IN THE CLASSROOM
Theodore J. Stolarz
Introduction
I was an elementary school student in the late 1930's and attended the
Ernest Prussing Elementary School in Chicago's far northwest side. It was
a rather new school then. It had an excellent reputation for achievement
and, as I recall, it was provided with the best of everything that could be
provided during the great depression. We had one of the first "adjustment
teachers" in Chicago and I remember taking a lot of tests that ultimately
resulted in my skipping a grade. But what I noticed about the adjustmentteacher'(spart from the fact that she was an attractive young woman) wasthat she used a stopwatch and a slide rule. It was the first time in my
life that I had seen such things. I remember that I had to build up my
courage to ask her about them and she demonstrated them to me. I had no
use for a stopwatch, but a ruler that could do arithmetic seemed like a
miracle. I had to have one.
.I saved nickels and dimes for what seemed an eternity and finally bought a
wooden slide rule with "painted on" numbers and 4 little instruction book.
The slide stuck in the rule like glue on damp summer days and fell, out if
you tilted it during the dry days of winter. I remember how disappointed I
was when I found out that it could not add or subtract. But I was the only
student in the 8th grade that had one and I carried it with me wherever I
went.
Later on, at Carl Schurz High School an algebra teacher caught me using my
slide rule on an examination. Her scolding made me feel as if I were
?cheating." When I later found time to explain to hetmy reasons for using
the rule she stated that it was a "crutch" and that I was doomed to going
through the rest of my life carrying a slide rule. I had a nightmare where
someone stole my rule and I was unable to multiply 3 by 4. The truth is,
however, that she was partially correct. I have always had one with me
until I purchased my Texas Instruments SR-52 calculator. My expensive
Pickett slide rule'now sits in my desk drawer at the college gathering
dust.
Many educators, with good reasons, fear the use of electronic calculators
by children in the elementary and secondary schools. I am sure that many
of you have heard their arguments. They are often difficult to answer.
Last year I was grading a problem one of my students handed in for a
statistics course I was teaching. About half way through the problem her
numbers started making no sense at all. I couldn't follow her logic. When
I asked her to explain the logic of her solution she stared at the paper
for a while and'then said, "I think my batteries were getting weak." I
117 112
confess to you that I did not know where to begin to help her. I was also
worried that a student might some day flunk out of college because a
transistor failed.
The Microelectronics Revolution
In reality, the "microelectronics revolution" is with us and the world will
never be the same as it was before it started. It is by no means over.
Whereas, in the past a teenager had to be able to add and make change to
get a job we now find that in the "fast-food" hamburger shops the
youngsters press buttons with pictu-res of hamburgers, milkshakes, and fried
potatoes on them and the microprocessor in the register totals up the bill
and indicates the correct change for the bills you offer in payment.
Within a decade or less most college students and many high school students
will have computers of their own with higher level languages such as BASIC
or PASCAL and graphics capabilities. The calculator is now virtually taken
for granted by children and versions'of Texas Instruments "Speak and Spell"
will be teaching vocabulary and spelling. The question is not, "Will ye
make use of calculators in the schools?," it is, "How will'we do so?" In
1971 I purchased one of the first "hand held" calculators (the Monroe Model
10) for $350. A much better machine can be purchased at Radio Shack or
K-Mart for about $10. I have in my home a personal computer system (the
Heath H-8) which is more powerful than the "MANIAC" computer which was used
to verify the mathematical equations for the first hydrogen'bomb which was
designed at Los Alamos, New Mexico, two decades ago. There is no turning
the world bac%.
The Programmable Calculator
We are concerned here, however, not with calculators and their uses, but
with programmable calculators. It is very important that we draw a clear
distinction between these two because, despite their obvious similarities,
they are-really very different machines. A calculator simply performs
operations that are familiar to the user and can be performed by the user
without the calculator. The programmable calculator introduces a new
mental process and A set of new concepts that must be learned. The
programmable calculator is a computer disguised as a calculator. The
"stored program" concept and the use of sequential steps following an
"algorithm" to solve a problem is a computer process. This 'difference
provides us with the opportunity to develop in our studem.s a method of
thinking that will prove to be of great value as they face the world of the
future.
118
113
Let us examine this difference by reference to the diagram below.
OAT*MGM ft y
rotin.RAM
Msnisty
ra a _a._ - -
AftiTH /Lecitc
Uri IT
(Lott(
r,14.44. *MAC 9
fuktvrior4S
INPUT 1 tort 14,1
(Ktf sumo)\DS?
1
1
ft. awe OM. em. we mass awe 4.11
A calculator contains the features shown inside the dashed lines in the
diagram. All actual calculations are performed by the arithmetic unit
which sends the results to a display (light emitting diodes or paper tape).
Input of numbers (data) is accomplished using the keyboard. Certain
keystrokes require the arithmetic unit to address preprogrammed subroutines
which extract square roots, calculate logarithms or trigonometric
functions, or in more complex machines convert degrees to radians, solve
rectangular to polar coordinates, etc.. The analogy to the central
processing unit, input/output functions, and subroutine branches is obvious
to anyone with a passing acquaintance with computer architecture. The
similarity is even greater when we recall the fact that all operations are
actually done in binary code using such standard computer techniques as
two's complement arithmetic and the rule of Boolean Algebra. All that need
be added to build a complete "digital computer" is to provide a program
memory which is sequenced by a "clock" which feeds the steps to the
arithmetic unit at a speed within the limits of its capacity and a data
memory which can store numerical data on a "destructive read-in,
non-destructive read-out" basis. All-programmable calculators have these
features and they differ only in memory sizes, speed, provision of remote
recorded program entry, and the "richness" of the instruction set provided
to manipulate the various "blocks" shown in the diagram.
The key point is this. The logic or "mental process" used to program the
programmable calculator is identical to that used to program any computer.
In the calculator, programming is done by defining a series of "keystrokes"
that will solve a problem for the user. In computer languages such as
"assembler languages" or so-called "higher level" languages such as ALGOL,
BASIC, APL, FORTRAN, COBOL, FOCAL, PASCAL, etc. we use a fixed set of
written instructions in a rigidly defined syntax to solve the problem. The
mental process used by the programmer is fundamentally the same. It
involves developing an "al,orithm" which breaks down the solution of the
problem into a series of seqUential steps which must be executed in an
exactly defined sequence. `There is no room for ambiguity in either case.
.> 119
114
Planning Algorithms
The key to developing programs that will solve problems using computers or
programmable calculators is skill in developing algorithms. The
programmable calculator is ideal for introducing this skill because the
user can concentrate on the problem of how to accomplish a goal without the
added confusion of learning the syntax of a computer language. This is not
immediately obvious. I had the unfortunate experience of doing a:1 this
backwards. My first efforts at building algorithms were with the IBM 1401
computer which had a FORTRAN compiler.. I had little prior experience with
computers and no instruction in their use. I acquired the IBM technical
manuals and spent some weeks before Icould get any idea of how to get
numbers into the card reader and results out on the printer. During most
of this time I had to operate the computer myself and the sheer size of the
thing-was unnerving. The few programmers around the computer did not know
FORTRAN and kept trying to correct my, syntax, not my strategy. Most of my
errors were in program flow, not syntax. The compiler caught most of my
syntax errors. The computer was faithfully doing what I told it to do, not
what "'I wanted it to do.
After about a year of this experience, I was writing many successful
programs, but my efforts tended to waste a good deal of computer time and
printer paper. At about this time we acquired an early programmable
calculator in the psychology department. It was made by the Monroe company
and we had to punch octal codes into blue cards to represent keystrokes,
which was somewhat confusing at first. But I found that solving problems
with this calculator gave me much greater skill in writing FORTRAN programs
for the large machine. No one prior to this took the time to teach me the
fundamental process of designing efficient algorithms. I was too worried
about losing control of the high- speed- printer which could empty most of a
box of paper before the computer operator could stop it or in counting card
columns to construct input format statements.
Later on in this conference when we see demonstrations of problem solutions
using programmable calculators we should keep in mind that the process we
are using includes a series of concepts that are necessary for the
efficient planning of algorithms. The programmable calculator can be used
to teach and/or illustrate these as we develop.this skill in our students.
The following is a list (not complete, of course) of the kind of concepts I,
am referring to.
1. The use of a "program." The key concept which students may notunderstand initially is that of using a "program" to solve a problem. The
programmable calculator has in common with the digital computer the fact
that using a supplied program allows the user to solve a problem which he
may not know how to give: Once a program is written it can be used many
times. A program I wrote for test item analysis in 1971 is, still widely
used at our university (and in many other places) to evaluate multiple
choice tests. It performs thousands of calculations using some rather
sophisticated algorithms. Most users are unaware of the statistical
niceties it contains. They do, however, understand the output it produces,
and that is all that matters. This is the "magic" of the computer;
programming languages are universal. Pre-written programs are available
for a]l programmable calculators and they are very useful.
2. Planning sequential steps for problem solutions. While some texts
start out with "flow charts" and diagrams which define an algorithm, the
student should first learn a fact of life. A computer can only do one
thing at a time. This is true of all computers, no matter how expensive or
sophisticated. We can, by clever programming, make a machine seem to do
several things at the same time. However, if we have one-central
processing, unit through which all data must flow, the computer has the
capacity of doing only dne thing at a time. This is very evident when
using a programmable calculator where tLe sequence of the keystrokes is
important and the task must be reduced to a finite number of steps done in
a certain order.
3. Planning the order of arithmetic operations.6
While I am sure that
the teachers of mathematics present here today teach their students that an
algebraic expression or formula is merely a convenient "shorthand" for
specifying the order of arithmetic operations, some of my students act as
if they do not know this. Planning the series of keystrokes needed to
solve an equation can reinforce the student's understanding of the
importance of this order. For example: a + (b/c) # (a + b)/c.
4. Memory storage and retrieval. Skill in manipulating memory is a
mark of the good programmer. The concept of depositing intermediate
results into a memory "bucket" for later use is not intuitively obvious.
Less obvious is the pladement of a constant like pi or e into a "bucket"
for later use on a repeated basis. This is easier to see on a programmable
calculator than on a computer using a language like BASIC where we might
say, "LET X2=2.712828." The beginner does not see that the computer will
reserve a fixed number of binary digits in memory to hold the value of the
variable designated as X2. Most programmable calculators allow the user to
add to a memory location or subtract a value from its contents directly.
This is excellent practice.
5. Planning,input/output operations. The output of a calculator is
typically a number (or a series of numbers) on a lighted display or on a
paper tape. The builder of the algorithm must plan when the program is to
display these numbers and he must "document" his program so the user will
be able to identify what the numbers mean. He or she must also plan the
program so that it will pause so that data can be entered into the sequence
via the keyboard so that program flow can proceed. This is a skill that
has a positive transfer to the learning computer programming in the
future.
6. Branching forward or backward.sequential the sequence does not alwaysSome parts,of a program may be executedexecuted many times to accumulate sums,easily accomplished by "labels" in manyallow the user to set "flags" which caninstructions which branch programs back
these techniques is very valuable.
121
While program execution isproceed in a single direction.only one time while others may beproducts, quotients, etc. This is
calculators. Some calculatorsbe tested .to bypass "GOTO"
over previous steps. Practice with
116
7. Conditional branching. Most programmable calculators allow tests3f the value in the "accumulator" or display register to see if it is,positive, negative, or zero before branching to a portion of the programoccurs. The test for "greater than, less than, or equal to" a certainvalue is easily accomplished. Such branching is so frequently used incomputer algorithms that it is actually difficult to write a program of anycomplexity that does not use this technique.
8. Forming repetitive "loops." Using the above concepts it-ispossible to teach the construction of programs which "loop" through aseries of steps for either a fixed number of times or until a desiredresult is accomplished. While this is planned for in programmables whichhave a "decrement memory and branch when zero" instruction, it can be donewith any'programmable which provides for the "+-0" branch test. Learningthe concept of "looping" is difficult for many students who first attemptit with "FOR-TO" loops in BASIC or "DO" loops in FORTRAN. It is one of themost widely used techniques in the construction of computer algorithms.
4A good exercise to teach this concept is to have the student write aprogram to obtain the square_root of a number using the method ofSuccessive approximations suggested by Newton. This is how many computerscalculate square roots.
9. Clearing storage. The fact that memory locations can holdft garbage"_from previous program steps and must be "cleared" or set to zeroif they are to be added to or subtracted froth will become painfully clearafter students write programs that "bomb out." I violated this rile theother day after over a decade of computer and calculator use and hundredsof successful programs in use all over the country.
10. Overlaying memory. It is an excellent builder of mentaldiscipline to fit a large program into aG,machine that is limited to a smallnumber of memory storage locations. Clearing and re-using memory locationsbuilds good programming skills. I find myself sometimes getting carelessin this regard when using my home computer which has 24,000 byes of memorywhich will soon be expanded to 40,000 bytes. Starting with a large machinecan develop bad habits.
11. Use of subroutines-Most programmables allow for construction ofsubroutines to perform repetitive chores with economy of program steps andmemory storage. This is a concept that takes some practice before itbecomes a matter of habit.
12. Indirect memory addressing. Some programmables allow for-branching to a program step specified in a memory location. This value canbe the result of calculations within the program. This is an advancedskill but it is within the capabilities of bright students. It provides avery powerful tool in the construction of some algorithms.
12211_7
13. Concern for program length and execution speed. There are, of
course, several ways to construct algorithms to solve any specific problem.
All of them are "correct" if they produce the desired results. The "best"
algorithm is the one that provides the result using the fewest program
steps or keystrokes. Since the "clock" in a programmable calculator is
relatively slow as compared to a computer "clock" this speed difference is
more obvious when using the calculator. The shorter program is also easier
to load since there is less chance for error.
The list is not complete. However, it does suggest how much a person can
learn from the use of a programmable calculator. The little machine that
fits into a pocket or purse is in fact an extremely complex marvel which
contains thousands of electronic components arranged so that they can be
instructed to solve a myriad of problems. In at least this one area, that
of teaching the building of algorithms, its use in the classroom is very
valuable. The positive. transfer of these learnings to future learnings
should be obvious. ,
Other Uses
While up to now my interests in computer algorithms, programming languages,
and digital electronics clearly have influenced my presentation of how
programmable calculators can be used effectively in teaching in elementary
and secondary schools, there are many other uses that can be found for them
that have educational value. It is a feature of this conference to explore
these potential uses through the exchange of ideas. The shared creativity
of many skilled educators can open up avenues and approaches that could not
be foreseen by a single person regardless how creative he or she may be.
Again I must reach into my 47n experiLnces which merely suggest how
fruitful the field is. I will illustrate a fe'w ideasthat I have. In most
of these the teacher does the "programming" but there is no reason why a
student or a group of students could not be involved in this effort.
1. Curve plotting. In teaching the concept of the "normal curve" to
my introductory statistics students I try to convey the idea that the curve
has no"standard" shape but is a plot of an equation involving certain
constants and variables. The constants are pi and e and the variables are
the mean, the number of scores, the standard deviation,-and the individual
scores in the distribution. By writing a program for the calculator which
solves for the height given the values of the variables previously
-Ifehtioned, I can plot the curve several times to show what effect the
variables have on the shape of the curve. Solving
S 2
-(X-M)2/2S2
123
118
for h given n, S, M, and X fifty or a hundred times'is a prospect no man
can face with equanimity. A short program for my Texas Instrumqnts SR-52
gives the values of h in a few seconds and allows the curve plotting to go
on. In a secondary school class the "feel" for simple equations employing
powers of 2 or 3 could be of value. It might also be useful to plot lines
1.4) selected points in linear methods where a program for a "least squares"
fit of a line to a set of points Is easily written.
2.. Building tables. I have written simple programs using the
formulas for compound interest to determine how much I must save, given
current bank interest to have enough money to purchase additional memory or
peripheral devices for my upputer at a certain point in time. I also
calculated depreciation schedules for the SR-52 for my income tax form. I
can.see possible uses for this ISind of programming in teaching business
mathematics or even consumer education to students who will be users to
credit in a few years.
3. Science education. I received a "lunar lander game" program with
the SR-52 and I spent a feu, hours playing with it. Most of the time I
impacted the moon at various velocities and theoretically killed myself
several times. I can see, however, that the acceleration of a falling body
could be demonstrated, or the trajectory of an artillery shell estimated.
The ease of plotting data points gives a better feel for the dynamic nature
of what is happening rather than the static feel one gets when it takes
considerable time to establish a single data point. I have also used the
metric conversion programs when my favorite scotch suddenly appeared In
'alf-liter bottles and I wanted to calculate the price per fifth gallon.
4. Mathematics classes. The combination of the availability of
trigonometric functions and programming capability helped me verify the
correct operation of a "Biorythm Plotting Program" I placed on my home
computer and the'computer system used by.our university. The sine fundtion
iqrused to generate the-curves, of course,, but verifying the plotted points
in the computer program helped "debug" the operation of the program and
ensure that the leap years were handled properly. I can see many possible
uses for this kind of programming in courses. in algebra, geometry, solid
geometry, trigonometry, etc. My lack of familiarity with the secondary
curriculum in these areas, of course, limits me in,anticipating uses.
I will leave the rest to you and I am eager to lee how much -I will learn
before tomorrow is over. Thank you for your kind attention and I hope that'
you will find this conference to be of value in our common goal of
improving the mathematics skills of our students. ,
This document may be obtained from EDRS as ED 193 026.
124.
119
(.7
THE_ CAA_ FORPROGRAMMABLE. CALCULATORf
JoHn J. WAvRlicWith all the attention being given to computers these days, another quite remarkable insfiliment
often gets tweriooked: the prograMmable calculator. I don't wish to open a debate about whether or notprogrammable calculators are computers. Let's assume that they are not. A programmable calculator hasmany computer-like features, but it is basically limited in its innut and output: it accepts numbers andoutputs numbers. Most computers can process words as well as numbers. This can be very nice for games(in one Star Trek version, if you accidentally destroy your own base, Mr. Spock will print you a messagetelling you that what you have done "is not only illogical, it's stupid!") and for non-numerical informationprocessing tasks. Some computers can produce graphics displays, some can be used to control other'ma-chines. Calculator's will only calculate. At the present time, though, they are better at this than modestlypriced computers. If "number-crunching" is your game the instrument of choice is a calculator ratherthan a modest computer. If factors of cost and portability are thrown in, we find that calculators form adistinct breed of machine that has itsAawn right to exist.
At the moment I find myself heavily involved with numerical work. I do have access to a largecomputer system but find myself doing much of the work with a programmable calculator. It may be apersonal quirk, but I find it easier to devise and test algoritms using the calculator. In some cases, though,the results that really interest me require, more storage and speed than the calculator can provide. In suchMel the procedures I use are designed with the aid of the calculator but the computer does the firm! work.It is true that computers run more qUickly'than calculators. Be sure, however, to count the time fromposing the problem to obtaining the solution! In many cases the computers intolerance for misplaced par-entheses and omitted semi-colons-will make the debugging process far more time-consuming. If you workwith a computer center, do you have to,drive there to drop off a deck of cards and come back later forthe results? How long does it -take to enter a program and data into the machine? Of course, if you'reinvolved in-long range weather forecasting a calculator won't do (but neither will a S600 home computer!).
There is another area in which programmable calculators are useful: education. I am working witha mathematics club for elementary school students in grader 4-6. In an effort to convey to students whatis involved in getting a computer to do what it does, demonstrated some simple programs with II Program-
msble calcu:ator. To my surprise, several of the students showed readiness to learn programming. Writingoriginal programs is an excellent problem-solving task for elementary school students. A pair of lourth gra-
ders who decided to write their own perpetual calendar program not only had :o do some research intocalendars and astronomy, but they also had to solve several technical mathematical and programming prob-lems. The calculator is also useful in allowing the exploration of areas that would be less accessible other-wise. A sixth grader, for example, used a random number subroutine to conduct experiments in proba-bility and statistics. The use of the calculator also has provided insight into computers. Students have abetter appreciation of what is involved in computer prograrnining and of what computers can and cannotdo. Some have found it quite easy to learn a computer language after their experience with the calculator.
Learning doesn't takeplace only at school. Indeed if a student has a great deal of interest in a spe-dal subject (like mathematics) it is unrealisticto expect any school to'provide the type of instruction sucha student really needs. Mathematically talented students have-needs, interests and abilities that no masseducation system can cope with. Such students should expect to do a great deal of their learning by pur-suing their interests at home. To parents of these students I would seriously recommend the purchase ofa programmable calculator. At a cost of about S100 one can buy what amounts to a personal laboratory
for work in mathematics and computer science.
Dr. Wavrik is an Associate Professor of Mathematics at the University of California at San
Diego. He is the author of "Fi:,ding the Klingon in Your Calculator", CALCULATORS/COMPUTERSMagazine, January 1978.
4
Reprinted by permission from Calculators/Computers 2: 63; April 1978.
a
125
P' 0
A SUMMER COURSE WITH THE Ti 57PROGRAMMABLE CALCULATOR
By ELI MAORUniversity of WisconsinEau Claire
Eau Claire, WI 54701
Let me wish that the calculating machine.lriviewof its great importance. -may become known in widercircles than is now the case. Aboi/e all, every teacherof_mathematics should become familiar with it. and itought to be possible to have tt demonstrated-in sec-ondary instructionr
Felix Kleinerrilerring to the newly inven mechanical calculator)in an address to high sc I teachers. Gottingen. 1908.
In the ew years since its appearance onthe arket, the programmable calculatoralready has opened up an entire new di-mension in numerical exploration. Hun-dreds of problems from number theory., al-gebra. and calculus. which until now haverequired the use of computer facilities (andhence the knowledge of a programminglanguage), Can now be programmed andrun on' one's own pocket calculator athome, on a trip. or while on vacationpro-viding many hours of reward and virtuallyan endless variety of mathematical topicsto explore.
Thetefore, when 1- was asked by thenewly founded Eau Claire Association for
- High-Potintial Children. Inc. to plan a spe-cial mathematics course as part of its sum-mer program for 1978. my instant choice'fell on the programmable calculator. Theproposal that subsequently .evolved calledfor a six-week. three-times-per-week coursebased on the Texas Instruments TI 57. Thechief philosophy behind the proposed prof- -ect was to offer an enrichment program forstudents of the upper elementary and jun-ior high school levels that would enhancetheir interest and motiviiion in mathe-matics; consequently, the syllabus put
stress on ideas and Concept's, rather than ontechr.,ques and
of
learning. The coursewas,to be one of fifteen courses-offered-bythe associationfoiits summer -program,ranging in topics-from biology, and astron-omy to art. Latin. and Greek mythology.
A grant enabled the association to, pur-chase seventeen calculators. which in turndetermine d- the number of participants ,inethe program: the plan called for every stu-dent to have his or her 'own instrument,thus eliminating some of the frustrationsthat are common at computer terminalswhere many users have to share the samefacilities. Two sections of the course wereofferedone for ages 8-11 and the otherfor ages -12-15. There were no entrance re-quirements, but the intention; was to offerthe course to particularly gifted and moti-vated children; and this indeed turned outto be the case.
The TI 57 programmable calculator waschosen chiefly because of its simplicity ofoperation. It has a modified algebraic logicknown as AOS (Algebraic Operating Sys=,tem). which riot only enables algebraic ex-pressions to be entered from left to rightexactly as written, but also "recognizes" the-hierarchy among algebraic operations. If,for example, we press the' keystroke se-quence 1 + 2 ,x 3 =. most calculators willdisplay 9 as the result. because they per-form the instructions sequentially as theyare entered, always completing the pre-vious calculation whenever a new opera-tion is encountered. The TI 57, on the otherhand, will display the correct result, 7, be-cause it "knows" that multiplication mustprecede addition. This feature...was consid-ered to be of major importance; because ofthe age level of the participants it was veryimportant to make the arithmetic as simple
Reprinted by permission from Mathematics Teacher 73: 99-106;
February 1980.127121
gts possil;1$ and to avoid,any complicationsdue to different :logics." Also, the pro-
.gramming features of the TI 57 are justabout right for the level of the students in-volved? the instructions are simple andstraightforward, eliminating the necessityof learning any sophisticated programminglanguage. This meant that we could "go'straight to businessconcentratinvon themathematics itself froth the very beginning
of the course.Perhaps a brief descr:ption,of the pro-.
gramming capabilities of the TI 57 wouldbe in order.' The instrument looks very'much like any other scientific calculatot,
but a special key denoted 1,111'.1 (';learn ")
transforms it into a "student" ready to re-ceive instructions from us. the "teachers"(this attitude, that the children: ate reallythe teachers of their own calCtilators. was"maintained throughout the entire course/.The calculator is capable of receivinrupIpfilly program instructions, which is morethan sufficient for most elementary. pur-poses. A "pause" key instructs the calcu-lator to display any intermediate results
during program execution, whereas theR/S ("run-stop") instruction halts the pro-gram at any desired step. There are two de-cision keys, x = t and x t, that comparethe current number x in the display-registerwith a prestored number r in the test-regis-ter and branch the program according tothe outcome of the comparison. Branching
can also be affected through the GTO ("goto") and RST ("reset") instructions; theformer diverts the program to a specified
location ("label"), whereas the latter sendsthe program back to the beginning. There
are several editing keys that enable one tomake corrections or changes in the prorgram without keying in the entire programagain. The instrument has eight memories,
irl,which numbers can be stored, added to.subtracted from, multiplied by, or diyided
by, as well aS exchanged with the displayvalue. Thus, the calculator has full "mem-
ory arithmetic" capabilities, a feature that
we constantly exploited in our programs.The display itself has an eight-digit capac:ity, but all calculations are internally done
-
with eleven digits that are then rounded offfoi displaying. Figure ) :aims the key-
board of the instrument.
r.
6
*;-'7.`".7 1 7:7;
.a.L.
;et4-8'.-;*:ear .%-darb
.... w_
; i .... . 1 .":'t -.w..- saw ....... :-....
I..... ........... Waalea.1 idilli.
,.I ."1.7..." U....rf I..L. 1.....71r. ::::. . . ro, I
"---=t-='s't it'. it.s..,1 2.411:.
t Ci. 7 1 1 y 1 2 i 3-.1 ....r.c- inr. .,...wir -p:*. 1
. -...-ol .
--., 4....T "'":". ........... _
-'-- ::,
Fig. I, The TI 57 prograRmable calculator.
Description.of the CourseThe course met for six weeks, three times
each week, in the seminar room of ourmathematics' department. There, sur-rounded by books and periodicals, the pu-pils were exposed to some of the flavor ofacademic life at a university. They werefree to check out any books they wished,and one of them asked tp take home thethirteen books of Euclid! Whether he ac-tually studied them I do not know, but theincident shows something about the curios-ity of these students to learn and explorenew ideas in mathematic%, a' curiosity thatpersisted throughout the entire course.
Each topic of the syllabus was used as an"excuse" to introduce a new concept, bringin some novel idea. or point out some high-lights from the history of mathematics(table 1), In this way the students were ex-
. posed to the spirit and flavor of mathe-matics even without having the more so-phisticated tools needed fora fuller grasp
128
122
of these concepts. For example, the multi-plication table was used to discover various"hidden"---mathematical. patterns, such asirithmetit progressions (any row or columnof the table), the progression of squares 1,
4, 9, 16, 25, ... along the main diagonal, orthe symmetry of the entire table with re-spect to this diagonal. The students them-selves discovered thisefeatures in responseto my questions, Sand they were fascinatedto find out, that the boring multiplicationtable has so many interesting secrets in it.
Next we learned how to "teach" the cal-culator to.generate some of these patterns.For example, to generate the progression of
TABLE )
Topics Covered in the Two Sections
Section 1 (agesI. bitroducticn: hot+ to use your TI 57 program-
%able calculator.2. Teach your calculator how to count (counting
by ones, twos, tens counting backwards).3. The multiplication 'able on your calculator (all
muhiples of a Liven integer).4 Sequences of numbers (all even numbers, all9 odd numbers. all squares).5. Number Patterns (sum of the first n integers.
sum of the first n odd integers).6. Arithmetic, geometric. and Fibonacci progres-
sions.7. The prime numbers.4. Introduction to algebra.,(letter numbers. arith-
metic sentences expressed algebraically).9. ;Ififinity and limits (limit of (n + ) )/n as n co.
sum of the decreasing infinite geometric pro-gression).
10. Games on the calculator (nonprogrammed andprogrammed games).
II. The number e.Section 2 (ages 12 -15):
1. Introduction: getting acquainted with your TI57.
2. A quick review of arithmetic.3. What is programming?4. Scientific notaticr.. '4oit, to write large numbers.S. Introduction to 1:Aebra.6. Number progressions (arithmetic, geometric.
and Fibonacci).7. The prime numbers.S. Other number patterns (Ulam's conjecture.
happy numbers. Pythagorean triples).9. EnlitS.10. Newton's iterative method: finding the square
root-of a-number.-II. Introduction to trigonoraeuy.11 The number sr.13. Calculator games.
129
squares we key in the following program("2nd" means an upper-case instruction).
LRN1 we
SUMORCL 0
x2
2nd PauseRSTLRNRSTR/S
Again, it was the students who suggestedthis program. When two different programswere proposed for the same task, we wouldwrite both on the blackboard and discusstheir relative merits or drawbacks; if bothprograms worked properly, we wouldadopt the one reqUiring fewer keystrokes.Ultimately, some of the youngsters becameso addicted to programming that theywould come up with suggestions for mak-ing even a very short program shorter yet!
We then moved to arithmetic and geo-,metric progressions and their sums. Wementioned some of the many cases wherethese progressions occur in daily life andnaturestaircases, mile signs along a road,the countdown prior to a rocket launch, thepetals of a flower, frequences of the musicalscale, and compound interest, to name buta very few. We then recounted some fa-mous anecdotes related to these progres-sions. The 'students were fascinated by thefamous story of how young Gauss foundthe sum of the first 100 integers in responseto his teacher's special assignment given lohim so that he would not be bored in class.Even more fascinating was the story about.the inventor of the game of chess: Whensummoned by the Shah of Persia and askedwhat reward he would like for his inven-tion, he merely requested to have one grainof wheat placed on the first square of thecheckerboard, two grains on the secondsquare, four grains on the third, and so onuntil the entire -board would be covered.The Shah, stunned by the modesty of thisrequest, immediately called for a sack ofgrain to be brought in, but it soon became
123
clear that not even the entire grain in thekingdom sufficed to fulfill the task. (Thename "chess," incidentally, is a distortionof the word "shah.") This beautiful legendnever fails to fascinate novices when theylearn how quickly the sum of the progres-sion 1 + 2 + 4 + 8 + grows. We wrote aprogram that displayed the partial sums ofthis progression and stopped after the re-
TABLE 2
1 -- Mo, MI, M2
Yes
Display M2
Increment Ma by I
Multiply M, by 2and keep product in M,
Add M, to M2and keep sum in M2
qui,ed number of terms had been reached.To make things simpler. we contented our-selves with a 5 x 5 checkerboard, so as toavoid entering into scientific notation, withwhich the younger class was not familiar.The program is presented here as a flow-chart (table 2) and in keystroke form (table3). In the former. Al,, i = 0, 1, 2 denotes thecontent of memory
The children were absorbed in anticipa-tion for the calculator to halt the programafter twenty-five steps and display the totalnumber of grains: 33 554 431. (The pro-gram can, of course, be considerablyplified if one dispenses with the halt in-struction:)
TABLE 3
LRNI RCL 2 2nd Lbl 2
STO 0 2nd pause RCL 2STO I I R/S
2nd Lbl I SUM 0 LRN
SUM 2 2 RST
RCL 0 2nd Prd I 25
2ndx=t RCL I x tt IGTO 2 GTO I , R/S
,Another subject that generated great cu-riosity was the prime numbers. Most of thepupils had learned about the primes inschool, but they were surprised to hear thatsome of the simplest questions about themare unanswered to this day: What is thenext prime after a given one? (In otherwords, is there a prime-producing for-mula?) Why are there so many "twinprimes" of the form (p, p + 2)? Are thesetwins finite or infinite in number? No oneknows. We then programmed our calcu-
..,_lators to generate the primes in increasingorder (this program is not simple. so I gaveit to the class). We let three calculators runthis program continuously for twenty-fourhours, and at the next session there was tre-mendous excitement as the childrencrowded around the three calculators inanticipation for the next prime to appear inthe display. The highest prime we reachedwas 12 479. The TI 57 is so constructed thatthe action inside the machine dimly showsup in the display even without the "pause"instruction, and so we could watch the ma-chine generating each integer and trying to
130
124
divide it successively by all smaller in-tegers. (One needs, of course, to test onlythe odd integers; also, it is snfficient to testthe divisibility only up to and including thesquare root of the integer, a fact that con-siderably,shonens the time of calculation.)Finally, we let the same program run alsoon the more advanced TI 58 coupled to thePC-10 12/A printer, which enabled us to printall the primes. We let that program rununinterruptedly for about seventy-sixhours, and the paper tape gradually beganto fill the room. When we eventuallystopped the program, it reached a totallength of some fifteen meters, to the amuse-ment of the children, reaching primes up to30 000. We then carefully folded the tapeand put it into a box for permanent storageso that no prime would ever be lost.
Another problem from number theorythat we explored was Ulam's conjecture:(a) take any positive integer, (b) if the in-teger is even, divide by two; if it is odd.multiply by three and add one; (c) do thesame with the new number, and so on; (d)ultimately you will reach 1. For example,beginning with 12, we get the following:
12--06)3--+ 10-05-016) 8-44-42-01
(Of course, from 1 we can go on to 4, 2, andthen 1 again; so we'll stop whenever wereach 1 for the first time.) That this is so forevery positive integer was conjectured byStanislaw Ulam (born 1909), but so far hasneither been proved nor refuted by a coun-terexample. The number of steps it takes toretch 1 varies irregularly from_ one numberto another and is difficultto predict; for ex-ample. it takes only 12 steps to arrive at 1from 106, but 100 steps to reach 1 from107, reaching numbers well into the thou-sands in the process. In programming thisprocedure into the calculator, we have to"teach" the machine how to distinguish be-tween an even and an odd number. This isdone by dividing the number n by 2 andtaking the fractional part of the quotient(there is a special instruction, denoted Int,that takes the integral part of a number; itsinverse, INV Int. -411 take the fractionalpart). This fractional part is then tested
131
against zero and the program branched ac-cording to the outcome (for details of thisprogram see Maor 1979). There was a greatcuriosity to test various numbers and findout how many steps it will take to reach 1in each case (the number of steps wascounted by the program and could be re-trieved at the end). We even played a gamewith this conjecture, by choosing a numbern and then letting everyone make a guess athow many steps it will take to reach 1; theone making the closest guess was the win-ner. The kids had special satisfaction intrying out very large numbers, such as99 999 999 or 12 345 678 and watching theup-and-down sequence of intermediatenumbers until the final 1 halted the pro-gram.
More than perhaps any other subject, thenotion of infinity, with its related limit con-cept, enchanted the students and turned ontheir imaginations. It is here that the pro-grammable calculator exhibits its full edu-cational capabilities. The limit concept isnne of the most abstract concepts in mathe-watics; and many students, even at the col-lege level, are deterred by it. This is per-haps because one cannot visualize thisconcept in one's imagination and so it re-mains a meaningless idea. With the pro-grammable calculator one can at once seehOw a progression tends to a limit as moreand more terms are calculated. A simpleexample is the progression 2/1, 3/2, 4/3,5/4, , whose general term is (n + 1)/n.(There was a problem with decimal frac-tions, with which some of the younger stu-dents were not yet familiar, so I gave a verybrief introduction to this topic.) The pro-gram for displaying the members of thisprogression is very simple:
LRN
SUM 0RCL 0
RCL 0
2nd pauseRSTLRNRSTR/S
(This program can be shortened to nine
4125
steps by writing (a + 1) /n = I + 1/n, but asmoscof the children were not fluent in al-gebraic techniques, I preferred the longerprogram). It was an exciting experience forthe kids to watch the numbers in the dis-play gradually apt., each 1; the questionthat loomed on everyone's mind was, Will
they ever reach 1? With the calculator. ofCOMIC, the answer is yes; when'the differ-ence between the current value and thelimiting value becomes smaller than thesmallest number the calculator can handle.the current number is rounded off to thetrue limiting value. But the students had riodifficulty in understanding that this is so onlybecause of the technical limitations of thecalculator (as also of a computer); they all
agreed that in theory the number I cannever actually be reachedonly ap-proached as closely as we wish. Thus, inOne stroke, they grasped the true nature ofthis important concept. Other explorationswith the limit concept were performed,such as the sum of the infinite decreasinggeometric progression 1/2 + 174-+ Mir+1/16 + , which was first explained interms of eating one-half a cake, then an-other quarter, then another eighth, and soOn ad infinitum. To :he question whetherone will ultimately eat up the entire cake,the class responded with a resounding no.
The last session of each week was de-voted to playing games. We played manygames, some of them nonprogrammable,such as Give-Take where the objective is toreach 999 999 by taking turns in addingand subtracting nuiribefs from the display(see Schlossberg and Brockman 1976, pp.
20-21), others programmable. such as Hi-
Lo or On Target (see "Making Tracks intoProgramming"). On other occasions wesolved simple crossword puzzles. using thefact that certain numbers in the display.when turned upside down, form variousletters and words, such as 14, which turnsinto hi (see Oglesby 1977).
In the upper section, algebra rather thanarithmetic was stressed. At this age levelthe students are mature enough to use someof the tools of algebra and do some moreadvanced mathematics. First we discusseithe so-called "scientific notation for repre-senting very large or very small numbers.using the EE key for this purpose. We de-signed a model of the universe in which theearth is represented by a cherrystone 3 mil-
limeters in diameter (our own sun will thenbe a football-size sphere some 38 metersaway). The class was then asked to place inthis model the nearest star, 4.3 light-yearsaway. This task made them acquaintedwith the notion of scale. After some strug-gling with the huge numbers involved. wefinally came up with the answer: the near-est star would be some 10 000 kilometersawayabout the distance from the equatorto either pole! That gave the class somemeans to visualize the enormous distancesamong the stars in our galaxy.
An interesting topic from number theoryis the Pythagorean triples. These are triplesof integers (a, b, c) such that c: = a= + b=that is, (a, b, c) form a right triangle with cthe hypothenuse and all other sides havingintegral lengths; (3, 4, 5) is an example. ThePythagoreans-were fascinated by suchnumbers and attached mystical significanceto some of them. The search for all suchtriples continued for many years until aprocedure was found to find them: take anytwo integers (u, v) such that u > v; then thenumbers (a, b, c) given by t
a=u=-- v2,b= 2uv,c =u2 +v2
form a Pythagorean triple, as can easily bechecked. (For a _proof that this gives allpossible Pythagorean triples, see Courantand Robbins, pp. 40-42.) It is easy to write
a program that,will display the members of
132
126
the triple for any choice of (u, v). and theclass was given this task as an assignment. Ithen asked them to modify this program sothat even if by negligence we interchange uwith v (i.e., u < v), the program will stillwork properly (this makes use of the ex-change key). The students enjoyed disCov-ering more and more of these triples. Theysoon realized that not all pairs (u, v) giveessentially new triples; for example, u = 2,v 1 gives the triple (3, 4, 5), whereas u =3, v = I gives the triple (8, 6. 10). which isnot essentially different from (3, 4, 5)thetwo triples represent similar triangles. Inorder to get only essentially' different trip-les, (u, v) must not both be, odd and mustpot have a common factor.
The final two sessions were devoted tothe number 17. This number always carrieswith it a certain mystique to those wholearn about it for the firsttime: why shouldas common and perfect a figure as thecircle be related to such a strange number?We first discussed the fact that r can onlybe defined as a limit, which in turn meansthat we can only approximate it. never ac-tually find its "true" value. We followedArchimedes' method of approximating thevalue of Tr by a sequence of regular in-scribed and circumscribed polygons ofmore and more sides. This method requiresa little elementary trigonometry, so we firstdefined the trigonometric functions sineand tangent. (These were very easily un-derstood by all students: it must be said, inthis connection. that the recent tendency ofmany textbooks to glorify trigonometry bydefining first the circular functions, thenthe..trigonometric_ functiOr_157.. 444 viewthem as separate entities. has done a lot tocomplicate a subject that really is verysimple in nature.) We then programmedthe method into our calculators for poly-gons whose number of sides increases in ageometric progression beginning with n = 3and doubling it again and again. The kidswere fascinated to see the numbers in thedisplay gradually approach the "truevalue of
We then examined some of the >othermethods av: ilable to calculate the value of
133
some of which mark milestones in thehistory of mathematics, such as the Greg-ory-Leibniz series
Tr
4
11111 3 5 7
(see Courant 1947, pp. 318 -19, 352, 440,443) or Euler's series
n2 111+ + + + - -
6 2 22 32 2
(see Beckmann 1971, pp. 148-49; Maor1976). The first series converges extremelyslowly and is of no practical value in com-puting ir (it is chiefly of historic interest,being one of the first applications of thenewly invented integral calculus), but ittakes a programmable calculator to ac-tually measure the rate of convergence; itturns out that 628 terms are needed to find77 to two decimal places (i.e., 3.14). The sur-prising fact is that the second series, whichone would expect to converge much fasterthan the first (due both to the fact that thedenominators increase as the squares of thenatural numbers and the fact that all termsare positive), actually converges almost asslowly as the first, requiring 600 terms toget 3.14. On the other hand, the series
774 1 1 1 1
=90 l .2' +34 44
and several others that can be obtainedfrom the Fourier expansion of various peri-odic functions (see Courant_ 1947, p. 446)converge extremely quickly; writing pro-grams for most of these series is very simple
and straightforward.We ended the course by having a little
party. during which a pie was brought withthe inscription "Tr = 3.14" on top, andeveryone had a piece of it. Needless to say,the children enjoyed this pie, and therewere many cheers and a lot of laughter.
The overall reactions to this first experi-ment were enthusiastic; participants ex-pressed favorable reactions on evaluationforms, and many parents bought program-mables for their children. Since then, fivemore courses have been offered, one ofthem a follow-up for students who had
127
taken the first courses and who wished todo some advanced work with the in-strument. Plans are also under way to forma calculator club where the students wouldmeet once or twice a week. exchange ideasand programs, and test themselves on theirown instruments.
ConclusionsAlthough the teaching of mathematics to
young children by using the programmablecalculator is an entirely new experience. itcan already be- said that it carries with itmany promising potentialities: it enablesthe teaching of programming conceptswithout learning a detailed programminglanguage, allows every student to use his orher own instrument (and take it home afterclass), and eliminates the usual commotionthat can be seen at computer terminals.Last but not least, this course demonstratedagain that even young children can learnand understand concepts from highermathematics, provided that these conceptsare presented to them at a level appropriateto their age. With its incredibly low price(340450 currently), there is little doubtthat the programmable calculator will havea marked influence on classroom instruc-tion in the near future. and we may well seeit change the entire pattern of mathematicseducation.
ACKNOWLEDGEMENTS
I should mention the invaluable help that I receivedfrom two of the upper-section students. Tracy Allenand Carl Etnier, who solunteered to serve as assistantsin. the lower section. Kann Chess's presence in thecourse was also of great help. Last but not least. Ishould mention the Hobbs Foundation for its finan-cial assistance in purchasing the. calculators and theEau Claire Associatioh for High-Potential Children.Inc. who started it all and made it possible to launch anovel program that. as it turned out. was very success-ful. -
REFERENCES
Beckmann, Pim A History of sr. Boulder. Colo.: TheGolem Press. 1971.
Courant, Richard. Dieirential and Integral Calculus.Vol. 1. New York: Inierscience Publishers. 1947.
Courant. Richard, and Herbert Robbins, What IsMathematics? London: Oxford University Press.1969.
134
"Making Tracks into Programming." (chap. 11. pp. 7-8). By the staff of the Texas-Instruments LearningCenter.-Texas Instruments. 1977.
Maor. Eli. "The History of a on the Pocket Calcu-lator." Journal of College Science Teaching 6 (No-vember 1976):97-99.
"Number Theory on the Programmable Cal-culator." Didactic Programming 1 (Spnng 1979).29-31.
Oglesby. Mice. "A Calculator Crossword Puzzle."Calculators/Computers Magazine 1 (October1977):38-39.
Schlossberg. Edwin. and John Brockman. The PocketCalculator Gamebook. New York: Bantam Books.1976.
128
i
o
Section VI
COMPUTERS
for
SCHOOLS
135
129
..-- VA.i . of % 1/ 7;-1,Z1
IClassroom,Computers:
4 Beyond the 3 R'sFred Bell
Since 1965. computer buffs, my-self included, have been promising arevolution in education becausecomputers are going to school. Butwhere is this revolution? Certainlythere has been at least a modestlearning revolution ; this is apparentfrom the many people who arelearning about computers and usingthem to learn other things. In spite ofthe fact that some worthwhile appli-cations are being done with coreputers in a few exemplary schools,this learning revolution has yet to takeplace in most schools. There hasbeen an evolution in school learning(at least in many schools) that can be'attributed, in part, to computer tech-nology, but no real revolution.
Before considering the potentialrevolutionary effects of personalcomputers upon education, it is help-ful to differentiate between schoollearning and out-of-school learning.
-The two are not always the same. Wetend to learn things away from schoolwhen we want or need to learn themand we do so in our ,own way and atour own speed. This kind of learninghas advantages and disadvantages.One advantage comes from highermotivation which encourages moreinspired and efficient learning. On theother hand, the tendency to avoid dif-ficult or uninteresting tasks mayFred Bell, Professor of Mathematics Educa-tion, University of Pittsburgh, Division ofTeacher Development, 4A01 Forbes Quad-rangle, Pittsburgh, PA 15260.
result in not learning some very usefuland important things. Consequentlyschools are useful in coercing stu:dents, hopefully in a friendly andinteresting way, into learning somethings that are good for them whichmay not be learned otherwise. Out-of-school ;earning can,be both good andbad, but so can in-school learning,which gets us to personal-computersand the education revolution.
ersonal Computers and Dollars
One of the big reasons whypersonal computers may catalyze arevolution in our schools is that theyare relatively cheap and should geteven cheaper. Any family that canafford two color TV sets can nowafford one color TV and a personalcomputer. Of course, any high schoolthat could scrape up $10,000 per yearfor each of 10 years from a $1,000,000per year budget could have had nearlyall of its students using a minicom-puter since 1969. (See James Saun-der, --Mathematics' Teacher, May,1978, pp. 443-447.) Fortunately, afamily's decision-making processesin buying a personal computer areless cumbersome than a echoes.Unfortunately for school students, asDavid Lichtman found (Creative Corn-puting January, 1979, page 48),educators are less enthusiastic about,the computer's role in society and itspotential for improving educationthan the general public.
But now, with low-cost personal
The classroom computer should,and can, go tar beyond rote com-puter-aided instruction by teach-ing the student to analyze, evalu-ate and develop complex skills:Perhaps, as a result, the long-awaited "revolution in education"'will be here sooner than predicted.
computers, good computer applica-tions may increase in schools. Home-cdrnputing enthusiasts have alreadybegun to Jake learning out of theschools and are putting some of itback into the home where it belongs.Conversely, as more and more per-sonal computers come to school,teachers can bring some of this good"street learning" back into theschools for the benefit of all students.Only $500 remaining in an equipment-and-supplies account at the end ofthe fiscal year can buy the first ofmany personal computers for studentand teacher use.
History shows that many tech-nological innovations that could bequite useful in promoting learning inschools do not get much use inschools until after they are commonin homes and on the streets; forexample, TV sets, audio recordersand hand-held calculators. Now thatpersonal computers are "on thestreets," we are beginning to seethem filtering into schools. But willthey be able to revolutionize educa-tion in schools? TV sets, audiorecorders, calculators, and even mini-computers, while affecting what goeson in schools, failed to revolptionizeeducation. Can vice expect the per-sonal computer to become a revolu-tionary agent? Yes, I think we can.
Personal Computers and Motivation
One of the most serious problemsin schools is that of motivating stu-
Reprinted by permission from Creative Computin& 5: 68-70; September 1979.
L37
13(1
dents; that is, making them want tolearn what teachers try to teach. Themotivation problem occurs becausesometimes teachers wait to teachthings (for good reasons) to studentswho do not particularly care to learn atthe time.. Perhaps the best hope formotivating students to learn in schoolisto pay attention to the nature of out-of-school learning.. It appears thatpeople learn non-survival things awayfrom school for several reasons: (1)People learn to make things work(such as cars and computers) becausethey like to have control over impres-sive machines. (2) People learn tobuild model airplanes, radios, book-cases, etc. because they find satis7faction in creating something fromnothing, or next to nothing. (3) Peopleteach classes, give speeches, andwrite articles because they like toshare their opinions and knowledgewith others and possibly influenceother people's opinions. Many thingsare learned because people enjoy therecognition and approval of otherpeople. (5) Other activities that are notnecessary for survival are carried outfor relaxation, enjoyment, and selfsatisfaction.
But why do so many students dis-like learning in school? First, stu-dents seldom have control over theacademic machinery of schools; thatis, -:the classroom learning environ-ment. Second, creating and buildingtangible things occurs all to seldomin most classes. Third, students'opinions tend to be overshadowed byteachers' opinions in many class-rooms. Fourth, many students getlow grades In school, which interfereswith their quest for recognition andapproval. Fifth, much of what stu-dents have to do in school is neitherrelaxing, enjoyable, nor self-satis-fying.
But how can a few personal com-puters in a classrodn solve thesemotivational problems for studentsand teachers? Well, computers andComputer-enhanced learning are noteducational panaceas, but they cangive students some real control overwhat they learn and how they-learn ILMaking a computer (an electronicmonster) do one's bidding is fun fOrmany people, in spite of the fact thatit is, at times, tedio JS and frustrating.Writing a computer program andmaking it do what it is supposed to dois creating something both aphysical and an intellectual creation.
of
Most people (including teachers andstudents) are impressed by goodinteractive computer games, simula-tions and tutorials, which providerecognition and influence for theircreators. Finally, messing around, ina meaningful way of course, with apersonal computer can be relaxingand enjoyable, in spite of manyminor, temporary frustrations andaggravations.
Therefore, we fine that personalcomputers in the hands of students inschool can remove some of the artifi-cial constraints of typical classtoomenvironments and replace them withsome of the personal freedomsinherent in many non-school learn* igsituations.
Personal Computers and Learning
What is learned in school? Eng-lish, reading, writing, arithmetic,Frelich, history, etc.? Yes, these aresome of the subjects that are taughtbut students should learn many otherthings that subsume all subjects.That is, students need to study eachsubject in a manner\ that permits themto function at all of the followingcognitive levels:
knowledgeunderstandingapplicationanalysissynthesisevaluationproblem solvingknowing how to learncreating knowledge
Schools are fairly good at impart-ing knowledge (i.e., -George Wash-ington was the first U.S. president")and understanding (i.e., '-!2 + 3 = 5because 2 marbles together with 3marbles is 5 marbles"). However,schools are only moderately suc-cessful at teaching applications (out-of-schoOl uses for each subject),analysis (breaking a skill or concep-,tual structure into its parts), synthe-sis (building complex skills or con-.ceptual structures from simplerthings), and evaluation (comparingskills and structures and makingjudgments about them). Schools andteachers have even less success atteaching students the skills andheuristic procedures of problem solv-ing, how to learn independently ofteachers and courses, and ways ofconducting the research and explora-tions that go into creating knowledge.
During the past 15 years we have
138131
demonstrateo, through many drama-tic examples, that computers can beused in schools to help teach know-ledge, understanding, and applica-tions of various subjectsthings thatwere being done fairly well withoutcomputers. This is the evolutionaryaspect of computers in education.But what about tne higher-levelcognitive activities, those things thatwe haven't been .able to teach verysucct.ssfully in school? Herein liesthe true power of computers (espe-cially personal computers) to reallyrevolutionize learning and teaching inschools.
.Writing a computer program re-quires analysis and synthesis of thesubject under consideration as wellas the program itself. A studentcannot write a program to tutorothers, play a game, simulate a situa-tion, or solve an exercise withoutanalyzing the topic being studied andsynthesizing it into a coherent teach-ing/learning program. The synthesisrequired in writing the programproperly and the analysis in debug-ging it provides additional practice atsynthesizing and analyzing. Sincemany non-tutorial computer pro-grams are higher-lever applications oftopics, the student programmer mustevaluate the approriateness of alter-native approaches to the topic and theprogram. When a student writescomputer programs to extend andclarify topics in school, the six stepsin problem-solving (posing the prob-lem, precisely defining the problem,gathering information, developing asolution strategy, finding the solu-tion and checking the solution) mustbe carried out. On the other hand,most so-called "problems" in text-books are really exercises for practic-ing skills, which require only one of,the six steps of problem solving;namely, finding the answer. Afterseveral years of working with peoplein Project Selo at the University ofPittsburgh, we found that many stu-dents and teacherS could tarry outindependent research of their ownchoosing in computer-enhancedlearning environments. That is, thesepeople were creating knowledge andlearning how to learn independent ofpeople who were labeled as theteachers and rooms that were calledclassrooms. ..
Now personal computers canbring the Solo concept of high-level,self-motivated learning out of theresearch-and-development laboratory
and put it in the hands of largenumbers of students and teachers inSchool classrooms.Carrying Out the Revolution
Even before the advent of personalcomputers (as early as 1972), thecomputer technology and col Isewareexisted for a revolution in teachingand learning in schools. Now per-sonal computers with their low costs,easy accessibility, total dedication tothe user, and person-en-the-streetpopularity May provide the long-awaited catalyst that is needed tomake some dramatic changes in howcomputers are used in schools. In afew years large numbers of studentsentering high school will be asfamiliar with a computer as they arenow with a TV set, probablY' more sosince they will have actively pro-grammed a computer, in comparisonto watching television Olassively.
As a consequence of the popu-larity of television, Americans areaccused of having become'spectatorsrather 'than participants in life.Personal computing certainly re-quires active intellectual participationon the part of the user. I have yet to
hear of anybne dozing off whilesitting in front of a personal com-puter.
For several years mathematicsteachers worried about whether kidsshould be allowed to use hand-heldcalculators in school. The popularityof calculators outside school quicklysettled that issue. Nearly every familyhad a calculator. Pre-school childrenplayed with them and studentsbrought them to school. Teacherscould not ignore calculators becauseit was impossible to keep them out ofschool; so now they are trying todetermine how best to incorporatecalculators and calculator-relatedskills into the school mathematicscurriculum. Even if people try to keeppersonal computers out of schools,they are going to fail. In a few years,when they are more efficiently pack-aged and even less expensive, per-sonal computers can fill the "lunch-
/box-technology" void created byschool-luncn programs. Instead of alunchbox, sttidentswill be carrying aPET or TRS-80 computer on a handleto school. When this time comes, anApple for the teacher will really help akid get a better grade in school. 0
e
139
132
Refsrences
Barstow, Daniel. "Computers and Educa-tion: Some Ouestions of Values," CreativeComputing 5: 116-119; February 1979.
Bell, Frederick H "Why Is Computer-Re-lated Learning So Successful", EducationalTechnology 14:15-18; December 1974.
Bell, Frederick H. "Can Computers ReallyImprove School Mathematics?", The Math,-mattes Teacher 71 :428.33; May1978.
Dwyer, Thomas A. "Soloworks: Computer-Based Laboratories for High School Mathe-matics," School Science and Mathematic',75:93; January 1975. ,
Lichtman, David. "Survey of EducatorsAttitudes Toward Computers," Creative Com-puting 5:48; January 1979.
Molnar. Andrew R. "The Next Great CrIsistrtAmerican Education: Computer Literacy,"THEJournal 5:35-38: July/August 1978.
GETTING STARTED IN A JUNIORHIGH SCHOOL: A CASE STUDY
By BILL ROBBINSRamsey Junior High School
Minneapolis, MN 55413
and ROSS TAYLORMinneapolis Public Schools
Minneapolis, MN 55413
Ramsey Junior High School 'has beenmakinginstructional-useof computers forover a dozen years. Although our efforts atgetting started seem like ancient history tous, we believe that our experience over the
t years, m4 benefit those ofjou who are get-_ ting started Today.
Ramsey has approximately seven hun-dred culturally diverse seventh- and eighth-grade students. About 41 percent of thestudents are from minority -groupsmostlyblack. The family income levels range fromvery low to very high. For example, 32 per-cent of the students qualify for free lunch,and Ramsey receives Title I funding toprovide instruction in reading and mathe-matics. In 1978 the student population, fac-ulty, and school program moved from Bry-ant Junior High School 'to the ?RamseyJunior High building as a result of declin-ing enrollment and the city desegregationplan.
During the 1968-69 school year, the firstcomputer teletype terminal was installed atBryant Junior High. It was donated to theschool by the nearby Bloomington-LakeFir3t National Bank. Bryant was the first-Minneapolis-jurtior-high-school-te-have- -a ---terminal, and within a year all other jmiorhigh schools in the 'city were similarlyequipped: Initially, the schools were fur-nished free computer time on a Honeywelltime-sharing computer. In addition, timewas purchased from Honeywell and othersources. ..
Today, Ramsey has two computer termi-nals that are used to access the MinneapolisPublic Schools' Hewlett-Packard 2000
time-sharing system and the Minnesota Ed-ucational Computing Consortium (MECC)Control, Data time-sharing system. Theseterminals are huused in a small computerroom adjacent to the media center. The ter-minals are mounted on wheels. and thereare telephone jack\ in every mathematicsclassroom so that the terminals can bemoved and used where needed. In addi-tion, Ramsey had two Apple II micro-computers, one of which is located in BillRobbins's classroom and the other ofwhich it used primarily by the music de-partment.
Computer users in Minneapolis receiveexternal support from a computer resourceteacher and computer teacher aide in theMinneapolis Public Schools as well as froma regional coordinator of the MinnesotaEducational Computing Consortium. In-service opportunities are provided by theschool system, MECC, and the Universityof Minnesota.
In recent years, the authors'served as di-rector and assistant director of the COM-PUTE project, which produced programstha,t allow teachers to have the computergenerate tests and worksheets keyed to 500different computation objectives. They alsoprovided leadership for a similar projectentitled Computer Generated MathematicsMaterials, involving 1000 mathematics ob-jectives in areas other than computation.
____Petsonal_Experlences andObservations of a Junior
High School Teacher
I am dividing the subject of computeruse in the junior high school classroom intofive areas: (1) utility, (2) demonstration, (3)drill and practice, (4) learning games andsimulations, and (5) programming. Theseareas are not mutually exclusive. I will usethe term computer to apply either to a mi-crocomputer or to a time-sharing terminal.
Reprinted by permission from Mathematics Teacher 74: 605-608;November 1981.
141
133
1
Utility
There are many applications that canmake life a lot simpler for a junior highschool teacher, ranging from the very largelibrary programs to the five-line programthat you write on the spot. I use programslike Compute to make up worksheets.Other programs help me to change numberbases;"do modular arithmetic; of do opera-tions on fractions. At theend of each mark-ing period, I write a short program to aver-age my grades, for that period. I could dothis on a calculator, but the computer auto-matically does the adding, dividing, round-ing, and incrementing for extra credit. Iteven assigns letter grades. It also gives me a,printout that I can use to check for errors.T;achers who use computers find that ideascome rapidly -for a variety of applicationswhere computers can make their work eas-ier and less tedious.
Demonstration
Where the computer has a video monitor(TV screen) fof output, it can be used forclassroom deinonstration. I usually connecttwo or three monitors to my Apple so theentire class can have a clear view.1 havelearned that pie charts, decimal-fractionequivalents, graphs, or just about .anythingthat can be drawn on a chalkboard or over-head projector, can-be quickly displayed onthe screenusually in.a more dramatic, in-teresting, and dynamic manner. There arevolumes of library programs available.(Most of the library programs that we useare available on Apple II diskette's. Personswishing to purchase these programs shouldwrite to the Minnesota Educational Com-puting-Consortium-2520-Broadway..Drive,_St. Paul, MN 55113, for information.) Inaddition, I like to write programs that arespecifically suited to mhy style of pmenta-
'tion. This sometimes takes a lot of time; butonce the prograin is written, it can be usedrepeatedly.
Drill and practice
A computer is ideally suited for drill andpractice for Title I or other remedial in-struction, Drill and practice can also be
1421
provided in a regular classroom setting. Ina small clasi (fifteen students or fewer),with a computer in the room, each studentcan spend part of most periods at the com-puter, practicing needed skills. In largerclasses or when the computers are housedin a remote location such as the media cen-ter, students who will receive drill andpractice on the computer have to be se-lected according'ccording io their needs.
Learning games and simulations
Computer 'learning games and simula-tions have high motivational value. Forsome students this can be the beginningand the ends of computer experience. Everyday I hear a student say, "I want to playthe computer." This may- be disturbing toanyone who is concerned with computerliteracy. However, I have observed that thecomputer can be used as a motivationaltool in any subject area. Many of the siltu-lations that are available on micro-computers or time-sharing systems haveeducational value in mathematics, sock.:studies, language arts, science, and otherareas. (An example of simulations is "Lem-onade" in which students simulate runninga lemonade stand. A number of simula:tions are available from the Minnesota Ed-ucational Computing Consortium.) More-over, for some students, playing games onthe computer leads to serious programminglater on.
Programming
The main obstacle to teaching computerprogramming in the classroom is usuallythe lack of availability of computers. One
would like to teach a three-week or five-week or x-week unit in programMing, tt eany other unit. Unfortunately, to accom-plish this effectively in a class of thirty stu-dents would require the availability of atleast ten computers.
The Minneapolis Public Schools are inthe process of purchasing sets of micro-computers that will be loaned to schools forspecified periods of time. In the meantime,I have found that 1 could get.by with a lim-ited number of computers by teaching pro-
gramming concurrently with other topics. Iintroduce an aspect of programmilig, as-sign a project, and continue to teach theother mathematics topics as usual. Eachday individual students go to the computerand type in their assigned programs whilethe others are doing their regular lessons. Ifa student makes an error, it will becomeobvious from the computer output. This ismuch better reinforcement than if I were tocorrect the program and point out the error.before the student goes to the computer.When all the students haiee had an oppor-tunity to complete the assignment, I inter-rupt the regular lesson for a day and in-troduce a new aspect of programthing andassign a related project. This is a satisfac-
s tory solution to a high student/computerratio. I do not like a batch method. becauseI feel that interaction between the studentand the computer is an absolute impera-tive.
Some of the concepts I teach are care offloppy 'disks, loading and running pro-grams, print statements, input, go to, if-then and read-data statements, for-nextloops, and microcomputer graphics. I startwith simple assignments in which studentsbegin by- writing programs that merelyprint their names and progress to assign-ments that involve problems, generation oftables, or creation of pictures using graph-ics capability.
My experience has been that after five orsix computer lessons the novelty.,wears off,and some of the students become less thanexcited about computer programming. Thisreaction can be expected. However, eachyear there are several students who are en-thusiastic. These students will generallycontinue on their own, with my help, by
_writing a simplega:meor modifyinga_gamethat is in the library. These students will bewaiting by the, door when I arrive in themorning, and they leave reluctantly when Ilock up in the evening. Some of them havewritten amazingly sophisticated programs.In following up on these students, I havefound that they do serious programming insenior high school and they tend to save upto buy their own microcomputers.
Tips for Getting Started
On the basis of our experience, we wouldlike to offer the following suggestions forpersons who are starting or expanding theiruse of computers in their schools.
Work cooperatively with others external to
the school.
There is a distinct advantage to purchas-ing hardware that is similar to that beingused by others in the area. Then experi-ences can be shared, software and pro-grams can be exchanged, and cooperativein-service efforts can, be arranged.
NCTM and its affiliated groups are in-cluding an increasingnuinber of computerworkshops in their programs. The Associa-tion for Educational Data Systems (.BEDS)and its affiliated groups are an-Other sourceof professional information (Associationfor 'Educational Data Systems. 1201 Six-teenth Street, N.W., Washington, DC20036] Nearby colleges and universitiescan help by. arranging in-service courses forcredit. There is great benefit in state br re-gional computer consortia:-
Build support within the schoO1:-
Most schools operate according tct thegolden rule:-Whoever hath the gold shallrule." Therefore, the support of the princi-pal is crucial. One of the best ways to getthis support is to show that there is a highdegree of interest and support on the 'partof students and parents. Demonstrations -and open house activities where studentscan show what they can do on the_ com- --
puler can build support of students, par-ents, administrators, and other faculty. Ex-clusive use of the computer by themathematics department _or by a single_ _ _teacher within the department can narrowthe support base. On the other hand, whenteachers realize that the computer has ap-plications in every discipline, the wholefaculty can become a base of support forpurchasing computers.
Be creative in seeking funds.
A local bank donated our first terminal
1431345
0
to us over twelve years ago. (Incidentally, itis still running.) Since computerwe highlyvisible, they are particularly suitible as do -'nations from businesses, PTAs, and othergroups. It is a good idea to mount on eachpiece of donated equipment a brass platewith the name of the donor and the date ofthe donation. Various fund-raising drivescan finance computer equipment. For ex-ample, at Ramsey we are using the pro-ceeds from student 'fund- raising drives topurchase microcomputers. In recent years,federal Title IV-B funds, which are avail-
. able to all public schools, are being used in-creasingly in many schools to purchasecomputer hardware and software. Title Ifunds are also soinetimes' used, but in thiscase the resources must be used for reme-dial instruction. Of course. there is alwaysthe regular school equipment budget.'These budgets have been used in the pastto supply the schobls with overhead projec-tors, motion picture projectors, videoequipment, and the like. By now, manyschools are saturated with such equipment,so in coming years the funds might be bet-ter used to purchase computers.
O
for all junior high students to have hands-on experience and to write and run at lep :ta few simple programs in BASIC.
Don't put of getting started
This is a time of accelerating change inboth software and hardware development.In such a time it can be tempting to waitfor anticipated improvements and possiblylower prices before getting started. How-ever, in times like these, such a strategycould cause you to wait indefinitely. Eventhough there are new developments aroundthe corner, the hardware you purchasetoday should be useful for its entire phys-ical life% An important reason for startingnow is to gain experience and knowledge.Persons who delay get left further and fur-
, ther behind. There are more and moreways that instruction can be supported byuse of computers, and you should make it apoint to learn about them. Finally, youshould get started so that you can help yourstudents develop the computer literacy theywill need to live in a computerized world.
Look aheadComputers are becoming increasingly
useful in allsUbject areas as well as in guid-ance, media service; and administration.Make pis for increasing use of computersby students at all grade levels and abilitylevers. In the computer field. with rapidlyaccelerating change, flexibility is the key.Where possible, purchase hardware thatsupports a large variety of software thatcan be upgraded, and that can run with var-ious input and output devices. Most impor-tant, when you look ahead; be sure to pro-,vide your students with the computerliteracy experiences that will prepare themfor the world in which they will live asadults. For many years now in Minneapoliswe have included questions on elementaryBASIC computer programming on oureighth-grade criterion-referenced test, andwe include computer literacy questions onour senior high school Basic MathematicalKnowledge Test. As you look ahead, plan
o
144136
lew
C
r.
GETTING STARTED IN A HIGH SCHOOL:A CASE STUDY
By HERB GESSHEL -GREENParkway Program
Philadelphia, PA 19103
Many teachers today are asking hovethey can get started teaching studentsabout the computer, how they can motivatethe students, and what kinds of activitiesthey can try. In this I will relate myiiersbnal experiences in these areas.
I was a mathematics teacher in a juniorhigh school before volunteering to teachcomputing. Alone with another mathe-matics teacher and a science teacher, Ijoim..t the. workshops for teaching cern-puter literacy that were offered by SylviaCharp, director of instructional systems ofthe ptiblic schools. Every secondary schoolwas.-eligible to rnceive a'computer terminaland support for a program of computer lit-eracy. We were taught how to use theBASIC language and how to operate theterminal on a large Hewlitt-Packard time-sharing system.
Our science teacher, John Di Lullo,played a major` role in establishing com-puter literacy clasAs at our school andthroughout the city by helping to developComputer Literacy: A Guide. This heftymanual is currently in use and contains de:tailed lesson plans, sample computergrams and runs, sources of films, and morethan sixteen hundred computer appli-cations in a wide variety of fields. ComputerLiteracy: A Guide is out of print; however,an updated mictocomputer version, Com-puter Literacy, is being published by thePennsylvania Department of Education(Harrisburg, Pennsylvania 17108).
,The other mathematjcs teacher, Joe Ga.riano, began ttuchitecomputer literacy iofourteen ninth-grade<elasses including aca-demic, general, and practical'arts majors.There was one terminal in his room toserve classes of about.thirty-seven students,
which met one class meeting each week.There was a problem in getting each stu-dent hands-on experience.
When I began to teach these courses, thestudents 'did a 'lot of seat work on flow-charts and writing simple programs. Stu-dents were highlyinotivated and wanted to"talk" to the computer. I hung copies of ,their programs in the room. Someone dis-covered the Snoopy program that prints acalendar and the cartoon of Snoopy atophis doghouse yelling. "Curse you, RedBaron." This printout hung on our walland copies found their way into the princi-
"s office.One program the students were assigned
that caused excitement throughout theschool was the rabbit program:
An ecologist places a pair of rabbits onan island. The is otherwise un-inhabited and the food supply is plenti-ful. There will be no predators duringthis experiment.
This particular breed of rabbits maturesin exactly two months. At the end cf thefirst two months, the female gives birthto a single pair of rabbits (one male, onefemale). Each pair of newborn rabbitstakes two months to mature, and theneach female gives birth to a single pair ofrabbits. After reaching maturity, each fe-male gives birth to a single pair of rab-bits (one male, one female) at the end ofevery month. This continues withoutmishap for 36 months.
Write a BASIC, program that will com-pute and print for' each of the 36 monthsthe number of pairs of rabbits. At theend of the 36-month period, print the to-tal number of rabbits present on the is-land. (From Computer Literacy: A Guide,Division of Instructional Systems,School District of Philadelphia, p. 168.)
Reprinted by permission from Mathematics Teacher 74: 610-612;
November 19$1.
145 137
o
As you can imagine there were many dis-cussions, a, guments, and guesses abouthow many rabbits there would be. Manyincidental concepts were learned alongwith new vocabulary. The answer,1.49304E+07; generated -, more questions.We took time to develop their understand-ing of scientific notation.
Students enjoyed learning how to playsome of the games from the library in C-ecomputer. Favorites were Tictactoe (the fa-miliar game of Xs and Os with the studentsworking furiously to beat the computer),Bagels (a logic guessing game in which thecomputer picks a three-digit number. thestudent guesses the number. and the corn-.puter tells whether each digit is in the rightposition, the wrong position. or not in thenumber at allsimilar to the gam; Master-minds), Guess (students guess a randomnumber between.-1 and 1000 chosen by thecomputer, the computer prints whethereach guess is too high, too low, or correctand then prints the number of guessestaken), Hangman (the familiar word guess-ing gamebut with the computer pickingthe secret word and printing out parts of'the gallows with each missed letter guess),Speed Drill (a timed drill with whole num-ber +, , x, + problems), and Findit (acrossword puzzle generated from studentinput of words). In addition, problems forcomputer solution were assigned, and stu-dents could reserve computer time by sign-ing a reservation sheet above the terminal.Time was available before, during, and af-ter school. Several gm' typing students en-tered and ran the programs during theirfree period. They put comments on the pa-pers when programs wouldn't run.
110971 I began to teach mathematicsand computing at a new "Parkway Programsnit it' Germantown. Naturally I ,lookedlir a terminal, only to find that it wastwelve miles away. Since this school usedcommunity space, I looked in the neigh-borhood and hit the jackpot. CommunityComputer Corporation. a computer time-sharing business, was owned by two educa-tional boosters, Walter Friedlich and ErniePhilips, who donated the use of their meet-
ing room to my basic mathematics andcomputer classes. What a great cyear thatwas with youngsters learning about com-puters from the inside of a computer busi-ness. Some of the programmers describedwhat they were writing and also explainedhow students could use the terminals toprogram the same computer that was con-trolling the production of steel via tele-phone lines at a mill twenty-five milesaway.
We used the CAMP textbook series(Computer Assisted Mathematics Prograr:,Hatfield, Johnson et al., Scott. Foresmanand Company, 1968) and Basic Basic byJames S. Coan (Hayden Book Company,1967) as resources.
Many ideas for programs came from thestudents. They wrote programs that printed
Many ideas fourograrnscame from the students.
tables of squares and cubes, the number ofheads in x flips of a coin, and grocery billsand change due. They also learned to corn=plete mathematics, homework by writingprograms. The class worked best in groupsof three to five students. -;
Two years later we 'Moved from the com-puter center to a reg, lar classroom. Fortu-nately, we were able to get one terminal atParkway. All my classes had an oppOrtu-nity to interact with library programs andio write and run some of their own pro-grams. One group of students wrote a com-puter program that printed "poetry" fromthe words input by others. Another grouporganized "speed drill" tournaments inwhich students competed to see who gotthe most problems correct from a set gener-ated by the computer.
This past year we used an Apple II mi-crocomputer. Students having difficultywith signed numbers in algebra could -getsome drill and instruction from Steketee's"Number Line" program (Steketee Educa-tional Software. 4639 Spru- St.. Phila-delphia, PA 19139). Prealgebra students
146 138
Worked on programs to generate sequencesand simulate a fast-food cashier. Studentsin the computer mathematics classes beganby dramatizing parts- of a computer andlater programmed a robot (portrayed byone ,member of their small group) to risefrom a chair, cross the room, touch thewall, return to the chair, and sit down.
Groups of four or #vestudents rotate class -timeuse of the microcomputer.
Radio Shack let us borrow a color com-puter to use in class for a couple of days.During that time we saw students we didn'tknow were still enrolled in our course.Everyone wanted to make the computerdraw color pictures and make music.
Groups of four or five students rotateclass-time use of the microcomputer.'Whenit's appropriate, they'll run a demonstrationprogram for the class. Usually the programwill include an input statement so that theclass can interact with it and make hypoth-eses. First, students guess the results, thenone team member enters the guesses andthe output is discussed excitedly by thewhole class. An elementary example is ashort program that takes a student's inputvalue and generates an output value for anequation like y = 4x + 7. Students try toguess the function. The best programshave, naturally, been those initiated by thestudents, although 'many texts offer ex-amples (see references). The students keepme on my toes by their constant questionsthat begin with "How can I get the com-puter to do thW?" Every day is different.
Last'Dectmber some of my students andI attended a microcomputer fair at TempleUniversity. They enjoyed working with theequipment and trying commercial pro-grams as well as their own. Some did re-,ports on the outing and on possible careersin the computer field.
I got started teaching about computersby volunteering for in-service train;;.` andthen for teaching computer-classes. There
were no problems getting students moti-vated to learn; they were eager! You knowsome of the activities we found successful.Now it is your turn to get started.
REFERENCES
Dwyer, Thomas. and Margot Critchfield. BASIC andthe Personal Computer. Reading, Mass.: Addison-Wesley, 1978.
Isaacson, Dan. DISCOVER THE MICRO-COMPUTER: As an Instructional Media Tool inElementary and Secondary Teaching: A Laboratoryfor In-Service and Pre-Service Teachers. Universityof
IOregon, Department of Computer and Informa-
tion Science, Eugene; OR 974p3. May 1980.
Schoman. Kenneth, Jr. The BASIC Workbook: Crea-tive Techniques for Beginning Programmers. 1977.Hayden Book Company, 50 Essex Street, RochellePark, NJ 07662.
14733
Section VII
COMPUTERS
and the
MATHEMATICS CURRICULUM
1491'10
BASIC Programming for GiftedElementary Students By James H. Wiebe
ie content of this article is the re-sult of a class taught through a summerprogram run by the Tempe (Arizona)P:rents Gifted Assotiation (TPG).During the summer of 1979, gifted stu-
dents from the Tempe ElementarySchool District and other interestedstudents from the fifth through eighthgrades were allowed to enroll in a vari-
ety ofclasses. One option was a coursein BASIC programming. Instruction inthe class was based on the bookBASIC for Beginners (Bitter and Gate-
ley 1978).Classes. met three hours a day, four
days a week, for a period of five weeks.,Half Of the instruction time was in re-lated-mathematical topics and half wasin BASIC programming. Studentsspent about three hoc.., a week enter-
ing and running programs on remote,teletype terminals connected by tele-phone line to Tempe Union HighSchool's PDP microcomputer. Sixty-
two students completed the course.The discussion and recommenda-
tions that follow are based on tech;niques used with and observationsfrom the fifth- and sixth-grade group.In most cases they are also1relevant tothe seventh- and eighth-grade group.
lamer Wiebe is currently assistant professor of
mrkulum and instruction at the, University ofSow;:estern Louisiana. His previous professional
experience Includes teaching mathematics at ele-
mentary, secondary,and college leveltin regularclassrooms and in clinical settings in California,
Arizona, and Zaire. Africa. In his present posnion
he teaches undergraduate and graduate elemen-
tary mathematics methods course.
ObservationsExperiences with the 1979 summerschool class in BASIC programmingindicate that it is feasible to teachBASIC programming to average andabove-average upper-elementary stu-dents. The programming structuresthat students,are able to master, how-
ever, depend on the student's aptitude,maturity, and interest in programming
Many students came into the coursewith no idea what it would be like.Their notions of computers were based
on television programs, popular movieslike Star Wars, and programmableelectronic games. AS a result somethought computer courses involved pri-
marily activities and games with thecomputer. It was also apparent thatsome of the students were hot prepared:for an intellectual challenge, especiallyduring the summer The majority ofthe students, however, soon became in-volved in the class.
One might expect that many stu-dents in a gifted program would have
received adyanced instruction inmathematics. As it turned out, how-
ever, all students in the programneeded supplementary mathematics in-struction. For example, only a few ofthe students in the course had been ex-posed to either decimal numeration orintegers. Supplementary concepts cov-
ered in the course included decimals,exponential notation, integers, begin-ning algebraic ideas (e:g.,, variables),bases other than base ten, and logicalstructures and notation (e.g., the differ-ence between and and or). In general,students mastered these concepts veryquickly.
At the beginning of the course manystudents wanted to go run 'their pro-grams immediately, hefore they hadproperly thought them through. It ap-peared that these students were notused to working problems through astep at a time. Rather, they were used
to taking "intuitive leaps' to solutions.As the course progressed, however, stu-dents began to see the value of havingtheir programs well designed andcoded before trying to enter them, es-pecially when they realized that theyhad only 45 minutes a session to enter,debug. and run their programs. Thus,an early course in computer program-ming might be good preparation forlater work in mathematics requiringstep-by-step solutions and proofs.
As for the type of -assignments thestudents were asked to do, the childrengenerally were not as interestedlong, complex computer calculations ororganizations of vast quantities of dataas they were in having the printer do"neat things. For example, they likedhaving the computer print out funnystateme: is, using the INPUT com-mand and having a friend interact withtheir program, hiving the primer printout sequences of numbers or -words,and hiving the computer print out de-
signs.
Reprinted by permission from Arithmetic Teacher 28: 42-44; March 1981.
11 141
Some students did not understandthe more complex concepts presentedin the course, such as imbedded loopsand subscripted variables. It is notknown if 'students would have mas-tered these topics if more time hadbeen spent on them. It is possible thatwith a more leisurely instructionalpace, with more examples and concreteexperiences, these students might havemastered the concepts. In any case, stu-dents who did not understand the morecomplex programming structures andtechniques seemed perfectly happy torun simpler programs of their own de-sign and to enter and run more ad-vanced programs used as examples inclass (rather than write advanced pro -grams bf their own).
RecommendationsThe following recommendations maybe of value to persons planning to be-gin BASIC' programming courses forelementary students.
1. Students should be provided withthe necessary inaihematics backgroundbefore or during the programmingcourse.
.1 As often as possible, use concrete,pictorial, real-life examples of the con-cepts presented. Using mailboxes formemory locations, and having studentswalk through paths drawn on the floorto demonstrate loops are examples.
3. Use games and activities to rein-force concepts. "Computer Baseball" isan example. In computer baseball, stu-dents are divided into teams. Whenstudents come to bat, they are given aquestion. If they answer it correctly ina given amount of time, they go to firstbase; if not, they are out. The gamelasts a predetermined number of in-nings. Questions may involv,e suchthings as the contents of a storage loca-tion after a given number of ',nesthrough a loop.
4. Assignments should be multi-leveled so that the more advanced stu-dents are challenged without theslower students becoming_ frustrated.For example, more advanced studentsmight be asked to develop their ownunicitte programs 'involving loops assubscripted variables, while slower stu-
dents arc asked to complete and runprograms involving the same conceptsbut using examples given by theteacher.
Introice the PRINT and INPUTcommands, string data, and mathemat-ical functions early:" Later introduceideas such as loops and subscriptedvariables.
6. Assignments should be in-trinsically interesting to pupils. "Reallife" applications of the computer, suchas data organization and scientific ap-plications, do not appear to be as inter-esting to pupils as writing programsthat draw designs, print. funny stories,or print out a lot of data from a shortprogram.
7. Have students flowchart and codeprograms before entering and runningthem.
8. Do not allow students to use theiron-line time with "canned" programs.They should write and run their ownprograms.
9. To show- the value of the com-puter and to keep pupils from usingprogramming shortcuts, assignmentsinvolving mathematical computationsshould be too complex to be done byhand
'10. If pupils are sharing terminals,be sure to monitor their on-line time. Ifyou do not, more aggressive studentswill dominate the terminals.
AssignmentsFollowing are a sauipling of assign-ments that upper elementary studentsmight find interesting and challenging.
1. Construct a flow diagram for us-ing a soft drink machine.
2. Run the following program fromstart to finish.
10 LET X = 25687520 LET Y = 153097
30PRINTX,Y,X+Y,X*Y40 END
(' means multiplication)
3. Write a BASIC program to double$0:62 "20; 26" tibia-. -Befare runningthe program, estimate what,the answerwill be.
4. Write a funny paragraph. Using
152 142,,
the PRINT statement, write a programthat will print your paragraph.
5. Write, enter, and run a programthat prints your initial in a large blockletter.
XXXXX XXX XXXXX70000C XXXXX XXXXXXXXXX XXXXXXX XXXXX
XXXXXXXX XXXXXX.XXXXXXXXX XXXX,OCXXXXXX XXXXX
6. Use the IF-THEN and GO TOcommands to write a program thatreads a number (X) in a line of data(someone's height, for exampl,t), makesa decision, then jumps to any prints astatement about that number, (for ex-ampli, if X'> 50, "YOU ARE TALL."
Draw a flow diagram for this pro-gram..
7. Write a program that someoneelse can run. Have the program ask theperson some questions (use the INPUTstatement). Use that information in asilly story. For example:
WHAT IS YOUR NAME?` JOHN
WHAT IS YOUR FAVORITEANIMAL?DOG .
WHAT IS YOUR FAVORITENUMBER?7
ONCE UPON A TIME THEREWAS A FAMOUS DOGNAMED HORTENCE AND HISFAITHFUL SERVANT JOHNWHO WAS 7 YEARS OLD....
Of course, your program should belonger and more interesting than this.After you have debugged the program,have a friend try it.
8. Run the following program.
10 LET X"= 220 LET Y = 225 PRINT 'I AM, AT LINE 25'30 IF X = Y THEN 1040 END
After 15 seconds, stop the execution byhitting the BREAK key. Draw a flowdiagram for this program.
9. Write a program of less than 10
lines using IF-THEN and GO TO thatprints out the numbers from 1 to 100next to each other.
10. Write a program that
A. Prints oi-ithe following:"Central Control of Space-ship Enterprise to all crewmembers: All units havebeen checked and are readyto blast off for Venus. Allpersonnel to stations. Securesafety belts. Ready."
B. Uses the IF-THEN, GO TOstatements and a counter toprint out the countdownfrom'10 to 1;
C. Prints"BLASTOFF';
D. And prints out a picture of arocket blasting off.
II. Write a program of less than 6lines using FOR TO, NEXT to printthe numbers 1 to 100, skipping everyother number.
12. Use TAB (I) and FOR TO,NEXT commands to make a geometricdesign.
13. Write and run a program thatuses the RND (X) command to simpelate a coin-flipping game.
ConclusionAlthough the class discussed in this pa-per was originally designed for giftedstudents, many of the students in theclass ha4 noi been identified as giftedby the ichbol distnct--they had signedup because they or their parents wereinterested. These students masteredmany of the concepts and procedurespresented in the class, successfully de-veloping and running their own,unique programs. Thus is appears that
153143
many of the concepts and techniquesin BASIC programming can be taughtto the average upper-elementary-school pupil, especially those who arehighly motivated.
Because of the present and futureimportance of the computer in our so-ciety, and because of the educational,benefits derived from learning to pro-gram, we should consider making be-ginning programming experiences astandard part of the upper-elementary-.school curriculum. A prerequisite is'that prospective elementary schoolteachers, especially those who plan toteach mathematics at the intermediatelevel, become proficient in a program-ming language: preferably BASIC, andthat they learn methods for, teachingelementary school pupils to program.'ReferenesBitter. Gary G.. and Wilson Y. Gateley. BASIC
for Beginners, 2d. ed. New York: McGraw-Hill, 1978. -
g,
Case and Techniques for Computers:'Using 1ComputersinMiddle School MathematicsBy Larry L. Hatfield
Computers are rapidly becoming ac-cessible to everyone. The costs of pur-charehaveContinuect to decrease, withthe recent miniprocessors and migio-procesiors representing pricing break-throughs, and inexpensive'rnicro-
Computers being promoted as personal,bomccomputers. Though much oftheir suggested usages to date relate tofamily leisure or home management,some vendors offer.packages for com-futcr-based games and drills involvingmathematical ideas. Instructional pro-
-.grams will become increasingly avail-abk as the marketplace develops. To-day's middle school students aregrow;ng up in a computerized society.Students probably feel more comfort-
-able(often excited and curious) withthe prospects of "everyman routinelyusing computers than do many adults,who stilEview computers as complexand futuristic.
Goals forMathematical Education
We teach mathematics,in a changing---siorld._The discipline of mathematics
and its applications are part of thiscontinual evolution and our per-spectives aboin the teaching and learn-ing of mathematics must reflect thischanging world. Our instructionalgoals and practices must be examined
As an associate professor of mathematics educa-
tad at the University of Georgia. Larry Hatfield isteadterand researcher. He has done mathemat-
tat with students of all ages and mathematicalsikocation with both presume and inservice teach-Hs. Cwtent interests include middle school teachereinnolon, problem solving. algorithmic learning,adaeloesmilical abilities, and computers in mathe-
MOW education.
0
frequently to assure that we are provid-ing for today's students an adequatemathematical education to function inthe society of, at least, the near future.With this in mind, the following ideasabout effective mathematics learningand teaching are summarized to sug-gest some viewpoints for consideringpossible applications of computers intomiddle school mathematics class-r00
I. Mathematics learning is primarilya person-centered, constructive pro-cess: students build and'modify theirknowledge from experiences with task-
, oriented situations characteristic ofmathematics. Students must experienceopportunities and develop feelings ofresponsibility for revising, refining, andextending their ideas as the ideas arebeing constructed. Instrttion basedon the concept of constructed ideas willallow and expect errors or short-comings in the student's responses atcertain stages; the processes of identi-fying and correcting these are integralto the learning. Computers can be effec-tive instructional contexts for such con-structive approaches to learning. For ex-ample, if the learning tasks involvewriting or understanding a computerprogram, the student may be called uponto build up a procedure, test it, find theerrors or inadequacies, correct or im-prove it, test it again, and pOssiblV refineor extend 1: to a more general procedure.
2. Solving problems is the most es-sential of all basic skills needed in ourcomplex, changing world, and learningto solve problems that involve,mathe-
- matics is the most fundamental goal ofmathematics education. Studying the
processes one uses in constructing a so-
lution to a problem may be of greatassistance in knowing generalized, per-haps heuristic, approaches to prob-lems. Various types bf computer usagescan feature.mathematical problem solo.ing and its study. Student-written com-puter programs can involve Considerableproblem solving. ,
3. Most school mathematics curri-cula put a major emphasis on al-gorithms. Children learn not only com-Outational procedures for variousoperations but also procedures for han-dling almost every mathematical taskthey encounter: finding solution sets,evaluating expressions, simplifyingfractions, factoring, finding specialnumbers (e.g., prime numbers, greatestcommon factors, square roots, and av-erages), completing geometric con-structions or transformations. renam-ing (e.g., fractions to decimals),graphing, checking (e.g., comruta-
lional results), estimating. and so on.In most treatments the procedures arepresented "ready-made." As a result,student learning of algorithms is oftenimitative behavior. If we believe thatmathematics ought to be a sensible re-sponse to a reasonable situation, thenmore attention should be given to help-ing children to construct procedures.Helping students to identify limitationsor revisions for their created al-gorithms can lead to significant think-ing. Computer programs are algorithms.
Middle school students can learn to con-struct and use computer proceduresformany mainline mathematical ideas.Such activity can further importantgoals related to algorithmic learning.
4. Effecting quality learning is acomplex combination of many factors.A reliable, knowledgeable source of
Reprinted by permission from Arithmetic Teacher 26: 53-55; February 1979.
155 144
help that ienerates. in a patient man-
ner. interactive reteaching. practicing.and testing for each student. alone withaccurate records of student responsesant; .:amulative progress. would greatly
assist the harried classroom teacher.Computers can he programmed to pro-videsuclrassistance to the teacher, andteachers, as novice computer program-mers, can readily leant to write and eval-
uate such computer programs.
InstructionalComputer Usages
Most of the exploratory usages of com-puters in school mathematics have oc-
curred in the past ten to fifteen years.Major attention has been given toteaching elements of computer lan-guages. such as BASIC or FORTRAN.and to engaging mathematics students
in writing and Using computer pro-grams for various mathematical topics.This use of the computer reflects theobvious interaction of the program-mable. numerical. -logical" machineand certain qualities or aspects ofmathematics. Several other types ofcomputer applications to teaching andlearning also have been identified andexplored. Let's briefly examine some ofthese usages of the computer beforeconsidering strategies for their imple-mentation. /
ProgrammingMathematics students of all agesthroughout the world are writing andexecuting their own computer pro-grams. Mathematics teachers cite sey-
eral purposes for students' engaging in
such programrhing tasks: (a) building"computer literacy" through firsthandexperiences with the capabilities andlimitations of programmable machines:(b) reinforcing a taught (noncomputer)procedure through students' analyzingits steps and restrictions in order toconstruct a computer algorithm (pro-gram): (c) illustrating the tge of a com-puter program as a dynamic problem-solving tool- -even a poor program. asan active object. can be executed. "de-bugged." and modified: (d) providingtransfer experiencei through students'using their knowledge to "teach" themachine to generate the'desired output:te) stimulating high levels ofstudentmotivation and persistence toward
more general, perhaps elegant, pro-
grams: and (f) emphasizing meiliods of
solving. discovering. and generalizing
by building programs. studying a pro-
gram's output.,and extending revised
programs toward algorithms aimed athandling entire classes of problems.
Middle school mathematics abounds
with situations that can be approached
as programmings tasks: (a) finding so-
lutions to open sentences: (b) testing
for properties of number systems: (c)
number theory topicsfinding prime
numbers. multiples. complete facto-rizations. greatest common factors,least common multiples. and deficient.
perfect. or abundant numbers: (d) solv-
ing measurement problemsperime-ter. area. volume. and angle relation-
ships: (e) manipulating fractions anddecimalsrenaming. simplifying. per-forming the four arithmetic operations.and ordering: and (f) solving appliedproblemsfinding averages, solvingproportions, calculating percents. and
figuring probabilities.
PracticingComputerized practice has been exten-sively applied to school mathematicsinstruction during:the past fifteenyears. Of course. the fundamental pur-
pose of a practice session at a computer
terminal is to rehearse for more auto-matic recall or recognition of certain
aspects of the ideas previously taught.Practicing programs usually do notdeal with explanations of mathematicalideas. The role of the teacher is to de-
velop the. background understandingsnecessary for effective practicing to oc-cur. Although of uneven quality. manypracticing programs do exist.
TutoringComputer-based tutorial instructioninvolves the use of the machine to pre-sent an introduction or review of ideas
which are not adequately known by the
student. The stored program attemptsto simulate a good tutor as it in-troduces. explains. characteriies, ex-emplifies. asks questions, accepts and
evaluates responses. diagnoses diffi-culties. prOvides feedback and rein-forcements. monitors performances.and selects appropriate placement intosubsequent lessons. Tutorial programsare similar in structure and function topractice programs but often contain
156'
conside my more detailed text to be
used in explaining the content of the
lesson. Thes are usually more complexand costly to prepare and to offer thanpractice programs. Although much at-tention has been given to the experi-mental dex elopment of tutorial pro-grams in certain federally-fundedprojects. this compu'ter usage has notbecome widespread. With loweredcosts for computers the prospect ofsuch private tutoring may become
more feasible.
Simulating and gamingAs a simulation device, the computer
can often be used to deal with other-
wise unmanageable phenomena. Theadvantages of computer -based simula-
tion include,the measurement and ma-
nipulation of variables that are difficultor dangerous to assess, dr opportunityto experiment when it is not otherwisepossible, the control of "noise" Zirrele-
vant variables) that might otherwiseobscure the item to be measured orstudied. and the compression of the
time dimension to allow long-term
events to be studied in short periods oftime. Among the contexts of possibleinterest to middle school mathematicsteachers are situations that use proba-bility models as well as other Physical
and social science applications ofmathematics. The following are someexamples of simulation programs: (a)
probability (flipping coins, rolling dice.blinded drawings). (b) economics(playing the stock market, purchasing
via loans. ruling a developing nation),
and (c) sciences (effects and control ofwater pollution, piloting a spaceship,
chemical or nuclear reactions, genetic
manipulations).) Gaming programs are simulationsfeaturing competitive settings in which
one or more players.can play, score,
and win. Hundreds ofcomputer-basedgames exist. with many appropriate toobjectives of middle school mathemati-
cal educition.
TestingThe computer has been programmed to
e as a test generator and adminis-
trator. Conceivably, each student could
be given a tailor -made examinationwhere the choice of objectives as well as
number and difficulty of,items for each
objective would be specified for eachstudent.
1 a
1 REM 144:11., Zi Ms. eateict,
10- -REM " FIND AU- FACTORS OF A NUMBER ""
20 INPUT "FACTORS FOR WHAT NUMBER", N
30 FORD = 1 TO n--Tar(n/D) THEN 6t)14tt- IF N/D < >
k
50 PRINT D;
10 NEXT D
70 GO TO a.°6- END
FACTORS FOR WHAT NUMBER? 121 ,2 3 4 '6 12
FACTORS FOR WHAT NUMBER? 35'-1 5 7. 35.FACTORS 'FOR WHAT NUMBER? 591 59
---FACTORS FOR WHAT NUMBER? 571 3 19 A7FACTORS FOR WHAT NUMBER? 1011 101FACTORS FOR WHAT NUMBER? 864.1 2 3 4 6 8 9 12 16 18 24 27 32 36 48 54
96 108 144 '216 288 432 864FACTORS FOR WHAT NUMBER? 971 97 -
FACTORS FOR WHAT NUMBER? 27* 3 9. 27
Program notes: -
This ts an example of a structured, incomplete programming task.
WWI 1 and 10 are simply remarks, ignored by the computer during execution.
--Una 30 through 60 set up a "loop" where the set of possible divisors 1 through N are tested.The condition in line 40 tests for even divisibility, skipping on to the next trial divisor (line 60) orprirsing adivisor (line 50).
Using Compute6in a School.
To many mathematiqs teachers theprospect of incorporating the use of acomputer into their curriculum appearsas an educational "future sliock." As arapidly growing number of teachers inbunslreds.of schools are finding, how-ever, the machines and accompanyingmaterials are oriented toward the nov-ice who has little or no programmingand computer background. The com-puters are simple to operate. beingquite "foolproof," and programminglanguages, such as BASIC, are easy tolearn, yet powerful and flexible enoughto satisfy the most advanced student.Numerous journals, oriented to the ed-ucational ow of computers, providefurthey background as well as practical
computer, run stored (ready-t -use)programs, and write and executsimple BASIC programs for severalmathematical topics taught in the \middle school curriculum. A sixth-grade teacher, for example, might de-cide to focus on two types of uses of thecomputer with his or her students dur-ing the first year of computer use,simple prograinmine tasks and storedpracticing programs. To develop theknowledge needed for students to writeTheir own programs, the teacher mightbegin by discussing and demonstratingcompleted programs: later the studentscan be given incomplete algorithmswhich they analyze, complete, and exe-cute. As students learzrthe BASIC lan-guage and how a program can be conestructed, they will become more able toinitiate their own programs for mathe-matical tasks.
The following is an example of stu-dent programming tasks for a'particu-lar unit to be studied:
72The whole number system
la) Find sums, differences, products,and quotients.
-(b) Siinplify number expressions.(c) Make lists of pairs, given a function
rule.(d) Find factors (see fig. 1):(e) Determine primeness.(f) Find all prime numbers in a given
interval.
classroom suggestions and actual pro-gram listings. A variety of resourcebooks for mathematics teachers pro,vides detailed suggestions fast. devel-oping student programming. Mostcomputer companies sponsor educa-tors+ "users groups" that offer informa-tive rrewsretters, meetings-, and a clear-inghouse service for exchangingcomputer programs a mong,members.And professional associations, such asthe NCTM, offer workshops at theirjneetings and written materials in theirjournals and supplementary pub-.lications to help teachers with com-puter-Oriented activities.
A school typically begins a programof computer use with limited access ito
a computer, a single terminal or com-puter.Through workshops. teach-ers can learn to operate a mini-
157 146
Summary
Computers can be used in educationalsettings and for instructional purposesin more ways than are generally real-ized; they can be much more than a
Means to individualizing testing ordrilling for competency in basic facts.Computers can be teaching aids thathelp to achieve the objectives formathematics learning identified earlier,in this article. When access to a
computer is available, students will' beable to Ilse the computer for pro-kramming the solutions to problems;for simulating situations in order totest hypotheses; for gaming, as a studyof probability and statistics; as well asfor practicing, testing, and tutoring.
The purpose of this article is to helpteachers become aware of the potentialfor this multiusage approach to com-puters.0
LET'S PUT COMPUTERS INTO THEMATHEMATICS CURRICULUM
By DONALD 0. NORRIS°too University
Athens, OH 45701
I propose a rather drastic restructuring ofthe traditional high school mathematicscurriculum for those students who plan tog0 to college. Whereas this proposal isaimed prirna:rily at the college-bound stu-dent, it should encourage everyone to studymore mathematics.
The heart of the proposal is to deleteplane geometry as a required course in thetraditional academic sequence and to re-place it with a year-long course in com-puter programming. My reasons for ,mak-ingthis :proposal are based on my-experiences as a college teacher. Sindents\ often study only two years of mathematicsin high school. They have usually studiedfirst- year algebra and plane geometry. Oneyear of algebra is grossly inadequate for thestudy of mathematics in college. Bear inmind that most college curricula requirethe knowledge of mathematics beyond 'thefirst-year:algebra level. For example,,engi-neering \and science majors require calcu-lus. Psychology majors and sociology ma-jors are usually required to take a .statisticscourse. All education majors take some col-lege-level mathematics. and business ma-jors are often required to take a calculus se-quence. By the time students get to college,many have forgotten most of the algebrathey did learn.
I believe plane geometry is in the cur-riculum for historical reasons. I can recallreading about ,how the formal reasoning,the use of logic, and the mental disciplinerequired to prove theorems in plane geom-etry would do wonders for the student's in-tellect. I donIt think such clairnt have everbeen proved. It is true that in the study of
There ispleiuy of room in the curriculum (weftsidrop the study of geometry.
plane geometry, one is introduced tomathematical thinking, that one learns therudiments of logic, and that one must bedisciplined; but very few people ever gointo a _discipline that uses mathematicalthinking, and there are certainly manyother ways toleam about logic that do notinvolve the study of.plane geometry.
I. would classify mathematical thinkingas a form of problem,solving. It involvesthe analysis of given information and thesynthesis of the information to discovernew facts. This type of reasoning is verydifficult. I know Of no better .way of in-troducing it' to students than through theuse of computers. In addition, computerprogram § must be very logically con-structed. They. are, in fact, a proof of thecomputability of some result. It goes withoutsaying that computers demand. discipline.One 'cannot deviate one iota from the pre-scribed rules of Syntax (the computer lan-guage) or else one is presented with an er-ror message.
Let me try to illustrate some of theseideas. In algebra we teach students how tosolve quadratic equations. I presume thatthe usual procedure is to show them how tocomplete a square on sample problems andthen to do it in general to derive the stan-dard quadratic formula
4 ± 4b174-a72aX as
If your students are like mine, they becomeglmsy-eyed somewhere in the middle of thedevelopment and stop listening. The real-ize that all they need do is to recognize aquadratic equation and to memorize thequadratic formula to solve it.
Now 'suppose you asked students whohave learned the quadratic equation to
Reprinted by permission from Mathematics Teacher 74: 24-26; January 1981.
159 147
write a prograM to,,print out the roots of aquadratic equation. They would have to doanalysis of the following sort:
I. Read in a, b, and c.2. Check whether a = 0, because.' in that
case they have a linear equation andcan't use the formula.
3. Compute the dis'aiminant, d = 4ac.
a) If d > 0, they will transfer to a partof the program where real roots arehandled.
b) If d = 0, they will transfer to a partof the program where repeated rootsare handled.
c) If d < 0, they will transfer to a partof the program where complex rootsare handled.
4. Print the results, making it clear whatcase they are in.
5. Provide the capability to repeat the pro-gram on a new set of data.
Compare the mental activity required tosolve a quadratic equation using the for-mula with the mental activity required toprogram the computer to use the formula.In the first instance the problem can be re-dueed to rote memorization, as it usuallyand this does notinvolve the use of logic orproblem- solving techniques-' except in avery meager way. However, the computerprogram required to solve this problem re-quires an understanding of the formula, theability-to distinguish all the possible casesthat might arise, and the ability to explainto an idiot (the computer) the precise in-structions it must follow to obtain the de-sired results.
As a second example, consider the prob-lem of solving a system of linear equations.Students usually learitto solve systems bysubstitution or elimination. Because, of thelarge amount of calculation involved, weusually restrict our attention to two- orthree-dimensional systems. Substitutionworks fine on two-dimensional,syStems, isunwieldy on three-dimensiOnal systems,and is an unmitigated mess for systems ofhigher ord4r. Elimination- is a good
160
method, but is usually taught in a non-algorithmic way. Since we restrict our at-tention to low-order systems, we attempt toteach our students to recognize from thestructure of the problems. which variableshould be eliminated. This approach ishopelessly inadequate for higher order sys-tems, and since students do not learn thealgorithmic approach(essentially Gaussianelimination) they are unable to tackle suc-cessfully higher order problems.
Now, if you want to program a computerto solve a system of linear equations, youmust have a thorough understanding of theelimination algorithm. You must under-stand that interchanging the equations docs---not affect the solution, or that multiplyingan equation by a constant does not changethe solution.. You learn, that the variablesare of no consequenteit is the coeffi-cients that determine' the solution. Youhave a natural vehicle ,for introducing ma7trix algebra and determinants, because youmust store the coefficients of the syttem ina two-dimensional array. Finally, the elimi-nation algorithm leads very naturally to theconsideration of special cases such as a rowof zeros, except in the last position, or arow of all zeros.
It requires a tremendousamount of problem-solvingability to make a comPutersolve a problem.
Programming the solution of a linearsystem is certainly not a trivial task andshould probably be one of the last topicstaught in a year-long course. But the payoffis large. A great deal of analysis and syn-thesis must be accomplished. In addition,the computer solution leads naturally to astudy of matrix algebra.
Of course, there" are many other prob-lems one could treat. Finding the roots of apolynomial by the bisection method is aneasy problem. Sorting a set of numbers intoascending or descending order is easy to dowithout the aid of a computer, but it isan
148
intellectual challenge to program a com-puter to do the scirt. It is easy to think of avariety of everyday-life problems that canbe programmed. For example, one mightwrite a checkbook balancing program or asimplified income tax computing program.
My point is this: It requires a tremendousamount of problem-solving ability .to makeit computersolvta problem. It requires thekind of analytical thinking we want ourstudents to Jearn. In addition, it is im-mensely practical to learn how to program.In the next ten years. it is likely that mostcollege students will learn how to programcomputers and will use. the, knowledgegained (the problem-solving techniques) intheir jobs.' With the advent of micro-coniputers such as the TRS-80 and Apple,which are very cheap and have tremendouscomputing capability. there is a revoltitioncoining in the use of computers. lv will notbe uncommon forhouseholds' to have mi-.crocoMputers7 TheY will be used to playgames and Maintain household financialrecords. People with programming skillswill have the ability to take full advantageof them. It behooves us to educate our stu-
dents its their uses and limitations. '-
Computers provide, feedback to the pro-vrammer that- is unlike the information wegive our students. Syntactical errors areli,filighted almost instantaneously. Thereis never any doubt about intentions ormeaning. The computer,accepts"Only what
it is prograniMed to accept, and nothingelse. Students must learn to follow therules, and if they don't, the computer pro-vides immediate feedback.
The analysis required to test and debug aprogram is very valuable. One must under-stand the structure of a problem in order toprovidtan adequate set of test data. Trying .to decide why the machipe did what it didinstead .of what you wanted it to do is avery .worthwhile experience in analyticalthinking.
My experience with junior and seniorhigh school studentt leads me to believethat the computer can serve as a device formotivating studenti to learn more mathe.;matics. Usually their first exposure to a
161
computer comes when they play a com-puter game. After the initial challenge ofgame-playing is over, they often want tolearn how to program their own games orhow to solve a mathematical problem.With a little direction from a teacher, theirmathematical horiions can be broa.4enedappreciably.
2500 years of Euclidmight be enough.
'In order to accommodate the use of com-puters. I propose the following curriculumfor college-bound students:
Freshman yearSophomore yearJunior year
Senior year'
Algebra I .
Algebra 2Computer program-
mingPrecalculus survey
Electives might inclUde plane geometry.calculus, or a higher level computer course.
,This: proposal was not, practical just afew years agb because the cost of comput-ers was exorbitant. That is no longer thecase. Over a period of a few years any highschool can acquire enough microcomputersto handle all student needs.
So let's .get going. It might take somecourage to drop good old Euclidean geoni-etry, but I think you wouldlike the results;2500 years of Euclid might be enough.
149
COMPUTER-ASSISTED INSTRUCTIONIS NOT ALWAYS DRILL
e' Microcomputers can help students understand random samplesand other statistical concept&
By JAMES W. HUTCHESONIlAuscoges County School District
Columbus, GA 31901
If you say you have computer-assistedinstruction in your school, many educatorswilLthink of drill work delivered by a com-puter. Some will think of simulation activi-lies in which students "run city govern-/meat'? or "mix 'dangerous chemicals."Others who have had computer-program-ming experiences will think of problemsolving. While teaching a statistics class, Idiscovered still another use of the computerto aid instruction.
Calculators have been used in my in-, troductory course in statistics to facilitate a
"number crunching" aspect of the course.Recently, a microcomputer was added tothe list of naterialt supporting the, course.Prior to the time the computer assisted inthe instruction, I had some difficulty inteaching a particular concept in sampling.Most students understood that
could be made from the sample tothe population when the size of the samplealmost equaled the size of the population.However, they were skeptical when thesample representetra smaller portion of thepopulation. / ;
Initially, the students examined the in-formation on table 1 and table 2.
Table 1 indicates the means of ten sam-ples of size 10 'drawn at random from apopulation of size 1000. Table 2 indicatesthe means of ten samples of size 50drawnfrom' the same population. The studentscompared the differences in the populationmean and standard 'deviation and themeans and standard deviations in the twodifferent size samples. The students found asmaller range of means on table 2 than on
table 1. They concluded that the trend of a
smaller range of sample means with largerand larger samples would continue. Mycurrent statistics class was treated to a simi-lar activity, but with more dramatic results.
TABLE 1Ten Samples of Size 10 Drawnfrom a Population of Size 1000
Samples MeansStandard
Deviations
10 16.4 11.210 19.9 12.010 20.1 15.410 20.4 . 8.610 23.7 16.410 25.5 10.710 26.5 17.610 27.1 14.010 27.3 `-% 14.6
10 29.0 15.6
Population 25.5 14.4
Instead of distlibuting the two tables tothe students; the class was allowed tochoose its own small sample size and largesample size. Summary sheets similar to thetables were generated on , the micro- =
computer and-display-ed= the screen (see
TABLE 2Ten Samples of Size 50 Drawnfrom a Population of Size 1000
Samples MeansStandard
Deviations
50 21.8 , 14.850 22.5 15.050 22.8 13350 26.1 14.750 26.3 15350 261 13350 27.4 14.950 27.5 13.050 27.5 14.250 28.2 15.3
Population 25.5 14.4
table I in the Appendix for the .program).This live demonstration sparked the stu-dents to"begin speculating on the effects of
Reprinted by permission from Mathematics Teacher 73c 689-691, 715;
December 1980.
163150
selecting even more sample sizes. A dis-\ cussion of random selection and samplesize-followed that-seemed. to involve more
dents than had ever been involved be-fore. The students selected their own*Ample Sizes and speculated on the range of
41e \means. Occasionally, a largethe size, resulted in an unusually large
range oUsaniple means. Initially, studentswere puzzled at the "error." Then one stu-dent remarked, "But. that's just the natureof random uumbers!!",This insight helpedOthers to internalize what, had happened. Ifelt most of the benefits of the lesson wereover. I was wrong.
Later in the course, we were discussingcontrol groups and experimental groups.Several students leinarked that it was pos-sible to select two samples at random fromthe same population and have the samplesbe significantly different. They each re-ferred to the computer activity that waspreviously cited, where samples were se-lected from the same population at ran-dom. The comment was made, "How doyou know we didn't accidentally select alarge sample and a small sample?" No sm-.
dent had recalled thi§ idea when I used thesummary sheets as handouts.
Perhaps computers can assist instructionby developing mathematical concepts thatare remembered better through a live dem-onstration. Other live demonstrations have
',been used where students predicted whatWould happen and then let the computerverify (or refute) their 'predictions.
Usually the development of linear re-gression has included a brief commentabout other predictiveyrocedures. The stu-dents understood that line that best fit ascattcrgram might not be straight. How-ever, I had no success in developing an in-tuitive feeling for multiple linear regres-sion. A computer program was used thatwas rtccessful in developing this intuitivefeeling. The rrogram generated three testsA; B, and C and produced correlation coef-ficients between Tests A and C and Tests Band C. The'user was promoted to pool TestA and Test B to form a new test, Test X.The computer indicated a correlation be-
tween Test X and Test C. The scores inTest X were formed by the equation
X = ( )A + ( )B.
The user supplied the weightings for Test Aand Test B. A copy of the computer pro-gram appears in table 2 in the Appendix. Asample run with coefficients appears intable 3.
TABLE 3
YOU TRIED X a. (3).4
TESTA TEST B
+ (4)B
TEST X TEST C1 2 11 5
1 10 83 9 45 15 .
3 6 33 86 16 82 12
0.615 0.703 0.692
IS THERE A BETTER CORRELATION WITH ADIFFERENT WEIGHTING?
After a few trials, all students seemed toknow which test to give more weight in thepooling process. I asked if the weightinghad to be'positive, and immediately severalsuggestions for negative weightings weretried by the students.
Following the computer activity, the stu-dents reported that the test with the highercorrelation should be given more weight-ing. I asked how the process could be ex-tended to Tests A, B, C, and D. A studentsuggested a formula for generating scoresfor Test X,
X ( )A + ( )B + ( )C
along with the proper order for weightingson Tests A, B, and C.
I was pleased with the results of the in-troductory activities and quite pleased thatthe total time for the computer activity anddiscussions was less than twenty minutes.
Computer programs such as these havehelped me to emphasize statistical thinkingand de-emphasize the arithmetic of statis-tics. Perhaps this shift in emphasis has beenthe most important aspect of using comput-ers in the classroom to aid instruction.
APPENDIX
Now: The TRS-80, Level II BASIC uses several func-
!Lon:unite on` pops 691, 713)
164t 1 51
TABLE I
1 REM .... JIM IlUTC1IFSON. TRS10. LEVEL II2 REM .... PROGRAM GENERATES POPULATION OF 1000 RANDOM 41$3 REM .... EACII 1.50. USER SELECTS SAMPLE SIZES4 REM PROGRAM RETURNS MEAN & SD OF 10 SAMPLES ORDERED5 REM ON MEANS ... PROGRAM RETURNS POPULATION MEAN A SD10 CIS20 PRINTTROGRAM IS LOADING 1000 NUMBERS FOR THE POPULATION"25 PRINT:PRINT30 PRINT"A SHORT PAUSE ...(ABOUT A MINUTE)..."35 S.051 =040 DIM AI(I000).M(10).S(10)501 OR I = I TO HUI55 Aq(1) - RW(50)60 PRINT6.475.1.A1(1)70 S = S + A141) SI = SI + A%(J)1280 NEXT I90 PR1NTg475.-POPULATION LOADED"1W (OR 1 - I TO 300 N EXT-C LS110 PRINrNow SELECT A SAMPLE SIZE...I0 SAMPLES WILL BE SELECTED"120 PRINT-FROM THE POPULATION AT RANDOM"130 PRINT PRINT-THE MEANS AND STANDARD DEVIATIONS WILL BE GIVEN"140 PRINT-111E COMPUTER WILL ORDER THE SAMPLES ON THEIR MEANS"
,150 PRINT.PR1NrHOW LARGE DO YOU WANT EACH SAMPLE TO Br; -
MO INPUT N170 CIS180 FOR I = I TO 10190 A - 0B-0
FOR K N210 X - A,I(RNI/(1000))220 PRINT4070.-SAMPLE-J.-SELECTED";K230 A -A +X 11-BOVX240 NI XT K250 M(1) A/N260 S(1) - SQR((B-Af2/N)/(N-1))270 NI XT I280 CI S PRINT-NOW ORDERING ON SAMPLE MEANS"290 FOR I TO 9300 oR K - 1+1 TO 103101E M(1K-M(K) THEN 350320 11 MI)) M(1) 84(K)M(K)113301I - S(1) S(1) - SOO.S(K) - H350 NEXT K360 NEXT I370 PRINT.PRINT"SAMPLE SIZE MEAN STANDARD DEVIATION"'380 FOR 1- I TO 10PRINTN.M(J),S(1):NEXT1390 PRINT.PRINrPOPULATION MEAN ";.S/1000,SQR((S1-S72/1000)/1000)400 PRINT-PRESS <ENTER> TO GENERATE OTHER SAMPLES";405 INPUT ?410 CLSGOTO 150
152
TABLE 2
10 REM JIM HUTCHESON. TRS110. LEVEL 1121) REM ... MULTIPLE LINEAR REGRESSION30 REM ... USER ENTERS WEIGHTIFIGS COMPUTER SHOWS PEARSON'S R40 REM ... USI.R 7 RI! S TO GET A BETTER R WITH BETTER WEIGHTINGS50 RUM ... PROMPT OF NEGATIVE WEIGHTINGS IS NEEDED60 REM70 REM80 DIM A(5).13(5).(15).T(5)90 CI S PRINTTOOLING SCORES IN A MULTIPLE LINEAR REGRESSIONES77
00 PRINEPRIN ilSTS ARE A.B.ANI) C"10 PRIM 'SAMPLE NIIMIII RS ARE ALREADY LOADED"20 SI 0S2.11S4-.0 515.-0 S6=0 S7=0 S8=030 FOR I 1 10 540 RI Al) A.B.050 Al))- A.1111)--13(71+,('6051 SPAS? S24.A'A
)70 S1- S31 /I S4 S4413'13180 SS. SS D.: S6. S6 I C'12190 S7- S7 + SK +VC200 NEXT 1210 DATA 1.2,C.2.18.3,9.15.3.6.0.6.16,12220 PRINTTLARSON'S R FOR:TEST A ANT) TEST C"230 RI (S7-S l'S5/5)/SOR(62-S 112/MS6-S5n/5))240 R 1=EN1(1000*kl+ 50000250 PRIM- P. I ".121
260 PR IN 1 PR INT"PEARSONS R FOK TEST B AND TEST C":70 R2 =1S8 S3' S5/S)/SOR((S4-S312/5)-:"4-S512/3))280 R2 INI(R2'10004 5) /I000290 NUN I- R2 ".R2300 PRIN r PR lArsuprosr. YOU FORMED A NEW SCORE. SAY X. 1Y"310 PRINE,TWILING 1111: SCOR1S IN TEST A WITH THE SCORES IN TEST r320 PR IN 1-1'01.s5 <I.N TI R> 10 CONTINUE":1NPUTJ310 CI S MUNI- psr A AND TIST C TEST 13 AND TEST C"340 PRIN 1'1 ADDER 1,TA1(24),K2350 PRINT PRIN 1-TI SE X WII.I. BE EORM:ID LIKE: X - (3)A + (4)B360 PR INIVIIAT WI imam .111OULD BE 1 r370 MIN 111-RE"380 PR IN I PRIN1-1 00K AT EH 0 TWO CORRELATION COEFFICIENTS AND"390 !TIN I"SlIGGI ST A WEIGH TING. THE COMPUTER WILL FORM TEST X"400 PR IN I-AND COMPUTE PI:ARSON'S R FOR TEST X AND TEST C'
-410 PR IN 1 PR I NT-PRI:SS <EN1 kR> TO CONTINUE" INPUT 3420 CI S PRIN1 I R2"410 PRIM R ER2440 PRINT INPI11"ENTER WEIGHING FOR TEST A",W1450 INPUT"! N I ER WEIGHING FOR TEST B".W2460 C1,0(1 C3.04701 OR 1 ,I1(1480 I-WI'A +W2'B(1)
T(1)-TSW CI4:1+TC2-C2+PT:C3.C3+(1):NEXT1510 CIS520 R1-(C3-SPC1/5/SGROC2-C112/5),(56-S5V/P)530 PR INT"YOU TRIED X - (":Wl,")A + (";W2.")B".PRINT540 PR INTILST A".TA11(16).-TESTB-;TAB(32),"TEST X":TA11(41);"TEST C"550 FOR I= IT05:PRINTA(1),B(4.T(J).C(J)NEXTJ560 R3-INT(R31000+.5)/1000570 PRINT.PRINTR I.R2.113580 PRINT PRINT-IS THERE A BETTER CORRELATION WITH A DIFFERENT WEIGHTINOr590 GOTO 440
153
...
COMPUTER-ASSISTED INSTRUCTIONIS NOT ALWAYS DRILL
(Continued from page 690)
tions that are different from;other versions ofBASIC.
I. CLS-clears the screen2. RND(X)-returns a random number I to X if X is
greater than I3. PR1NT6-prints at a position on the screen. The
positions are 0-1023
The programs are written in an "endless loop-
I
166
%
style. The user is required to depress the BREAK key.This style was chosen over the more conventionalEND statement or the escape to END question forMO reasons:
1. The END statement required the author to antici-pate the number of attempts a particular userwould need. This varied too often to be practical.
2. The escape question would normally appear.; t theend of the usc:'s sample selection or weigl ring.The Students in the statistics class made too manytyping errors to risk an additional user input. Fur-ther. the escape question tends to break the train ofthought when the students are concentrating onthe output of the computer.
154
4,
(
D
Section VIII
COMPUTER LITERACY
167
155
COMPUTER L1TERACY-WHAT IS IT?
A
By DAVID C. JOHNSON-University of London
London, England $W6 4HR
RONALD E. ANDERSONUniversity of MinnesotaSaint Paul, MN 55113
THOMAS P. HANSENand DANIEL
. K1 A"="IMinnesota Educational Computing Consortium
Saint Paul, MN 55113
The May 1978 Mathematics Teacher in-cludes an important position statement re-garding "Computers in the Classrooin"(prepared by the Instructional Affairs'Committee and approved by the NCTMBoard of Directors) which states thatan essential outcome of contemporary education iscomputer literacy. Every student should have first-hand experiences with both the capabilities and thelimitations of computers through contemporary appli-cations. Although the study of computers is in-trinsically valuable. educators should also developanawareness of the advantages of computers both in in-terdisciplinary problem solving and as an instruc-tional aid.
Whereas such a statement reflects. a com-mendable concern for implementing some-thing called computer literacy, it provideslittle guidance in explicating what this area.of study should include. The NationalCouncil of Supervisors of Mathematics, intheir 1977 position paper on basic Mathe-matical skills (Feb. 1978 My), do an ex-cellent job of describing the many impor-tant components of school mathematics.Their statements of rationale pay particularattention to the fact that "the availability ofcomputers and calculators demand a rede-
This report is based on research supported by theNational Science Foundation under Grant No.SED77-18658. Any opinions. findings. conclusions. orrecommendations expressed therein) are those of theauthors and do not necessarily reflect the views of theNational Science Foundation.
fining of the priorities for basic mathemati-cal skills." They go on to include computerliteracy as one of ten basic skill areas:
It is important for all citizens to understand whatcomputers can and cannot do. Students should beaware of the many uses of computers in society, suchas their use in teaching/learning, financial transac-tions and information storage and retrieval. The-rnynique" surrounfitng crsmvters is disturbing andcan put persons with no understanding of computersat a lisadvantage. The increasing use of computers bygovernment. industry and business demands anawareness of computer uses and limitations.
It is clear that the mathematics educationprofession is advocating that some sort ofcomputer literacy experience should beprovided for all pupils. (Note that manyothers, outside the 'field of mathematicseducation, are also calling for action in thisarea: e.g., see the paper by Molnar [1978],which speaks to the importanceoriduca---tion regarding computers and their im-pact.) However. we need to go beyondthese geneial"statements and be more ex-plicit in our statements of possible cora-'Orients (experiences) and desired out-comes (objectives).
AThe National Science Foundation,through its in Science Educationprogram, has awarded a grant to the Spe-cial ProjectsDivision of the Minnesota Ed-ucational Computing Consortium (MECC)to explore the impact of precollege educa-tional programs designed to increase com-puter literacy. and the effects of human-Computei interaction within the context ofinstructional computing environments. TheMECC Computer Literacy Study (May1979 MT) is concerned with such questionsas the following: (1) What is the impact onstudent knowledge, attitudes, and skills ofvarious approaches to the development of
For information on two computer literacy programs.see "New Programs" in this issue.Ed.
Reprinted by permission from Mathematics Teacher 73: 91-96; February 1980.
169
156
computer literacy? and (2) What impactdoes using a computer as part of the in-structional process in science educationhave on student attitudes and beliefs aboutcomputers? An important initial phase ofthe study was to search the available litera-ture (articles, curriculum materials. tests,and so on) and attempt to pull together thevarious views espoused, explicit and im-plicit, into some coherent set of statementsthat reflects what is implied by the phrasecomputer awareness or computer literacy.
If one peruses the literature in this area.it soon becomes apparent that different au-
, thors place priorities on very different as-pects of computing. On one extreme 'wefind the emphasis placed on knowledge andconcepts related to hardware and progiam-ming, and at the other the emphasis is onan awareness of applications and issues.This is not to say that the literature can beeasily divided ujto two disjoint sets, as mostauthors espouse some combination of theseideas. Whereas most authors (includingthose deyeloping course outlines at theschool level) identify the material to be in-cluded, explicitly or implicitly, they gener-ally indicate their recommendations for ex-clusion through omission. Thus it is oftendifficult to_ascertain whether there is reallya debate regarding topics or if the curricu-lum recommendations merely reflectinter-_,ests and local concerns. However. since theviews appear to be so diverse. the projectelected to develop a conceptual frameworkthat attempted to incorporate as much aspossible the views of all. That is, the result-ing description should describe computerliteracy or awareness in its broadest sense.This flexible, general approach is a some-what less absolute view. and it was felt thatsuch an approach would enable us to assessstudent knowledge and attitudes across awide range of possible topics as well asfocus on particular aspects of hardware.software, application,, and issues. In addi-tion, such an approach should provide abasis for further discussion regarding thepriority that might be given to particularaspects of computing.
As a first step, information on a wide
range of courses was collected (in theUnited States and in Great Britain)through personal contacts. responses torequests for information that appeared invarious journals, and communications withthe institutions cited in the HumRRO ex-emplary institutions" book (Human Re-sources Research Organization,1977). (More complete information oncourses and materials is available in the fi-nal report of the project.) One indication ofthe extensive nature of this -collectionphase" is the fact that responses in the formof course descriptions,- course objectives.curriculum guides, or evaluation in-struments were received from about fiftyschool districts throughout the country.Whereas many of these were similar. therewere sufficient differences (both with theoutlines and other text sources) to justifythe earlier decision to attempt to explicatethe concept in a broad sense and not at-tempt to survey common practice or reporton the set of common recommendations.
In addition to course descriptions. nearly2000 test items relating to computers werecollected, judged for quality, and cate-gorized. The course descriptions and testitems were then used to develop a rathercomprehensive list of the topics and objec-tives covered in the courses. These weregrouped under six main headings: Hard-ware -(H): Programming and Algorithms(P); Software and Data Processing (S): Ap-plications (A): Impact (I): and Attitudes.Values, and Motivation (V). The first fiveheadings deal with cognitive outcomes andthe sixth is concerned with attitudes. or theaffective domain. The list of objectives wassubjected to many revisions and then circu-lated to twelve "outside experts" represent-ing the professional societies. industry, andeducation, for reactions and validation. Onthe basis of feedback from these individ-uals. the list was revised once more and thecurrent version is presented in this paper.
Before moving to the actual statements itis important to provide some informationabout format of the lists. The objectives arereally informational objectives. Whereassome are stated rather specifically, explic-
170
-15 7
itly designating a desired outcome,- for themost part they are not behavioral but rep-resent guides' for the construction of testitems. Since subsequent research will in-volve groups of students in schools, therewas also a need to try to reduce the set to
-------some=sma&er subset that could be assessedin a reasonable-periocLof time. Thus thelisting contains a number orstarred_state-ments; these are referred to as "core objec---tives" and as such represent the set to beused in the research. In general, the outsidereviewers suggested that, whereas the corewas appropriate for the research task, thereis a need to extend this set, as most of thestatements so designated are only represen-
qative of the lower levels of cognitive -skillsand understandings. Thus the readershould not attach the idea of "minimumcompetency" to this core set, but ratherrecognize that this is only minimal for de-scribing what might be called computerawareness under the condition that thesubsequent assessment would be manage-able within 'the constraints imposed, by thelarge-scale study.
Computer Literacy ObjectivesCognitive
. .
Hardware (HZ'
*H.1.1 Identify the five major componentsof a computer input ttquipment,memory unit, control unit, arith-metic unit, output equipinent.
*H.1.2 Identify the basic operation of acomputer system: input 'of data orinformation, processing of data orinformation, output of data or in-formation.
111.1.3 Distinguish between hardware andsoftware.
- .
D_enotes core objectives.a. Note that the cOclingis HHardware. PProgramming
and Algorithms. 5,-,Software and Data Processing. AAp-plications. and IImpact. Also. for each statement the firstdigit after the letter refers to a cognitive levelI indicating alow level. generally a skill or knowledge of facts. and 2 stand-ing for a higher level otunderstanding. requiring some analy-sis and/or svothesis. The final digit is merely a count of itemswithin each level. Whereas no moray is intended with the fi-nal digit, there has been an attempt to place the ideas in somesort of logical sequence.
'H.
1,4 Identify how a person can access acomputer: for example,
1. via a keyboard terminala. at site of computerb. at any distance via telephone
lines-2. via punched or marked cards3. via other magnetic media (tape,
diskette)
1.5 kiiognize-thelapid growth of com-puter hardwaresiii-ce-the-1940s.
*H.2.1 Determine that the basic corn-ponents function as an inter-connected system under the controlof a stored program developed by a,person.
*H.2.2 Compare computer processing andstorage capabilities to the 'humanbrain, listing some general similar-ities and differences.
Programming and Algorithms (P)
Note: The student should Le able to ac-coMplish objectives 1.2-2.5 when the al-gorithm is expressed as a set of English Ian-,guage instructions and is in the form of acomputer program.
P.1. Recognize the definition of "al-gorithm."
'P.1.2 Follow and give the correct outputfor a simple algorithni
'P.1.3 Given a simple algorithm, explainwhat it accomplishes (i.e., interpretand generalize).
'P.2.1 Modify a simple algorithm to ac-complish a new, but related, task.
P.2.2 Detect logic errors in an algorithm.P.2,3 Correct errors in an improperly
functioning algorithm.P.2.4 Develop an algorithm for solving a
specific problem.P.2.5 Develop an algorithm that can be
to solve a \set of similar prob-lems.
Software and Data Processing (S)
S.1.1 Identify the fact that we communi-
caw with computers through a bi-nary code.
S.1.2 Identify the need for data to be or-ganized if it is. to be useful. °
S.1.3 !den* the fact that information isdata that has been given meaning.
S.1A Identify the fact that data is acoded mechanism for communica-tion.
S.1.5 Identify the fact. that commnica-don is the transmission of informa-tion via coded messages.
S.1.6 'den* the fact that data process-ing involves the transformation ofdata by means of a set of 'pre-defined ides.-- .
S.117 Recognize that a computer needsinstructions to operate.
S.1.8 Recognize that a computer gets in-structions from a program writtenin a programming language.
S.1.9 Recognize that a computer is ca-pable of storing a program anddata.
S.1.10 Recognize that computers processdata by searching, sorting, delet-ing, updating, summarizing, mov-ing, and so on.
S.2.1 Select an appropriate attribute forordering of data for a particulartask.
S.2.2 Design an elementary data struc-ture for a given application (thatis, provide order for the data).Design an elementary coding sys-tem for a given application.
S.2.3
Applications (A)
A.1.1 Recognize specific uses of comput-ers in some of the following fields:
a. medicinebi, law enforcementc. educationd. engineeringc. businessf. transportationg. military defense systemsh. weather prediction
172
i. recreationj. governmentk. the libraryI. creative arts
A.1.2 Identify the fact that there are manyprogramming languages suitablefor a particular application forbusiness or science.
*A.1.3 Recognize that the following activi-ties are among the major types ofapplications of the computer:
a. information storage and re-trieval
h, simulation and modeling
c. process controldecision-mak-ing .
d. computatione. data,processing
*A.1.4 Recognize that computers are gen-erally good at information-process-ing tasks that benefit from the fol-lowing:
a. speedb. accuracyc. repetition
A.1.5 Recognize that some limiting con-siderations for using computers areas follows:
a. cost .'b. software availabilityc. storage capacity
A.1.6 Recognize the basic features of acomputerized information system.
*A.2.1 Determine how computers can assistthe consumer.
"*A.2.2 Determine how computers can assistin a decision-making process.
A.2.3 Assess the feasibility of potentialapplications. .,.
A.2.4 evelop a new application.
Impact (I)1
1.1.11 Distinguish among the followingcareers:
a. keypuncher/keyoperator
159.
A
e
b. computer operatorc. computer programmerd. systems analystC. computer scientist
'1.1.2 Recognize that computers are usedto commit a wide variety of seriouscrimes, especially "stealing moneyand-steaiing information.
*1.13 Recognize that identification codes(nunpers) and passwords are a pri-mary means for restricting the useof computer systems, computerprograms, and data files.
1.1.4 Recognize that proceituroc for de-tecting computer-based crimes arenot well developed.
*1.1.5 Identify some advantages or disad-vantages of a data base containingpersonal information on a largenumber of people (e.g., the listmight include value for researchand potential for privacy invasion).
1.1.6 Recognize several regulatory proce-&fres; for example, privilege to re-view one's own file and restrictionson the use of universal personalidentifiers that help to insure theintegrity of personal data files.
'1.1.7 Recognize that most "privacy prob-lems" are characteristic of large in-formation files whether or not theyare computerized.
*1.1.8 Recognize that computerizationboth 'increases and decreases em-ployment.
'1.1.9 Recognize that computerizationboth personalizes and imperson-alizes procedures in fields such as . *V.5education.
9.1.10 Recognize that computerization canlead to. both greater independence'and dependence on one's tools.
*1.1.11 Recognize that. whereas computersdo not have the mental capacity,that humans do, through tech -'niques such as artificial in-
tion set and do many of the infor-mation-processing tasks that hu-inans do.
* 1.1.12 RecOgnize that alleged "computermistakes" are usually mistakesmade by people.
* 1.2.1 Plan a strategy for tracing and cor-recting a computer-related error,such as a billing error.
1.2.2. Explain how computers make pub-. lie surveillance- more feasible.
'1.2.3 Recognize that even though a per-son does not go near a computer.he or she is affected indirectly be-cause the society is different inmany sectors as a consequence ofcomputerization.
1.2.4 Explain how computers. can beused to effect the distribution anduse of economic and politicalpower.
-Computer Literacy ObjectivesAffective
Attitude, Values, and Motivation (V)6
'V.1 Does not feel fear, anxiety, or in-timidation from computer experi-ences.
'V.2 Feels confident about his or her abil-ity to use and control computers.
'V.3 Values efficient .information process-ing provided that it does not neglectaccuracy, the protection of individualrights, and social needs.
'V.4 Values computerization of routinetasks so long as it frees people to en-gage in other activities and is notdone as an .end in itself.Values increased communication andavailability of information made pos-sible through computer use providedthat it does not violate personal rightsto privacy and accuracy. of personaldata.
V.6 Values economic benefits of comput-erization for a society.
telligence, computers have beenable to modify their own instruc-
b. The coding scheme: V.I -v.9. is merely for recordingpurposes and ts not intended to convey any pnorities or hier-archy.
V.7 Enjoys and desires work or play withcomputers, especially computer-as-sisted learning.
V.8 Describes past experiences with com-puters with positive-affect words, likefun, exciting, challenging. and so=on.
V.9 Giyen an opportunity, spends somefree time using a computer.
SummaryTheo lists of ,objectives in the previous
section could- use further description. Inparticular, note that A.2.1 and A.2.2 speakQn the nritinrk of .2ccictirIg the Cnttetimet rtrin decision making. In their phrasing, theseobjectives tend to suggest the positive as-pects of computing. Of course, they mustalso include aspects of the liniitations of themachine, in particular those aspects of de-cision making that r-late to ethical andmoral considerations.
One further point deserves mention here.Whereas the objectives are intended to pro-vide a broad perspective, there is one areathat has been omitted. This is the detail ofearly history (e.g., Babbage and Hollerith).This was a conscious decision based on dis-cussions among the investigators and inputfrom the review panel and other knowl-edgeable perions. This is not to suggest thatthe topic is not interesting or motivating,but rather that it reflects the bias that ::uchknowledge is not really functional or .pre-requisite to an understanding of the otherareas.
The lists of objectives in the previoussections are quite extensive and, of course,complete coverage is probably not feasiblewithin any single course and even less fea-sible as an addition to the school mathe-matics arriculum. However, if we viewcomputer literacy in this broad sense, it isnow appropriate to "pick and choose," ifyou will, and allocate selected activities/ex-periences to various points in the totalschool curriculum, including mathematics.It may also be desirable to allocate a por-tion of some, year to a course called com-puter awareness or computer literacy andinclude those objectives that require some
174
reasonable time for study. Such decisionmaking is a job for those inost suited forthe taskthe teachers in the school.
Whereas these lists should provide abasis for such decisions; much more re-mains to be done. In particular. it shouldbe noted that the language used in many ofthe statements will probably need to berephrased if it is to be communicated toyounger students. Also. the objectives willneed to be translated into a form appropri-ate for assessment. The research project hasdeveloped items for those statements desig-nated as core and will disseminate informa-tion about the tests in the near future. Ofcourse, the assessment is most appropriatefor determining the effectiveness of a givencomputer literacy curriculum, and at pres-ent, this is a relatively undeveloped area.
Reaction to the ojectives is welcome: thestatements are all open to revision and youmay feel we have omitted some imnortantpoints. The project has also produced otherreports including further descriptive infor-mation on the research: the results of astatewide (Minnesota) survey of all second-ary school science, mathematics, and busi-ness teachers (as well as selected teachers inother disCiplines); and summaries of se-lected references. Questions should be di-,rected to Computer, ,Literacy Study. Minne-sota Educational Computing Consortium.2520 Broadway Drive, St. Paul, MN 55113.
REFERENCES
"Computer Literacy Study." Mathematics Teacher 72(May 1979):390.
"Computers in the Classroom.** Mathematics Teacher.71 (May 1978):468.
Human Resources Research Organization. AcademicComputing Directory: A Search for Exemplary Insti-tutions Using Computers for Learning and Teaching.Alexandria. Va.. 1977.
Molnar. Andrew R. "The, Next Great Crisis in Amen-,can Education: Computer Literacy." The Journal:Technological Horizons in Education 5 (July/August197E):35-39.
"Position Statements on Basic Skills " MathematicsTeacher 71 (February 1978):147-52.
A bibliography of selected references and instruc-tional matinals is\aiailable from the project.
16.E
;i
COMPUTER LITERACY--WHAT SHOULD IT BE?
By ARTHUR LUEHRMANNComputer Literacy
1466 Grizzly Peak Blvd.Berkeley, CA-94708
The way things are is not necessarily theway they Should be.
The February 1980 Mathematics Teachercontains a significant report of researchconducted for thP NatiAnni,Sriinre Frmn-dation Ily John Son et al. (1979) at the Min-nesota Educational Computing Con-sortium (MECC). The report, titled"Computer LiteracyWhatis It?" has al-ready generated a great deal of discussionamong teachers and others attempting to'understand the, place of computing in theprecollege curriculum. At the 1980 annualmeeting of the National Council of Teach-ers of Mathematics one of the best attendedsessions was a report ou this project by oneof the MECC authors (Klassen).
The goal of defining computer literacy isof great importance.'Much hangs on it. Ifthe public decideSthat the subject is worthteaching in its Schools, then it faces anequipment bill of about S1 billion in theUnited States alone. It faces the additionalcosts of curriculuin development, ofteacher, training, and of the assessment ofstudent achievement. It is obvious that theway computer literacy is defined will have aprofound effect on the public's willingnessto 'support the teaching orcomputing in
their schools.The Specific concern that prompt& the
present article is that many readers of theFebruary MT report are making the mis,take of seizing upon the list of sixty-threeitems (which occupies half of the reportand is given the boldface title "ComputerLiteracy Objectives") and interpreting thelist as providing an objective measure ofwhat computer literacy should be.
Whereas the organization and physicalappearance of the MT article may invitesuch an interpretation, and, indeed, the au-thors may wish us to do so, they also ex-plain clearly that their methodology indzAwing up the list results in little morethan an empirically gathered' collection ofthe educational objeCtives actually found inexisting computer courses in 1978-79.Team, readers should not conclude thatthere is any moral imperative to teachthose things. In the latter half of this com-munication I will go a step farther than theMECC authors. I will argue that fully four-fifths of these empirically discovered objec-tives should not be used in any signfficantdefinition of computer literacy. More of thatlater.
This observed tendency to ,interpret theMECC list of objectives as authoritative,imperative, or official is aggravated by thepublication and dissemination of an im-pressive 1979 MECC document bearing thetitle "Minnesota Computer Literacy andAwareness Assessment," and a credit lineindicating NSF grant Support. Who couldblame a casual reader for inferring that,these fifty-three multiple-choice contentquestions provide an authoritative, officialdefinition of what a course in computingshould be aimed at? "At last," one canMost hear the harried teacher say, "now Ican see ,whether I'm teaching the right
Equiptnent could cost41 billion in theUnited States.
things." In fact, as I will soon argue, theteacher who is teaching toward that test in-strument is teaching the wrong things.
I do not take the MECC authors to task
Reprinted by permission from Mathematics Teacher 74:, 682-686;
December 1981.
175
162
for this public confusion about the resultsof their project, but only call attention to
- the fact-that-confusion-does-exist-aboutwhat a course in computing should be. TheMECC test instrument was developed pur-posely to find out whit students are learn-ing in today's computer literacy courses. Itis very good for such empirical work andcertainly should not be interpreted as set-ting a standard or a goal. Furthermore. theauthors of the MT article "precede the list ofobjectives with this extremely forthright ca-veatMost of,the statements (designated as core objectives)are only representative of the lower levels of cognitiveskills and understandings. Thus the reader should notattach the idea of "minimum competency" to this coreset, but rather recognize that'this is only minimal fordesaibing what might be called computer aware7nest (p. 93)
What Is Wrong with theMECC Objectives?
Little or nothing, if we use them to de-fine computer awareness, as the authorswarn us that we should. Everything, if theyare to define computer literacy.
The argument proceeds,by analogy. Lit-eracy in a language meads the'ability toread and write, that is. to do somethingwith language, not merely to recognize thatlanguage is composed of words, to identifya letter of the alphabet, or to be aware ofthe .pervasive role of language in society.Literacy in mathematics means the abilityto add numbers, 'solve equations, and soonto do mathematics, not merely to rec-ognizeognize that numbers are written as sets ofdigits of to identify a fraction by its appear-ance or to be aware of the vocational ad-vantages of being able to do mathematics.
By analogy, computer literacy must alsomean the ability to do computing. and notMerely to recognize; identify, or be awareof alleged facts abourconiputing. The basicflaw in attempting to apply the MECC ob-jectives as a standard fot what should, betaught in a computer. literacy course is thatof the sixty-three objectives given, onlytwelve require that the student be able todo anything. (Significantly, eight of thenine objectives in the "Programming and
Algorithms",section fall into this categoryalong with three of the thirteen in :Soft-ware and Data Processing.") The otherfifty-one objectives involve ,nothing morethan student acceptanceof hearsay knowl-edge about computing, such as might be ac-quired from a book or from being told bythe teacher.
This category of knowledge. the lowestin P,lato's hierarchy, is essentially verbal.Its acquisition involves the student mainlyin encoding words and remembering themwhen appropriate stimulus is presented.It is qualitatively different from the knowl-edge that comes ,from experience: doingwriting, doing mathematics, or doing com-puting.
.:Consider the following objective (1.I.1.1in the MECC list): "Identify the five majorcomponents of a `computer: input equip-talent, memory unit, control unit. arithmeticunit, output equipment" (Johnson et al.,93). This "core" objective isstested for byitem 68 in the MECC assessment in-strument: "In addition to input and outputequipment, computers contain": for whichthe correct response is "memory unitsrcon-
Computer literacy is theability to dd computing.
trol units, arithmetic units" (MECC 1979).Clearly, the student who has read. andmemorized the classical definition of acomputer will ace this item.
Yet we should all be worried by the eactthat months and years can go by in the lifeof a professional computer scientist withoutany need to remember or make use of oreven reflect on the_fact that somewhere inthe bowels of a computer lies an arithmeticunit and somewhere else (a few micronsaway on the same chip, nowadays) lies acontrol unit, and that they are. logically atleast, distinct. He or she could create an en-tire management information system or anairline reservations system or a numericalanalysis package without ever calling onthat piece of knowledge or putting it to use.
176 163
Except for a few who work at or near thehardware level, people who do computingrarely exercise that bit of textbook knowl-edge.
An even, greater cause of worry about themisuse of objectives like this'one comes outof direct experience in working with thou-sands of children and adults who -are ac-tively-gaining firsthand experience in usingcomputers. After only a few hours of such
Performance objectivesare needed.
laboratory experience, they know enoughtoscore near the top on the handful of testitems based on the dozen MECC objectivesrequiring that the student be able to docomputing. Yet these same students do nothave the foggiest idea about what an arith-metic unit is or a control unit is or what thedifference, is. (They do know about input,output, and processing, by the way, becausethey experience these things and have aneed for words td describe them.)
Yet these doers of computing have a'knowledge qualitatively superior to thatofthe hearers about computing. The Chineseproverb says it well: "I hear, asd I forget. Isee, and I- remember. I do, and I under-stand." The doers should not be punishedby misapplication of tests. and objectivesderived from classes where hearsay knowl-edge is the principal commodity.
I have criticized one objective at length,but the same criticism: applies to four-fifthsof the rest in the MECC list. Here are a fewrepresentative ones, taken only from thestarred "core objectives" in the February1980 MT article:
H.1.5 Recognize the rapid gro4/th of corn;puter hardwareare since the 1940s.
S.1.6 Identify the fact that data procesSinginvolves the transformation of databy means of a set of predefinedrules.
A.1.4 Recognize that computers are gener-ally good at information-processing
177
tasks that benefit from the following:a) speed, b) accuracy, c) repetition.
A.2.1 Determine how computers can assistthe consumer.
1.1.8 Recognize that computerization bothincreases and decreases employ-ment.
Altogether there are eleven objectivesthat start with" "Identify," twenty-one thatstart, with "Recognize," and three that startwith "Determine." In every such case littlereal change occurs if these verbs are re-placed by the single word Remember. In-deed. the correspbnding test items on theMECC assessment instrument require onlythat the student, remember the "right" setof words that goes with that objective.
To make clear the contrast betweenhearsay knowledge and knowledge thatcomes out of practice, consider the verydifferent flavor of the following sampling ofthe twelve MECC objectives not in thehearsay category:
P.1.2 Follow and give correct output for asimple algorithm.
P.2.1 Modify a simple algorithm to accom-plish a new, but related, task.
P.2.4 Develop an algorithm for solving aspecific problem.
S.2.2 Design an elementary data structurefor a given application.
Any course in computer literacy ought to_4spend four-fifths or more of its time and ef-fort on performance objectives, such as theones in this latter group. Going lurther,,Iwould argue that it is intellectuallyproper to inculcate beliefs and values abouta subject that do not arise out of the stu-dent's direct experience with the content ofthat subject. If I were writing about mathe-matics or reading and writing, there wouldbe little disagreement about this point.Readers of this journal, for example, wouldbe properly outraged if they were asked tospend four out of five days working onJohnny's and' Janey's beliefs and valuesabout. the s 'Eject of mathematics and tospend the other day teaching them to do
164
mathematics. However much we want ourstudents to remember facts about, and feelgood about, mathematics, we know thatthese beliefs and values will be short livedif our students go out into the world withpoor ability to do mathematics.
Th. Nub of the Problem,When one remembers the goals and the
methodology of the MECC study, it is easyto see liow that particular list of objectivescame into being. The study's goal was toassess :the impact on student knowledge,attitudes, and skills of various approachesto computer literacy." The authors tell usthat they defined computer literacy asbroadly as possible, so, as not to leaVe outany approach. Their methodology, then,was to collect "course descriptions, courn:objectives, curriculum guides (and) eva:aa-don instruments" from existing courses ofstudy in various school districts. The result-ing list of objectives is a somewhat cleaned-up union set of all the stated and impliedobjectives of actual courses then in place,(1978-79).
That constraining phrase, "then inplace," is the nub of the problem that arisesif one is looking for guidance as to whatshould be taught in a computer literacycourse today and in the decade ahead.
The vast majority of precollege comput-ing courses then in place were woefully un-prepared to give students significant practi-cal experience. Thirty students in a classoften had combined access to a single com-puter or terminal for an hour a day. In anentire semester each student inight experi-ence perhaps only two hours of direct use.No one shc-:d be surprised, then, that 80to 90 percent of the student's time in such acourse was spent reading and listening to'textbook information about computers andcomputing. Teachers have to teach some-thing, after 411, and the MECC study showsclearly that what they were teachifig for themost part was computer awareness.
It takes much more eauipment to be ableto turn out literate users of computing. Thefirst equipment plateau is reached at abouteight computers; available an hour a day
for each' thirty-stueent computer class' inthe school. This arrangement allows hathe students to work in pairs in The com-puter lab while the other half is receivinginstruction in the classroom. Alternate-clayrotation between lab and classroom giveseach student pair about thirty hours ofcomputer practice. The next plateau occursat about fifteen computers per school. Lo-gistics are the same as with eight comput-
.ers, but each student can work alone in the
I am familiar with several schools thathave reached the first plateau and one en-tire school district nearby that is well onthe way to reaching the second plateau. Al-though this situation is attainable and addsless than 1 percent to the operating cost ofa school, nevertheless, it is still.an unusualone to find today. Declining equipmentcost and increasing public awareness of theintellectual and vocational value of being aliterate computer user will in due courseenable schools to do the job property.
ConclusionsThe MECC Computer Literacy Study
should not be misinterpreted. It gives us Itwell-documented snapshot of the staie ofcomputer education as actually practiced in1978 -79. What we see in the snapshot is aclassroom with a lot of reading and listen-ing, a little seeing, and hardly any doing.
The MECC study gives little guidance,however, .as to what the learning objectivesof a precollege computer literacy courseshould be. Readers of the study deserveblame if they turn to the published MECCobjective list as an ideal or if they use theMECC assessment instrument as a 'stan-dard test of anything more than elementarycomputer awareness.
A significant challenge rentains, then: todefine learner objectives for a course thatwill turn out literate doers of computing,and then to embody those objectives in apractical curriculum intended for wideadoption.
Although I have well-formed ideas abouthow to rio that, the pages of this journal areprobably not an appropriate place to pre-
sent subjective judgments. Suffice it to saythat such a curriculum will put primaryemphasis on the direct interaction betweenthe computer and the student, with alearner goal of mastering wholly new ana-lytic, expressive, and problem-solvingskills.
Computing belongs as a regular schoolsubject for the same reason that reading,writing, and mathematics are there. Eachone gives the student a basic intellectualtool with wide areas of application._ Eachone gives the student a distinctive means ofthinking about and representing a problem,of writing his or her thoughts down, ofstudying and criticizing the thoughts ofothers, and of rethinking and revisingideas, whether they are embodied in aparagraph of English, a set of,mathemati-cal equations, or a computer program. Stu-dents need practice and instruction in allthese basic modes of expressing and com-municatilig-ideas,Mere awareness-of thesemodes is not worth the time it takes away
from teaching the creative and disciplineduse of these fundamental intellectual tools.
REFERENCES
Johnstin. David C., konald E. Anderson, Thomas P.Hansen, and Daniel L. Klassen. "Computer Liter-acyWhat Is It?" Mathem'attcs Teacher 73 (Febru-ary 1980):91-96.
Minnesota Computer Literacy and Awareness Assess-ment. St. P. Minn.: Minnesota EducationalComputing Consortium, 1979
179 16,6
g. IN DEFENSE OF A COMPREHENSIVE VIEWOF COMPUTER LITERACY- -A REPLY TO LUEHRMANN
By RONALD E. ANDERSONMinnarga Educational Computing Consortium
. Saint Paul, MN 55113
DANIEL KLASSENSt-Olaf College
Northfield, MN 55057.
and DAV* C. JOHNSONUniversity of London
London, England SW6 4HR
In Arthur Luehrmann's critioue of ourFebrtrtry 190 Mathematics Teacher articleon computer- literacy objectives, he pro-poses a rather narrow view of litricy. De-spite his claim, there are two (not just one)generally accepted definitions of literacy.One is, as he points out, the ability to com-mingate, for-example, reading and writ-ing; and the other, which he neglects, is thestate of being informed, "cultured," andwell versed. Whereas the first is a subset ofthe second, both definitions are commonlyused. It is not surprising that the term com-puter literacy shares the semantic ambiguityof language literacy. The narrow view isthat computer literacy is simply a matter ofdoing things with a computer. The compre-hensive view is that computer literacy is anunderstanding of computers that enableson% to evaluate computer applications aswell as to do things with them.
The comprehensive view of computer lit-eracy, is consistent with the long-estab-
The authors wish to thank Catherine Dunnaganwho gave helpful comments on versions of this paper.
This ankle was prepared with the support of Na-tional Science Foundation-Grant No. SED-18987.Aity opinions, Endings, conclusions, or recommenda-tion$ expressed are those of the authors pnd do notaece.ssarily reflect the views of the National ScienceFouudatiun or anyone else.
lished tradition of scientific literacy and re-lated formulations such as technologicalliteracy, geographic literacy, and economicliteracy, to name, only a few. Scientific lit-eracy is generally defined as the knowledgeabout science that the layperson needs tofunction effectively. Scientific literacy re-fers not only to learning scientific facts butalso to one's understanding of the implica-tions of science and science- society issues.Thus it is not surprising that we often seecomputer literacy equated with "computersand society" and courses on'the social roleof computers.
The comprehensive view of computer lit-eracy is also consistent with the current rec-ommendations Of the National Council ofTeachers of Mathematics (1980). In A.rtAgenda for Action, they recommend that "acomputer literacy course. familiarizing thestudent with the role and impact of the.computer, should be a part of the generaleducation of every student." This iew ofcomputer literacy is also in accord with theliterature on the conceptualization of corn-puter literacy. For example. Moursund(1976), Rawitsch (1978), and Watt, (1980)all define computer literacy in a broad,comprehensive fashion.
Luehrmann's attempt to define computerliteracy as simply "doing computing" andhis attempt to belittle knowledge of com-puter systems by calling it "hearsay knowl-edge" neglect the semantic ancestry of theterm and close the door on a broader un-derstanding of computer technology. Thereare very good reasons for people to be ableboth to communicate with computers andto be knowledgeable abdut them.
In defining computer literacy it is usefulto distinguish it from computer science.The most succinct distinction is to say thatcomputer literacy is that part of Wmputer
Reprinted by permission from Mathematics Teacher 74: 687-690;
December 1981.
181
167
science that everyone should know or beable to do. Both language literacy and sci-entific literacy are commonly defined interms of the layperson and his or her needs.Likewise, computer literacy should bethought of as the knowledge and skills theaverage citizen needs to know (or do) aboutcomputers. This obviously implies that stu-dents should be taught more than simplyhow to operate or program a machine.They also need to know how computerscan be productit ply used and what the con-sequences of computerization are. Thusmatters of computer literacy should betaught not only in the mathematics depart-ment but in science and social studiescourses as well.
Qrdinary people, old and young alike,ht.ve very mil, practical needs for com-puter understauding. For example, pebpleneed to know enough about computer sys-tems so as not to be intimidated by a com-puterized billing error, people need toknow whether to acquire computer equip-ment for home or work; people need tolearn how to evaluate when computer ap-plications are helpful and when "they areharmful; and so forth. Some of these thingscan be learned as a byproduct of learningto write simple BASIC programs, but mostof this type of useful knowledge cannot belearned that way. Indeed we would arguethat most of what every ordinary citizenneeds to know about computers will not belearned from learning how to program.
Instructional ObjectivesIn evaluating the instructional objectives
we proposed for computer literacy, Luehr-mann claims that they are merely a de-scription of what was taught two years ago..In fact, whereas we do not claim that theyrepresent a perfect set. our list of objectivesis much more. The list is an evolving con-ceptual structure and smorgasbord forcomputer literacy. The objectives obviouslydeserve ongoing refinement and updating.In the absence of any alternative sets of ob-jectives, we constructed a list that wouldguide us in the construction of test items.This accounts for the omission of "doing"
or computer usage objectives. More re-cently these objectives have been revised totake into account changing technology andbroader concerns. The revised objectives,which appear in Anderson and Klassen(1981), are designed to serve as a concep-tual framework for curriculum planningand development., Whereas we were origi-nally constrained by what might be mea-sured with paper-and:pencil tests, recentefforts allow us to add- behavioral or psy-chomotor activities that are central tolearning about computers but are hard tomeasure except by observation.
Objectives are generally specified at oneof three different levels: (1) broad learninggoals, (2) informational objectives or de-sired learning outcomes, or (3) behavioralobjectives for a specific activity. We haveaimed our objectives at the second level,the informational level, but many of theseobjectives can be `used as behavioral objec-tives as well. Because we intended that theobjectives would be worthwhile guides forcurriculum materials, we used behaviorallanguage including such terms as recognizeand identify. Whereas some' of the objec-tives deal with factual information that re-quires the exercise of recall, many of theseobjectives require a thorough understand-ing of concepts and principles. In most in-stances it would be much more appropriateto substitute the word understanding forrecognize. Consequently, Luerhmann'sclaim that most of the objectives are .Livialis totally unfounded.
Furthermore, we know from both ourMinnesota assessment of computer literacy(Anderson, Krohn, and Sandman, 1980)and the National Assessment (Carpenter etal. 1980) that a great many junior and sen-ior high school students have basic mis-conceptions about computers. Eyen amongthose who have taken computer program-ming classes we find a great deal of misin-formation. Luerhmann suggests that ourobjectives and associated test items are ofsuch a low level that after a few hours ofhands-on computer experience a typicalstudent would be able to "score near thetop." The empirical facts are just the oppo-
site. From the 1978 National Assessment ofMathematics we know that many, if notmost, of the students who had taken one se-mester of computer programming stillcould not read simple flowchart (Ander-son 1980). In addition, the study, whichtested some fifty computer classes (Klassenet al. 1980), found that the average per-formance on programming items was only30 percent correct for those students whocompleted courses where the instructortaught computer programming. A morethorough statistical analysis in this studyrevealed that the number of hours allo-cated for hands-on computer activities did
Computer literacy isnot the same ascomputer science.
not contribute as much to computes learn-ing as other factors such as the type ofcourse and time spent on computer topics(Klassen et al. 1980).
Luerhmann prOposes that low-levelknowledge about computers should becalled computer awareness. We would ac-cept such a definition but only if computerliteracy- is defined to include computerawareness as well as computer program-ming and. other essential ingredients ofomputer literacy., We believe that com-
puter literacy must encompass the follow-,ing domains:
programming and algorithmsskills in computer usagehardware and software principlesmajor uses and application principleslimitations of computerspersonal and social impactsrelevant values and attitudes
From this taxonomy, which is based on ourrevised list of objectives for computer liter-acy, it is obvious that we place a significantand strong emphasis on the experiential as-pects of computing as well as on thoseareas traditionally called compute; aware-ness. The comprehensive approach to com-
183
puter literacy requires that all these do-mains be included 1 curriculum dealingwith computer literacy. We do not assumethat each domain or each objective will begiven equal weight, as Leurhmann seems toassume. We certainly hope that those usingour objectives will "pick and choose" fromthe entire list and will weigh those that theychoose.
Negative Aspects ofthe Narrow View
The narrow view claims that specifictypes of computer experience and com-puter. programming are the only importantcomponents of computer literacy. Thosepersons who promote this philosophy mayunwittingly promote mindless or meaning-less "doing" as well as constructive experi-ences. Without adequate direction, stu-dents who are actively "doing computing"'may in fact be learning nonrigorous proce-dural thinking, acquiring misconceptions,and otherwise wasting valuable instructiontime. The typical "doing computing" 'ap-proach consists of teaching students towrite a few program_ s in the BASIC lan-guage. As PaPert (1980) Paints. out, thislearning strategy often results in studentlearning that stifles creativity and sup-presses motivation. It can also lead to awk-ward or poor algorithmic thinking.
The development of the BASIC lan-guage during the 1960s paved the way forthe narrow view of computer literacy.BASIC was offered as the single instruc-tional language that everyone was encour-aged to learn. The numerous developmentsin computer science and computer tech-niques during the 1970s have made thisview obsolete. Now there is a wide diversityof computer systems' in place, and few ofthem require the user to write simpleBASIC programs. If teachers now focus onusing microcomputers to replicate the ap-proach of the 1960s, students will not befully prepared to deal with the many large,complex systems of the 1980s, either as pro-grammers or as end users.
Another serious problem is that in thebrief time of a typical course, the student
169
can learn very little about problem solvingand algorithms. It is also difficult to pro-vide experiences' with a variety of lan-guages and a variety of computing systems.Students may complete typical instructionwith limited and inefficient strategies for
Hands-on computerexperience does notguarantee computerliteracy.
problem solving. And if the student onlylearns about a single computer system, heor she can develop a narrow conception ofcomputing and may have difficulty intransferring those skills to other computerenvironments.
The solution is not to provide studentswith computing experiences per se. The so-lution is to provide students with construc-tive computer experiences. It is not easy toprovide these constructive experiences. Thedesign of instruction is not an easy task.Those-that argue-for-the-narrow -view ofcomputer literacy should stop telling uswhat computer literacy should nor be andadvise us on the specifics of what it shouldbe. We need many-minds to work on whatcontributes to the "ability to do comput-ing" and how we can move toward an on-going state of computer literacy in schoolsacross the country.
BIBLIOGRAPHY
Anderson, Ronald E. "National Computer Literacy,1980." Paper presented at the National ComputerLiteracy Goals for 1985 Conference, 18 December1980.
Anderson, Ronald E.. and Daniel L. Klassen. "AConceptual Framework for Developing Computer.Literacy Instruction." AEDS Journal (Spring 1981).
Anderson, Ronald E., Karl Krohn, and RichardSandman. "User Guide for the Minnesota C01:13puter Literacy and Awareness Assessment." St.Paul, Minn.: Minnesota Educational ComputingConsortium, 1980.
Carpenter,'Thomas P., Mary Kay Corbitt, Henry S.Kepner, Jr., Mary Montgomery Lindquist, andRobert E. Reys. "The Current Status of ComputerLiteracy: NAEP Results for Secondary Students."Mathematics Teacher 73 (December 1980):669-73.
184
Johnson, D. C.. R. E. Anderson, T. P. Hansen, andI?. L. Klassen. **Computer LiteracyWhat Is It?"Mathematics Teacher 73 (February 1980):91-96.
Klassen, D. L., R.'E. Anderson, T. P. Hansen, andD. C. Johnson. "A Study of Computer Use and Lit-eracy in Science Education, Final Report." St. Paul,Minn.: Minnesota Educational Computing Con-sortium, 1980.
Moursund. D. "What Is Computer Literacy?" Crea-tive Computing 2 (1976):55.
Papers, Seymour. MindstonnsChildren, Computers,and Powerful Ideas. New York: Basic Books, 1980.
National Council of Teachers of Mathematics. AnAgenda for Action: Recommendations for SchoolMathematics of the 1980s. Reston, Va.: The Coin-cil, 1980.
Rawitsch, D. G. "The Concept of Computer Liter-acy." MAEDS Journal of Educational Computing 2(1978):1-19.
Watt, D. H. "Computer Literacy: What Schot4:Should Be Doing about It." Classroom ComputingNews (December 1980).
170
1411XPLAINING COMPIEJTER RELATEDCONCEPTS & TERMINOLOGY
Harold W. Lawson, Jr.
Have you.ever tried to explain computer systems to someonewho knows nothing or very little about them? Where do youstart? Do you first explain the binary number system. Booleanalgebra and/or computer logic? Perhaps you begin by showingthe classic CPU. memory. I/O block diagram and explaineach of these constituents. including stored programs. instructionsets, etc. Maybe you skip all of these low level details andexplain flowcharting and/or some relatively simple high levelprogramming language such as Basic.
Regardless of where you begin. you soon recognize thatalthough you may be somewhat successful (after sufficienteffort), the person being taught has no idea about computersystems and computing as a whole and many questions remainabout the "miracle of the computer." On the other hand. itseems, on the surface. to be difficult to Rive a completepicture of computer systems right from the beginning.
After several years of experience with a "process oriented"approach we can remove the doubt and say that it is indeedpossible (and desirable) to introduce computer systems conceptsand terminology right from the beginning in an integratedmanner. This approach has resulted in the boor UnderstandingComputer Systems* and the purpose 51-thii- article is topresent the basic concepts used in this approach. As with allendeavours of this nature. the success depends upon properstructuring of the presentation. The approach used is highlypictorial, and terminology is introduced bit-by-bit in a logicalfashion. Each chapter finishes with a summary. work list andproblems. The entire vocabulary introduced is presented atthe end in the form of a glossary.
But enough about the structure of the booklet us considerthe approach.
Introducing Processes and Systems"People learn best when new concepts are presented in termsof what they already know."'
The process is used as the LCD (lowest common denominator)for explaining the constituents of computer systems. At thebeginning, g, it is necessary to select widely known model processesthat "can be used as the basis for association with computersystem related processes. To show the example used. wetake, as a verbatim quotation from the book, the introductorymodel process descriptions.
'Understanding Computer Systems. by Harold W. Lawson.Jr., Lawson Publishing Company. Linkoping. Sweden. ISBN91-7372-333-9. Similar versions of this paper have been previouslypublished in Computer Age. issue 13. 1960 under the title"Understanding Computer Systems" and in Mikrodatorn. Nr.
2, 1981 under the title -Act Forklara Begrepp thornDatatekniken." Permission of these publibhers has been grantedfor publication in Creative Computing.Harold W. Lawson. Jr.. Linkoping Lnisersay. S3I &I Linkoping.Sweden.
We begin this education process %ith a pictorial representationof a well-known real life process (task) faced by many of us.
A Process.Note that the concept of process and task are synonymousand thus in all further references to process related concepts.the word "task" may also be used.
Let us now consider this real life process of washing thedishes in the form of an abstraction which shows the majorelements of this process as follows:
DIRTY DISHES --01-DETERGENT
WATERDISH CLOTH --1
WASH CLEAN DISHES
A Process Abstraction.
Dirty dishes, detergent. water and dish cloth are processinputs and clean dishes are a process output. The processwhich we have shown here can only be carried out. executed.when we apply a processor.
A Processor.Let us complement this single process by introducing a
second process, thus creating a system of cooperating pro-cesses.
A Second Process
Reprinted by permission from Creative Computing 7: 92-102; October 1981.
185
In explaining control, we consider signals to processesandrelated control {mechanisms. For example. the followingIllustrates the producer-consumer relationship ofasynchronouscontrol where the dishrack provides a perfect example of abuffer. The concept of synchronous control is then explainedand compared to asynchronous control.
FOODOVINO 1170140I CORMUNG
IMMO PSOCITS 'ROMsOK to IFROCZND OR TO PROCSIO
0:118141.CtSCUTS*
OTT
FILL BIIIITILII num
Asynchronous Control of Multi-processors
In finalizing the control notions, an important analogy ismade by presenting the process as a machine with lights(corresponding to states) and pushbuttons as illustrated in the
following illustration:
.sown IL&CNOTIl
Duerr tstosts--4 L <m) _....$,....., is mess* 'Baca
OM CLOTSWATILA PI' CI --i o
oF. 8OZT51141XTo.
WA=
,\ INITIAL-i, ef.'. sgitat. Wrilit TICrnIT
ITAIIT---. 1A Process as a Machine Controlled by Push-buttons
_ Programming ConceptsAfter considering- the WASH process af-a pushbutton
machine. it is a s'mple matter to present the logic of a process
as the following state transition diagram.
IDLE
gimes:lugoelopletod
State Transition DiagramTo introduce the basics of programming. a modified form
of "dimensional flowcharting" as described by Witty' is utilized.This flowcharting format is accomplished by using threedrawing directions for indicating sequences of actions. selectionof alternative actions and refinement of larger actions into asequence of actions. These rules are easily remembered fromthis picture.
SELECTION
REFINEMENT
SEQUENCE
Flowchart Conventions
These concepts are motivated step-by-step but we can see
as the final example. the use of the rules for program-like
representation of the WASH process with declarations andprocedures appears in Figure I.
i, .,.....
rwatt MINK IIIIMISSINOT. INTIM MO 41101114 111 M. !f4 t. SIONT. IL1411 MP% MMS
R110IMMWILINI
1
IONA 11114at. IP ~JIM ma& somas. lune ilarposanier
Irosaimum
\NI we ghins111111VIIMINIff
IRMO
SIAM
11111111411111 IR IPOSIMM
1881P..-11.11111118 MA IN Mn MOM
?NWT Ma WM IMOWIN MS61111MR I. SIMI IMMOIII 0/?-INITIS Mb MP
4.
Figure 1. Flowchart of Process Logic
Your intuition should help in understanding this representation
with the knowledge that the ground symbol (--r--) is used to
show a sequence termination and the asterisk () is used toshow the end of a repetitive sequencecontrolled by a condition
at the beginning of the sequence.At this point; enough has been presented indirectly about
computing concepts in the context of the model processes
that the student has achieved a basic understanding and
confidence. The scope of the material described thus far has
permitted the following vocabulary of concepts and terms to
be understood-(presented- here in groupings of individual
chapters).process (task) process inputs
process outputs proceis execution
processor system
cooperating processes uni-procesior
concurrent processes multi-processors
interrupt process priority
process creation process suspension
process initiation process termination
process resumption processor assignment
parallel processes
data informationinformation processing system data processing system
alphabet storage
Poninput port
output port storage process
transmission channels
simplex duplex
half-duplex
control asynchronous control
speed (time) independent synchronous control
speed (time) dependent signals
error conditions buffer
system constraints producing process
consuming process resource
monitor process resource ownership
clock signal process initialization
process (machine) state
1 'Dimensional Flowcharting." Robert W. Witty, SoftwarePractice and Experience. Vol. 7. 553-584 (1977).
187
Note: Implicit in this description is the definition of a systemas a collection of interrelated processes.
We can continue the general abstraction of this system ofcooperating processes by introducing an abstraction for thesecond process, namely the drying process. in the following
manner.
An Abstraction of a System of Cooperating Processes
lithe processes are to be carried out by a single processor.susi-processor, then this single processor must be assigned toboth the WASH and the DRY process as indicated in thefollowing picture.
A Uni-processorNote that if the dishrack becomes full during WASH
exeetation, then the single uni-processor must be alternatedbetween the execution of the processes WASH and DRY.Alternatively, we could assign a processor to each process.that is, the processes are executed concurrently (at the sametime) by multi-processors.
Ontelf OMNI*
Multi-processorsDuring execution of a process by a processor. let us say the
WASH process, the processor could be interrupted by a high
priority process such as the following:
A High Priority ProcessConsequently, a new process is created which we can represent
abstractly as follows:
Lt BABYWET DIAPER
CLEAN DIAPER. .POWDER
XHANGE DRY BABY
An Abstraction of the New Process
188
In the case of a single uni-processor. the processor temporarilysuspends the process it is currently executing and initiatesexecution of the CHANGE process. After termination of theCHANGE process. the processor returns to. resumes. executionof the prncess that was suspended.
Alternatively, in the case of multi-processors. the processorthat acknowledges the interrupt (let us say the processorserving the WASH process) could assign another processor(the processor assigned to DRY or another`available processorsto execute the CHANGE process parallel with the ongoingWASH and/or DRY processes as follows:
Cooperating Process
At this point, we have introduced several important computerrelated concepts in terms that all can understand. The flavorof this example should fibw be remembered as we browsethrough, in a highlight fashion. the further presentation ofconcepts-and-terminology, While the remaining, descriptiQPof the approach will be terse. you can certainly use yourimagination to guess how the details Of the approach arepresented in the book.
Data Flow and ControlHaving introduced the basic process and system concepts.
it is easy to build on the example to include data and controlaspects. Data can be made analogous to objects processed bythe processes of our example leading to the following viewwhere the alphabet of objects is concretely declared.
ALPHAS= OF MIDIS /COP, SAIICSSI, SALAD DISH. ICUS PIIVI
raote
Cs,tIN SERC:2) 9 CD TO
DIRRRACK
:'ration of Process Related Objects (Data)flow of objects within a process is explained in terms
of . lassical data processing steps: namely. input take objectfrom counter), process (wash object) and output {place objectin dishrack). The transmission of objects between processes(over channels), including "storage processes" is indicated bythe following, from which. for example. the explanations ofterms such as simplex. half-duplex and full duplex are easil!,developed.
CLINNIL 8701401 eltAPIOrt 11110011111
.0111/114lt. -1 1 . DST
Transmission of Objects tDatatOver Channels
174
state transitions state transition diagramfinite state machine local storagestate indicators state variablessequence selectionrefinement program (algorithm)flowcharts (flowdiagrams) repetitive sequencedeclinations procedures
Now many of us mastered so many concepts in so short aperiod of time when we began. In classical approaches toeducation (including self education) in this area. many ofthese concepts are learned only after many hours. days.weeks, months (dare we say years) of struggling with details.It is not_us....immon that the details that have been learnedare then a deterrent to gaining a clear picture of these morebasic, general concepts:.
Introducing the Real ThingWe now begin to consider. more concretely. the data
processed by computer systems by introducing the terminaland the computer system in the following, now familiar.frameidork.
MEMORY
4EHIL1.J
COMPUTER
SYSTEM
CHANNEL
Terminal Connection to a Computer Systemirektyto convince-the-student that-by-pushing buttons:-
data is transmitted across the channel and placed in a memorywhere it is processed. and results are sent back for display on
- the terminal screen. At this point. we still treat the computeras a "black box." The terminal process is used to transmitdata back and forth.
In making "data" a more concrete concept. the nature ofnumbers of base 2 and 10,is presented including an easy-to-comprehend algorithm for the process of converting a binarynumber to a decimal number as indicated in Figure 2.
_Imam POSITION VALtIR0SWART NVIIIIZR SSIIKAL NIIIIBIT
INACCIIIIIILATILD SW[
START- --). i
sior1011 001f1111111 10111101 TO OSTINAL\TrAllatIONS
-.401/111'N'
1 SWART 11190111116 1111110L. IITAITT. MICMAC NI71110131
MTN Wittman;4. WIT. PONTION TOUTS. AOCTITIMATED In
i11100111/1111rs Noma, STOAT
t141ToNCSVITINATINI row
/TON NINON? NONSIS
I.'PT POSTON TOM TO 1 WITT- A MAST 010I1 'TIMI .*
OMIT 1111:47 LON 01015 DIOff
! i1 I
ITTLTSPLT NTT NITS POSITION TACT{111111 /NM TO ACCTNOLATID SON
4.
h111.117LT POSTON SALTS IT I
1NVSITST 1,010111TLATD STY TO TISCIVAL WINOS*
Figure 2. Binary to Decimal Conversion Process
189175
After having mastered the WASH processarom the earlierexample. this process doeS-not in any way seem strange andcan be traced by following the dimensional flowchart andindicating resulting changes of the process state variablesgiven within the process.
Further details on electronic representation of signals.precision of arithmetic data and the ASCII character set areeasily introduced. Word processing systems and the represen-tation of programs as text are followed by the presentation ofbask data structure concepts including arrays.
The More Conventional ViewThe stage has now been set to accept the computer system
as simply a collection of cooperating processes connected bychannels as seen in the following picture. ,
B B
'enna. '.....
....tsoesst. anta-.. POIMOTTPre - if& L.MISPIIMM-"7, =mum.= re.r11? Demosmyr, .
tio imamCZ, =3
The Cooperating Processes pf a Computer System
These constituent parts and their general properties are noweasy to introduce. In fact, the CPU viewed as a process iseventually portrayed as follows:
wra spasms 191.2 *MOT
4.-
1
nocsee sasoar--UO "Ta SOTZara 017-4
IMMOBT MOM= . 1160 DATA TO
001Mg LOCATION
rwrzta 1./tni 3p
PIMPIMAL LOOMSI wostnto
MILMIT.t..Z. asoirms I/0party, - C.. = nani"""
°Thor,
The Central Processing Unit
A CPU state transition diagram and CPU process diMensiOnalflowchart provide an easy guide to understanding CPUoperation. A brand name independent simple instructionrepertoire and assembly program presentation then makesthe point about concrete programs and their execution.
Memories and Peripheral DevicesMemories of the core arid semiconductor type are presented
along with the structure of a flip/flop element as' a processwith its related process dimensional flowchart as shown inFigure 3,
norms IMOMFIN
OOMMATIMSiNtI ANTI
CM; MI M. Mt OM MAT& RAM OMPLIMMT
RAM %mum=e I III. M 0071.011111
1PNCIIMP1
MITIMM4:1 M-MTII MT TM M..-111M MOOLIMMT
OR Of"ITT 11P 001071.111111111. eN
MITTOt OM Mr: lit air aorruarrn OTTTITT tipTS MAIM TO Mall COMPLAMMT TO MATO
#1i
*miry Pit ancurinonTO MATT COLBUMT
4.
Figure 3. A Flip/Flop Process Description
Now that we are convinced that the process concept canbe used is a'common denominator to explain both hardwareand software concepts. Storage and peripheral devices (tapes..disks. A/13 and D/A converters. printers. etc.) are then describedin a rather conventional manner.
190
Digital Processes and Hardware ConstructionEven the smallest of computer elements (gates) are presented
as processeswith related process logic as indicated in the. following eiample.
The principles of computer arithmetic are considered.including the view of a binary adder in process form. Thenotions of combinational and sequential circuits are convenientlyintroduced along with a basic explanation of clock signalsand timing.
These digital building blocks and their eventual encapsulationinto integrated circuits. printed circuit boards and chassis areeasily described with this background.
Putting it AU TogetherThe process concept has enabled us to present basic computer
system concepts. the notion of programming and the compo-sition of computer hardware. Putting this all together. wethen create the notion of "architecture." starting with basicdelineation between hardware and software architecture asfollows.
USERS
APPLICATION SYSTEMS
System AnalystsApplication Programmers
DATA BASE SYSTEM
Datadlase Programmers
LANGUAGE TRANSLATORS
Compiler Writers
OPERATING AND FILE SYSTEMS
System Programmers
TARGET SYSTEM ARCHITECTUREMachine Language Progiammers
MICRO ARCHITECTURE
Arteroprograrrunsts
.
, \'MARE
AlICHITXCTUNIC
HANDWARX
COMPUTER CIRCUITSARCIUTLCTURE
Logic Designers
INTEGRATED CIRCUITS
Circuit DesignersSolid State Physicists /
..
191
A further key concept. namely that of "processor-memorypairs" as illustrated in the following provides the keystone todescribing thp various architectural levels.
MEMORY
PROGRAM
oDATA
PROCESSOR
'PROCESSOR-MEMORY PAIR
The Basis for Constructing Programmed Processes
The architecture of the target system (prozrammer's machinelanguage machine) can then b'e described abstractly as nestedpairs of processors and memories as indicated in the follow-ing.
TILOWT *ritzy
112C10 22CMTINCTWilli irsomoses
usamtv I , c-zrnso1 F-7 ....zolmig I,.., 21310at ..,..LORI 112 ; 111010. 1........4
222 11,011
Architecture of the Target System.
This nesting is bulb upon to describe further the placementand role of operating and file systems. language translatorsutility systems, and data bases. and culminates in the followingview of the relationship between ttle application system andthe lower leyel systems of cooperatine processes.
APPLICATIOM SYSTEMS
r minion: -1
PROGRAM I,
-DATA 1'
DATA BASS SYSTEM
MX11ol:T i ,rLANGVAGZ 1I- -
PROGRAM
DATA 1 ;:ti
f AND
L-
r
.
fi
uter ArchitectureApplication S}
Note: the dotted lines around the memory simply deno:e thatit, is not a separate physical-memory but is a program and datarepresentation in a common physical memory. .
Further architectural concepts such as multiport memories.DMA ctiannels and- common busses (highways) can be
-introduced in a natural manner as architectural alternatis es.Finally, the concepts of network architectures and relateaterminology round off his brief (but comphehensive) introductionto the "core" concepts and terminplogy of computer systems.
Experience with the ApproachThe approach described here has been tested and proved
to be successful for a variety of audiences. For those seek=a starting point in learning about computer systems (amateursand professionals to be) the approach permits an examinationof the nature of the forest before examining the leases of thetrees. As an appreciation (computer literacy) vehicle theapproach has orketi successfully fog jotirnalists. politicians,
177
co.
'and administrative and technical people. We have also beensuccessful in removing the fear of secretaries who are learningto rise word processing systems. Terminal operators of allvarieties would certainly benefit from the understandingprovided here. They get an idea of where they fit into thesystem that they are using and some idea of what is happeningin the computer.
The demystification of the computer is a must if we hopeto achieve any degree of harmony between technology andthe citizenry of the world. ThiS approach. which has been oris in the process of being translated into a variety of languages..-provides a step in the right direction.
Some Final CommentsIt is important to note that the approach is described in a
completely product independent manner. therefore. theapproach can be used to complement the real details'bf anyproduct. Teaching assistance materials. including visual aids.are available and are quite useful for presenting the approachto small as well as large eroups. The approach is well suited tobeing supported by animation and plans are being made toprovide and animated video tape and animation via softwarefor commonly available microcomputers.
Understanding Computer Systems can be ordered in singlecopies by sending a check in US dollars for 515.25 (517.25 forairmail) to LAWSON. Gustav Adolfsgatan 9. 382. 20. Linkoping.Sweden. Make checks payable to LAWSON. All other inquirescan be directed to- the sane- address. 0
192
178
aL
CUPERTINO SCHOOL DISTRICT
DEVELOPS COMPUTERLITERACY CURRICULUM
,Cupertino Union School District
10301'Vista Drive "Cupertino, CA 95014
Carl W. Krause, Superintendent
Editor's Note: A. more microcomputers appear in schooli, more school districts or education agencies find
themselves codifying goals and objectives in their computer literacy curriculum. TCT is pL. ,Dishing the Cupertino
Union School' District Computer Literacy Curriculum to promote the exchange of computer ,education goals and
objectives. TCT solicits submission of similar curriculum plans, suggestions, critiques of this document. and reac-
tiors to TCT publishing such material.The Cupertino Union School District and TCT grant permission to reprcquce all or part of document provid-
ed credit 15 given.
The Cupertino Union School District (K-8) has an
average daily 'atteniance of 13,000 pupils. Althoughthe districtls in the heart of Silicon Valley, developingand implementing a, computer literacy (awareriesS)curriculum has taken several, years As Bobby Good-son, District Computer Resource Specialist, says:
"The success of a program like this, introducedthroughout a district, is dependent upon a well-devel-oped inservice program with wide participation thatgives teachers a gcod foundation to build upon. Wehave reached this point because we have taken time(over three years) and worked in stages. I think a dis-trict.would have difficulty instituting such a programas a complete package. People need to be trainedand veady with an explicit curriculum in hand if theprogram is to be truly successful."
District personnel and parent'volunteers began work-
ing with microcomputers in 1977. Financial supportcame from donations, small amounts from existingbudget accounts in schools, _and title monies (U.S.government funds for special purposes such as pro-
grams for the gifted or handicapped). In 1981, theschool board allocated capital expenditure funds topermit widespread piloting of the computer curriculumduring the 1981-1982 school year. Two junior highschools will start the year with twelve microcomputerswith dual floppy disk drives and one printer. Twomicrocomputers will be on a cart along with a 25-inchcolor monitor. Each elementary school that applies andis selected will receive five microcomputers, one ofwhich will be on a cart with a 25-inch color monitor.Implementation dates will be staggered in mei' thatstaff may finish sufficient training and have adequatesupport as the program begins. Most schools plan tohouse the microcomputers in their media centers, but
one plans to use them in a Specific Skills LearningCenter.
Next spring the district curriculum committee willevaluate the computer literacy (awareness) curriculumand its implementation by pilot schools. The super-intendent will then be in a position to make furtherrecommendations to the board.
COMPUTER LITERACY CURRICULUM
PHILOSOPHY & GOALS OF COSI)
All students.will have anopportunity to becom familiar with the operation of a microcomputer. They will be-
dome aware of the widespread use of computers in the world around them. They will be aware of hoth the com-
puter's capabilities and limitations.
Re,!rinted by permissiOn from CmputIng Teacher 9: 2734: September 1981.
1931 79
IMPLEMENTATION OF THE COMPUTER CURRICULUM
In grades 3-6, the use of a microcomputer will be part of the normal school activities. It will be available for usein the classroom as part of the regular curriculum and by students for individual use. Most computer curriculumobjectives at this level can be met by routine classroom use.
In grades 7-8, a specific course in Computer Aw4eness and Introouctory Programming will give students someof the knowledge needed to make wise educational choices in high school and eventual career choices. The com-puter curriculum objectives can be met ,n such a course and/or with the inclusion of computer related topics inmath, social studies and science classes. .
This curriculum. a guide to the ideas students should have an opportunity to learn about computers, has beenprepared with the help of many people. 1.1,ithout the support of the Board, administrative staff, teaching staff, andinterested members of our community, this curriculum would not be possible.
The principal writer was Bobby Goodson, with the assistance of Jenny Better, Judy Chamberlin, Ron La Mar,Barbara Mumma, Jerry Pnzant, Richard Pugh and Cheryl Turner.
Symbols used indicate the level at which given objectives and activities are introduced (I) and reviewed or rein-forced (R).
IN SOCIAL STUDIES, STUDEN1S WILL:K-3 4-6 7-8
1. Become familiar with a computer.
101 Become familiar with a microcomputer through its use in the class- I R R
MOM.
102 Use a prepared program in a microcomputer. I' R
103 Describe the historical development of computing devices. I R
104 Tell about the,history of Silicon Valley. I
, 2. Describe how computers affect our lives.
201 Explain ways computers affect our lives. I R
202 List several ways that computers are used in everyday life. I R R
203 Identify ways that computers are used to help consumers. I R
204 Illustrate the importance of the computer in modern science znd I R
industry.
205 Identify career fields related to computer development and use. I R
206 State the value of computer skills for future employment. I
207 Define the term "data base." I
208 Describe some advantages and disadvantages of a data base of I
personal information.
209 Describe problems related to the "invasion of privacy." I
210 Describe.ways in which computers are used to commit a wide variety I
of crimes and how these crimes are detected.
3. Describe how computers are used by social scientists.
301 Describe how computers are used by sociologists and other scientists.
302 Describe how computer simulations are used in problem solvingsituations.
303 Identify ways in which computers help make decisions.
304 Explain how computer graphics are used in engineering, science,art, etc.
,
194
1 SO
I
I R
305 Explain how ..,:nputers are used as devices for gathering and process-ing data.
306 List several sampling techniques and statistical methods used in thesocial sciences.
307 Describe computer applications such as those consisting of informa-tion storage and retrieval, process control, aids to decision making, computa-tion and data processing, simulation and modeling.
IN LANGUAGE ARTS, STUDENTS WILL:
4. Define and spell basic computer terms.
401 Define (and spell) basic computer terms.
5. Tell about a person or an event that influenced thehistorical development of computing devices.
501 Tell about a person or an event that influenced the historicaidevelopment of computing devices.
6. Describe how computers are used ininformation and language related careers.
601 Explain the meaning of "word- processing."
602 Use a computer as a word-processor.
603 Describe some of the ways computers are used in the informationand language related careers.
IN SCIENCE, STUDENTS WILL:
7. Define "computer" and "program."
701 Tell what a computer is and how it works.
702 rescribe the historical development of computing devices as relatedto other scientific devices.
703 Know the characteristics of each generation of computers.
704 Differentiate among computers, calculators and electronic games
705 Differentiate between analog and digital devices.
706 Differentiate among micro-, mini -, and main frame computers andidentify the five major components of any computer.
707 Define (and spell) basic computer terms.
708 Define software and hardware and list two examples of each.
709 Define "computer 'program."
710 Explain why a comptker needs a program to operate.
711 Define "input" and "output" and give an example-of each.
712 Recognize the relationship of a program, or input, to the result,,or output.
195
181
K-3 4-6 7-8
R R
R
I R
I R
I R
I R
I R
I R
I R
I R
713 Explain the basic operation of a computer system interms of the inputof data or information, the processing of data or information, and the outputof data or information.
714 Evaluate output for reasonableness in terms of the problem to besolved and the given input.
715 Recognize the need for data to be organized to be useful.
716 Describe how computers process data (searching, sorting, deleting,updating, summarizing, moving, etc.).
717 State what will happen if instructions are not properly stated in theprecise language for that computer.
718 List at least three computer languages and identify the purposesfor which each is used.
719 State that BASIC is one of the languages used most commonly bymicrocomputers.
720 Explain the existence of several variations in BASIC.
8. Explain how computeis are used by scientists.
801 Describe the computer's place in our growing understandingof science.
802 Show how a scientist would use a computer.
803 Explain how computers are used in predicting, interpreting andevaluating data.
804 Show that computers are best suited to tasks that require speed,accuracy and repeated operations.
805 Describe situations which limit computer use (cost, availability of soft-ware and storage capacity).
806 Identify common tasks which are NOT suited to computer solution.807 Explain how computer models are used in testing and evaluatinghypotheses.
9. Use a computer to accomplish a simple task.
901 List several fundamental BASIC statements and commands.
902 Differentiate between random computer commands and computerprograms.
903 State the difference between system commands and programstatements.
904 Use a prepared program in a microcomputer.
905 Create a simple program in BASIC.
IN MATHEMATICS, STUDENTS WILL:
10. Exnlain that thesfesign and operation of a computeris based on standard logic patterns.
1001 Explain that a computer design is based on standard logic patterns.
196 182
K-3 1 4-6 7-8
R
R
R
R
R
R
R
1002 State the meaning of "algorithm."
1003 Explain what is being accomplished.by a given algorithm.
1004 Follow and give correct output for a given algorithm.
1005 Describe the standard flow chart symbols.
11. Demonstrate how a computer could be used toaccomplish logical or arithmetic tasks.
1101 Read and explain a flow chart.
1102 Draw a flow chart to represent a solution to a proposed problem.
1103 Order specific steps in the solution of a problem.
1104 Translate mathematical relations and functions into a computerprogram.
1105 Use thp computer to accomplish a mathematical task.
1106 Evaluate output for reasonableness in terms of the problem to besolved and the given input.
1107 State that data must be organized to be useful.
1108 Describe the techniques computers use to prc ass data such assearching, sorting, deleting, updating, summarizing, mming.
1109 List several ways computers are used to process statistical data.
1110 Explain the statement: "Computer mistakes" are mistakes madeby people.
If these activities are combined into a single Junior High Computer Literacyelective, they could be regrouped with the following objectives:
THE STUDENT WILL:
A. Develop a vocabulary of common computer terms.
401 (707) Define (and spell) basic computer terms.
705 Differentiate between analog and digital devices.
701 Tell what a computer is and how it works. ,
704 Differentiate among computers, calculators and electronic games.
706 Differentiate among micro-, mini-, and main frame computers andidentify the five major components of any computer.
: 708 Define software and hardware and list two examples of each.
711 Define "input" and "output" and give an example of each.
207 Define the term "data base."
B. Be familiar with the history of computing devicesand the development of computers.
103 Describe the historical development of computing devices.
702 Describe the historical development of computing devices as relatedto other scientific devices.
197
183
K-3 4-6 7-8
R
R
R
R
R
R
R R
R R
I R
I R
I R
K-3
501 Tell about a person or an event that influenced the historicaldevelopment of computing devices.
104 Tell about the history of Silicon Valley.
703 List the characteristics of each generation of computers.
C. Develop an understanding of how computers are used.
201 Explain ways computers affect our lives.
202 List several ways that computers are used in everyday life.
203 Identify ways that computers are usectto- help consumers.
302 Describe how computer simulations are used in problem solvingsituations. .363 Identify ways in which computers help make decisions.
803 Explain how computers are used in predicting, interpreting andevaluating data.
807 Explain how computer models are used in testing and evaluatinghypotheses.
1109 List several ways computers are used to process statistical data.
306 List several sampling techniques and statistical methods used inthe, social sciences.
601 Explain the meaning of "word processing."
801 Describe the computer's place in our growing understandingof science.
204 Illustrate the importance of the computer in modern scienceand industry,
304 Explain how computer graphics are used in engineering, science,art, etc.
305 Explain how computers are used as devices for gathering andprocessing data.
307 Describe computer applications such as thine consisting of informa-tion storage and retrieval, process control, aids to decision making, compu-tation and data processing, simulation and modeling.
804 Show that computers are best suited to tasks that require-speed,accuracy and repeated operations.
805 Describe situations which limit comp;:ter use (cost, availability ofsoftware and storage capacity).
208 Describe some advantages and disadvantages of a data base ofpersonal information.
209 Describe problems related to the "invasion of privacy."
210 Describe ways in which computers are used to commit wide varietyof crimes and how these crimes are detected.
D. Learn about computer-related career opportunities.
205 Identify career fields related to computer development and use.
198
184
R
7-8
R
R
R
R
R
ry
206 State the value of computer skills for future employment.
301 De Scribe how computers are used by sociologists and other socialscientists.
603 Describe some of the ways computers are used in the informationand language related careers. -
802 ShoW how a scientist would use a computer.
E. Gain a non-technical understanding of how computers function.
1001 Explain that a computer design is based on standard logic patterns.
713 Explain the basic operation of a computer system in terms of the inputof data or information, the processing of data or information, and the outputof data or information.
716 Describe how computers Vocess data (searching, sorting, deleting,updating, summarizing, moving. etc.).
1108 Describe the techniques computers use to process data such assearching, sorting, deleting, updating, summarizing, moving.
715 (1107) Recognize the need for data to be organized to be useful.
1110 Explain the statement: "Computer mistakes" are mistakes madeby people.
806 Identify common tasks which are NOT suited.to computer solution.
709 Define "computer prograin."
710 Explain why a computer needs a program to operate.
712 Recognize the relationship of a program, or input, to the result,.or output.
714 (1106) Evaluate output as to its reasonableness in terms of the problemto be solved and the given input.
717 State what will happen if instructions are not properly stated in theprecise language for that computer.
718 - List at least three computer languages and identify the pUrposes forwhich each is used.
719 State that BASIC is one of the languages used most commonly bymicrocomputers.
720 Explain the existence of several variations in BASIC.
F. Learn to use a computer,
101 Become familiar with a microcomputer through its use in theclassroom.
102 (904) Use a prepared program in a microcomputer.
1002 State the. meaning of "algorithm."
1003 Explain what is being accomplished by a given algorithm.
1004 Follow and give correct output for a given algorithm.
1005 Describe the standard flow chart symbols.
1101 Read and explain a flow chart.
199
1&5
K-3 J 4-6 7-8
R
I,.
1
R
R
R
R
R
R
R
R
R
R
1102 Draw a flow char+ to represent a solution to a proposed problem.
1103 Order specific steps in the solution of a problem.
901 List several fundamental BASIC statements and commands.
902 Differentiate between random computer commands and computerprograms.
903 State the difference between system commands and program,statements.
905 Create a simple program in BASIC.
1105 Use the computer to accomplish a mathematical task.
i104- Translate mathematical relations and functions into a computerprogram.
602 -- Use a computer as a word-processor.
.,
l'
i
186200
K-3 4-6 7-8
I R
I R
I R
I R
I
I `RI R
I
I
-.
END
Section IX
COMPUTER SIMULATIONS
in
MATHEMATICS
201 187
Computer Simulations in Mathematics EducationBy Janice L. Flake
Janice L. Flake. Associate Professor of MathematicsEducation, Florida State University, Tallahassee
Computer simulations can be used to maketheory more relevant. Such simulations canhe used for making either: (1) mathematicaltheory relevant for the youngster, or (2)educational theory relevant for the teachereducation student.
Background
A simulation is a working analogy of asituation, such as a "flight Simulator" or"classroom simulator." Through gamingactivities, the student can manipulate theenvimnrnent to study the interactions ofvarious conditions. The idea is to put thestudent in the middle of a simulated problemsituation; through manipulating the situa-tion, he is to solve the problem.
Such activities should provide fc- thefollowing stages of learning: (1) exploratoryinvestigations. 12) formulating of principles,and (3) operating from principles. Thus, thesame simulation could be used to: (1)introduce the theory, (2) provide for avenuesfor discovery of principles, and (3) providefor reinforcement of the principles.
There are four basic 'components in asimulation gamy: (1) abstracton of anenvironment this is the model, perhapsconsisting of a system of models, (2) a seriesof rules for how the model behaves, ormodels interact this is the simulation, (3)the freedom for the student to interact withthe simulation to develop his or her ownstrategiesthis is the game, and (4)"reality" feedbackthis is what makes itcome "alive."
A simulation study begins with thedevelopment of a eustom-made model. Sucha model may employ a systems approach,where a s stem is a group of interde-pendent elements acting together to ac-
complish a predetermined purpose. For ex-ample, several educational theories, e.g.,teaching strategies and questioning be-haviors, can be written into the model.
Rationale
Too often theory is given primarily in theabstract form, and students fail to see itsrelevance. SimUlations can put the studentinto a concrete example of the theory., As thestudent manipulates the simulated environ-ment, he should gain insights as to how thetheory relates.
Having the facilities of an interactivecomputer system allows for immediatefeedback. Hence, the student can explore anumber of possibilities within a matter of afew minutes. Through his exploratory inves-tigations, he can find some patterns forbehavior of the model. Those patterns canhelp him to become aware of relationships.
In addition, the 'students tend to regardwhat tley are doing as play. There are anumber of strong supporters of the merits ofplay in schools (Piaget, 1951; Bruner, 1975;Caplan and Caplan, 1974).
Simulations can also be used to studyindividual behaviors. The way the studentmanipulates the environment can give a
great deal of information concerning his orher thinking patterns. This information canunobtrusively be :stored and used to studythe child's thinking patterns. and thus canlead to research.
Examples
The following examples show how suchsimulations could be used throughout thecurriculum. Most of the examples relate to acomputer system such as PLATO: This
Reprinted by permission frum Contemporary Education 47: 44-48; February1975.
203
system offers the facilities of graphics andanimated motion. The- visual effects addmuch to the dynamics of the situation, i.e.,they bring much "action" to the student.AtVitional features that can be used for littlechildren are audio facilities, and the touchpanel, where the child simply touches thescreen and the location is recorded.
Example 1
At present I am working with some 6
year olds, trying to teach them the concept of
X.
'(linear measure. T, e literature suggests thatyoungsters of this age have difficultyfocusing on lengths. Thus, we are usingCuisenaire rods tc "build roads." Thestudents are trying to find who can make thelongest road. This activity has kept thechildren's attention for several days.
A simulation of this activity can bedeveToped for PLATO, where. again, thechildren are trying to make the longest road,or perhaps help a playmate cross a ravine toget to a friend. (See Figure 1.1
Ak10.117
Figural.
204 186'
A
Within certain constraints, e.g., timelimits, the student is to decide which replicasof rods he wishes to use. Using the touchpanel, the chiid can "move" the rods to formhis road. If the youngster is successful ingetting the road built, then the playmateruns across the road to his friend. If theyoungster uses an ineffective strategy anddoes not build an appropriate road, then theroad tumbles into the ravine.
Through manipulation of this ,game, theyoungster will find an algorithm of puttingthe longest rods first, then the next longestrods, and so on. Hence, giving the youngsterthe freedom to manipulate the game canallow him to go through the various stages oflearning and build his own strategies.
The use of the simulated activity is moredynamic than the straight use of the blocksbecause. (1) more controls can be set and agreater amount of constraints can be placed,(2) the youngster can receive dynamicfeedback, which, hopefully, will keep his orher attention, and (3) accurate records of thechild's attempts can be kept. Through the,study of such records one can tell if the cladis improving his strategy.
Such a program could be used with first orsecond graders. The use of the actual rods,'though, should be a prerequisite to the-use ofthis program. Piaget (1956) warns that it isthe' actual '-' "action" of manipulating theblock's that helps build concepts.
Example 2The number line is rich with possibilities
for simulations. Through taping trips, basicoperations of addition, subtraction, multipli-cation, and division can :le evolved. One canfirst build ideas of these basic operationswith whole numbers, then carry the sameidea through to the treatment of rational
'numbers.PLATO has the additional feature that a
trail can be left showing the path that hasbeen traveled. Thus, if one were travelingfrom 3 to 5, the 'student could observe themotion of an object traveling from 3 to 5 witha traifigft, showing the traveled path. Thesame program can also be written at a moreabstract level Where no trail is left.
The use of the number line can allow for amuch more consistent introduction to frac-tional numbers than is normally done. Oncethe child has the unit iteration idea, he cantreat a number such as 1'.;' as 3 one-fourth units. This would avoid the oftenconfusing use of many different interpre-tations of fractional numbers, such as partsout of parts, parts of a whole, and so on.
Caution should be used, however, in usingthe number )ine prematurely. The numberline is based upon perception. PLATO is ahighly perceptual instrument. Piaget (1956)warns that a chill's conception of space doesnot evolve out Of perceptions alom,. The childneeds considerable actions on objects (ma-nipulative aids) to build concepts. Theextremely important concept of linear mea-sure should be a prerequisite for using the.number line to illustrate operations onnumbers. I would not want to use thenumber line with `children until they haveinternalized the coordination of the referencesystem.
Example 3"Rate-distance-time" problems often give
youngsters difficulty. Consider a simulationof two cars driving on two intersectinghighways. Allow the child to manipulatevarious values tor rate, distance, and time tosee if he or she can make the cars avoid,eachother or crash into each other. The actionthen is actually carried out. To accentuatethe effects of one of the variable:,. one mightgive fixed values to two of them, and,Vary thevalues of the other.
This program could be used at the fourthgrade level or higher.
Example 4
The ideas of quadratic, functions can beillustrated through the use of motionproperties. For example, suppose one shootsa basket with a basketball. One can vary theinitial velocity and initial angle until he findsthe right combination.
Such a problem could be used at theexploratory investigation stage of learning tointroduce a unit on quadratic functions. Thenit could he used to help build properties ofquadratic functions. Later, it could be used
205 9
to reinforce the properties, where thestudent is opera'ting at the application ofprinciples level of learning.
Example 5
The ideas off line reflections can beillustrated through a simulated billiardsgame. The student is to "shoot" a, targetfrom a starting location, and reflect hissimulated ball off of each of the walls. (SeeFigure 2.)
The student sees the action carried out,'and can try over again and again until he hasdeveloped a strategy that works. .Thestarting positiori and the 'target can berandomly placed, allowing more flaibility.
Example 6
Calculus is rich .with many applicationswhere the use of graphics is, very effective.Problems of m4rimum/miqimum motion,instantaneous veiocity/ and area/volume area few of the possibilities that could' besimulated very easily.
Example 7
An example o educational theory beingmade relevant fog' teacher education internscan be in the fcirm of interns "teaching"
. li
. /
Figure 2.
simulated classes-The intern is to "ask" theclass a question, get a simulated response,react to that response, "ask" a nextquestion, and so on. Through such question-ing behaviors, the intern is to "teach" the'class a principle or concept.
-,Such a simulation is described in more
detail in other publicationt; (Flake, 1973,1974). In this simulation articulation ofseveral educational models was employed:(1) lesson planning, (2) Henderson's movesand strategies of teaching mathematics(Henderson, 1963, 19G7), (3) Wills' approachto problem solving (Wills, 1967), (4) asimplifiedlearnipg theory, and (5) variousquestioning behaviors.
Additional Examples1 The mathematical examples given aboveate rather simple; far more complex prob-lems can be.created. For example, the stuAyof ecology, economics, traffic problems usinglinear programming techniques, agricultural'
?/planning, and flight patterns all lendthemselves to interesting simulation . Theeducational simulation briefly mentioned in'example 7 is an example of a far morecomplex simulation, where some cdm-ponerits of a s classroom intera tion arestudied.
i
t.
206 19j
Capable secondary level students canbecome quite proficient in programming,and delight in creating their own simula-tions. This is an excellent experience forthem, because in order to create a simulatedenvironment, one must carefully understandthe environment that is to be simulated.Summary
Some principles have been givenlconcern-ing how one could use simulations .as a partof the mathematics curriculuml.examples have been given. These are only afew of the many possible examples that couldbe give p. It is hoped that simulations cangive rqlevance, motivation, and a deeperunderstanding of the theory, as well as leadto usefid research.
Lett;t get some "action" into mathematics,and h4lp mathematics become more rele-vant!
REFERENCES
Bruner. J. S. "Play is Serious Busineis," PsychologyToday, 1975.
Caplan, Frank and Theresa Caplan. "This Issue,"Theory into Practice, 1074. Excerpts from The Powerof Play. Doubleday and Company, Inc.
Chorafus, Dimitris N. Systems and, Simulations.Acialenia Press: Nev. York. 1965.
Hake. J. L. "Interactive Computer SimulationsANew Component to Teacher Edtration ERIC EDOS8 ti5:1. 197.1.
. "Interactive Computer Simulations for TeacherEducatilin, Edocational Technology. Publication inprocess
The Use of Interactive Computer Simulationsfor Sensitiiing Mathematics Methods Students toQuestioning Itohataur, Ph.D. thesis, a Universityof Illinois, 1973.
Henderson, K. B. A Model for Teaching Mathe-matical Conceins,' The Mathematics Teacher, 6011967), 573-577.
. "Research in Teachitig Secondary SchoolMathematics, Ilundbook on Research on Teaching,ed. N. L. Gage. Chicago: Rand McNally and Com-pany, 1963.. pp. 1007-1030.
Kazermier, B. H. and D. Vuysje, eds. The Conceptand thl Role of the Model in Mathematics andNatural and Social Science. New York: GordenRid Breach, Science Publishers. 196i.
Piaget, J. Play, Dreams and Imitation in Childhood.' New York: W. W. Norton and Company, 1951.
' The Child's Conception of Space. London:Routledge and Kegan 'Paul, 1956.
Wills, Herbert. "Tramifer of Problem Solving AbilityGained Through Lt:arning by Discovery," Ph.D.thesis. University of Illinois, 1967.
207'
192
PROBABILITY SIMULATIONIN MIDDLE SCHOOL
By GLENDA LAPPANand M. J. WINTERutettigan State UniversityEast Lansing, MI 48824
By middle-school age many studentshave formed notions about the likelihoodof an event happening. These notions areoften incorrect, and yet firmly fixed in theminds of students. Studying probability asa sq of theoretical rules. in our experience,is very unlikely to change any of these pre-conceived notions. Teaching probabilitythrough concrete experiments holds muchmore promise.
We will describe tivo "situations" thatare real to studentsbasketball and cerealbox prizes. Each situation is simulated by aspinner against an appropriate back-ground. The "realness" of the physical situ-ation is not lost; exclamations of "Boy, youmissed again," and "Now you've only gotto get the princess," show the ,nvolvementof the students as they experiment andgather data.
Basketball SimulationThe situation simulated here (proposed
f'dy the Mathematics-Methods Project at In-diana University) is the one-and-one foulshot situation in basketball. A player mak-ing a free throw from the foul line is givena second shot only if on the first, the ballgoes through the hoop. Thus, a basketballplayer shooting a one-and-one can score 0,1, or 2 points.
Suppose we know a player shoots with,.60 percent accuracy. We ask students, "In25 trips to the foul line for a one-and-one,how many times will the player get 2points, 1 point, and 0 points?" Most stu-dents guess that 1 point will happen moreoften than either 0 points or 2 points.
The attempt to make a foul shot can besimulated by a spin against the background
tr
shown in figure 1. If the first "throw" re-sults in a basket, another spin is made. Onegroup of students obtained the data in tableI. At first, there appears to be little patternin the results. However, after doing foursets of twenty-five shots, most students ac-knowledge that I point is not occurring asfrequently as 0 or 2.
144°= 0.4 x 560°
216°= 0.6ic 3CO°
Fig. I. Background for player who shoots with 60percent accuracy
- From a mathematical point of view, thisis a very interesting situation because the-theoretical probabilities of the events 0points, 1 point, and 2 points are obtainedby multiplying probabilities, which is al-ways harder for students to understandthan situations where probabilities areadded. The theoretical expected frequen-cies, computed from the probabilitiei of abasket, 0.6, or a miss, 0.4, are given in table2.
Combining the class results and writingthe frequency fractions as decimals give re-sults close to these theoretical values. Be-cause the numbers 0.4 and 0.36 are close, itis often difficult to detect a difference be-tween the frequencies for v points and for 2points.
Reprinted by permission from Mathematic,; Teacher 73: 446-449;
September 1980.
209
193
TABLE 1One Group's Results of Four Trials
Poir is Trial 1 Trial 2 Trial 3 Trial 4 Tr ' Frequency
32
0 7 10 11 4 32Tr) = 0.32
211 4 3 4 10 21 160 4, 0.21
2 14 12 10 11 4747x0.47
... 0.47
We can observe a different player whohas probability 0.7 of making a free throwby using a new background for the spinner(see fig. 2). The theoretical expected fre-
TABLE 2Theoretical F...44uencies for 60 Percent
Points . Probability
0 0.4 0.4
1 0.6 x 0.4 0.24
2 0.6 x 0.6 0.36
..
quencies for 0, 1, and 2 points are given intable 3. The students' results will now al-most certainly show a clear distinction be-tween the frequencies for 0 points and for 2points.
TABLE 3Theowtical Frequencies for 70 Percent
Points Probability
0 0.3 . 0.31 0.7 x 0.3 me 0.21
2 0.7 x 0.7 0.49
After simulating the results for two play-ers, many students will make reasonable
. prediciions of the results for a superstarwho shoots with 90 'percent accuracy. Afollow-up with older or more able studentswould be an investigation of the theoreticalfrequencieswhy probabilities are multi-plied, and so on.
In the previous activity, each group's ex-perimental results agreed, on. the whole,with the averages of the class results. In thenext activity, the class results arc usuallyquite different from the results of any stu-
dent. This emphasizes the point that proba-bilistic statements are statements aboutwhat will happen in the long run, overmany trials of the experiment.
Star Wars Simi/int:onThe goal for the students is to accumu-
late a complete set of six Star Wars modelsby "buying" cereal boxes. One model iscontained in each box. (We are assumingthere are so many boxes available that eachbox is equally likely to contain any one ofthe mc, 'els.) The students' guesses of howmany boxes they would, on the average,have to buy to get a complete set are usu-ally too small. We can simulate purchasinga box by a spin using a background for thespinner that reflects the situationsixequally likely outcomes (see -fig. 3). Thestudents keep count of how many boxes arebought in collecting a complete set of mod-els. We had each group record their data
108°a 0.3x 360°
Basket
210'
/ 94
252' u 0.7 x 360)
Fig. 2
GI
for collecting four complete sets of models(see table 4).
The theoretical or expected number ofboxes is obtained using the fact that if anevent has probability p, on the average itwill take 1/p trials for it to occur. For ex-ample, the average number of tosses of a
Fig. 3
coin until heads occurs is 1/0.5 = 2. For themodels, the probability that the second boxcontains a different figure from the, first is5/6. Once two distinct models have beencollected, the probability of a box having amodel different from both is 4/6. The ex-pected number of boxes is thus the sum of
1 I
41
3
1
2+
11 +
5+ + +
6 6 6 6 61
... 6(-6 + +4 +3 +2+3 11= 14.7.
The students listed the numbers in thelast column (total number of spins) on theboard. They varied widely: on a list oftwenty-five numbers. there may be onegreater than 45 and another as small as 8.The average will be close to 15. A dis-cussion of the significance of this averagewill provide a background for the ideas ofdispersion and deviation. How representa-tive is the average? Are half the numberslisted within 2 or 3 of this average? Are 2/3of them within 5 or 6 of the average?
Related investigations would be for thestudents to construct new backgrounds forthe spinner:, and compute the averagenumber of boxes needed if there are two,three, four, five, six, seven, or eight differ-ent models in the set. The expected aver-ages for these numbers are given in table 5.For seven different figures the averagenumber of boxes is
7(1 + + 1 + 1 + 1 + 1 + l) = 18.15.7 6 5 4 3 2
If the students graph the number of modelsversus the average number of boxes, theymay be able to predict the average fortwenty-five differen t models (baseballcards) by extrapolating from the graph. Af-ter this much hands-on simulation, itwould be appropriate to replace the spinnerwith a computer program to do the simula-tion and to keep track of the results. Asample output from the BASIC program atthe end of this article is given in table 6
TABLE 4
Set*1
1
n 2
3
4
,DarthVadar
081Kenobi
II NI11 1111
NI 1111
III 1
R2D2 C3P0 Luke PrincessTotal Number
of Spins
NI I I& 29
1 1111 III 20
1 I 1111 24
1 I NI /3
TABLE 5Approximate Expected Averages
Number of models 2 3 4
Approximate expected average 3 5.5 8.3
S 6 7 8
11.41 14.7 18.15 21.74
211U15
(both have been modified for publicationpurposes).
TABLE 6Sample Computer Output
RUNBASEBALL CARDS. THERE ARE
N CARDS IN THE SERIES, ENTER N25FROM HOW MANY TRIALS DO YOUWANT TO COMPUTE THE AVERAGE?1069 CARDS151 CARDS137 CARDS108 CARDS108 CARDS88 CARDS133 CARDS141 CARDS48 CARDS101 CARDS
AVERAGE NUMBER OF CARDS IS 108.4RUNBASEBALL CARDS. THERE ARE
N CARDS IN THE SERIES. ENTER N25FROM HOW MANY TRIALS DO YOUWANT TO COMPUTE THE AVERAGE?1082 CARDS198 CARDS74 CARDS192 CARDS67 CARDS86 CARDS140 CARDS56 CARDS90 CARDS127 CARDS
AVERAGE NUMBER OF CARDS IS 111.2
Conclusion
Both of these activities were designed toinvolve students in exploring a probabilis-tic situation about which their intuitionwas faulty. Each of these is rich from thepoint of view both of mathematical think-ing and processes and of the potential forspin-off in interesting and challenging di-rections. Making conjectures, modeling asituation. gathering data, and validatingthe conjectures should bea part of everystudent's mathematical experiences.
J
212
APPENDIX
BASIC Program for Baseball Cards
100 DIM C000)110 PRINT " BASEBALL CARDS. THERE ARE115 PRINT "N CARDS IN THE SERIES. ENTER N"120 INPUT N130 PRINT "FROM HOW MANY TRIALS DO YOU135 PRINT "WANT TO COMPUTE THE AVERAGE.*'140 INPUT K150 S160 FOR 1 TO K170 = 0180FOR1 =ITON190 CO) =200 NEXT ./210 X = INT (N*RND( +220 C(X) C(X) + I230 IF C(X) = I THEN 250240 GO TO 210 ,.250 D=D+ 1260 IF D = N THEN 280270 GO TO 2102801 =0290 FOR = I TO N300 T T + C(1)310 NEXT320 PRINT T; "CARDS"330S=S+T340 NEXT I350 PRINT "AVERAGE NUMBER OF CARDS LS-WK360 END
196
