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Harvey, WayneImproving the Teaching and Learning of Algebra Usinga Visual Approach. Rethinking Algebra in SecondaryMathematics Education. Reports and Papers inProgress. Report No. 91-2.Education Development Center, Inc., Newton, MA.Center for Learuing Technology.15 Dec 9120p.Publications, Center for Learning, Teaching, andTechnology, Education Development Center, 55 ChapelSt., Newton, KA 02160.Viewpoints (Opinion/Position Papers, Essays, etc.)(120)

EDRS PRICE MF01/PC01 Plus Postage.DESCRIPTORS *Algebra; Computer Assisted Instruction; Computer

Uses in Education; *Courseware; *CurriculumDevelopment; Curriculum Enrichment; *Functions(Mathematics); Instructional Materials; MathematicsEducation; *Mathematics Instruction; Protaem Solving;Secondary Education; *Secondary School Mathematics;Teaching Methods; Transformations (Mathematics);Visualization

IDENTIFIERS Function Supposer; *Graphing (Mathematics); ProblemPosing; Representations (Mathematics)

ABSTRACTThis paper considers how the algebra curriculum in

secondary mathematics might be reformulated by rethinking both thecontent of algebra and the approaches to teaching and learningalgebraic concepts. It considers how the concept of function can bemade a central theme of the algebra curriculum and further suggeststhat computer software, when designed properly and used withappropriate materials, can provide an opportunity for engagingstudents in significant mathematical inquiry. After a discussion ofthe multiple representations of the concept of function, severalfacets of curriculum innovation involving computers are presented.Some lessons learned from cognitive research invol:ing students'interactions with a variety of graphically based softwareenvironments are given. Three examples of software tools growing outof research findings involving visual representations of functionsare presented; they are: "The Function Supposer"; "The FunctionComparator"; and "The Function Analyzer." Examples of classroommaterials developed to engage students in problem-solving andproblem-finding activities and to take advantage of new opportunitiesfor learning and teaching made available by the new computer softwareare presented. Finally, new direction' in software environmentsbeyond visualization are explored. One of these, a tool called"controlled dynamic phenomena" is described as an appLoach that willbe a catalyst for new intuitions and understandings of functions. (32references) (MDH)

Improving the Teaching and Learningof Algebra Using a Visual Apprlach

Rethinking Algebra in SecondaryMathematics Education

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Publications in the series Reports and Papers in Progress grow out ofthe work of the over twenty projects of the Center for Learning,Teaching, and Technology (111) of Education Development Center.LIT was established in 1982 as the Center for Learning Technology tostudy and develop applications for information technologies. Earlywork explored new and productive uses in education formicrocomputers, video, CD-ROM, and multime,lia. Building on thiswork with technology, a talented and skilled staff of fifty currentlyengage In research and policy, product desigrl and development,professional development programs, and dissemination. LTT nowrepresents a wide-ranging capacity that spans al! levels from earlychildhood to college and in-service, a variety of omtent domains, andlearning and teaching in and out of schools.The Center for Learning, l'eaching, and Technology is part ofEducation Development Center, Inc. (EDC), an international,nonprofit, research and development organization with offices inNewton, Massachusetts, and Washington, D.C. For over three decades,EDC has been recognized as a leading center for curriculum reform,training, research, and institutional development. EDC worksthroughout the United States and around the world to solve a widerange of educational, health, and social problems.

To receive a list of current publications, please write to

PublicationsCenter for Learning, Teaching, and TechnologyEducation Development CenterSS Chapel StreetNewton, MA 02160

Improving the Teaching and Learning of AlgebraUsing a Visual Approach

Rethinking Algebra in Secondary Mathematics Education

Wayne Harvey

This paper considers how the algebra curriculum in secondarymathematics might be reformulated by rethinking both the contentof algebra and the approaches to teaching and learning algebraicconcepts. The author considers how the concept of function can bemade a central theme of the algebra curriculum and further suggeststhat computer software, when designed properly and used withappropriate materials, can provide an opportunity for engagingstudents in significant mathematical inquiry.

Igebra is a central focus in the secondary mathematics curriculum as it iscurrently structured. For many students, their first year of algebra is also their

last year of mathematics. For those students who do continue, geometry often is castas a kind of interlude between algebra 1 and algebra 2a break from the routine, andone that is not destined to ?lay much of a role in the courses that follow. The secondyear of algebra is partly a refurbishing of algebra 1 skills that have remained unusedfor a year, supplemented with greater emphasis on graphs of higher-order polynomials

4

AHARVEY

and transformations on these graphs (transla-tions and dilations). Other topics, which varyfrom text to text, often include preparations fora second preparation (precalcul us) before calcu-lus.

Despite its position as the central thread in themathematics curriculum, little in the way of anorganizing idea behind the algebra curriculum isapparent to students or teachers. The lack of anorganizing idea leads many to talk about thealgebra curriculum as a list of skills to be learned,almost as if they were reciting the chapters in atypical algebra textbook. To students, and oftenenough to teachers as well, the algebra curricu-lum has no internal coherenceit appears as acollection of techniques for manipulating ex-pressions.

The NCTM Curriculum and Evaluatfon Standardsfor School Mathematics (1989) present us withthe beginnings of an appropriate resolution tothis incoherence. One section in the Standardsis devoted to a central theme in secondarymathematics: Functions (Standard 6 in "Cur-riculum Standards for Grades 9-12").

The concept of function is an importantunhying idea in mathematics. Functions,which are special correspondences betweenthe elements of two sets, arc commonthroughout the curriculum. In arithmetic,functions appear as the usual operationson numbers . ; in algebra, functions arerelationships between variables that repre-sent numbers; in geometry, functions re-late sets of points to their images undermotions such as flips, slides, and turns;and in probability, they relate events totheir likelihoods. (p. 154)

There is lively interest now in how studentscome to understand functions, how well theytranslate among symbolic, graphical, tabular,and other representations of these functions;and what role functions play in the overallpicture of a secondary mathematics education.

One underlying assumption of this paper, then,Is that the concept of function can and shouldbe a central theme of the algebra curriculum. Itis clear that the notion of function is at thefoundation of elementary calculus, and a facileunderstanding of and ability to use both graphi-cal and symbolic representations of functionsare critical for work in almost all scientific

disciplines. Unfortunately, few high schoolstudents (or even college students completingcalculus) are able to identify equivalence be-tween algebraic and graphical representationsof functions, interpret graphs accurately, ordevelop an intuitive understanding of func-tions and their representation as graphs (Fey1984; Clement 1985; Goldenberg 1987;Goldenberg & Kliman 1990; Eisenberg & Dreyfus1991).

The NCTM Standards (1989) also emphasizethatin the shift towards developing students'understanding of functions and their variousrepresentationsappropriate uses of computersand graphing calculators should be explored.

Computing technology provides tools, es-pecially spreadsheets and graphing utili-ties, that make the study of function con-cepts and their applications accessible toall students in grades 9-12.The integation of ideas from algebra andgeometry is particularly strong, withgraphical representation playing an im-portant connecting role. Thus, frequentreference to graphing utilities will be foundthroughout these standards; by this wemean a computer with appropriate graph-ing software. . . (p. 125)

But the Standards do not give us insight intowhat °appropriate" graphing software mightbe. Although there is widespread agreementthat computers can and should play a signifi-cant role in a new algebra curriculum, there isnot similar agreement about what constituteseffective uses of computers in the teaching andlearning of algebra. This is an important area forfuture research, and this paper describes somepreliminary attempts to build and exploresoftware environments for studying algebraicconcepts which may inform future work. Thefocus of this work has been on firnction andrepresentation.

"Function" and "Representation"

Functions may be described by a string of sym-bols such as frx) = x2 - 3x + 6, represented by agraph or table, explained in words, or imple-mented as a computer procedure. Only suchmanifestations are available for us to manipu-late and study. Functions, themselves, are anabstractionat the heart of the matter, but

invisible. It is likely that viewing functionsfrom more than one of these perspectives willultimately build a more robust concept thancan leadily be built from one perspective alone;that assumption seems to be the basis of theNCINI's position as stated above. Since it is tootime consuming or too limiting to create andwork with most representations by hand, com-puters are necessary tools for fully interactingwith and manipulating multiple representations.

To make sense out of multiple representationsof a single underlying objectthe functionunder considerationone must be able totranslate readily among the representations and,more importantly, reconcile the different infor-mation provided by the different representa-tions so as to understand the common abstrac-tion underlying all of them. Students' difficultyin doing this is well attested (Clement 1982;Kaput & Sims-Knight 1983, Eisenberg & Dreyfus1991; Goldenberg & Kliman 1990).

Software tools that use multiple linked represen-tations hold great potential for helping studentsunderstand the concept of function. The use oflinked representationswhere changes in onerepresentation are automatically and immedi-ately reflected in alternative representationscan contribute significantly to the develop-ment of a flexible understanding of the notionof function. This method can help studentslearn to see these representations as alternativeviews of the same underlying objectthefunction.

Of course, software alone will never bring aboutmajor improvements in the teaching andlearning of algebra. No single ingredient, actingalone, will suffice. However, software can con-tribute to and support changes in teaching andlearning algebra. To do so, software must bedeveloped in the context of ongoing cognitiveresearch, must address teacher support needs,and must be accompanied by materials andactivities that effectively draw upon the peda-gogical principles of the tool.

The Many Facets of Curriculum Innova-tion Involving Computers

At Education Development Center (EDC), anearly attempt at achieving this kind of integra-tion was carried out in geometry, with The

IMPROVING ME TEACHING AND LEARNING OF ALGERRAA

Geometric Supposer software (Schwartz&Yerushalmy 1985-1991) and curriculum series(Chazan & Houde 1989; Yerushalmy & Chazanin press).1 The software has been accompaniedby project activities and teacher support mate-rials, including video. All are designed to be partof a comprehensive approach to m eeting teacherand student needs. This approach has beenadvocated by a variety of researchers (for ex-ample, Dugdale 1981; Kelman et al. 1983; Fey1984; Kaput 1986) and Is also embotiled in theVisualizing Algebra software environmentsveloped by me and my colleagues at EDC andpublished by Sunburst Communications (Harveyet al. 1989; Mark & Harvey 1990; Harvey et al.1990).

These initial efforts have suggested the effec-tiveness of integrating the following strands ofwork as we rethink the way algebra is taught andlearned in secondary school classrooms:

conducting research on students' interpreta-tion of, and interactions with, visual envi-ronments for exploring functions(Goldenberg & Harvey 1989-1991;Goldenberg & KIlman 1990; Goldenberg1988; Goldenberg et al. 1987)developing multiple representation softwareallowing students to manipulate functionsvisually (Harvey et al. 1990; Harvey et al.1989; Goldenberg Ez Harvey 1989-1991)developing project-based curriculum mate-rials approaching algebra learning visually(Harvey et al. 1989; Goldenberg & Harvey1989-1991)working with teachers in high school class-rooms to integrate these new approaches toalgebra teaching and learning into the exist-ing curriculum (Ruopp 1990-1992)

This paper describes some of our research effortsand findings, discusses some of the softwareenvironments that grew out of this research,and, finally, presents some examples of thecurriculum materials that are so critical forsuccessfully integrating software into the class-room. Our efforts to work with teachers to assistthem in adapting to such new approaches arediscussed elsewhere (Chazan & Houde 1989;Yerushalmy et a). 1990; Yerushalmy & Houde1985).

3Y

&HARVEY

Some Lessons from the Research

While developing, testing, and refining ouralgebra software, and in preparation for devel-oping appropriate curr cular materials for usewith such software, we performed some basiccognitive research involving students' interac-tions with a variety of graphically based soft-ware environments.2 Teachers, mathemati-cians, and mathematics education specialistshelped analyze transcripts which recorded stu-dents' work with software prototypes and algebraactivities designed specifically to elucidate someof our conjectures about students' understand-ings ard misunderstandings. While much ofthis work is described in detail elsewhere(Goldenberg & Kliman 1990; Goldenberg 1990;Goldenberg 1988; Goldenberg 1987; Goldenberget al. 1987), included below are a few examplesof how the research informed the developmentand revision of the software and led to newinsights into the design of curriculum andteaching methods.

Our work has shown students to have some veryweak notionssometimes very incorrect no-tionsof the function concept. These previousconceptions often determine to a great extentwhat students will see, or not see, when inter-acting with visually based software.

For example, we have found that students'perceptions when they view graphs may differfrom those of their teachers, and the strategieswith which students act on their perceptions areoften limited Orconfused (Goldenberg & Klim an1990; Goldenberg 1990; Goldenberg 1988;Goldenberg et 31. 1927). One example of thisperceptual confusion is described below.

When students look only at the symbolicrepresentation of a function such as asecond-order polynomial, they often eas-ily recognize that the domain is unlimited:any value may be plugged in for x and avalue for y may be com puted. On the otherhand, their visual impression of the graphof the same function often takes prece-dence over symbolic analysis and leadsstudents to reason .. . as if the domain isbounded somewhere within the extremesof the domain depicted in the graph. in agraph such as the one in f the figure oppo-site), x looks like" it will never grow be-yond roughly ± 10. (Goldenberg 1988,p. 162)

V-4

This is the kind of research finding that hasimportant implications for the design of edu-cational graphing software and related curricu-lum materials. In this case, it demonstrated theneed to place greater emphasis on scaling issues.We took this into account as we designed TheFunction Analyzer software, described in the nextsection.

Perhaps the most important conclusion of ourresearch is that educational uses of graphing toolsplace very different demands on the softwarethan scientific or engineering uses do.3 This isto be expected. When engineers and scientistsuse gra phers. they are often interested primarilyin the behavior of a particular function. Stu-dents, too, must deal with particular functions,but most of the educational value Is in thegeneralizations students abstract from the par-ticulars. The shape of -2x2 + 30x - 108 is of nospecial educational consequence, but it mayserve as a data point about any of several broaderclasses: a particular family of quadratics (forexample, ones that differ only in the constantterm); more generally all quadratics; still moregenerally all polynomials; or even all functions.

This difference in purpose has particular im-plications for the user interface. If software isintended for student use, it must be easy tomodify functionsnot just through a generaleditor of algebraic expressions, but through a"smart" editor that understands the syntax ofthe expression and allows a student easily tofind and increment or decrement some param-eter in the expression. For example, when astudent is interested in exploring the role of the

-20 x2 x - 1Illusion of Constrained Domain

linear coefficient in a function defined asfrx) = x2 - 6x + 3 a manipulation which, con-ceptually, involves merely stepping up from -6in incrementsa smart editor saves having toattend to delete keys, spaces, signs, and poten-tial typographical errors. All of our algebrasoftware tools allow for this kind of direct ma-nipulation of coefficients In expressions, therebypermftting students to devote more attention tothe nature of the experiment and ..5 outcome.

Our research also showed, however, that experi-ments of this kind can lead to unexpectedconfusions in a student's understanding offunction and, more specifically, the student'semerging concept of vari able (Goldenberg 1988).For students to interpret graphical representa-tions of algebraic expressions correctly, theymust understand the meaning of the variableand of the parameters of the function thatsurround it (for example, coefficients in alolynomial). To the naive student, there is little

difference between the a, b, c, and x in thedefinition frx) = ax2 + hx + c. All four letters'values seem equally "variable." Yet, as thefunction is defined, only x is its variable. Fur-ther, when a student studies quadratics, thisform is a stand-in for an entire class of functions.As students explore the effects ot varying thevalues of a, b, and c on the graph of the functionfrx)x2+bx+c, they are really studying not f(x),whose variable, x, is numeric and whose outputis numeric, but some different kind of function,F(a,b,c), whose three variables a. b, and c arenumeric and whose output (a particular qua-dratic function) is visualized by its graph. Theconcept of variable is already difficult for studentsto learn; yet, here we see variables and constantsswitching roles as students experiment withgraphing. Our latest software experiments at-tempt to address these concerns by includingsome new ways for students to interact withgraphical representations. This work is describedat the end of this paper in the section "NewDirections."

As noted earlier, any particular function is notwhat algebra is about. Rather, it is only asstudents abstract important features of wholefamilies of functions that they develop the kindof knowledge and intuition on which they canbuild. Helping students make such abstractionsis the role of th'.! teacher, assisted by appropriate

IMPROVING THE TEACHING AND LEARNING OF ALGEBRA*

curriculum materials. Consider the taskof help-ing students to understand the essential notionof variable when studying functions likeF(12) x2 + bx + 2 whose domain variable is areal number but whose range elements arethemselves functions with both domain andrange in the real numbers. In the context of theconventional curriculum, these new objectsfunction-valued functionsmay seem terriblyabstruse, but the ideas are apparently quitenatural and are even reflected in much of ourlanguage. When we make statements like "ax2+ 3 generates an upward-opening parabola withits vertex above the origin when a is positive,"we are, in effect, describing a non-numeric"value° (the shape of the picture) of a functionof a. Curriculum and teaching approaches cancapitalize on such intuitive understandings.

In general, where the research we conductedshowed that capabilities provided by the com-puter can lead to unexpected pitfalls In students'understanding, we have tried both to take thesepitfalls into account in our software design andto develop curriculum materials and identifyteaching strategies that address the pitfalls.

Software Environments for Studying theConcept of Function

In collaboration with algebra teachers, math-ematidans, cognitive researchers, and softwaredevelopers, we have designed several soff aretools that use multiple linked representations.In testing our software with students, we havefound reason to be optimistic that students canacquire significantly deeper understandings ofthe function concept by working more directlywith carefully designed visual representationsof functions. This sectfon describes three suchsoftware tools and provides examples of theiruse.

The Function Analyzer. The Function Analyzerprovides tools for the exploration and manipu-lation of functions as expressions or graphs.This software allows one to manipulate thefunction expression or the function graph, ex-amine the values of the function, plot points onthe coordinate plane, or change the scale of thecoordinate plane. Functions are plotted onthree related grids (figure 1, page 6): (1) the largeview on the left side of the screen, (2) the"zoomed out" view on the upper right portion

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Figure 1. The Function Analyzer.

of the screen, and (3) the "zoomed in" view onthe lower right portion of the screen. By pre.senting each function at three different scales,the software focuses attention on the importanceof scale in interpreting the graphical representa.Hon.

One can use the software to plot a variety offunctions and then change them in a number ofways. One can vary the symbolic parameters of

a function, for example, changing the -6 In thefunction x2 - 6x + 3 to the values -6, -4, -2, 0, 2,4, and 6. This generates a family of curves thatprovides some insight into the role of the linearterm in the quadratic function. As the coefficientof the linear term is varied, the parabola momup one side of the screen and then down theother side, without changing shape. Figure 2shows the results of this sequence of graphs.

-30

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Figure 2. The family of functions, fb(x) = xi + bx + 3.

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IMPROVING THE TEACHING AND LEARNING OF ALGEBRAA

We have been especially interested in allowingstudents to manipulate representations of func-tions other than symbolic ones. To this end, wehave designed the software to allow students tomanipulate function graphs directly by trans-lating, stretching, and reflecting them. Trans-lating a graph (figure 3, opposite) correspondsin the symbolic world either to substituting x +constant for x or to adding a constant to ffx).Stretching a graph corresponds either to substi-tuting ax for x or to multiplying /Ix) by a con-stant. Reflecting a graph corresponds either tosubstituting -x for x or to multiplying fiX) by -1.These features allow students to explore func-tions as mathematical objects that can be ma-nipulated and acted upon rather than strictly asprocesses that serve to generate values.

Of course it is important for students also torecognize function expressions as recipes forgenerating values. in an effort to more clearlyrelate the values of x to the values of fix), and thepair of values (x, rim to a point on the graph, thesoftware provides a table of values as a thirdrepresentation of function that is linked to thegraphical representation by both color codingand a special bar on the graph (figure 4, below).The values table displ a) s t he values for x, f(x), andg(x). A vertical bar connecting the current x-value with its corresponding function values isdisplayed in the graph window.

#01) = (00-6)5)04-6).2)

,IMMQ 8: 2Figure 3. Translating the graph of f(x) (x+.5)(x+2).

The Function Supposer. Whereas The FunctionAnalyzer is an environment in which studentscan explore individual functions, their variousrepresentations, and function families, a secondsoftware environment The Function Supposer

allows students to compose new functions byadding, subtracting, multiplying, and dividingfunctions, both symbolically and graphically(Harvey et al. 1990). For example, figure 5 (page8) shows two functions being combined to forma third function. The operation, in whichh(x) = f(x) g(x), is displayed both symbolicallyand graphically.

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Figure 4. The display of function values.

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Figure 5. Building a new function in The Function Supposer.

in the past, operations on functions were al-most universally taught and learned in environ-ments that made use of symbolic representa-tions only. Solving equations, factoring poly-nomials, and finding roots of functions havetraditionally been thought of as topics to beexplored in a world of symbols. It is possible,however, to develop new understandings ofthese activities when they are examined in TheFunction Supposer environment.

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f(H) = 2,28-3

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Consider the function in figure 6 (above). Whenstudents WC only at Its symbolic representa-tion, 14) = x2 + 2x - 3, it is not immediatelyoovious to them that (x-1) and (x+3) are factorsof this function. However, if they look at thegraph of the function and understand the sig-

nificance of the x intercepts, the factors aremore evident because the graph crosses the xaxis at x = 1 and x =-3. Even more convincingevidence can be derived by overlaying the graphsof the presumed futors (figure 7, below). Hereis visual meaning for the fact that (x-1)a nd (x+3)are factors of this function. Note that the rootsof f(x) and g(x) match those of h(x). A functionh represented as a product of functions f and gcannot have the value zero unless one of for ghas the value zero.

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f(H) =904) = H*3hC8) = (8-1) 04.3)

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Similarly, one can take a graphical approach totactoring polynomials and gain new insightsinto the behavior of functions. Consider, for

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IMPROVING THE TEACHING AND LIMNING OF ALGERRAA

f(H) = H3 -2H2 -5H*6

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Figure 8. Dividing polynomials.

example, the function f(x) = x3 - 2x2 - 5x + 6.

Figure 8 (above) shows what happens when onetiles to "factor" this by g(x) = x + 4. it is inter-esting to note that the graph of the resultingfunction, h, has the same three roots as theoriginal function, f. it is also noteworthy thatthe function g does not have a root in commonwith f. These and other features of these graphsmay be obvious to those with more mathemati-cal expertise, but such observations can be quiterevealing for many students.

Another interesting exploration is to note therelationships among the shapes of the graphs,as well as their relative maxima and minima.Students come to expect that when a cubicpolynomial is divided by a linear, the resultvught to look something like a quadratic. But inthe view shown in figure 8, the graph of h(x)does not in fact look much like a quadratic.Students can be encouraged to explore the causesfor the surprise appearance of h(x); such explo-ration may lead them to conjecture that thescale in figure 8 is hiding some of the importantdetail of the graph of h(x).

It is interesting to ask how these graphs mightchange as the divisor, g(x), gets closer and closerto a factor of f(x). Figure 9 (page 10) shows animage in which g(x) is a true factor.

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Notice again the relationships among the rootsof the functions under consideration. And no-tice how the appearance of h(x) changed fromthat in figure 8 (partially approxinnting a pa-rabola as one looks further away frm the rootof g(x) at x = -4)4 to its parabolic appearance infigure 9.

The Function Comparator. Also important inthe analysis of functions is an understanding offunctional comparison. Existing curriculausually frame the study of functional compari-son as the study of equations and inequalities,and use problems in this area (usually of the"solve for x" sort) to lead students to practicesymbol manipulation. An example might be:"Find the set of all x that satisfy the comparisonx2-2> x." Students familiar with such problemswill successfully manipulate the symbol- andfind the solution set: (xl x < -1 or x > 2). Whilepractice may increase students' computationalfluency, it is not clear what insight they derivefrom learning to solve such problems. Mycolleague Judah Schwartz has proposed to keepthe function concept central to such problemsby reframing the proHem as a comparison oftwo functions. !n this example we would inter-pret the problem as the following comparison:frx) >g(x) where f(x) = x2 - 2 and g(x) = x. Now,

12

AHARVEY

f(N) = H3 -2H2-5H.6

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Figure 9. Factoring a cubic.

Figure 10. Comparing functions.

when we graph f and g, the solution set is visu-ally represented and may become more under-standable: it is the set of all x where the valuesof f(x) are greater than the values of g(x) (seefigure 10, above).

Curriculum Materials for Exploring theFunction Concept

Alongside the previously described softwaretools, we have developed a set of classroom

materials that take advantage of the new oppor-tunities for learning and teaching made avail-able by the tools. We have developed thesematerials with the aid of teachers interested inencouraging their students to build mathemat-ics ideas through group and individual projectwork. Our goal has been not only to engagestudents in interesting problem-solving activi-ties, but also to engage students in interestingproblem-findingactivities. Mathematio is, after

13

ail, more about learning to recognize worth-while questions than it is about learning tosolve questions that have already been answered.In designing these materials, we therefore asked,How can we encourage students to ask interestingmathematical questions and to defend anddiscuss their mathematical ideas?

Our approach has been to provide teachers witha variety of materials to motivate students toengage in and discuss mathematics in theclassroom. The intent is to provide enoughstructure in the activity so that explorationscarried out by students can be productivethatis, contain important mathematical contentand also build on previously learned conceptsand skills. At the same time, we want studentsto be raising at least as many questions basedupon their own observations as are raised ex-plicitly in the materials themselves. The follow-ing two examples suggest the range of materialswe have been creating.5

Example 1: Exploring Transformation Points.The first project asks students to use theirknowledge about functions and transforma-tions to complete a table relating function ex-pressions, critical points, transformations, anddrawings of transformed function graphs. (SeeTable 1, page 12, for examples from such atable.) The objectives of this project are asfollows:

Explore how critical points in function graphschange through various graph transforma-tions. For example, do the roots of functionschange when the graph is stretched? trans-lated?Investigate the similarities and differencesin various graphical transformations.Explore how different classes of functionsare affected by graphical transformations.Practice translating between the graphicaland symbolic representations of functions.

The project first challenges students to fill in atable, using The Function Analyzer as a tool toassist them in their explorations, and then asksthem to discuss their findings. The table providessufficient structure for students to understandthe challenge they face; the software providessufficient support to help students find possiblesolutions for the task at hand; and the classdiscussion provides the opportunity for stu-

4

IMPROVING THE TEACHING AND LEARNING OF ALGEBRAA

dents to recognize that their particular line ofthought was not necessarily the same as that ofother students.

Commentary. This project can motiv stu-dents to clarify their knowledge of ft, .onsand transformations and requires them to thinkabout transformations in different ways. Itreinforces the relationships among the variousrepresentations of functions: as expressions, astables of values, and as graphs.

Notice also that the project is designed to allowmany different possible solutions. In somecases, there are no possible answers (for ex-ample, in #2, there is no special invariant pointwhen translating -3x, and similarly in #7). Theproject can lead a class into lively discussions inwhich students generalize the effects of trans-formations and the behavior of functions undervarious graphical transformations. In particu-lar, students can explore how points on a graphare affected by translations, stretches, or flipsand, most importantly, come to understandwhat aspects of a graph are invariant under suchtransformations.

Even more important, however, are the newquestions students begin to raise (and thenexplore) as they compare their approaches. Forexample, in #6 students are provided with threeroots and are asked to make a sketch and find anexpression for a function having those threeroots. As students compare their sketches, theyfind that not all are alike, but they also find thatthe sketches tend to have some similarities. Thestudents' function expressions will also be dif-ferent in some ways and similar in other ways.New questions are raised about how many pos-sible solutions there may be and whether In factthere is a way (using parameters in the expres-sion) to capture the entire family of solutions.6

Example 2: What's Missing? This secondproblem asks students to complete a simpletable of calculations; but rather than usingnumbers (or even algebraic expressions) as theoperands, student3 are asked to think aboutcombining graphical objects. The goals of thisproject include the following:

Explu,,, how operations on functions, whenviewed graphically, affect the shapes of thegraphs.

liv

Table 1, Example 1: Exploring Transformation Points

0Function

ExpressionPoints Transfor-

mation DrawingsValue 1 Position

1 x2 (0,0) WiMiliffirlikkhKVIim,dmem-3x Translate

x 2)(x + 3) roots

(0,-7) rintercept

(5,0)

Wan(-5,0)(0,0)(7,0)

x-axisreflection

10 anyto

11 l*

on

In the Points column, identify points on the function which do not change under thetransformation listed or implied by the drawing.

V12

15

Learn to recognize transformations of graphsas operations on the functions representedby those graphs.Investigate properties of graphical transfor-mations by experimenting with their effectson graphs.Practice reading and manipulating graphs ofsimple polynomials.

Students are asked to use their knowledge offunctions and binary operations to help theirfind the missing element(s) in various graphs offunctional relationships.

In the examples in Table 2 on page 14, studentsare asked to make predictions about the missinginformation before they check their predictionswith The Function Supposer.

Commentary. This project is an unusual recastingof the kinds of problems that are often found inthe present algebra curriculum. Consider howthe project would look if we took a more tradi-tional approach:

Solve for h(x)Complete the calculations in the

table below.

f(x) g(x)-x x - 4 =

x2 + 2x/2 2x3 - x3 - 2x =

h(x)

In this form the intent of the problems isto havethe student exercise symbolic manipulationskillsin effect, performing as an algebraic cal-culator. Whatever increased skill may come ofsuch practice, little else is going on.

By contrast, in the visual alternative presentedearlier, the variety of strategies for thinkingabout the problemsincluding strategies thatmake use of symbolic manipulationskeepsthe problems rich. For example, a studentmight think about problem #1 (f(x) g(x) = h(x))in the following way:

The function h has to be zero when eitherfor g is zero, since h is the product of f andg. Therefore, Ii must be zero twice. Be-tween these two places both f and g arenegative; thus h must be positive. Giventhat h is the product of two linear func-

IMPROVING THE TEACHING AND LEARNING OF ALGEBRA&

tions, I will conjecture that the graph of Itis an upside down parabola.

Problem #4 leads to even more interesting dis-cussions and new questions. Some students wills'T that frx) is a cubic and that h(x) is linear, andtherefore conclude that division is necessary.But this reasoning presents a quandary becausedividing a cubic by something to get a linearfunction requires that the divisor be quadratic.And g(x) is certainly not quadratic. Other stu-dents may reason that if h(x) Is linear and g(x) iscubic, then, given division as the operation, //x)must be a fourth-degree polynomial. Could thisbe possible? That in itself is an interestingdiscussion. And for all of these students someproblem-posing activities are suggested: howcould the problem be changed so that it can besolved with division as the operation? In fact,with the aid of The Function Supposer, somestudents will find that subtraction does indeedwork as a solution to the problem.

New Directions

Our efforts to encourage and support curricu-lum reform in algebra will continue to focus onexamining students' understandings, and mis-understandings, of algebraic concepts and howwe might best use computer environments toprovide more visual approaches to mathemat-ics learning. But it is increasingly clear thatreform of the algebra curriculum must be con-sidered in the broader context of reform of theentire secondary mathematics curriculum. Wesee a strong connection between the dominanceof syntactic/linguistic, non-visual mathematicsinstruction and the decontextualization ofmathematics learningthe fragmentation ofideas, the answering of questions nobody hasasked, the learning of discrete and disconnectedfacts and procedures, and the failure to see howone mathematical idea relates to another. Tech-niques out of context rarely mean much tostudents, and are hard to apply to novel prob-lems. The consequence: students who d )n'tlike mathematics and students who cannot domuch with it.

Of course, mathematical visualization is not, byitself, a solution. Tools for visualization areoften made so context specific that they con-tribute to the isolation of mathematicl ideas.

16

*HARVEY

2

3

4

IFT-4

7.

F.-

$

f(x)

I11

-el

$

f(x)

f(x)IS.

I$

-6.

f(x)

Table 2. Example 2: What's Missing?

*

+

6

..

-8,

e

g(x)

0)

o

.3 $

g(x)

,MIN,Orel

iMMII

-AM..

4.10

Aar

h(x)

h(x)

h(x)

1 / 0

1 7

h(x)

Furthermore, visualization is not always easy.We argue, however, that much of the difficultycan be eliminated by building tools that providestudents with the right kind of dynamic controlover the visualizations they are using.

To this end, we are exploring an entirely newkind of software environment in which weintroduce controlled dynamic phenomena. Forexample, we have been pursuing ways to de-velop students' understanding of function asmapping, and in particular, mappings from R toR. Most graphing software allows one to type ina function expression (for example, fix) = x + 2)and then graphs the specified function in thereal planethat is, in R2. However, the dy-namic-interactive properties of computer in-terfaces permit alternative graphical represen-tations that offer different views of functionalvlationships. For example, instead of using thetraditional perpendicular system of axes forgraphing f(x) = x + 2, one can represent thismapping from R to R on two parallel numberiines. With a single value of x plotted on onenumber line, and its image, fix), plotted on theother number line, a student can use a mouse tomove the x-value on its axis, causing the image,f(x), to move simultaneously on its parallel axisaccording to the functional relationship speci-fied (see figure 11).7

n Line

1(H) Line I Oa a /11

Figure 11. Mapping x f(x) = x - 2 using a parallelnumber line Dynagraph.

This kinesthetic approach to investigating func-tions is a catalyst for new intuitions and under-standings. The simple function IN) = x - 2 canbe teen more obviously as a "subtracting of 2"from the value of x when investigated on theparallel axes than when graphed in the usual R2plane. Various calculus concepts such as limitsand rate of change become more directly ob-servable. In fact, the behavior of functions with

(

IMPROVING ME TEACHING AND LEARNING Of ALCEBRAA

asymptotes, for example frx) 1/(1-x), can sup-port dramatically new intuitions about infinity.As a student moves the plotted x-value acrossthe asymptote boundary, its Image, fix), shootsoff the screen towards positive infinity and thenInstantly back onto the screen from negativeinfinity. By moving the x-value back and forthacross this boundary, one gets the distinct im-pression that positive and negative infinity are"connected back there somewhere," therebyencountering a rich topological idea.

The development of this kind of softwav allowsus to ask some other kinds of questions: Whatdo computers inspire us to do that we might nothave done before? What new questions can beasked of students? What new options areavailable for the sequence in which algebraicskills or concepts are presented? What newcontent becomes interesting to teach? How dostudents misinterpret representations? Doesstudent interaction with the representation af-fect these misinterpretations? What are theeffects of allow1n7 students to manipulate thegraphic representations directly?

It is imperative that we focus our attention onhow to engage more students in more math-ematical inquiry and mathematical explorationin the classroom. Although the teacher is cen-tral to any such changes, we also need to pro-vide teach2rs and students with new kinds ofmaterials, new tools, and new approaches.

Teachers and students need projects and prob-lems that lend themselves to a multitude ofapproaches and a variety of learning and teach-ing styles. In addition, students must feel thatthey can make progress on the project, canperceive avenues to explore the problems, andcan find ways of assessing their progress. Andonce a solution to a problem is found, thatshould not necessarily be the end of the student'sinvestigation projects should lead not only tomany different kinds of solutions, but also tootIt2r new projects.

That is, we want problems that lead more toinvestigations than to answers. All too oftenstudents work on problems that are best calledexercisesactivities designed to have studentsapply previously learned techniques to ques-tions that yield single correct answers. Whenstudents have previously learned the relevant

18

mathematics, they usually find such exercisessimplc; otherwise, they tnay find the exercisesimpossible. Although there is certainly a placefor such mathematics exercises in the curricu-lum, this approach alone will instill in studentsonly a dictionary knowledge of definitions, rules,and arbitrary tricks.

There is a need for much more research withindividual students to help guide developmentefforts, but we are convinced that the story linefor the algebra curriculum in grades 7 through12 will ultimately have to be entirely rewritten.The current curriculum was conceived of duringa time when paper and pencil were the onlytechnologies available in the mathematicsclassroom. Now we are finding that with the useof computers and calculators we can set moreambitious goals than simply doing a better jobat teaching the old curriculum. We can begin toformulate a mathematically rich and coherentapproach to teaching and learning algebra.

Acknowledgements

I wish to acknowledge the visions and ideasgenerated from years of collaboration with E.Paul Goldenberg of EDC, Judah Schwartz ofHarvard University, and Michal Yerushalmy ofHaifa University, Israel. I would like to thankJim Hammerman, Glenn Kleiman, MarleneKliman, and Faye Ruopp for their thoughtfulcomments on earlier drafts of this paper. I

would also like to acknowledge June Mark andLeigh Peake for their work on the graphics andediting. Finally, special thanks to Ilene Kantrov,who was willing to read countless drafts of thispaper and still always provide constructivesuggestions on content and style.

Notes

1. This series of four programs provides tools forintroducing basic geometry concepts and for encour-aging student investigation of the relationships amonggeometric shapes, elements, and constructions. TheGeometric Supposers are designed for grades 8-12 andinclude the preSupposer, Triangles, Quadrilaterals, andCircles. A Problems and Projects curriculum guide ispublished for each program. Other available mate-rials include teaching guides for integrating thesoftware and project materials into commonly usedtextbook curricula and a series of videotapes byteachers. All materials are published by SunburstCommunications, Pleasantville, N.Y.

V-1

2. The research reported here was conducted undera subcontract from the Educational Technology Cen-ter, Harvard Graduate School of Education, and wasfunded in part by the United States Office of Educa-tional Research and Improvement (Contract No. OEM400-83-0041).

3. This message is detailed in Goldenberg, 1991.

4. This approximation to a parabola is even moreapparent when the obscured portions of h are madevisible at appropriate scales.

5. These materials are extracted from a number ofpublications now distributed by Sunburst Commu-nications. These include Visualizing Algebra: TheFunction Analyzer, The Function Supposes: Explonstionsin Algebra, and Pmblems and Projects for The FunctionAnalyzer. Credit for these materials goes to JuneMark, Jim Hammerman, and Michal Yerushalmy.

6. In fact, all functions of the form frx) .7a(x+5)x(x-7)have the three roots indicated. But while this familyof functions captures all the cubic polynomials thathave the three roots, there are infinitely many morefunctionshigher-order polynomials or non-poly-nomial functionsthat also have those roots.

7. Credit for this idea of parallel number lines as analternative to perpendicular axes goes to Phil Lewis.

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