Schultz Article

7/31/2019 Schultz Article

1/6

448

DISCUSS WITH YOUR COLLEAGUES

Why Representations?

James E. Schultz and Michael S. Waters

growing number of factors seem to influence theteaching and learning of mathematics. These fac-tors include developing conceptual understanding,preparing students for higher-level mathematics,applying mathematics to real-world settings, effec-tively using more technology, and adapting to dif-ferent learning styles.

The NCTMsPrinciples and Standards for

School Mathematics (2000) document embracesmuch of the earlier Curriculum and EvaluationStandards of 1989. A notable difference is theinclusion of a new process standard that addressesrepresentations (p. 67):

Representation Standard

Instructional programs from prekindergartenthrough grade 12 should enable all students to

create and use representations to organize, record,and communicate mathematical ideas;

select, apply, and translate among mathematicalrepresentations to solve problems;

use representations to model and interpret physi-cal, social, and mathematical phenomena.

After discussing the importance of representa-tions from several points of view, this article illus-trates the importance of this new process standardwith regard to a number of contrasting criteria pro-posed by the authors:

1. Which representation best promotes conceptualunderstanding?

2. Which representation best generalizes to higher-level mathematics?

3. Which representation best applies to findingapproximate solutions?

4. Which representation best applies to findingexact solutions?

5. Which representation is best for a given type oftechnology?

6. Which representation best suits the learningstyle and comfort level of the student?

Although all the listed criteria reflect issues thatare likely to be important to teachers, the last

two are particularly important for students toconsider when learning to make choices aboutrepresentations.

THE IMPORTANCEOF MULTIPLE REPRESENTATIONS

Concern has been growing about the role of repre-sentations in teaching mathematics. For example,

writing inProblems of Representation in the Teach-ing and Learning of Mathematics, Kaput (1987, p. 20)argues that representation is underrepresented,citing research that suggests that representationaffects such issues as estimation sense, rote versusmeaningful learning, and procedural versus concep-tual knowledge.

More recently, Greeno and Hall (1997, p. 362)maintain that forms of representation need not betaught as though they are ends in themselves.Instead they can be considered as useful tools forconstructing understanding and for communicatinginformation and understanding. They underlinethe importance of students engaging in choosing

representations: If students simply completeassignments of constructing representations informs that are already specified, they do not haveopportunities to learn how to weigh the advantagesand disadvantages of different forms of representa-tion or how to use those representations as toolswith which to build their conceptual understanding.

Greeno and Hall (1997, p. 365) further suggestseveral ways in which research shows that repre-sentations enhance problem-solving ability. Repre-sentations often match the process of solving theproblem, providing a kind of model of the studentsthinking as they work. . . . Students often constructrepresentations in forms that help them see pat-terns and perform calculations, taking advantage of

Jim Schultz, [email protected], teaches mathematics edu-cation at Ohio University, Athens, OH 45701. MikeWaters, [email protected], was a graduate student inmathematics education at Ohio University when this arti-cle was written. The authors share an interest in curricu-lum development and technology.

A

MATHEMATICS TEACHER

Concern hasbeen growing

about

the role of

representa-

tions in

teaching

mathematics

Copyright 2000 The National Council of Teachers of Mathematics, I nc. www.nctm.org. All rights reserved.

This material may not be copied or distributed electronically or in any other format without written permission from NCTM.


2/6

Vol. 93, No. 6 September 2000 449

the fact that different forms provide different sup-ports for inference and calculation. Citing otherresearch, Brenner et al. (1997, p. 66667) contendthat attempts to teach problem representationstrategies for mathematical problem solving havefocused on teaching students ways to translate thewords of a problem into other modes of representa-

tion using diagrams, pictures, concrete objects, theproblem solvers own words, equations, numbersentences, and verbal summaries.

Emerging technologies also affect the use of mul-tiple representations in the classroom. Demana andWaits (1990, p. 218) contend that the ability of stu-dents to operate within and between different rep-resentations of the same concept or problem settingis fundamental in effectively applying technology toenhance mathematics learning.

FIVE REPRESENTATIONS FORA GIVEN MATHEMATICAL SITUATION

Although the idea of representation applies to

almost any topic in the curriculum for grades K12,this article explores using five different representa-tions to solve two linear equations with twounknowns, a topic that may be developed over aperiod of years. The reader may wish to reflect onhis or her own representations and answers to thesix questions above before reading further.

Solve this pair of simultaneous equations:

3x +y = 9

x + 2y = 8

Representation A: Concrete

Concrete materials, such as pens and notebooks,

shown in figure 1, can be used to facilitate the fol-lowing representation:

Suppose that three pens and one notebook cost$9, whereas one pen and two notebooks cost $8.If each pen costs the same amount and each note-book costs the same amount, find the cost of each.

With the information as shown, the cost of eachitem is not apparent. One way to solve the problem

is to make the number of notebooks the same inboth situations and compare the difference in thenumber of pens and their costs. Begin by doublingthe amounts in the first situation, as indicated infigure 2. The first situation doubled has the samenumber of notebooks as the second situation. It alsohas five more pens, which cost ten more dollars. See

figure 3. Since each pen costs the same amount,each pen costs $2. Use this fact and the first situa-tion to find that the cost for each notebook is $3.

Other examples using this concrete representa-tion can be found in such curricula as ComparingQuantities, a sixth-grade algebra unit fromMathe-matics in Context (Kindt et al. 1998). For a thor-ough discussion of several different strategiesdeveloped by sixth graders to solve systems usingconcrete representations, see Multiple Strategies =Multiple Challenges (Meyer 1999).

Discuss: Concrete representations seem toappear in elementary textbooks but rarely in highschool algebra textbooks. Should high school text-books give more concrete examples and illustratethe use of manipulatives to solve problems?

First situation:

costSecond situation:

cost

Fig. 1Pictorial representation for solving

the system 3x + y = 9 and x + 2y = 8

cost

Fig. 2The first situation in figure 1, doubled

cost

Fig. 3The difference between the first situation doubled

and the second situation

Representation B: Tables

Tables facilitate the guess-and-check, or trial-and-error, approach. Figure 4 shows the table createdby an eighth-grade student, Bethany, who arrivedat a correct solution in four tries. Figure 5 shows asimilar approach using a spreadsheet. Tables canalso be generated using graphing calculators. Beginby solving the original equations fory and creatinga table of values:


3/6

450 MATHEMATICS TEACHER

3x +y = 9, soy = 9 3x.Use Y1 = 9 3X.

x + 2y = 8, soy = 4 0.5x.Use Y2 = 4 0.5X.

Make a table showing X, Y1, and Y2. Try variousvalues ofX to find when Y1 = Y2. See figure 6. Thesolution isx = 2 andy = 3.

The use of the table feature of graphing calcula-tors to solve simultaneous equations can be foundin such textbooks asAlgebra 1 (Schultz, Kennedy,et al. 2001).

Representation C: Graphs

Graph Y1 = 9 3X and Y2 = 4 0.5X, obtained as inrepresentation B. The solution is the point of inter-section of the two lines. See figure 7. The solutionisx = 2 andy = 3. The traditional graphical andalgebraic representations can readily be found inmost algebra textbooks.

Fig. 4Handwritten table representation

for solving the system 3x + y = 9 and x + 2y = 8

Fig. 5Spreadsheet table representation

for solving the system 3x + y = 9 and x + 2y = 8

Fig. 6TI-92 table representation

of Y1=93X and Y2=40.5X

Fig. 7TI-92 graphical representation of

Y1=93X and Y2=40.5X

Representation D: Algebraic

The reader is undoubtedly familiar with severalapproaches to solve this system algebraically. Theaddition-and-subtraction-with-multiplicationmethod, which was used informally in the solutiongiven in representation A, can be shown alge-braically, as in figure 8.Another method is to usethe substitution method for the equations obtainedin representation 5.

3x + y = 9 6x + 2y = 18 x = 2 x + 2y = 8 x + 2y = 18 y = 3.

5x = 10

Fig. 8The addition-and-subtraction-with-multiplication

method, shown algebraically

Representation E: Matrices

To represent this system of equations using matrixmethods, use matrices

a = 3 1 ,1 2z = x ,y

and

b = 9 .8Then the solution to az = b isz = a1b, which agraphing calculator handles easily. See figure 9.


4/6


more than two linear equations by graphing tech-niques involves graphing in three or more dimen-sions, and solving two nonlinear equations by usingmatrices involves more advanced methods. SeePeressini, Sullivan, and Uhl (1988).

Criterion 3: Which representation best applies

to finding approximate solutions?Somewhat ironically, the algebraic representationthat is commonly emphasized in mathematicsclasses is typically limited to the make-believeworld of textbooks and often does not lend itself toequations whose coefficients are generated by real-world data. But equations with unfriendly co-efficients can readily be handled using graphs ormatrices. Examples abound in everyday situations.For example, consider the following problem:

A consumer estimates that her long-distance com-pany charges an average of 10.24 cents per minutewith no monthly fee. Another long-distance com-pany has a plan in which all long-distance callsare 7 cents per minute, with a monthly fee of$4.95. Should the consumer switch long-distancecompanies?

Of course, the answer depends on how manyminutes of long-distance calls the consumer makeseach month. To obtain the answer, solve the systemof equationsy = 0.1024x andy = 4.95 + 0.07x. Agraphing calculator can handle this problem easilyby graphing Y1 = 0.1024X and Y2 = 4.95 + 0.07Xwith an appropriate viewing rectangle and theintersection feature. See figure 10. So if the con-sumer makes 153 minutes or more of long-distancecalls each month, then she would be wise to switch

long-distance companies.

Fig. 9TI-92 matrix representation

for solving the matrix equation az = b

The use of the matrix feature of graphing calcu-lators to solve simultaneous equations can be foundin such textbooks asAlgebra 2 (Schultz, Ellis, et al.2001).

CHOOSING THE REPRESENTATION

ACCORDING TO THE CRITERIAEach of these representations has its place in sixththrough twelfth grades, depending on the criteriathat are applied. The six criteria given at the begin-ning of the article are considered in the following:

Criterion 1: Which representation best promotesconceptual understanding?

The five representations were given in order fromthe most concrete to the most abstract, as judged bythe authors. In particular, the first representationcan be thought of in terms of real-world objects fa-miliar to the student. Such representations, in whichstudents actually manipulate concrete objectspossibly algebra tilesor refer to the pictures ofobjects, could be given in sixth grade or earlier.Representation in a table to summarize guess-and-check methods also serves as a bridge to the moreabstract methods. Unfortunately, far too many stu-dents never experience anything but the more ab-stract representations while learning mathematics.

Criterion 2: Which representation bestgeneralizes to higher-level mathematics?

Two types of generalizations are considered:

Solving more than two linear equations withmore than two unknowns

Solving two nonlinear equations with twounknowns

Interestingly, each of these generalizations isbest approached by a different representation. Solv-ing more than two linear equations using matrixmethods and technology is an easy matter. Usinggraphing technology to solve two nonlinear equa-tions is also relatively simple. By contrast, solving

Fig. 10TI-92 graphical representation

for solving y = 0.1024x and y = 4.95 + 0.07x

using the trace or intersection feature

Many

students

never

experience

the concrete

representa-

tions while

learning

mathematics

A graphing calculators zoom feature enables theuser to apply a graphic representation rathereffortlessly to find approximate solutions with ahigh degree of accuracy. Alternatively, its matrixfeature can also be used with minimal effort.


5/6

Criterion 6: Which representation best suits thelearning style and comfort level of the student?

The choice here, of course, rests with the student.

But unless students are familiar with the variousrepresentations and have had opportunities tolearn to choose and create suitable representa-tionsmuch like problem solving involves knowinghow to apply various computational techniquesand choosing the ones that apply to a given situa-tiontheir ability to make appropriate choices islimited.

Discuss: Do students with different ability lev-els prefer different representations, or do all abilitylevels benefit from the same representations? Inother words, do higher-ability students preferabstract representations over visual or concrete rep-

resentations? What representations do your stu-dents prefer?

CONCLUSION

Thus, representations are significant not only inthe realm of skills but in developing concepts andproblem-solving ability as well. Moreover, studentsbenefit by learning to choose the representationsthat apply to a given situation. This article illus-

452 MATHEMATICS TEACHER

Each of these methods has prerequisites. Thegraphics approach requires that the user representeach equation in Y= form and select an appropri-ate viewing rectangle, whereas the matrixapproach requires that the user represent equa-tions in matrix form. When using technology, newskills emerge to replace old skills, as in rewriting

an equation in a certain form, selecting an appro-priate viewing rectangle, or representing equationsin matrix form.

Discuss: Under criterion 3, the authors mentionthat using technology for the graphing approach re-quires that students have such new skills as alge-braic manipulation and the ability to estimaterange and domain. What skills become less impor-tant when using graphing calculators instead ofpencil-and-paper graphing?

Criterion 4: Which representation best appliesto finding exact solutions?

An exact solution, such asx = 2/7 andy = 5/6, isgenerally best found when using algebraic ormatrix representations, whereas solutions generat-ed by tables or graphs are often approximate.Examples of systems of linear equations for whichexact answers are required do not readily presentthemselves; however, nonlinear systems requiringexact answers occur frequently, such as in solvingoptimization problems in calculus.

Discuss: How do you handle approximateanswers from students using graphs, as opposed toexact answers obtained by algebraic manipula-tions? Are either acceptable? Do you impose a level

of accuracy for approximate answers?

Criterion 5: Which representation is best for agiven type of technology?

Technology greatly enhances our ability to solveequations. Technology that can solve systems of lin-ear equations has been present for many years.Symbolic manipulators that readily handle virtuallyall symbolic manipulation are now commonplace andcan be accessed using the World Wide Web (for ex-ample, Maple is available at www.maplesoft.com)or by using handheld devices (for example, Deriveon the TI-89).

Depending on the choice, technology can also be

used as indicated in the following examples:

If a spreadsheet or a graphing calculator with atable feature is used, then representation B(tables) is a logical choice. See figures 5 and 6.Some graphing calculators facilitate generatingtables by both using an initial value and an in-crement and using a more flexible ask feature.

If a graphing package on a calculator or comput-

er is used, then representation C (graphs) is thenatural choice. See figures 7 and 10.

If technology having matrix capabilities is used,then representation E (matrices) is the choice.See figure 9.

Finally, if a symbolic manipulator (computer-algebra system) such as Maple, Derive, or Math-

ematicasome of which are now available inhandheld versionsis used, then representationD (algebraic) is a suitable choice. See figure 11.

Discuss: What are the implications for represen-tations when computer-algebra systems that do allthe algebraic manipulations come into greater use?

Fig. 11TI-92 symbolic-manipulator representationfor the solution to 3x + y = 9 and x + 2y = 8


6/6


trates how several important criteria may influencea choice with regard to just one of many topics inthe school mathematics curriculum. The ideas fromthis article could also be applied to a wide range ofother topics, such as representing fractions, per-cents, areas, and probabilities. In addition, otherbenefits from familiarizing students with alterna-

tive representations range from checking theirwork to developing a deeper appreciation for con-nections within mathematics.

This topic and these criteria support the notionthat as part of attaining true mathematical power,students should be acquainted with multiple repre-sentations for a given situation. In summary, theauthors suggest that these illustrations support therepresentation standard as a key feature of theNCTMsPrinciples and Standards for School Math-ematics (NCTM 2000).

Discuss: How can the authors suggestions beapplied to teaching such other topics as quadraticequations, logarithms, and trigonometry?

REFERENCES

Brenner, Mary E., Richard E. Mayer, Bryan Moseley,Theresa Brar, Richard Duran, Barbara Smith Reed,and David Webb. Learning by Understanding: TheRole of Multiple Representations in Learning Alge-bra.American Educational Research Journal 34(winter 1997): 66389.

Demana, Franklin, and Bert K. Waits. EnhancingMathematics Teaching and Learning through Tech-nology. In Teaching and Learning Mathematics inthe 1990s, 1990 Yearbook of the National Council ofTeachers of Mathematics, edited by Thomas J.Cooney and Christian R. Hirsch, pp. 21222.

Reston, Va.: National Council of Teachers of Mathe-matics, 1990.Greeno, James G., and Rogers P. Hall. Practicing

Representation: Learning with and about Represen-tational Forms.Phi Delta Kappan 78 (January1997): 36167.

Kaput, James J. Representation Systems and Math-ematics. InProblems of Representation in theTeaching and Learning of Mathematics, edited byClaude Janvier, pp. 1926. Hillsdale, N.J.:Lawrence Erlbaum Associates, 1987.

Kindt, Martin, Mieke Abels, Margaret R. Meyer, andMargaret A. Pligge. Comparing Quantities. InMathematics in Context: A Connected Curriculumfor Grades 58, edited by the National Center forResearch in Mathematical Sciences Education and

Freudenthal Institute. Chicago: EncyclopaediaBritannica Educational Corp., 1998.Meyer, Margaret R. Multiple Strategies = Multiple

Challenges.Mathematics Teaching in the MiddleSchool 4 (May 1999): 51923.

National Council of Teachers of Mathematics (NCTM).Curriculum and Evaluation Standards for SchoolMathematics. Reston, Va.: NCTM, 1989.

. Principles and Standards for School Mathe-matics. Reston, Va.: NCTM, 2000.

Peressini, Anthony L., Francis E. Sullivan, and JerryJ. Uhl, Jr. The Mathematics of Nonlinear Program-ming. New York: Springer-Verlag, 1988.

Schultz, James E., Paul A. Kennedy, Wade Ellis, Jr.,Kathleen A. Hollowell, and Irene Jovell.Algebra 1.Austin, Tex.: Holt, Rinehart & Winston, 2001.

Schultz, James E., Wade Ellis, Jr., Kathleen A.Hollowell, Paul A. Kennedy, and Martin Engelbrecht.

Algebra 2. Austin, Tex.: Holt, Rinehart & Winston,2001.

The authors wish to thank Sid Rachlin for offeringseveral helpful suggestions in the preparation ofthis manuscript.

Schultz Article

Documents

Transcript of Schultz Article