A Technological Environment for Promoting MathematicalReasoning: A Dynamic Approach to Geometric Proof

Jill VincentDepartment of Science and Mathematics Edcuation

The University of Melbourne UniversityVictoria 3010

Australia

jlv@alphalink.com.au

Keywords: geometric proof, dynamic geometry software..

Introduction

Euclidean geometry and geometric proof have occupied acentral place in mathematics education from classicalGreek society through to twentieth century Westernculture. It is proof which sets mathematics apart from theempirical sciences, and forms the foundation of ourmathematical knowledge, yet the latter part of thetwentieth century witnessed the demise of both Euclideangeometry and proof in school mathematics curricula inmany countries. Research indicates that students often failto understand the purpose of mathematical proof, andreadily base their convictions on empirical evidence orthe authority of a textbook or teacher. A major large-scalesurvey of above average Year 10 students in the UK(Healy and Hoyles 1999) has shown that many students,even those who have been taught proof, have little idea ofthe significance of mathematical proof, are unable torecognise a valid proof, and are unable to construct aproof in either familiar or unfamiliar contexts.

More recently, however, there are indications that at leastin some countries there has been renewed interest inproof. Recent changes to mathematics curricula in manycountries have emphasised the need for students to justifyand explain their reasoning, prompting renewed debateamongst mathematics educators concerning the inclusionof proof in school mathematics. The debate has also beendriven by the development and introduction into schoolsof dynamic geometry software, such as Cabri GeometryIITM and The Geometer’s Sketchpad®. With in-builtEuclidean geometry tools, precise measurement tools, anddrag and trace facilities, these dynamic geometryenvironments have the potential to transform the teachingand learning of geometry. Concern has been expressed,however, that dynamic geometry software may becontributing to a data gathering approach to schoolgeometry, where empirical evidence is becoming asubstitute for proof. In many classrooms it appears that

Copyright © 2003, Australian Computer Society, Inc.This paper was presented at the IFIP Working Groups 3.1and 3.3 Working Conference: ICT and the Teacher of theFuture, held at St. Hilda’s College, The University ofMelbourne, Australia 27th–31st January, 2003.Reproduction for academic, not-for profit purposespermitted provided this text is included.

visual and numerical feedback from dragging screenfigures is usurping the role of proof as verification, withlittle or no attempt by teachers to introduce students todeductive reasoning. Noss and Hoyles (1996) note that inthe UK, for example, there is little evidence of teachersexploiting the powerful new ways of approachinggeometry offered by dynamic geometry software. Instead,traditional geometry exercises have been adapted for thecomputer and geometry is being reduced to pattern-spotting in data generated by dragging and measurementof screen drawings, with little or no emphasis ontheoretical geometry: “school mathematics is poised toincorporate powerful dynamic geometry tools in ordermerely to spot patterns and generate cases” (Noss andHoyles 1996: 235).

Hölzl (2001) asserts, however, that the problem lies withthe way dynamic geometry software is used, rather thanwith the software itself:

The often mentioned fear that the computer hinders thedevelopment of an already problematic need for proofis too sweeping. It is the context in which the computeris a part of the teaching and learning arrangement thatstrongly influences the ways in which the need forproof does — or does not — arise

(Hölzl 2001: 68-69)

Dynamic Geometry Software as a Bridge toDeductive Reasoning

In the quest for a motivating, visually-rich context inwhich to introduce Year 8 students to geometric proof,my attention was drawn to mechanical linkages, orsystems of hinged rods. Found in many commonhousehold items, as well as in ‘mathematical machines’from the past, mechanical linkages are frequently basedon simple geometry such as similar figures, isoscelestriangles, parallelograms or kites, and offer a wealth ofgeometry appropriate for secondary school mathematics.Dynamic geometry software models of the linkagessimulate the action of the actual linkages, creating acognitive bridge between the physical linkages andtheoretical geometry. The accurate measurements,animation, and tracing of loci which are possible with thesoftware contribute to a conjecturing environment wherestudents can engage in argumentation and deductivereasoning. Proof then assumes the multiple roles ofverification of the truth of conjectures, understanding ofgeometric relationships, and explanation, that is, giving

insight into why a particular linkage works the way itdoes.

Figure 1. Cabri Model of Tchebycheff’s Linkage

In a research experiment with Year 8 students,Tchebycheff’s linkage for approximately linear motion(see Figure 1) was used to set the scene for the need forgeometric proof. The linkage consists of three rigid bars,AC, BD and CD, with lengths five, five, and two unitsrespectively. Points A and B are fixed, with the distanceAB equal to four units. When CD rotates, the midpoint, P,of CD, moves along an almost linear path. The studentsfirst constructed the linkage from plastic geostrips, andconjectured that P moved in a straight line. When thestudents then dragged the Cabri linkage, their realisationthat the path was not in fact linear, and their astonishmentat seeing how little the path actually deviated from astraight line, was sufficient to convince them that visualand empirical evidence could not be trusted.

Other linkages that the students explored included carjacks, enlarging and copying pantographs, and folding

tables. Operating the physical linkages wasfound to represent a significant motivationalaspect, at least for some students. The linkagesalso set the geometry in real contexts, andgenerated the visual data on which the studentsbased their preliminary ideas about how eachlinkage worked. These ideas generally related,though, to the mechanical action of thelinkage, rather than to its geometry, and couldbe regarded as pre-conjectures—tentativespeculations without any firm supporting

evidence. In the case of the enlarging pantograph, for

example, the geostrip model (see Figure 2a) providedevidence that the pantograph enlarged. However, becausethe manually-produced image was inexact, the studentswere unsure of the precise enlargement factor, and henceof how the geometry of the linkage was related to theenlargement. When given access to the Cabri model, thestudents were able to use the Cabri Trace facility (seeFigure 2b) to simulate the behaviour of the actualpantograph. The accurate empirical feedback thenallowed them to produce a valid conjecture linking theoperation of the pantograph to the similar trianglesunderlying its construction.

In each of the linkage tasks it was the accuracy of thefeedback from the Cabri models that gave credence to thestudents’ conjectures, engendered the confidence to seek

(a) Plastic Geostrip Model (b) Cabri Model

Figure 2: Enlarging Pantograph

Figure 3. A Year 8 Student’s Written Proof for the Enlarging Pantograph

geometric explanations for their conjectures, andsupported the students’ deductive reasoning. Through thissoftware-supported argumentation the students developeda sense of ownership of their reasoning which facilitatedthe logical ordering of statements in the subsequentwritten proofs. Figure 3, for example, shows a typicalproof for the enlarging pantograph with an enlargingfactor of two.

Dynamic feedback also played a key role as arbitrator inresolving cognitive conflict, and in provokingmetacognitive thinking, as illustrated by Anna’s responsewhen Kate made a false conjecture because she hadfocused on a non-generic position of the Cabri linkagemodel: “But remember we measured these and theyweren’t always the same … see … remember?” Thestudents’ use of Cabri was by no means restricted todragging, and it was in fact their competence in the use ofthe software tools that allowed them to exploit itscapabilities. The observed purpose of the pantograph, forexample, led naturally to the use of the Cabri Tracefacility. In other linkages explored by the students,tabulation of angle measurements was also used, andmanipulation of screen drawings prompted students toadd construction lines to assist them in identifyinginvariant properties or relationships. The students’ priorfamiliarity with the features of Cabri was therefore animportant aspect of their successful use of the Cabrilinkage in helping them to produce their conjectures andto construct their proofs.

Rather than eliminating the need for proof, theconvincing evidence and the unique opportunities forexploration and discovery provided by Cabri provokedintense argumentation and established a culture ofproving in this Year 8 Mathematics class. Usedappropriately, then, dynamic geometry software has thepotential to transform the teaching and learning ofgeometry and geometric proof.

References

HÖLZL, R. (2001): Using dynamic geometry software toadd contrast to geometric situations—A case study.International Journal of Computers for MathematicalLearning 6: 63–86.

HEALY, L. and HOYLES, C. (1999): Technical reporton the nationwide survey: Justifying and proving inschool mathematics. London, Institute of Education,University of London.

NOSS, R., and HOYLES, C. (1996). Windows onmathematical meanings: Learning cultures andcomputers. Mathematics Education Library, 17.Dordrecht, The Netherlands, Kluwer Academic Press.