Elsewhere in this issue Ferdinand Verhulst described the discussion of the interaction of anal-
ysis and geometry in the 19th century. In modern times such discussions come up again and again. As of 2014, synthetic geometry will not be part of the Dutch `vwo ¿ mathematics B¿
programme anymore. Instead, the focus will be more on analytic geometry. Mark Timmer and Nellie Verhoef explored possibilities to connect the two disciplines in order to have students look at analytical exercises from a more synthetic point of vie

Timmer, M.

Verhoef, N.C.

Elsewhere in this issue Ferdinand Verhulst described the discussion of the interaction of analysis and geometry in the 19th century. In modern times such discussions come up again and again. As of 2014, synthetic geometry will not be part of the Dutch 'vwo - mathematics B' programme anymore. Instead, the focus will be more on analytic geometry. Mark Timmer and Nellie Verhoef explored possibilities to connect the two disciplines in order to have students look at analytical exercises from a more synthetic point of view

Timmer, Mark

Verhoef, Neeltje Cornelia

University of Twente Research Information

1 11 1Mark Timmer and Nellie Verhoef Increasing insightful thinking in analytic geometry NAW 5/13 nr. 3 september 2012 217Mark TimmerFormal Methods & ToolsUniversity of TwenteP.O. Box 2177500 AE Enschedetimmer@cs.utwente.nlNellie VerhoefELANUniversity of TwenteP.O. Box 2177500 AE EnschedeN.C.Verhoef@utwente.nlIncreasing insightful thinking inanalytic geometryElsewhere in this issue Ferdinand Verhulst described the discussion of the interaction of anal-ysis and geometry in the 19th century. In modern times such discussions come up again andagain. As of 2014, synthetic geometry will not be part of the Dutch ‘vwo – mathematics B’programme any more. Instead, the focus will be more on analytic geometry. Mark Timmer andNellie Verhoef explored possibilities to connect the two disciplines in order to have studentslook at analytical exercises from a more synthetic point of view.Although analytic geometry is a wonderfultechnique to prove a variety of theorems inEuclidean geometry in a convincing and easymanner, it rarely provides many insights. Sec-ondary school students often apply it withoutany consideration of what they are actuallydoing. We conjecture that this leads to frag-mented understanding. Rather than develop-ing an overall picture of the geometric con-cepts the students are working with, the ana-lytic and synthetic geometry remain isolateddomains. This results in limited understand-ing of the mathematical structures at hand,and a limited set of techniques and strategiesfor solving exercises from these different do-mains. Analytic geometry becomes an end initself; students manipulate formulas withoutany feeling for the underlying concepts.Additionally, an analytical approach mightsometimes even be much more cumbersomethan a synthetic argument. By using ana-lytical techniques for dealing with geomet-ric figures, students sometimes forget aboutthe properties of these objects, resulting inlengthy, unnecessary calculations.In the context of the first author’s Master’sthesis for his mathematics teaching degreeat the University of Twente, we tried to em-phasise the underlying concepts of syntheticgeometry when covering a chapter on analyt-ic geometry. This was often accompanied byvisualisations using the GeoGebra computerprogramme. The overall goal was to providestudents a richer understanding of geome-try [12]. More specifically, we were hopingfor them to develop richer cognitive units [2].That way, students understand better how dif-ferent representations of geometric conceptssuch as ellipses relate, and are able to quick-ly switch between them. Hence, they mightwork more efficiently when solving exercisesfor which a purely analytical approach is un-necessarily difficult.We already extensively discussed the les-son series and research project that resultedfrom the ideas above in a previous article [13].Here, we elaborate more on the theoreticalbackground regarding cognitive units and vi-sualisation of geometric objects. Moreover,we discuss the way in which the results ofthis research project were put into practice asa workshop during the National MathematicsDays (NWD).Underlying school mathematicsOur research primarily focused on the ellipse.This mathematical object can be defined asfollows.Definition 1. An ellipse is a set of points thatall have the same sum of distances to twogiven focus points.Definition 2. An ellipse is a set of points thatare equidistant from a circle (the directrix cir-cle) and a point within that circle.The first definition is illustrated in Figure 1,the second one in Figure 2.It is not hard to see that these two def-initions coincide. In Figure 1, by definitionF1P1 + P1F2 = F1P2 + P2F2; let this constantbe r . In Figure 2,MPi +PiF equals the radiusof the circle, for both point P1 and P2, and allother points on the ellipse. The ellipse con-sisting of all points that are equidistant froma point F and a circle with centre M and ra-dius r , therefore coincides with the ellipseconsisting of all points with cumulative dis-tance r to M and F . Stated differently, Mand F are the focus points of the ellipse inFigure 2.Placing an ellipse in a Cartesian coordinatesystem with the focus points on the horizontalaxis (see Figure 3), we can show that it coin-cides with the set of points (x,y) such thatx2a2 +y2b2 = 1. Here, a is half of the length ofthe horizontal axis, and b half of the lengthof the vertical axis. In Figure 3 this yieldsx225 +y29 = 1. Interestingly, such an analyticalrepresentation relates in several ways to theP1F1F2P2Figure 1 Equal cumulative distance to two focus pointsMFP1P2∗∗◦◦Figure 2 Equal distance to circle and point2 22 2218 NAW 5/13 nr. 3 september 2012 Increasing insightful thinking in analytic geometry Mark Timmer and Nellie Verhoef−6 −5 −4 −3 −2 −1 1 2 3 4 5−3−2−11230eF1 F2Figure 3 An ellipse in a coordinate systemsynthetic definitions discussed above. For in-stance, 2a corresponds to the radius of the di-rectrix circle, and 2√a2 − b2 is the distancebetween the focus points.We expected proficiency in such conver-sions between the analytical and the synthet-ic domain to increase understanding and in-sight, helping students solve exercises moreeffectively and efficiently.Theoretical frameworkIn this study we investigated students’ cogni-tive items with respect to geometric objects.In particular, we assessed the effects of ateaching method based on visualisation andsynthetic geometry on these units. Hence,this section provides an overview of the theo-ry regarding cognitive units and visualisation.Cognitive unitsThe human brain is not capable of thinkingabout many things at once. Complicated ac-tivities such as mathematical thinking there-fore have to be made manageable by abstract-ing away unnecessary details and focusing onthe most important aspects [2]. The term cog-nitive unit originated from this idea:“A cognitive unit consists of a cognitiveitem that can be held in the focus of atten-tion of an individual at one time, together withother ideas that can be immediately linked toit.” [10]The ‘cognitive item’ mentioned here couldbe a formula such as a2 +b2 = c2, a fact suchas 10+3 = 13 or a mental image of an ellipse.The connectivity between cognitive items andrelated ideas depends on the degree of under-standing. For instance, most people wouldprobably immediately relate 3+4, 4+3 and 7,and hence have strong connections betweenthese cognitive items. They can then be con-sidered as a single cognitive structure: a cog-nitive unit.Barnard and Tall emphasise the impor-tance of rich cognitive units, having strong in-ternal connections between different objectsor representations of objects, and leading topowerful ways of thinking. In our case, sever-al different characterisations of the ellipse areconsidered. Initially, such characterisationswill probably not be strongly connected in thestudents’ brains. Later on, rich cognitive unitsmight develop, allowing the students to per-ceive the characterisations as different repre-sentations of the same object. This is expect-ed to yield more efficiency and understand-ing.Compression to rich cognitive units. Richcognitive units do not develop out of thin air.At first, a student will have a fragmented un-derstanding of a new concept. Then, severaldifferent approaches might be needed to ob-tain a full understanding. However, once aconcept has been fully understood, a signif-icant mental compression can often be ob-served. Thurston explains how this results ina complete mental perspective — although atfirst obtained by a long process — to be easilyused as part of a new mental process [11].The notion of compression is applied onthe one hand for the compression of knowl-edge into small cognitive items [3], and onthe other hand for the way in which differ-ent cognitive items are coupled into strongly-connected cognitive units [10]. Since bothprocesses yield richer cognitive units, we donot distinguish between these two meanings.Causing compression. In order to inducecompression, brain sections have to be con-nected to such an extent that addressing oneof them also activates the others. After all,this makes the combined knowledge and un-derstanding of these sections function to-gether as a single cognitive structure [10].More specifically, compression can bebrought about in several different ways [9]. Astudent could categorise concepts or performthought experiments, leading to connectionsbetween properties of those concepts. Re-peatedly practising certain procedures untilthey are automated may also yield rich cog-nitive units. Finally, compression can be in-duced by abstraction: introducing symbols ornames. Gray and Tall indeed indicate thatwe can only effectively talk about phenom-ena once they have been given a name [3].As this compresses them to a cognitive unit,it enables us to think about them in a moresophisticated manner.VisualisationIn this study, the underlying concepts fromsynthetic geometry were often visualised us-ing GeoGebra, a computer programme for dy-namic geometry [4]. The geometric objectsunder consideration indeed perfectly fit dy-namic visualisation. For instance, we can eas-ily use an equation for an ellipse and a slid-er determining its parameter a, to teach stu-dents this parameter’s effect on the ellipse.Scientific literature indicates that visual-isation may improve mathematical under-standing, although this does not necessaryhas to happen. Stols explains how the use ofIT — more specifically, GeoGebra and Cabri 3D— only positively affects geometric insights ofstudents that did not have much understand-ing yet, and even then only marginally [8]. Herecommends to deploy applications such asGeoGebra to improve visualisation skills andconceptual understanding, and enable stu-dents to discover important relations. Howev-er, these programmes should not be expectedto improve reasoning skills. We indeed onlyused GeoGebra for visualisation and to ob-serve connections between concepts.Langill also describes that software likeGeoGebra should mainly be used as a sup-plement to non-technological sources, suchas books [6]. She noticed that distance mea-suring and point dragging are among themost powerful applications of dynamic geom-etry. Therefore, we indeed combined visuali-sations with additional exercises, and exten-sively applied dragging and measurements toillustrate geometric properties.Other researchers confirmed that tech-nology can help students discover connec-tions between different representations of thesame concept, but also noticed that it shouldnot be deployed too early [1]. They foundthat visualisations should be linked directly toknowledge that the students already possess,to avoid frustration and misconceptions. Wetherefore only used GeoGebra to clarify con-cepts the students were already familiar with,avoiding this pitfall.Despite the potential merits of dynamic ge-ometry software, it is still not used very often.Stols and Kriek report that a negative attitudetowards the added value of such software,as well as a lack of confidence in their owntechnical skills, prohibit teachers from usingapplications like GeoGebra [14]. Zhao, Pugh,Sheldon and Byers also reached this conclu-sion, and observed that teachers have to takesmall evolutionary steps when introducing ICTin the classroom; a revolutionary approachwould only lead to failure and frustration [15].In this study, GeoGebra was only used bythe teacher. Obviously, it is also possible tohave the students play with the application.Although this is indeed expected to help stu-dents discover geometric theorems [7] or un-derstand geometric transformations [5], we3 33 3Mark Timmer and Nellie Verhoef Increasing insightful thinking in analytic geometry NAW 5/13 nr. 3 september 2012 219only applied GeoGebra for demonstrations.After all, we did not focus on developing newgeometric skills, but more on the applicationof available geometric knowledge in the con-text of analytic geometry.This studyWe performed our study in a vwo 5 mathe-matics D class at the Stedelijk Lyceum Kot-tenpark in Enschede. Since this class con-sisted of only four students (for privacy rea-sons all addressed by ‘he’ in this article), wewere able to observe the students in muchdetail and question them individually. Theresearcher taught Chapter 14 of the Getal &Ruimte vwo D4 method. This chapter coverssymmetry, parametric equations and differ-ence quotients, based on parabolas, ellipsesand hyperbolas.We tried to encourage the students to fo-cus on connections between synthetic andanalytic geometry in three different ways:(1) by giving additional explanations — oftenaccompanied by GeoGebra visualisations —to make students aware of what they are do-ing, (2) by discussing how several analyticalexercises from the book can be solved moreeasily using geometric reasoning, and (3) byintroducing a number of new exercises for thestudents to practice these skills on. We referto [12–13] for an extensive description of thelesson series.Semi-structured interviews before and af-ter the lesson series have shown quite a dif-ferent effect on each of the four students. Forone of them, the focus on synthetic geome-try seemed to work out poorly. This studentshowed only limited knowledge and insight,both before and after the lesson series. Hepreferred to rely on an analytical approach,and already declared upfront to rather just cal-culate than think of a smarter way to solve anexercise. Additionally, he often indicated tonot have much confidence in his own mathe-matical understanding, explaining his prefer-ence for structured rules and procedures.The other three students were much moreenthusiastic, and showed a positive attitudetowards the new way of approaching analyt-ic geometry. They most liked the feeling ofdeeper understanding, as well as the simplic-ity to achieve results. One student indeedshowed considerably more insight during thepost-test. He switched rapidly between dif-ferent representations of the same concept,for instance by using symmetry for an ana-lytical exercise and by combining both def-initions of the ellipse in a smart manner.Additionally, he often first took a momentto think before relying on calculations, andshowed growth in his associations with geo-metric concepts.The other two students showed slight-ly less progress, but still improved visibly.They were able to identify more representa-tions and more often applied geometric con-cepts such as symmetry. Interestingly, it ap-peared that some insights were present, butonly surfaced after considerable encourage-ment. This indicates that certain connectionsbetween cognitive items have been made,but also that more practice is needed to en-able fast switching between the accumulatedknowledge from different domains.National Mathematics DaysTo share our findings with a larger group ofteachers, we conducted a workshop duringthe most recent National Mathematics Days(www.fisme.science.uu.nl/nwd). There ap-peared to be quite some interest in our top-ic; teachers were happy to discuss a moreinsightful manner of working with analyticgeometry.After a short introduction of the subject,the teachers were asked to work on some ofthe exercises the students also tried to solveduring their post-test. They intensely calcu-lated and discussed, and appeared to pursuemany different approaches. We found thatthey did not always fully use all available da-ta and possible connections to other repre-sentations. The determination to solvethe difficult exercises, however, was inspi-ring. Such an attitude would benefit everystudent!The teachers asked many questions aboutthe translation from our ideas to the class-room: how can we make students followour approach, combining different represen-tations and thinking before computing? As wementioned before, frequent practice seemsto be key. The workshop participants werepleased to hear and experience a creative wayof addressing synthetic geometry in the cur-rent mathematics curriculum.More details on the lessons and exercis-es can be found in [13]. For an extensivedescription of the research project, we referto [12]. Both articles, as well as all materialused at the NWD, can be found at http://fmt.cs.utwente.nl/˜timmer/research.php. kReferences1 M. Alagic, Technology in the mathematics class-room: Conceptual orientation, Journal of Com-puters in Mathematics and Science Teaching22(4), 2003, pp. 381–399.2 T. Barnard and D.O. Tall, Cognitive Units, Con-nections and Mathematical Proof, Proceedingsof the 21st Conference of the International Groupfor the Psychology of Mathematics Education,1997, pp. 41–48.3 E. Gray and D.O. Tall, Abstraction as a naturalprocess of mental compression, MathematicsEducation Research Journal 19(2), 2007, pp. 23–40.4 M. Hohenwarter and J. Preiner, Dynamic Mathe-matics with GeoGebra, Journal of Online Math-ematics and its Applications 7, 2007.5 K.F. Hollebrands, High school students’ under-standings of geometric transformations in thecontext of a technological environment, Journalof Mathematical Behavior 22(1), 2003, pp. 55–72.6 J. Langill, Requirements to make effective use ofdynamic geometry software in the mathemat-ics classroom: a meta-analysis, 2009 (unpub-lished).7 R.A. Saha, A.F.M. Ayub and R.A. Tarmizi, TheEffects of GeoGebra on Mathematics Achieve-ment: enlightening Coordinate Geometry Learn-ing, Procedia Social and Behavioral Sciences 8,2010, pp. 686–693.8 G.H. Stols, Influence of the use of technologyon students’ geometric development in termsof the Van Hiele levels, Proceedings of the 18thAnnual Conference of the Southern African As-sociation for Research in Mathematics, 2010,pp. 149–155.9 D.O. Tall, A theory of mathematical growththrough embodiment, symbolism and proof,Annales de Didactique et de Sciences Cogni-tives 11, 2006, pp. 195–215.10 D.O. Tall and T. Barnard, Cognitive Units, Con-nections and Compression in MathematicalThinking, 2002 (unpublished).11 W.P. Thurston, Mathematical education, Noticesof the American Mathematical Society 37(7),1990, pp. 844–850.12 M. Timmer, Rijkere cognitieve eenheden doorhet benadrukken van synthetische meet-kunde tijdens de behandeling van analytis-che meetkunde, Master’s thesis, UniversiteitTwente, 2011.13 M. Timmer and N.C. Verhoef, Analytischemeetkunde door een synthetische bril, NieuweWiskrant 31(4), 2012, pp. 13–18.14 G.H. Stols and J. Kriek, Why don’t all mathsteachers use dynamic geometry software intheir classrooms?, Australasian Journal of Ed-ucational Technology 27(1), 2011, pp. 137–151.15 Y. Zhao, K. Pugh, S. Sheldon and J.L. Byers, Con-ditions for Classroom Technology Innovations,Teachers College Record 104(3), 2002, pp. 482–515.

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