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ED 139 621 SE 022 316
AUTROP Blakeway, Edward G.TITLE Drill, Practice, and Testing Applications for the
Reluctant Learner and Teacher of Mathematics:Elementary and Secondary.
"PUB DAT? Oct 76NOTr 2/p.; Paper presented at the annual conference cf *he
National Association of Users of ComputerApplications to Learning (Portland, Oregon, October
30, 1976) ; Not available in hard copy due to marginallegibility of original document
RDRS PR''CrDESCRTPTORS
ABSTRACT
MF-s0.83 flus Postage. RC Not Available from EDFS.Algebra; Arithmetic; *Computer Assisted Instruction;Computers; *Elementary School Mathematics; ElementarySecondary Education; Geometry; *Instruction;*Instructional Materials; Mathematics Education;*Secondary School Mathematics; Student Motivation
One-hundred ,twenty packages of computer programs,
each package dealing with a single matmematical topic, are developed.
Each package consists of three programs; the first gives the student
a problem set, the secon,d allows students to check answers, and the
third composes a test. Topics range from addition of non-negaticainteger's and other elementary topics to coordinate geometry, the
tnomial expansion, and other upper-level subjects. Experienceindicates that even reluctant learners are motivated to use the
system which is easy for both students and teachers to operate.
(SD)
********************************v**************************************Documents acquired by ERIC include many informal unpublished
* materials not available from other sources. ERIC makes every effort *
* to obtain the best copy available. Nevertheless, items of marginal
* reproducibility are often encountered and this affects the guality
* of the microfiche and hardcopy reproductions ERIC makes available
* Vi3 the ERIC Document Reproduction Service (EDRS). EDPS is not
* responsible for the quality of the original document. Reproductions *
* supplied by EDRS are the best that can be made from the original.
***************************************4*******************************
Drill, Practice, and Testing Applicationsfor the Reluctant Learrer and Teacher of Mathematics:
Elementary and SecondarY
Edward G. Blakeway. School Computer Education Center
Department of Mathematics and Science EducationNorth Carolina State University
Raleigh, N. C.
U S DEPARTMENT OF WEALTH.EDUCATION IL WELFAIUENATIONAL. INSTITUTE Of
EDUCATION
7ottS DOCUMENT p,AS BEEN REPRO-DUCED ExACILT As RECEdED FROMT.E PERSON OR OROALRA,ZATtONORiOrs-
TiNo T POt NTS OF v,E osa OP, %IONSSTATED DO NOT NECESSARLT REPRE-SENT OFF iC AL 14,.ONEL .NSTITUTE OFEC,CA'10. POSYCN OS POtCv
The story you are about to hear originated in a local need. A number
of chool systems in North Carolina had gotten together to form a non-profit
corporation called School Computer Service Corporation to provide economical
computing for the instructional program ani there was a need to supoiement
the practice exercises available to schools that,were offering a course usino
the text A Second Course in algebra and Triocnometry With Computer Programming.
Sets of three computer programs, written in the BASIC language, were
produced. Under each topic the three programs were as follows:
' The first provided students with a set of problems. These
were randomly produced so thArno two students were likely
to get the same set of problems.
' The second checked the work that the student had done by
providing answers to his set of problems.
The third tested and graded his knowledge of the topic.
Again, no two students were likely to get the same problem.
Originally it was thought that teachers might develop their own "sets
of three." Consequently, a pattern was developed that might be followed by
a teacher. Experience has shown that teachers do not have the time tha; is
needed for this work. However, the development of a "set of three" has, been
csfc) 2t.14
used very successfully as a project in college student seminars ane teacher
workshops where participants had very little Previous experience of program-
ming.
Over the past four years there has been an expansion to cover topics
ranging from simple arithmetic through most high school algebra topics.
There are now over 120 "sets of three." The programs are used in elementary
schools, junior and senior high schools, and even in some colleges. Over
the years, experience with the programs has resulted in refinement and per-
sonalizing of the programs as well as in instant feedback of the correct
answer, following an incorrect one in the tests.
Success has been par'Acularly noteworthy among slow or reluctant
learners. Many such students have been completely turned around as e, result
of using these programs. The following is an illustration of this type of
success:
A teacher of first year algebra found that she had a group
of students in her tenth grade class who showed total disinterest
even to the point of not turning in any'work (including tests).
It had been felt that they should have a mathematics course and
this was the only one available at the tenth grade level.
They were introduced to the terminal and became fascinated
to the point that they would voluntarily go to the terminal room
to work on their algebra. These students were often found working
on their algebra for 45 r4nutes at a time and completely unsuper-
vised. Prior to this, it is unlikely that any of them would have
voluntarily worked for one minute on mathematics. 1.1ey felt success,
were making good grades, became eager to participate in class, and,
3
in adlition, began to clean themselves up and dress more attrac-
tively. They soon eliminated their poor grades of the early
weeks of the year and passed their course.
These students had not only felt success (evidently an
element that had been lacking) but had gained some self-respect
and, in taped interviews with some of them the following fall,
they al7 strongly recommended that more teachers make use_of
the computer, and felt that without it they would not tIve
passed their course. One who was interviewed volunteered the
information that he thought that he would have forotten.every-
thing bj, the fall but this was not so. He had run some of the
tests and found that he was able to work the problems.
What happened to these students? Before getting involved
with the computer it is unlikely that anyone would have given
them jobs of responsibility. Following the computer experience,
they were used as aides in the terminal rcom to assist their
peers, and did a very fine job.
EAperlence has shown that, unfortunately, many teachers are reluctant to
turn this type of student loose in the terminal room. Provided the orienta-
tion is handled carefully and the topic used in the orientation is simple
enough for the student to succeed, There need be no fear.
Here is a list of the toPics covered with the number of "sets of thrce"
(varying in difficulty) under each topic:
4
4
Addition of Non-Negative Integers
Subtraction of Non-Negative Numbers
Multiplication of Non-Negative Integers
Division of Positive Integers
Finding All Pairs of Factors ofPositive Integers
Addition of Decimals
Subtraction of Decimals
Multiplication of Decimals
Division Of Decimals
Least Common Denominator
Greatest Common Divisor
Reduction of Fractions to Lowest Terms
Conversion of Fractions to Decimals
Addition of Fractions
Multiplication of Fractions
Percentages
Simple Interest
Addition of Signed Numbers
Subtraction of Signed Numbers
Multiplication of Signed Numbers
Division" of Signed Numbers
Addition of Algebraic Expressions
Subtraction of Algebraic Expressions
Rounding Off Numbers
Writing Equations in Form y=ax + b
Finding Slone and Y-intercept GivenLinear Equation
Finding Linear Equation Given Slopeand Y-intercept
Grouping Symbols
Solving:Linear EquationsLinear InequalitiesSystems of Two Linear EquationsSystems of Three Linear Equations
Products of BinomitAs
Products of Trinomials by Binomials
TOPICS
Number
9
3
3
3
1
5
4
4,
1
1
2
1
3
1
1
1
1
1
1
1
1
2
2
3
1
1
1
1
2
1
1
Division of Polynomials
Factoring:TrinomialsDifference of TWO SquaresSum of Two CubesDifference of Two Cubes
Exponents
Quadratic Equations:Solving by FactoringSolving by Completing the
SquareUsing the DiscriminantVertex, Symmetry, Critical
Value
Synthetic Substitution orDivision
Analytics:Coordinates of Mid-Point
of Line SegmentSlope of Line Segment
Joinirg 2 Given PointsSlope of Perpendicular toLine Joining 2 Points
Equation of Pero. Bisectorof Line Segment
Circles:Finding Center and Radius
Given EquationFinding Equation Given. Center and Radius
Ellipses:Equation, Intercepts, Foci
Complex Numbers
Logarithms
Arithmetic Sequences
Geometric Sequences
Probability
Rth Term of BinomialExpansion
Determinant of 3 x 3 Matrix
Rational Roots of CubicEquations
Trigonometric Ratios
Cartesian Products
Permutations
Combinations
Number
4
3
1
1
1
2
2
2
1
3
1
1
1
1
1
1
1
6
5
1
1
1
1
1
2
1
1
1
1
5
Now let us/take a specific set, the first in a series ori multiplication
of non-negative integers.-
Here is a RUN of the problem program. You will note that the in-
--,
structions are brief and include the name of the check prcgram to be called
on completion of the work. The student will have been informed that the
asterisk is used to indicate multiplication.
MULTn1 10:45 'JED. 10-20-76
FP1D THE FOLL?"PIC 10 PR7DU.CTS, THE*1 CALL '4ULTC1*** A"!D
CHrCY:''71.7:7 Nqc"JERS.
1. 5 * 2 2. 4 * R 3. 6 * 4. 7 * 6 5.
6. 6 * 5 7. 1 * 4 * 4 O. R * 1e. *
6
RUNS of the check and test programs follow.
The check program (MULTC1) has been modified here to fit on one page
(checks only 8 of the answers). In the °heck program you will note that the
_student has the option of requesting instructions or not. The first time a
student runs a program he is encouraged to answer "Yes"; thereafter, of
course, he does not require the instructions. The comments at the end of -
the program are self-explanatory.
In the third in the set, the test, you will note the personalizing
that occurs_here. You may also note that the student is spoken to by name
when he gives an incorrect answer - to soften the blow? This is'hardly
necessary if the answer is correct! A very important feature of the test
is the instant feedback of the correct answer when a mistake has been made -
a feature that is not available under more traditional test conditions.
In this way the test is also a learning experience.
7
XULTC1 11:02 "ED. 10-20-76
THIS PROGRAM WILL CHECK YOUR ANSwERs TO THE IP.1BLEMS
THAT WERE GIVEN TO YOU BY THE PR1GRA MULTP1*** .
DO YOU NEED INSTRUCTIONS?YES
IN ANSWER TO THE WESTION 'WHAT ARE THE FACTORS IN THEFIRST PROBLEM?', IF THE PROBLEM YERE 5 * 7 "OU V1ULD TY'E5,7 AND PRESS THE RETURN KEY. THE COMPbTER WOULD THENGIVE YOU THE CORRECT ANSWER.
WHAT ARE THE FACTORS IV THE FIRST PR7BLEM?5,2
YOUR PRODUCT SHOULD BE: 10
NEXT PRIBLEM?4,q
YOUR PRODUCT SHOULD BE: 32
NEXT PROBLEM?6,9
YOUR PRODUCT SHOULD BE: 49
NEXT PROBLEM?7,6
YOUR RRODUCT SHOULD BE: 42
NEXT PROBLE4?9,3
YOUR PRODUCT SHOTTLD BE: 27
NEXT PRIBLEM?6,5
YOUR PRODUCT SHOULD BE: 30
NEXT PROBLEM?1,4
YOUR PRODUCT SHOULD BE: 4
NEXT PRIELEM?7,4
YOUR PRODUCT SHOULD BE: 29
IF YOU HAVE DONE WELL ON THESE PROBLEMS AND FEELCONFIDENT ABOUT FINDING PRODUCTS OF THIS TYPE THEN CALLMULTT1*** THIS IS A TEST. IF YOU NEED MIRE PRACTICECALL MULTP1*** FIR HIRE PROBLEMS.
8
8
MULTT1 6:06. WED. 10:-20-76 .
THIS PRO.GRAM WILL TEST YOU TO FIND OUT HOw VELL YOU CANFIND PRODUCTS OF NUMBERS.
WHAT IS. YOUR NAME?J9E
DO YOU NEED INSTRUCTIONS, JOE?YES
WHEN THE COMPUTER GIVES,YOU A PROBLEM, JOE, YU MAY USE APIECE OF SCRATCH PARER T3 FIND THE INFORMATION THAT THECOMPUTER ASKS FOR. THE.COMPUTER WILL KEE? COUNT IF _YOURSCORE AND'WILL GIVE YOU A GRADE AT THE END. YOU cHOULD THEN
-TAKE THIS GRADE TO YOUR TEACHER.
FIND THE FOLLOVIN_G PRODUCT THEN TYPE YorR ANSwEP ANDPRESS R7TURN:
tORRECT. VERY
NEXT PROBLEM:CORRECT. VERY
GOOD.
GOOD.
10
10
POINTS.
POINTS.9
NEXT PROBLEM: 2
YOUR ANSWER IS NOT CORRECT, JOE.
TRY ANOTHER PROBLEM: 4
CORRECT. VERY GOOD. 10 POINTS.
NEXT PROBLEM: 2
CORRECT. VERY GOOD. 10 POINTS.
NEXT PROBLEM:CORRECT. VERY GOOD. 10 POINTS.
NEXT PROBLEM: 6
CORRECT. VERY GOOD. 10 POINTS.
NEXT PROBLEM:CORRECT. VERY GOOD. 10 POINTS.
NEXT PROBLEM:CORRECT. VEPY GOOD. 10 POINTS.
NEXT PROBLEM: 9
CORRECT. VERY G/I1D. 10 POINTS.
* ?72
* 3 ?27
* 4 ?5IT SHOULD BE g
* 5 ?20
* 3 ?24
*
*
6 ? 36
?32
* 7 756
* 2 71q
YOU HAVE COMPLETED TIE TEST AND YOUR SCORE. IS 90
IF THIS IS'NOT A SATISFACTORY GRADE, JOE, SEE YOuR TEACHEROR CALL MULTP1*** AGAIN.
9
Now let us take a quick look at some other sets.
First: Addition of algebraic expressions. In this set, Joe scored
100 on the test and you see how the computer congratulates him.
ADD'IPR LI :16 10-20+76 0
ADD THE ALGEBRAI C EX737vS S I C2'15 T7HE Llm I 3LE1S,THE,I CALL ADD'ICK*** AMD CHECK r1T:R. AMcT;ERc.
1. -3A + -97-;-5A + -1S
3. -5YT +
-2Y r 2 +
5. - 2A +5A +
+
-5Y + -5-3Y +
15-43
1 0
2. 3A + -3: 4.
2A + -SE + 2C
-2
6 -4Xt 2 + -4Y + -2,-3`f, 2 + -1Y + 1
1Y 2 + 1" +
r,
ADDNCK 11:10 vED. 10-20-76
THIS PROGRAM "PI. CHECK THE A'JS'ERS TO THE 6 PROBLEMS,BRE 1!-IVEN BY THE PROGRAM
D7 YOu NEED INTRUCTIONc?YES
IN ANSWER TO THE QUESTION 'YHAT ARE THE COEFFICIENTcPROBLEM', 17', IF THE RR1BTEM vERE:
3A + -4F-2A + 1B
YOU 'JOULD TYPE 3,-4,-2,1 r/N THAT ORDEF1 ATDRETURN. THE COMPTER OULD THEN GII'E YOU THr CORRECTANSWER. /
1 -1
TZHAT ARE THE COEFFICIENTS
YOUR ANSvEP SHOULD BE:
WHAT ARE THE COEFFICIENTS
IN
I
PROBLEM 1 ?-3,-2,-5,-1
-3 A + -3 B
PROBLEM 2 ?3,-3,-3,2,-5,
YOUR ANS"ER SHOULD BE: 5 A + -1 5 + -1 C
WHAT Ar;E THE COEFFICIENTS IN PROBLEM 3
YOUR ANSVERJSHILD BE: -7 Xt2 + + -2
WHAT ARE THE COEFFICIENTS IN PROBLEM 4
YOUR ANcVER SHOULD BE: -P x12 + 5 X + 2
:./HAT ARE Tf-IE COEFFICINTS Il PROBLEM 5 ?-2,1,5,-4,-5,
YOUR ANSWER SHOULD BE: -2 A + 2 B
VIAT.,ARE THE COEFFICIENTS IN PROBLEM 6 ?-4,-4,-2.-3,-1)
YOUR ANSWER SHOULD BE: -6 MI2 + -4 M - 1
IF YOU HAVE DONE WELL ON THESE PROBLEMS AND FEELCONFIDENT ABOUT FINDING SUMS OF ALGEBRAIC EXPR IONS
THEN CALL ADENT1*** . THIS IS A TEqT ON THIS TYPE
OF PROBLEM. IF YOU NEED MOPE PRACTICE RECALL
ADDNPR*** FOR MOPE PROBLEMS.
.e""
ADDNTI
1' 11
11:23 WED. 10-20-76
THIS PROGRAN'WILL TEST YOU TO FIND 1UT Hlv WRLL YOU OAN
ADD ALGEBRAIC EXPRESSIONS.
VHAT is yOUR
DO yOu NEED INSTRUCTIONS, J3E?N1
ADD THE FOLLO"ING ALGEBRAICEXT3RESSI1NS THEN TYPE THE
COEFFICIENTS Itq YOUR ANSWFR AND PRESS RETURN:
1. + 5R-5A + -4B
/NAT ARE THE COEFFICIENTc IN YOUR AN574ER?-7',1
YOUR AVVER IS CORRECT. 20 POINTS.
VEICT nROBLEM: 2. 5A + -33 + 2C
-1A + -5B + -2C
WAT ARE THE COEFFICIENTS IN YOUR ANSVER?4,-S,0
CORRECT. VERY GOOD. 20 POINTS.'
NEXT PROBLEN':- .3 -4Xt2 + -3X + 3
1X*2 + 3X + 1
4P.
COEFFICIENTS',-3,0,4
CORRECT. GOOD WORK, JOE. 20 POINmS.
NEXT PROBLEMi,
COEFFICIENTS70A-2,-9
CORRECT.
1Xt2 + -3X + -4-1:p2 +. 1X + -5
GOOD(WORK, JOE. -41!20 POINTc-.
NEXT PROBLEM: 5 IXT2 -5X +-3Xt2 5Y + 5
COEFFICLENTS?-2;0,3
CORRECT.. GOOD WORH. JqE. 20 POINT.'
YOU 'HAVE COMPLETED THE TEST AND YOUR SCORE IS 100
12.. EXCELLENT WORK, 40E
Ar I
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Lastly: Find the rational roots of second and third degree polynomial
equations. In this set you will note that thc check program provides tne
student with informatioi (the "possible" roots) leading to the actual"
roots.
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19
These "sets of three" are used very effectively in relation to class
assignments
1) where the test is handed in;
2) with students requiring additional practice;
3) voluntarily by students preparing for a classroom test. If a
good score can be made on the "set of three" test, then it is
likely that the student will do well on his classroom test.
If a student does not do well on a "set of three" test, he can always
take the test again (he will have a different set of problems). There is no
objection to this because he is getting more practice. In fact there is much
motivation here. Students, dissatisfied with their grade, will repeat the
test until they have made 100 or some other grade that they.regard as satis-
factory.
And now a word about the teacher ln relation to programs such as these.
Many schools having access to a computer have "reluctant teachers" - teachers
who are reluctant to make use of the computer. This, of course, is usually
due to insecurity and lack of knowledge. The "sets of three" can be used
very effectively with the absolute minimum of training - in fact, as long
as the student can log on, call a program, and loq off, the teacher is in
business. Teachers normally would want to know how to do this themselves and
usually to know a little more about what is going on. But, with the minimum
of training for the teacher, these Programs will support any text, teacher,
or student. There is no requirement to follow a prescribed course determined
by the programs - as in computer managed instruction. Any teacher can make
immediate use of these "sets of three" or this MATHPACK - as the package is
being called.
2 0
20
Ideally, these programs can free the teacher to teach while assignment
of problems (each student having his own), checking of work, assigning tests
(each student having his own) and grading, are handled by the computer with
a few learning experiences thrown in (for examole, instant feedback of cor-
rect answers.) And why "ideally"? Only because most schools would require
more terminals than are currently available.
- 2 1
