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Symbolic-Graphical Calculators: Teaching Tools for Mathematics Thomas P. Dick Mathematics Department Oregon State University Corvallis, Oregon 97331 -4605 The recent development of hand-held symbolic-graphical calculators represents the opening of one of the most exciting chapters in the ongoing story of technology and its impact on mathematics education. For several years, mathematics educators have been intrigued with how powerful computer software might alter how mathematics is taught and learned. Graphic plotters, symbolic algebra systems, and numerical analysis packages are some broad classes of software which have held promise for use in the mathematics classroom. The problem of access has delayed fulfillment of that promise. Few schools can boast of the economic resources which would allow every mathematics student to have full access to apersonal computerforboth classworkand homework. More often it is the case that at most one or two classroom computers are available, and they are used by the teacher for demonstration rather than hands-on use by the students. The question of access is rapidly being rendered moot. Affordable hand-held calculators with the capabilities to graph functions and relations, manipulate symbolic expressions including symbolic differentiation and integration, compute with matrices and vectors, and perform high-precision numerical integration and root-finding of functions will provide the reality of mathematics classrooms where every student has tools rarely available on mainframe computers 20 years ago. The National Council of Teachers of Mathematics (NCTM) has called for increased attention to the systematic use of both computer graphing to develop conceptual understanding and computer-based methods for solving equations. In terms of students’ personal access to technology, theNCTM’sC^rnc^uw and Evaluation Standards for School Mathematics (1989) makes the following underlying assumption explicit for grades 9-12, "Scientific calculators with graphing capabilities will be available to all students at all times" (p. 124). The availability of powerful calculators plays a fundamental role in thecurrentcalls forreform in the mathematics curriculum. Rather than being just tools to be used in the mathematics classroom, this technology is helping redefine what school mathematics should be. As the Mathematical Sciences Education Board’s influential document Everybody Counts (1989) states: The ready availability of versatile calculators and computers establishes new ground rules for mathematics education. Template exercises and mimicry mathematicsthe staple diet of today’s tests-will diminish under the assault of machines that specialize in mimicry. Instructors will be forced to change their approach and their assignments. It will no longer do for teachers to teach as they were taught in the paper-and-pencil era. (p. 63) So the availability of symbolic-graphical calculators raises some primary, important questions: 1. How will symbolic-graphical calculators affect what students learn and how they leam mathematics? 2. How will symbolic-graphical calculators affect what teachers teach and how they teach mathematics? 3. How will symbolic-graphical calculators affect how both students and teachers perceive mathematics? Many distinctions can be made between algebraic and geometric reasoning, and mathematicians themselves find it convenient to classify themselves either as primarily algebraic or as primarily geometric thinkers. The insights derived from skilled manipulation of meaningful symbols and notation are of a quite different nature than those gained through the interpretation of a visual model. (To be sure, symbolic and visual reasoning are highly complementary mathematical activities, and the communication to students of the power of using both is a major goal of mathematics education.) For the purposes of the discussion in this article, it seems best to consider separately the impact of the symbolic and graphical capabilities of this new generation of hand-held calculators on the mathematics classroom. Symbolic Calculators: Redefining Basic Computational Skills Strictly speaking, even a 4-function calculator is a symbolic calculator, in the sense that numeral digits and decimal points are symbols for inputting and displaying numerical information. The point to be made is that the term symbolic should be interpreted in its broadest sense to include numerical, algebraic, vector, matrix, and any other types of notation used to convey mathematical entities. In this section, the term symbolic calculator will be used in a slightly more restrictive sense to refer to any calculator with the capability to formally manipulate symbolic expressions involving variables. As for what students leam and teachers teach. the availability Volume 92(1), January 1992

Symbolic-Graphical Calculators: Teaching Tools for Mathematics

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Symbolic-Graphical Calculators: Teaching Tools forMathematics

Thomas P. Dick Mathematics DepartmentOregon State UniversityCorvallis, Oregon 97331 -4605

The recent development of hand-held symbolic-graphicalcalculators represents the opening of one of the most excitingchapters in the ongoing story of technology and its impact onmathematics education. For several years, mathematicseducators have been intrigued with how powerful computersoftware might alter how mathematics is taught and learned.Graphic plotters, symbolic algebra systems, and numericalanalysis packages are some broad classes of software whichhave held promise for use in the mathematics classroom.

The problem of access has delayed fulfillment of thatpromise. Few schools can boast of the economic resourceswhich would allow every mathematics student to have fullaccess to apersonal computerforboth classworkand homework.More often it is the case that at most one or two classroomcomputers are available, and they are used by the teacher fordemonstration rather than hands-on use by the students.

The question of access is rapidly being rendered moot.Affordable hand-held calculators with the capabilities to graphfunctions and relations, manipulate symbolic expressionsincluding symbolic differentiation and integration, computewith matrices and vectors, and perform high-precision numericalintegration and root-finding offunctions will provide the realityofmathematics classrooms whereevery student has tools rarelyavailable on mainframe computers 20 years ago.

The National Council ofTeachers ofMathematics (NCTM)has called for increased attention to the systematic use of bothcomputer graphing to develop conceptual understanding andcomputer-based methods for solving equations. In terms ofstudents’ personal access to technology, theNCTM’sC^rnc^uwand Evaluation Standards for School Mathematics (1989)makes the following underlying assumption explicit for grades9-12, "Scientific calculators with graphing capabilities will beavailable to all students at all times" (p. 124).

The availability ofpowerful calculators plays a fundamentalrole in thecurrentcalls forreform in themathematics curriculum.Rather than being just tools to be used in the mathematicsclassroom, this technology is helping redefine what schoolmathematics shouldbe. As theMathematical SciencesEducationBoard’s influential document Everybody Counts (1989) states:

The ready availability of versatile calculatorsand computers establishes new ground rulesfor mathematics education. Template exercisesand mimicry mathematics�the staple diet of

today’s tests-will diminishunder the assault ofmachines that specialize in mimicry. Instructorswill be forced to change their approach andtheir assignments. It will no longer do forteachers to teach as they were taught in thepaper-and-pencil era. (p. 63)

So the availability of symbolic-graphical calculators raisessome primary, important questions:

1. How will symbolic-graphical calculators affect whatstudents learn and how they leam mathematics?

2. How will symbolic-graphical calculators affect whatteachers teach and how they teach mathematics?

3. How will symbolic-graphical calculators affect how bothstudents and teachers perceive mathematics?

Many distinctions can be made between algebraic andgeometric reasoning, and mathematicians themselves find itconvenient to classify themselves either as primarily algebraicor as primarily geometric thinkers. The insights derived fromskilled manipulation of meaningful symbols and notation areof a quite different nature than those gained through theinterpretation of a visual model. (To be sure, symbolic andvisual reasoning are highly complementary mathematicalactivities, and the communication to students of the power ofusing both is a major goal of mathematics education.) For thepurposes of the discussion in this article, it seems best toconsider separately the impact of the symbolic and graphicalcapabilities of this new generation of hand-held calculators onthe mathematics classroom.

Symbolic Calculators:Redefining Basic Computational Skills

Strictly speaking, even a 4-function calculator is a symboliccalculator, in the sense that numeral digits and decimal pointsare symbols for inputtinganddisplaying numerical information.The point to be made is that the term symbolic should beinterpreted in its broadest sense to includenumerical, algebraic,vector, matrix, and any other types of notation used to conveymathematical entities. In this section, the term symboliccalculator will be used in a slightly more restrictive sense torefer toany calculator with the capability to formally manipulatesymbolic expressions involving variables.

As forwhatstudents leam and teachers teach. the availability

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Symbolic-Graphical Calculators

of symbolic calculators could and should have an impact on thecontent of the mathematics curriculum. In many ways, thechanges will simply continue the natural evolution which startedwith the first widespread availability of numerical calculators.

Mathematics classroom teachers have all heard the refrain"Why do we need to know this stuff?" from their students. It isa healthy question whose answer will change in light of hand-held calculators with the capability to perform many ofthe tasksthat traditionally have occupied time in the mathematicsclassroom. Just as the numerical paper-and-pencil algorithm forderiving square roots has virtually disappeared from the highschool curriculum, itseems likely thatpaper-and-pencil algebraicskills such as trinomial factoring or complex fractionsimplification will seem archaic by the beginning of the 21stCentury. On the other hand, the mental skill of estimating asquare root has retained its importance, particularly as a monitorof the reasonableness of results obtained through use of thecalculator. Similarly, as the power of calculators continues toincrease, so will the importance of mental skills.

The NCTM’s position statement (1986) on calculators in themathematics classroom was written before the appearance ofthe first hand-held symbolic calculators. It emphasizes theimportance of mental skills and places in perspective thecalculator as a computational tool to be used intelligently toimprove mathematical reasoning and problem solvingand to aidstudents in understanding mathematics and its applications. Assuch, the statement could easily be adapted to serve as aguideline for symbolic calculators as well. Towards that end,here is avery slightly edited position statement, with the italicizedadjective symbolic added in the appropriate places.

A Proposed Position Statement on SymbolicCalculators in the Mathematics Classroom

Symbolic calculators could free large amounts of the timethat students currently spend practicing symbolic computation.The time gained should be spent helping students to understandmathematics, to develop reasoning and problem-solvingstrategies, and, in general, to use and apply mathematics.

To use symbolic calculators effectively, students must beable to estimate and to judge the reasonableness of results.Consequently, an understanding ofoperations and a knowledgeofbasic facts are as important as ever. The evaluation of studentunderstanding of mathematical concepts and their application,including standardized tests, shouldbe designed to allow the useof the symbolic calculator.

It is recommended that all students use symbolic calculatorsto:

1. concentrateon theproblem-solving process rather than onthe symbolic calculations associated with problems;

2. gain access to mathematics beyond the students’ level ofsymbolic computational skills;

3. explore, develop, and reinforce concepts includingestimation, computation, approximation, and properties;

4. experimentwithmathematical ideas and discoverpattems;and

5. perform those tedious symbolic computations that arisewhen working with real data in problem-solving situations.

Symbol Sense:Mental Skills are Still Important

Estimation and number sense are mental skills important touse as monitors of numerical calculator computations. Forsymbolic calculator computations, symbol sense will bejust asimportant. For example, how does one estimate or check theexpansion of (2a2 + 3fc)4 if it is performed on a symboliccalculator? Here are some simple questions which could beanswered mentally to check on thereasonableness ofthe result:

1. What should the degree of the resulting polynomial be?2. How many terms should it have? What are a couple of

terms that must appear?3. What should the value of the result be for a = 1 and b =

O? for a=0 and b= I?Substitution of simple values as a check on symbolic

calculations has the highly desirable side benefitofcontinuallyreminding students that algebraic expressions can carry aninterpretation and need not be just literal strings of symbolsto be moved about according to arbitrary rules. Another sidebenefit should also be mentioned. Symbolic calculatorsnaturally demand the use of unambiguous notation. Thus, torealize the savings in time and to harness the power ofcomputation that a symbolic calculator can provide, studentswill need to pay more, not less attention to understanding themeaning of the symbols and notation they use.A nice analogy could be drawn between computational

skills in mathematics and carpentry. Skill in making cuts witheither a handsaw or an electric saw does not a carpenter make.Knowing when and where and how to make the proper cuts isthe key knowledge, as long as some appropriate tool is used.Similarly, skill in using either a written algorithm or a slide-rule or a calculator in and of itself cannot be a proper objectivefor mathematics education but knowing when and where andhow to apply appropriate mathematical tools should be.Symbolic calculators are new tools which simply lengthen thelist of computations which are best left to a machine. Inclassrooms where conceptual understanding and problem-solving skills are the primary mathematics educationalobjectives, the availability of symbolic calculators shouldbring about only subtle shifts in emphasis. They will not bringabout a new revolution; they will carry on logically thatrevolution started by the first hand-held calculators.

Graphical Calculators:From Ends to Means

A calculator which can act as a symbolic manipulatormerely extends our notion of the grasp of hand-held machine

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computations. In contrast, a calculatorwith graphics capabilitiesopens up truly different ways of approaching mathematics.This point cannot be overemphasized; graphical calculatorsprovide us with an opportunity to change our approach tomathematics instruction enough that students of the 1990s mayview mathematics in ways fundamentally different than theirpredecessors.

Thatratherboldproclamation requires some clarification. Itis not suggested that teachers have neglected graphing inmathematics instruction. On the contrary, the graphing offunctions and relations has rightfully occupied a prominentplace in the mathematics curriculum for many years. It isrecognized that the human mind seems particularly well-suitedfor quickly capturing and recalling entire visual images, so thepower of graphs to provide complex information in a singleglance should be exploited whenever possible. To paraphrasethe old proverb, "a picture is often worth infinitely manyordered pairs."

Unfortunately, graphing functions and relations the old-fashioned way-by computing several ordered pairs and thenplotting the corresponding points-requires much time andenergy. Students quickly come to view a graph as an arduoustask to be completed rather than as a useful aid in mathematicalanalysis. Textbooks often reinforce that perception by treatinggraphing as an application of algebra or calculus, separating itas if it were a field like physics or engineering. A graphicalcalculator makes it possible to view the graph of a function asa first step instead of a last one. Graphical calculators promiseto unleash the power of these visual representations to fulladvantage in our mathematics instruction.

Graphing Sense

For many of us, the symbolic stimulus of a function such asy = x1 will actually trigger a mental image, in this case that ofa certain parabola. That image, in turn, allows us to extractsalient features about the function y = x1’, to literally see that itis an even function by the symmetry of its graph about the y-axis. The position and shape of the graph in our mind’s eyewarns to reject numerical computations which result in negativefunction values or more than one extremum. This graphingsense provides a powerful aid to use in our analysis of thefunction’s characteristics.

Unfortunately, many students never truly develop thisgraphing sense; that is, this propensity and ready willingness toutilize graphical information to interpret and inform relatedmathematical activities. Students in advanced collegemathematics course work will routinely ignore or fail to makeuse of simple graphical information to monitor their symbolicornumerical computations (Dick, 1988b). Forexample, calculusstudents will blithely accept -2 as a reasonable result of thecomputation of

^dxx2

after readily acknowledging that a definite integral representsarea under the curve. This type of error will occur even in theface of a correct graph of y = 1/x2 that is produced by thestudent.

The view of a graph as a powerful interpretational tool andinformational device can be obscured by students’ perceptionthat graphing is something you do to a function as a task whollyunrelated to other tasks. By providing a means of obtaininggraphs quickly and easily, graphical calculators can turn thatperception around. Threenew avenues ofteaching mathematicsare opened by this technology:

1. Graphing as an exploratory activity. Students withgraphical calculators have the opportunity to observe patternsin graphs, form conjectures based on those observations, andthen subsequently test those conjectures. For example, studentscan graph 15 or 20 exponential functions of the form y = a^within a fewminutes with graphical calculators. Theseexamplescan be used to formulate rules about the influence of theparameters a and b on the graph. In turn, these rules can betested by using them to predict the behavior of a new functionof the same general form and checking the prediction with thegraphical calculator. Similarapproaches canbeusedeffectivelythroughout the graphing curriculum, as with functions of theform y = Asin (Bx + Q + D.

The time spent graphing by plotting individual points byhand can obscure the main ideas. The time saved by usinggraphical calculators allows thefocus toremain ontheconnectionbetween a function and its graph without being divertedunnecessarily to the process of generating the graph.Furthermore, these time savings can be invested in increasedattention to the mathematical concepts under discussion. Evensurprises on the graphical calculator can be valuable launchingpads themselves for discussion of important concepts. (Whydoes my calculator refuse to graph (-S)"?)

2. Graphing as a problem-solving heuristic. Graphicalcalculators allow graphing to be used in the initial stages ofproblem analysis. Just think of how many problems can bereduced to finding the roots or extrema of a particular function.The graph of the function provides immediate and convincingevidence as to where to look. Notice how the tables haveturned. Instead of analyzing the function for roots, extrema,asymptotic behavior, etc. in order to graph the function, thegraph can be used to help analyze the function in problemsolving. Insteadofcomputing functional values toplot individualpoints to include in ourgraph, important values canbe taken offthe graph. For example, a visually obtained estimate of a rootcouldbe fed to anumerical root-finder(included on thegraphical

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calculator, of course) as a first approximation seed. Instead offactoring a polynomial in search of its roots, its roots can be usedto determine its factors.

3. Graphing as a monitoring device. Students fail to usegraphical information to monitor their work because of theperception ofgraphing as a time-consuming and unrelated task.(And few of us would consider graphing by hand a complicatedfunction or relation simply because it arises in a computation.)Graphical calculators should makegraphinga function or relationas natural as a quick mental estimateofa numerical computation."Let me take a look at its graph first" should become the sloganof the mathematics classroom of the 1990s.

New Graphing Skills

Graphical calculators remove the constraints which haverequired teachers and textbooks to concentrate on artificiallynice examples and exercises. (It comes as a surprise to somestudents that not all polynomials have small integral roots.)With that freedom come new constraints. Any graphical devicenecessarily has a finite viewing screen, and that inherentlimitation gives rise to the need for graphing skills which havelonggone underemphasized. That littlewindow on thegraphicalcalculator can be thought of as a porthole to the Cartesian plane.Students will need to leam the navigational skills to position thatwindow for thebest scenic views ofthe graph. Awareness oftheimportance ofscaling on axes and the range ofvisible coordinatevalues demand the attention they deserve when a graphicalcalculator is being used.

Comparisons of graphs will especially require attention toscaling. Comments such as, "A line with slope 1 rises at a 45°angle" or "This parabola is skinnier than that one" requirescaling information to have any real meaning. In this respect,graphical calculators will force teachers to deal with some topicsfrom which students have been protected by textbooks andscale-worry-freeexamplesand exercises. Thenew skills neededfor intelligent useoftechnologyshouldbeviewed as opportunitiesto underscore the limitations of technology while emphasizingmathematical understanding.

Symbolic-Graphical Calculators:Concerns and Conclusions

When hand-held electronic calculators first becameeconomically accessible to all students, there was much concernover theirproper role in the mathematics classroom. Specifically,not a few people expressed worry that students would fail todevelop essential mathematical skills in an environment wherecalculators were readily available. The most popular argumentposed againstcalculator use in the classroom seems based on theanalogy of "brain as muscle." That is, one develops mentalcapacities in the same way one develops physical capacities-through exercise, and just as driving a car does not provide the

physical benefits of running, neither does using a calculatorprovide the same mental aerobics that a paper-and-pencilalgorithm does.

The main idea in the argument against calculators of anytype is that ifa black box does your mathematical work foryou,then it stands to reason that your brain’s mathematical muscleswill atrophy. In particular, opposition to the use of4-functioncalculators in the elementary classroom has sometimes welledup from fears that students will replace blind dependency on amachine for developing their basic arithmetical skills. Whilemuch of that worry has been dispelled by the extensivemathematics educational research on calculators (Hembree &Dessart, 1986), controversy overcalculator usecanbeexpectedto continue (Dick, 1988a). Opposition to the use of symbolic-graphical calculators in secondary and even college classroomswill inevitably arise out of similar fears for students’developmentofbasic algebraic andgraphing skills. (Ironically,some of the same people who have long championed theabolishment ofteaching the long division algorithm in favor ofusing calculators will be among those voicing fears regardingsymbolic-graphical calculators.) The message of this articlehas been that such an argument simply confuses the means ofmathematics education with its ends.

Just as numerical calculators have called into question theteaching ofsome numerical paper-and-pencil skills, so too thecapabilities of symbolic calculators will naturally call intoquestion the teaching ofsome symbolic manipulation skills. Inthis case, a shift in emphasis will be subtle. Some topics willbe covered more quickly than before and/or extended a littlefurther in terms of their application simply because of thesavings in time and energy afforded by the new technology. Inthe case ofgraphing, thechange should be dramatic. Graphicalcalculators allow graphing to be used as a tool for exploration,a heuristic for problem-solving, and a device for monitoringcomputations easily and effectively.

Initially, some teachers may wonder, "Will this omnipotentmachine make me obsolete?" After spending some time usingsymbolic-graphical calculators to solve mathematics problems,the refrain should change to, "What a great machine, but it isof little use unless you know what to do to solve the problem."Thatrealization underscores the limitations ofany technologicaldevice. In terms of Polya’s (1957) problem-solving steps, ittakes people to understand a problem, devise a plan for solvingit, and interpretand evaluate that solution. Apowerful calculatoror computer can only help us carry out that plan.

To focus on the important goals of problem solving andconceptual understanding, the response of mathematicseducators can continue to be intelligent adaptation ofemergingtechnology to the classroom and not a reactionary pose againstthat technology. People, not black boxes, do mathematics.Symbolic-graphical calculators are new teaching tools for themathematics classroom. They provide students and teachers inthe 1990s both more time and power for thinking

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mathematically, calculators in precollege mathematics education: A meta-References analysis. Journal/or Research in Mathematics Education,

77. 83-89.Calculators in the mathematics classroom. (1986). NCTM National Council of Teachers of Mathematics. (1989).

position statement. Curriculum and evaluation standards for schoolDick, T. P. (1988a). The continuing calculator controversy, mathematics. Reston, VA: Author.

Arithmetic Teacher, 35, 37-41. National Research Council. (1989). Everybody counts - ADick, T. P. (1988b). The development of graphical sense, report to the nation on thefuture ofmathematics education.

Paper presented at the annual meeting of the International Washington, DC: National Academy Press.Conference on MathematicsEducation, Budapest, Hungary. Polya, G. (1957). How to solve it (2nd ed.). Princeton, NJ:

Hembree, R., & Dessart, D. J. (1986). Effects of hand-held Princeton University Press.

Summary of the Annual Meeting and Board of Directors* MeetingsSchool Science and Mathematics Association

Tulsa, Oklahoma-Marriott Hotel-October 17-19,1991

Past-Presidents present included Joe Kennedy, Ed Jones, Peggy House, and Dorothy Gabel. Lloyd Barrow and Pat Blossercompleted service as Dircctors-at-Large for 1988-1991. Dorothy Gabel completed a year as Past-President, following two yearsin the office ofPresident. New board members are President-Elect Jerry Beckerand Directors-at-Large Jane Jamsen and MelfriedOlson.

The 1991 George G. Mallinson Distinguished Service Award was presented jointly to Edward and Sue Jones. The 1991Excellence in Integrated Mathematics and Science Award was presented to ArthurWiebe. The establishment ofa new award wasapproved as the Annual Integration of Mathematics and Science Article Award for articles appearing in School Science andMathematics.

The Policy Committee recommended and the Board approved retaining the policy which excludes ex officio members ofcommittees from voting on committee decisions.

The 1990-1991 Preliminary Financial Report showed that SSMA realized a profit for the year, due mainly to the success ofthe Cincinnati convention. The Science/Mathematics Education Endowment Fund was established during the year with$10.000.00 from theconvention profits and$2,500.00 from theGeorgeFern Company. TheSSMAmembershipremains constant.

The Tulsa convention had approximately 900 registrants. The number of SSMA members attending was relatively small.Approval was granted to advertise the UPLINK videoconference videotape in the journal and newsletter. It is expected that afinancial loss will occur this year for the convention.

The SSMA Marquette convention is October 8-10,1992. Bob McGinty will be the Program Chairperson. A call for paperswill appear in the SSMArrt newsletter.

The Journal Editor reported the Association purchased two Macintosh computers and a laser printer with desktop publishingsoftware to move away from the time and expense of typesetting articles. Articles in School Science and Mathematics havemaintained a high quality and are often quoted in otherjournal articles. Submission of articles via computer disk is working well.

The Publications Committee was charged with the responsibility of beginning the process of selecting a new Journal Editor.The SSMArrt Newsletter Editors will complete their term in June 1992. Norma Hemandez, Jim Milson, and Buena Milson ofthe University of Texas at El Paso were approved as Newsletter Editors effective July 1992 for a 3-year term.

The Executive Committee will consist of Bob McGinty, Jerry Becker, Donna Berlin, Jane Jamsen. Larry Enochs, and DarrelFyffe for 1991-1992. The Spring 1992 Board meeting was scheduled for Chicago on May 2 and 3.

Members who wish a completecopy ofthe Minutes oftheBoard ofDirectors’ Meetings should contact theExecutive Secretary.

Submitted by,

Darrel W. FyffeExecutive Secretary

Volume 92(1), January 1992