Iliada Elia*, Athanasios Gagatsis*, Régis Gras**

Transcription

1 7 0 0 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function Ilid Eli*, Athnsios Ggtsis*, Régis Grs** *Deprtment of Eduction, Uniersity of Cyprus, P.O. Box. 07, 78 Nicosi, Cyprus ilid@ucy.c.cy, ggtsis@ucy.c.cy **LINA, École Polytechnique, Uniersity of Nntes L Chntrerie, BP 00 0 Nntes cedex regisgr@clu-internet.fr Résumé. L ojectif de cette recherche est d identifier les difficultés qui concernent l conersion d un mode de représenttion d une fonction à un utre mode(p.e. grphique, symolique, erl) et d'exminer le phénomène de comprtimenttion, c est-à -dire l'incpcité à coordonner u moins deux modes de représenttion du concept. Nous ons proposé deux tests à 8 élèes de qutrième (élèes de - ns) et à 0 élèes de seconde (élèes de -7 ns). Les deux tests concernent les mêmes reltions lgériques mis ils proposent différentes conersions et en prticulier l représenttion- source de conersions. L emploi de l nlyse sttistique implictie sur les tleux de données montre une sence de ressemlnce et d'impliction entre les tâches des deux tests pour les deux groupes d âge. Ces résultts montrent insi que les différents types de conersion entre représenttions du même contenu mthémtique sont trités de mnière distincte soulignnt l existence du phénomène de comprtimenttion. Introduction: Representtions nd the concept of function Nowdys the centrlity of representtions in teching, lerning nd doing mthemtics seems to ecome widely cknowledged. There is strong support in the mthemtics eduction community tht students cn grsp the mening of mthemticl concepts y experiencing multiple mthemticl representtions (e.g., Sierpinsk 99; Lesh et l. 987). An importnt eductionl ojectie is for students to use efficiently rious forms of representtion in communicting with one nother. Multiple representtions support students reflection, help students keep trck of their ides nd inferences nd lso fcilitte them to orgnize their work. In this pper, the term representtions in restricted sense is interpreted s the tools used for representing mthemticl ides such s tles, grphs nd equtions (Ggtsis et l. 00). The representtionl systems re fundmentl for conceptul lerning nd determine, to significnt extent, wht is lernt (Cheng 000). By trnsltion or conersion process, we men the psychologicl process inoling the moing from one mode of representtion to nother (Jnier 987). The ility to identify nd represent the sme concept in different representtions, nd flexiility in moing from one representtion to nother, re crucil in mthemtics lerning, s they llow students to see rich connections, nd deelop deepen understnding of concepts (Een 998). A concept of fundmentl importnce in the lerning of mthemtics, which is considered to e mjor focus of ttention for the mthemtics eduction reserch community is function (e.g., Engelidou et l. 00; Ggtsis nd Shiklli 00; Sfrd 99; Sierpinsk 99; Vinner nd Dreyfus 989). The understnding of functions does not pper to e esy, gien the diersity of representtions relted to this concept (Hitt 998). Ech one of the rious representtions of the notion of function points out different spect of the concept nd ll these together contriute to glol representtion of it, nd none of them seprtely cn descrie the I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

2 7 0 0 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 notion entirely (Kldrimidou nd Ikonomou 998; Ggtsis nd Shiklli 00). Tht is why some students difficulties in the construction of concept re linked to the restriction of representtions when teching. Thus, s Sierpinsk (99) pointed out, it is importnt to proide students with rod spectrum of wys of representing functions in order to preent them from identifying ny of these with functions. The stndrd representtionl forms of the concept of function re not enough for students to construct the whole mening nd grsp the whole rnge of its pplictions. Mthemtics instructors, t the secondry leel, trditionlly he focused their instruction on the use of lgeric representtions of functions. Eisenerg nd Dreyfus (99) pointed out tht the wy knowledge is constructed in schools fors mostly the nlytic elortion of the notion which deteriortes the pproch of function from the grphicl point of iew. A reson for this is tht in mny cses the iconic (isul) representtions cn cuse cognitie difficulties, ecuse perceptul nlysis nd synthesis of mthemticl informtion presented implicitly in digrm often mke greter demnds on student tht ny other spect of prolem (Aspinwll et l. 997). Furthermore, most instructionl prctices limit the representtions of functions to the trnsltion of the lgeric form of function to its grphic form. Kldrimidou nd Iconomou (998) showed tht techers nd students py much more ttention on lgeric symols nd prolems thn on pictures nd grphs. Sfrd (99), on the other hnd, found tht students re unle to ridge the lgeric nd grphicl representtions of functions, while Mrkoitz et l. (98) osered tht trnsltion from grphicl to lgeric form ws more difficult thn ice-ers nd tht the exmples gien y the students were limited in the grphicl nd lgeric form. Sierpinsk (99) mintins tht students he difficulties in mking the connections etween different representtions of functions (formuls, grphs, digrms, nd word descriptions), in interpreting grphs nd mnipulting symols relted to functions. These kinds of ehiour cn e seen s n indiction for the existence of comprtmentliztion. The prticulr phenomenon reels cognitie difficulty tht rises from the need to ccomplish flexile nd competent trnsltion ck nd forth etween different kinds of mthemticl representtions (Dul 00). Vinner nd Dreyfus (989) referred to the notion of comprtmentliztion in roder sense nd not just within the context of representtions. Specificlly, these reserchers suggested tht comprtmentliztion rises when n indiidul hs two diergent, potentilly contrdictory schemes in her cognitie structure nd pointed out tht inconsistent ehior is n indiction of this phenomenon. In the present study we re focused on the more restricted mening of comprtmentliztion, which refers to representtions. Specificlly, the present pper ims t identifying the difficulties tht rise in the conersion ck nd forth etween different modes of representtion (i.e., grphic, symolic, erl) of the mthemticl concept of function nd exmining the phenomenon of comprtmentliztion which my ffect in negtie wy mthemtics lerning. Preious empiricl studies he not clrified comprtmentliztion in comprehensie or systemtic wy. The existence of this phenomenon in students ehiour could not e uncoered thoroughly or erified y students success rtes neither y students protocols in tsks inoling different representtions. These types of dt cn just e seen s n indiction of the existence of comprtmentliztion. Thus, we theorize tht the implictie sttisticl method of nlysis, which reels the similrity connections nd implictie reltions etween students responses in the dministered tsks, cn e eneficil for identifying the existence of comprtmentliztion in students ehiour. Our sic conjecture is tht comprtmentliztion ppers when the following conditions pper: First, students del inconsistently or incoherently with different types of trnsltion of the sme mthemticl knowledge from one mode of representtion to nother; nd second, success in one mode of representtion or type of conersion of concept does not entil success in nother mode of representtion or type of conersion of the sme concept. Method. Prticipnts The smple of the study inoled two groups of students in Greece. The first group consisted of 8 students of Grde 9 ( yers of ge). The second group inoled 0 students of Grde ( yers of ge). I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

3 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00. Instrument Two tests were constructed nd dministered to the prticipnts. The first test (A) consisted of six tsks in which students were gien the grphic representtion of n lgeric reltion nd were sked to trnslte it to its erl nd symolic form, respectiely. The second test (B) consisted of six tsks (inoling the sme lgeric reltions with test A) in which students were sked to trnslte ech reltion from its erl representtion to its grphicl nd symolic mode, respectiely. For ech type of trnsltion, the following types of lgeric reltions were exmined: y<0, xy>0, y>x, y=-x, y=/, y=x- sed on relent reserch of Rymond Dul (99). The former three tsks correspond to regions of points, while the ltter three tsks correspond to functions. Ech test included n exmple of n lgeric reltion in grphic, erl nd symolic form to fcilitte students to understnd wht they were sked to do, s follows: Grphic representtion Verl representtion Symolic representtion y It represents the region of the points hing positie sciss. x>0 x x y TAB - An exmple of the tsks included in the test. Dt nlysis For the nlysis nd processing of the collected dt sed on students performnce in the tsks, Grs s implictie sttisticl method hs een conducted y using computer softwre clled C.H.I.C. (Clssifiction Hiérrchique Implictie et Cohésitie) (Bodin et l. 000). Grs s Implictie Sttisticl Model is method of nlysis tht determines the similrity connections nd the implictie reltions of fctors (Grs 99; Grs, Brind nd Peter 99; Grs et l. 99; Grs et l. 997). For this study s needs, similrity digrm nd n implictie digrm were produced from the ppliction of the nlysis for ech ge group of students. The similrity digrm llows for the rrngement of the tsks into groups ccording to the homogeneity y which they were hndled y the students. The implictie digrm contins implictie reltions, which indicte whether success to specific tsk implies success to nother tsk relted to the former one (see the ook: Grs et l. 99). Results The mens nd occurrences of ech tsk of the two tests mesuring students ility to trnslte lgeric reltions from one mode of representtion to nother re shown in Tle nd Tle respectiely for ech ge group. I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

4 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 Grde 9 Test A Test B N=8 Grphic Verl Grphic Symolic Verl Grphic Verl Symolic Occurrence Men Occurrence Men Occurrence Men Occurrence Men V: y< V: xy> V: y>x V: y=-x V: y=/ V: y=x TAB - Occurrences nd mens of Grde 9 students for the tsks of Test A nd Test B (from the tle of occurrences) Grde Test A Test B N=0 Grphic Verl Grphic Symolic Verl Grphic Verl Symolic Occurrence Men Occurrence Men Occurrence Men Occurrence Men V: y< V: xy> V: y>x V: y=-x V: y=/ V: y=x TAB - Occurrences nd mens of Grde students for the tsks of Test A nd Test B (from the tle of occurrences) Een though students in Grde performed etter thn students in Grde 9 in ll of the tsks, some common remrks cn e otined, indicting tht similr ptterns hold s regrds students ilities to trnslte lgeric reltions from one mode of representtion to nother. In oth grdes the first tsk nd the second tsk of the two tests (y<0 nd xy>0) were the esiest tsks, while the sixth tsk nd the fourth tsk (y=x- nd y=-x) were the most complex tsks. Furthermore, students of oth ges chieed higher outcomes in the conersions strting with erl representtions reltie to the conersions strting with grphic representtions. In ddition, ll the conersions from the grphic mode of representtion to the symolic mode of representtion ppered to e more difficult thn the conersions from the grphic mode of representtion to the erl mode of representtion. Students perceie the ltter type of conersion more esily in leel of met-mthemticl expression rther thn leel of mthemticl expression. In fct, students re sked to descrie erlly (in text) property perceied y the grph. On the contrry, the conersions from grphicl form to the symolic form entil mstering of lgeric concepts concerning equlity or order reltions nd lso using efficiently the lgeric symolism. Figure presents the similrity digrm of the tsks of Test A nd Test B sed on the inry responses of students of Grde 9. I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

5 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 Test A Test B FIG. - Similrity digrm of the tsks of Test A nd Test B ccording to Grde 9 students responses Note: The symolism used for the riles of this digrm (nd the digrms tht follow) is explined elow.. stnds for Test A, nd stnds for Test B. The first numer fter stnds for the numer of the tsk in the test i.e., : y<0, : xy>0, : y>x, : y=-x, : y=/, : y=x-. The second numer stnds for the type of conersion for ech test, i.e., for Test A, : grphic to erl representtion, : grphic to symolic representtion; for Test B, : erl to grphic representtion, : erl to symolic representtion. Two distinct similrity groups of tsks re identified in Figure. The first group inoles similrity reltions mong the tsks of Test A, while the second group inoles similrity reltions mong the tsks of Test B. This finding reels tht different types of conersions mong representtions of the sme mthemticl content were pproched in completely distinct wy. The strting representtion of conersion, i.e., grphic or erl representtion, seems to influence students performnce, een though the tsks inoled the sme lgeric reltions. The corresponding implictie digrm of Grde 9 students (Figure ) is in line with the similrity digrm nd the oe remrks. In prticulr, one cn osere the formtion of two groups of implictie reltions. The first group inoles implictie reltions mong the tsks of Test B nd the second group inoles implictie reltions mong the tsks of Test A. It cn e sserted tht crrying out conersion of function with prticulr strting representtion did not necessrily imply success in the conersion of the sme function with nother strting representtion. For exmple, students who ccomplished the trnsltion from grphicl representtion of n lgeric reltion to its erl representtion were not utomticlly in position to trnslte successfully the sme lgeric reltion from its erl representtion to its grphicl form. Oerll, sed on the reltions included in the similrity nd the implictie digrms for students in Grde 9, it cn e inferred tht there ws comprtmentliztion etween students responses t the tsks of the first test nd the tsks of the second test, which inoled conersions of the sme lgeric reltions ut different strting modes of representtion (i.e., grphic nd erl respectiely). Students higher success I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

6 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 rtes in the tsks of Test B, i.e., conersions strting with grphic representtions, reltie to the tsks of Test A, i.e., conersions strting with erl representtions, proide further eidence for their inconsistent ehiour in the two types conersions. Test B Test A FIG. - Implictie digrm of the tsks of Test A nd Test B ccording to Grde 9 students responses The similrity reltions within ech group of riles re lso of gret interest since they cn e seen s indictions of students wy of understnding of the prticulr lgeric reltions. Specificlly, two clerly seprted su-groups re identified within the group of Test A. The first su-group consists of the tsks, nd, while the second su-group inoles the tsks, nd. Tsks nd represent functions, while tsk corresponds to region of points inoling function. Students hndled tsks, nd differently form the other tsks of Test A, ut coherently etween them. Tsks nd represent regions of points, while tsk refers to constnt function, which represents horizontl line, prllel to the scise xis. Its grphic form is dissimilr to the grphic form of the other functions of the test. Tht is perhps the reson why students delt differently with this tsk reltie to the other tsks inoling functions (, nd ). The ltter tsks were pproched consistently y the students, influenced y their functionl chrcter. Further eidence for the oe distinction of tsks is gien y students performnce t these tsks. Specificlly, students success rtes in the tsks, nd were higher thn their success rtes in the tsks, nd. To sum up, the oe remrks indicte tht the wy students crry out conersion of n lgeric reltion from grphic representtion to nother representtion is influenced y the conceptul (i.e., type of reltion: inoling function or not) nd the perceptul fetures (grphic form) of it. As for the similrity group of Test B, two su-groups re lso distinguished. The first su-group inoles tsks nd which correspond to regions of points. Tsks,, nd form the second su-group proly I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

7 8 0 0 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 due to the fct tht they inole functions. This distinction of tsks is supported y the students different success rtes in the tsks nd nd the tsks,, nd. Students performed clerly etter in the former tsks reltie to the ltter tsks. Moreoer, within this su-group strong similrity reltions re identified mong the tsks inoling the sme type of conersion, i.e., erl to grphic representtion, or erl to symolic representtion. It cn e inferred tht the wy students crry out conersion of n lgeric reltion from erl representtion to nother representtion is influenced y the conceptul chrcteristics of it s well s the trget of the conersion. Although students in Grde chieed higher success rtes to the tsks reltie to students in Grde 9, congruent results were ttined y the ppliction of the implictie sttisticl nlysis on the dt concerning the former group of students. From the similrity digrm shown in Figure two distinct similrity groups of tsks re formed. The first group inoles similrity reltions mong the tsks of Test A, while the second group inoles similrity reltions mong the tsks of Test B. This finding reels tht students of Grde delt inconsistently with conersions strting with grphic representtions nd conersions strting with erl representtions, lthough they inoled the sme mthemticl content. The former type of conersion ppers to e more difficult thn the ltter one (see Tle ). Thus, like ninth grders outcomes, eidence is proided for the existence of comprtmentliztion in eleenth grde students ehiour when deling with different types of conersions of lgeric reltions. Similrly to the similrity digrm for ninth grde students, within ech similrity group strong connections re identified mong tsks inoling reltions of functionl type (i.e., tsks,, nd ), indicting students consistent ehiour when trnslting mthemticl reltions inoling functions. Tsks nd re lso ssocited to ech other, indicting the homogeneity y which they were hndled y Grde students, since oth tsks represent regions of points, not inoling function. Test A Test B FIG. - Similrity digrm of the tsks of Test A nd Test B ccording to Grde students responses The implictie digrm shown in Figure is in ccord with the similrity digrm. I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

8 8 0 0 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 Test B Test A FIG. - Implictie digrm of the tsks of Test A nd Test B ccording to Grde students responses Two seprte groups of implictie reltions re identified. The first group inoles tsks of Test B, while the second group inoles tsks of Test A. The formtion of these reltions indictes tht success in conersion strting from grphic form does not imply success in conersion of the sme mthemticl reltion strting from erl mode of representtion. Thus, further support is proided for the phenomenon of comprtmentliztion. In line with the similrity digrm, direct implictie reltions pper mong tsks -, which inole functions or lgeric reltions with functionl chrcter, s well s mong tsks nd, corresponding to regions of points. Compring Figure which corresponds to Grde students to the corresponding Figure, which concerns Grde 9 students, we osere the following:. The implictions etween the riles concerning functions re stronger in the cse of Grde students in reltion to Grde 9 students, which re expected, s the former students receie systemticlly instruction on functions.. In the cse of Figure (Grde ) there re not implictions etween functions (,, ) nd region of points (, ). On the contrry, in Figure, success in tsks inoling region of points is presupposition for succeeding in tsks inoling functions. Thus, the ppliction of the implictie sttisticl nlysis reels the difference of the epistemologicl chrcter etween knowledge of Grde students nd Grde 9 students on functions. This difference cn not e uncoered y the corresponding similrity digrms. I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

9 8 0 0 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 Discussion The im of the present study ws to identify nd inestigte the phenomenon of comprtmentliztion s regrds the understnding of the function concept y using the implictie sttisticl method of nlysis. Despite eleenth grde students higher performnce reltie to ninth grde students performnce in the conersions of lgeric reltions from one mode of representtion to nother, there ws congruent structure of the similrity nd implictie digrms for oth ge groups, indicting tht the phenomenon of comprtmentliztion exists irrespectie of students ge. For oth ge groups, success in one type of conersion of n lgeric reltion did not necessrily imply success in nother mode of conersion for the sme reltion. Lck of implictions or connections mong different types of conersion (i.e., with different strting representtion) of the sme mthemticl content is the min feture of the phenomenon of comprtmentliztion nd indictes tht students of oth ge groups did not construct the whole mening of the concept of function nd did not grsp the whole rnge of its pplictions. As Een (998) supports, the ility to identify nd represent the sme concept in different representtions, nd flexiility in moing from one representtion to nother llow students to see rich reltionships, nd deelop deepen understnding of the concept. This inconsistent ehior cn e lso seen s n indiction of students conception tht different representtions of the sme concept re completely distinct nd utonomous mthemticl ojects nd not just different wys of expressing the mening of prticulr notion. In other words, students my he confused the mthemticl oject of function or other lgeric reltion with its semiotic representtion (Dul 99; 000). The differences mong students scores in the rious conersions from one representtion to nother, referring to the sme lgeric reltion or function proide support to the oe findings s well s the different cognitie demnds nd distinctie chrcteristics of different modes of representtion. For exmple, students greter difficulty in crrying out conersion of reltion strting from grphic mode of representtion rther thn erl mode of representtion, which ws osered in the study of Mrkoitz et l. (98) s well, my e due to the fct tht perceptul nlysis nd synthesis of mthemticl informtion presented implicitly in digrm often mke greter demnds on student tht ny other spect of prolem (Aspinwll et l. 997). The grphic register functions effectiely only under the conentions of different mthemticl culture. Due to students poor knowledge of this new culture, grphic modes of representtion re difficult to decode nd work out. It cn e sserted tht the support offered y mthemticl metlnguge is more fundmentl thn the id gien y the grphic register for crrying out trnsltion from one mode of representtion to nother. Besides the effect of the type of conersion (modes of representtion) nother significnt fctor tht hd n impct on students chieement is the type of mthemticl reltion, inoled in the conersion. As shown y the similrity digrms nd students success rtes, lgeric reltions of the sme conceptul chrcter (i.e., reltions inoling cler connection etween x nd y, or reltions hing x nd y independent from one nother) were pproched consistently y the students. Conclusiely, the phenomenon of comprtmentliztion illustrted tht the expected semntic congruence does not pper neither in the cse of Grde 9 students nor in the cse of Grde students, despite the fct tht the ltter group of students ttended more systemtic nd rigorous teching of lger. Tht is ecuse lerning cn e ccomplished through de-comprtmentliztion (Dul 00). Therefore, the use of multiple representtions in mthemtics lerning nd in this cse lerning of functions, the connection, coordintion nd comprison with ech other nd the conersion from one mode of representtion to nother should not e left to chnce, ut should e tught nd lerned systemticlly, so tht students deelop the skills of representing nd hndling flexily mthemticl knowledge in rious forms. I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

10 8 0 0 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 Références Aspinwll, L., Shw, K. L. & Presmeg, N. C. (997). Uncontrollle mentl imgery: Grphicl connections etween function nd its deritie. Eductionl Studies in Mthemtics,, 0-7. Bodin, A., Coutourier, R., & Grs, R. (000). CHIC : Clssifiction Hiérrchique Implictie et Cohésie- Version sous Windows CHIC.. Rennes : Assocition pour l Recherche en Didctique des Mthémtiques. Cheng, P.C.H. (000). Unlocking conceptul lerning in mthemtics nd science with effectie representtionl systems. Computers nd Eduction,, Dul, R. (99). Registres de Représenttion Sémiotique et Fonctionnement Cognitif de l Pensée, Annles de Didctique et de Sciences Cognities,, 7-. Dul, R. (00). The cognitie nlysis of prolems of comprehension in the lerning of mthemtics. Mediterrnen Journl for Reserch in Mthemtics Eduction, (), -. Eisenerg, T., & Dreyfus, T. (99). On the reluctnce to isulize in mthemtics. In W. Zimmermnn & S. Cunninghm (Eds.), Visuliztion in Teching nd Lerning Mthemtics (pp. 9-). United Sttes: Mthemticl Assocition of Americ. Engelidou, A., Spyrou, P., Eli, I., & Ggtsis, A. (00). Uniersity students conceptions of function. In M. Johnsen Høines & A. Berit Fuglestd (Eds.), Proceedings of the 8 th Conference of the Interntionl Group for the Psychology of Mthemtics Eduction (Vol., pp. -8). Bergen, Norwy: Bergen Uniersity College. Een, R. (998). Fctors inoled in linking representtions of functions. The Journl of Mthemticl Behior, 7(), 0-. Ggtsis, A., & Shiklli, M. (00). Aility to trnslte from one representtion of the concept of function to nother nd mthemticl prolem soling. Eductionl Psychology, (), -7. Ggtsis, A., Eli, I., & Mougi, A. (00). The nture of multiple representtions in deeloping mthemticl reltions. Scienti Pedgogic Experimentlis, 9 (), 9-. Grs R. (99) Dt nlysis : method for the processing of didctic questions, Reserch in Didctic of Mthemtics, Selected ppers for ICME 7, L Pensée Suge, Grenole. Grs R., Brind H., Peter P. (99) Structurtion sets with impliction intensity in E. Didy, Y. Lechellier, O. Opitz (Eds) Proceeding of the Interntionl Conference on Ordinl nd Symolic Dt Anlysis - OSDA 9, Springer, Pris. Grs R. nd ls (99) L impliction sttistique, Collection ssociée à Recherches en Didctique des Mthémtiques, L Pensée Suge, Grenole. Grs R., Peter P., Philippe J. (997) Implictie sttisticl nlysis, Proceedings of IFCS 9, Springer- Verlg, Tokyo. Hitt, F. (998). Difficulties in the rticultion of different representtions linked to the concept of function. The Journl of Mthemticl Behior, 7(), -. Jnier, C. (987). Representtion nd understnding: The notion of function s n exmple. In C. Jnier (Ed.), Prolems of Representtion in the Teching nd Lerning of Mthemtics (pp. 7-7). Hillsdle, NJ: Lwrence Erlum. Kldrimidou, M., & Ikonomou, A. (998). Fctors inoled in the lerning of mthemtics: The cse of grphic representtions of functions. In H. Stenring, M.G. Brtolini Bussi & A. Sierpinsk (Eds.), Lnguge nd Communiction in the Mthemtics Clssroom (pp. 7-88). Reston, V: NCTM. Lesh, R., Post, T., & Behr, M. (987). Representtions nd trnsltions mong representtions in mthemtics lerning nd prolem soling. In C. Jnier (Ed.), Prolems of Representtion in the Teching nd Lerning of Mthemtics (pp. -0). Hillsdle, NJ: Lwrence Erlum. Mrkoitz, Z., Eylon, B. nd Bruckheimer, M. (98). Functions tody nd yesterdy. For the Lerning of Mthemtics, () 8-8. I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function

11 8 0 0 Troisièmes Rencontres Interntionles Terzo Conegno Internzionle - Third Interntionl Conference A.S.I. Anlyse Sttistique Implictie Anlisi Sttistic Implicti Implictie Sttistic Anlysis Plermo (Itly) -8 Octore 00 Sfrd, A. (99). Opertionl origins of mthemticl ojects nd the qundry of reifiction - The cse of function. In E. Duinsky & G. Hrel (Eds.), The Concept of Function: Aspects of Epistemology nd Pedgogy (pp. 9-8). United Sttes: The Mthemticl Assocition of Americ. Sierpinsk, A. (99). On understnding the notion of function. In E. Duinsky & G. Hrel (Eds.), The Concept of Function. Aspects of Epistemology nd Pedgogy (pp. -8). United Sttes: The Mthemticl Assocition of Americ. Vinner, S., & Dreyfus, T. (989). Imges nd definitions for the concept of function. Journl for Reserch in Mthemtics Eduction, 0 (), -. Summry The present pper ims t identifying the difficulties tht rise in the conersion from one mode of representtion of function to nother (i.e., grphic, symolic, erl) nd exmining the phenomenon of comprtmentliztion, i.e. deficiency in the coordintion of t lest two modes of representtion of the concept. Two tests were dministered to 8 students of Grde 9 nd 0 students of Grde. Both tests consisted of the sme lgeric reltions, ut differed in the types of conersions tht they inoled nd more specificlly their strting mode of representtion. The ppliction of the implictie sttisticl method of nlysis to the dt showed lck of connections nd implictions mong the tsks of the two tests for oth ge groups. This finding reels tht different types of conersions mong representtions of the sme mthemticl content were pproched in distinct wy, indicting the existence of the phenomenon of comprtmentliztion. I. Eli, A. Ggtsis, R. Grs, Cn we trce the phenomenon of comprtmentliztion y using the implictie sttisticl method of nlysis? An ppliction for the concept of function