How Pythagoras theorem is taught in Czech Republic, Hong Kong and Shanghai: A case study

Transcription

1 Anlyses ZDM 00 Vol. 34 (6) How Pythgors theorem is tught in Czech Republic, Hong Kong nd Shnghi: A cse study Rongjin Hung, Frederick K.S. Leung, Hong Kong SAR (Chin) Abstrct: This pper ttempts to explore certin chrcteristics of the mthemtics clssroom by investigting how techers from three different cultures, nmely, the Czech Republic, Hong Kong nd Shnghi, hndle Pythgors theorem t eighth grde. Bsed on fine-grined nlysis of one lesson from ech of the three plces, some fetures in terms of the wys of hndling the sme topic were reveled s follows: the Hong Kong techer nd the Shnghi techer emphsized exploring Pythgors theorem, the Shnghi techer seemed to emphsize mthemticl proofs, while the Czech techer nd the Hong Kong techer tended to verify the theorem visully. It ws found tht the Czech techer nd the Hong Kong techer put stress on demonstrting with some degree of student input in the process of lerning. On the other hnd, the Shnghi techer demonstrted constructive lerning scenrio: students were ctively involved in the process of lerning under the techer s control through series of deliberte ctivities. Regrding the clssroom exercises, the Shnghi techer tended to vry problems implicitly within mthemticl context, while the techers in the other two plces preferred vrying problems explicitly within both mthemticl nd dily life contexts. ZDM-Clssifiction: D10, E50, G40 1 Bckground Mthemtics eduction in different countries is strongly influenced by culturl nd socil fctors tht shpe gols, beliefs, expecttions, nd teching method (An et l., 00), nd cross-culturl comprison cn help reserchers nd eductors understnd explicitly their own implicit theories bout how techers tech nd how children lern (Stigler & Perry, 1988). It will lso help eductors to reflect upon their own prctices, nd recognize wys of improving their prctices (Bishop, 00). Recently, interntionl comprtive studies in mthemtics hve focused on clssroom teching (Leung, 1995; Lee, 1998; Stigler & Hiebert, 1999). For exmple, the TIMSS Video Study (Stigler & Hiebert, 1999) nd the on going TIMSS- R Video Study im t identifying different ptterns of clssroom instruction in different cultures by exmining representtive smple of eighth-grde mthemtics clssrooms in ech prticipting country. The Lerners Perspective Study, on the other hnd, ims t exmining lerning nd teching in eighth grde mthemtics clssrooms in more integrted nd comprehensive fshion (Clrke, 00). However, these studies do not py due ttention to how prticulr mthemtics topic is hndled in different cultures. Since techer must hndle or present certin mthemtics topic in certin wy so tht students cn understnd the content, study on how the sme topic is tught in different cultures will highlight the similrities nd differences mong different cultures. This study intends to cpture the fetures of mthemtics clssrooms in the Czech Republic nd Chin (Hong Kong nd Shnghi) by investigting the wys techers tckle Pythgors theorem in the three different plces. Methodology.1 Source of dt Firstly, eight Hong Kong videos nd five Czech videos on Pythgors theorem (t grde 8) were selected from the TIMSS-R Video Study. Secondly, eleven Shnghi grde 8 lessons on Pythgors theorem were videotped by the reserchers in Mrch 001 following the method used in the TIMSS-R Video Study (see website for detils). Bsed on thorough nlysis on ll the Hong Kong nd Shnghi videos, the ptterns of teching the theorem in these two cities were identified (Hung, 00). Then one representtive video on the first lesson of the unit on Pythgors theorem ws chosen from ech of the three plces. For the Czech Republic, unfortuntely none of the five videos ws the first lesson in the unit, nd so one video which ws the erliest lesson in the unit ws chosen. Totlly, three videos, their trnscripts, nd relevnt documents such s the techer questionnires, lesson plns, copies of textbooks nd worksheets constituted the dt for this study. The bckgrounds of the three lessons re shown in Tble 1 (see next pge). The Tble indictes tht the clss size in Shnghi ws the lrgest, while the techer in Shnghi ws the lest educted. The Hong Kong techer nd the Shnghi techer mjored in mthemtics, but the Czech techer mjored in science.. Dt nlysis The English trnscripts of those videos tken from the TIMSS-R Video Study were lredy vilble. The Shnghi videos were trnscribed verbtim in Chinese by the techers who delivered the lessons, nd were then trnslted into English. Through wtching the videos, nd reding their trnscripts nd relevnt documents gin nd gin, the three lessons were compred. The dt nlysis focused on the following spects: Structure of the lesson; Wys of teching Pythgors theorem; Ptterns of clssroom interction. How the first two spects re nlyzed will be illustrted in the next section. The wy of describing the clssroom interction will be discussed here in more detil. 68

2 ZDM 00 Vol. 34 (6) Anlyses 6 Czech Republic (CZ) Hong Kong (HK) Shnghi (SH) Rnk nd score in TIMSS 4 th N/A 1995 t grde (4.9) 588(6.5) Rnk nd score in TIMSS 15 th 4 th N/A 1999 t grde 8 50(.7) 58(4.3) Clss size Durtion of lesson (minutes) 45 40* 45 Eductionl level of techer Bchelor in Mthemtics Bchelor in Diplom 3 in Mthemtics nd Physics Mthemtics Tble 1: The bckgrounds of the three lessons According to Sfrd (001), lerning is nothing else thn specil kind of socil interction imed t modifiction of other socil interctions, nd communiction should be viewed not s mere id to thinking, but s lmost tntmount to the thinking itself. In this study, the fetures of clssroom interction were investigted from two spects: Firstly, the feture of questioning will be mesured quntittively. Secondly, the ptterns of interction will be investigted qulittively. The techers questions were clssified into three ctegories: 1. Requesting simple yes or no s response, sking students to confirm n nswer, or questions of mngement [Yes /No];. Asking students to nme n nswer without sking for ny explntion of how the nswer ws found, or sking for the solution to n explicitly stted clcultion [Nme]; 3. Asking for n explntion of the nswers given or procedures crried out, or sking question tht requires students to find fcts or reltionships from complex situtions [Explntion]; 4. Asking students for generlizing or evluting n nswer [Generliztion]. Only the first three types of questions were found in the three lessons, nd exmples for the three types of questions re illustrted below: Question 1: Hve you herd of Pythgors theorem before? [Yes/No] Question : Wht re the reltionships between, b, nd c. [Nme] Question 3: Why is the qudrngle t the center squre? [Explntion] 3 Results 3.1 The structure of the lesson It ws found tht ll the lessons in this study, by nd lrge, include n opening, introducing the new topic (except the CZ one), justifying, prctising, nd ending, s shown in Tble (see next pge). The Tble shows tht the Czech techer spent reltively high proportion of time on orgnizing nd revision, while the Hong Kong techer nd Shnghi techer spent substntil mount of time on justifiction of the theorem. All three techers emphsized clssroom exercises. The following discussions will be focused on introducing the theorem, justifying the theorem nd pplying the theorem. 3. Wys of teching Pythgors theorem 3..1 Introduction of the theorem Hong Kong lesson. After reviewing how to identify the hypotenuse of right-ngled tringles in different positions, the techer pointed out this lesson ws bout Pythgors theorem. Through demonstrting the re reltionship between two digrms s shown in Figure 1(1) nd Figure 1(), Pythgors theorem ws found nd proved. EINBETTEN A b B b C c b 1 The dt on the results of TIMSS 1995 nd TIMSS 1999 re from those reports (Mullis et l, 1997; 000). The informtion relted to the techers is from the techer questionnires collected by the TIMSS-R Video Study. It is common prctice tht there re two consecutive lessons without brek between them. The lesson selected here consisted of two sections. The first prt focused on introducing Pythgors theorem which took round 60 minutes, while the second prt ws devoted to clssroom exercise, which lsted round 0 minutes. This study only focused on the first prt. 3 This is document issued by n eductionl institution in minlnd Chin, such s college or university, testifying tht the recipient hs successfully completed prticulr course of study which usully requires two or three full-time yers fter finishing Grde 1 in Secondry schools. Figure 1(1) Figure 1() Minly through techer s demonstrtion, nd occsionlly through questioning students, the techer explined tht the re of Figure 1(1) is: A+B+4 b/, while the re of Figure 1() is: C+4 b/. By tking wy the four congruent right-ngled tringles from both lrge squres which re of the sme re, the students found the reltionship: A+B=C nd Pythgors theorem. 69

3 Anlyses ZDM 00 Vol. 34 (6) Plce Segment Description Durtion (Minutes) Czech 1 Evluting the previous written test nd collecting exercise 6.4 Republic (CZ) book Reviewing the previous lesson Introducing new proofs of the theorem; 1. Explining the theorem 4 Exmples nd exercises; demonstrtion using physicl 19.9 objects 5 Summry nd ssignment.1 Totl 44 Hong Kong (HK) 1 Distributing worksheets 1.7 Reviewing relevnt knowledge Introducing the proofs of Pythgors theorem Exmples nd exercises Summry.5 Totl 57 Shnghi 1 Estblishing sitution for lerning the new topic 1.5 (SH) Doing experiment nd mking conjecture 7. 3 Proving nd explining Pythgors theorem Exmples nd exercises Summry nd ssignment 1.1 Totl 45 Tble : Overview of the three lessons Shnghi lesson. The Shnghi techer strted the lesson by sking the questions wht is the reltionship between the three sides of tringle? nd wht is the reltionship between the three sides of right-ngled tringle?. The techer then motivted students to explore the unknown feture, which ws ctully the topic for this lesson. After tht, the clss ws divided into groups of four (two students in the front row turned bout, so the four students round desk formed group); nd ech group ws ssigned to do the following tsks: (1) Drwing right-ngled tringle ABC, C=90 ;() Mesuring the lengths of three sides, b nd c; (3) Clculting the vlues of, b nd c ; nd (4) Guessing the reltionship mong, b nd c. Bsed on drwing, mesuring, clculting nd communicting, the students were sked to mke conjecture on the reltionship between the three sides of right-ngled tringle, i.e. + b = c Discussion The forementioned description indictes tht the Hong Kong techer nd Shnghi techer tended to explore the theorem through deliberte ctivities. 3.. Justifiction of the theorem All the three techers tried to justify the theorem by different methods. The following describes how the techers proceeded with the justifiction. Czech lesson. Pythgors theorem nd one proof of the theorem hd lredy been introduced in the previous lesson. In this lesson, the Czech techer further brought in new proofs. Firstly, the techer introduced proof through demonstrting model similr to the digrms in Figure 1. The students were sked to identify ech prt of the figures nd explin the reltionship between the two digrms (s shown in Figure 1(1) nd Figure 1()). Finlly, the students were convinced tht the sum of the res of the squres constructed from the djcent sides (of right-ngled tringle) is equl to the re of the squre constructed from the hypotenuse, nd this ws clled the Pythgors theorem by the techer. Then the students were sked to red prgrph in the textbook entitled venture prdise in which three proofs were introduced bsed on three digrms (similr to Figure ), nd students were sked to complete the proofs fter the lesson. Hong Kong lesson. The techer orgnized cuttingnd-fitting ctivity to verify the theorem. The students were sked to ply gme by following the instructions on the wll chrt s follows: (1) Two squres (one whole squre (A) nd nother (B) mde of 4 pieces I, II, III, IV)) re ttched to the two shorter sides of the tringle s shown in Figure () Fit the 4 pieces of puzzles cut from the squre B, together with the squre A to mke squre (C) on the longest side. Bsed on this gme, the students found the reltionship mong the res of the three squres: A+B=C, nmely, Pythgors theorem + b = c. Shnghi lesson. In the lesson, two proofs were explored by the following steps: specilizing to n isosceles right-ngled tringle, nd then generlizing to A C c b II B I III IV Figure 70

4 ZDM 00 Vol. 34 (6) right-ngled tringle. A. Specilizing: isosceles right-ngled tringle A group ctivity (in groups of four students) ws orgnized: fitting squre by using four congruent isosceles right-ngled tringles. Students were sked to present their digrms by projector, s shown in Figure 3(1) nd Figure 3(), nd explin the resonbleness of the digrms. Then the students were sked to clculte the re of the squre (let the hypotenuse be c, nd the djcent side be ) in different wys, nd the eqution: + = c ws rrived t. Thus, the theorem ws proved for n isosceles right-ngled tringle. c Figure 3(1) Figure 3() Through exploring the specil cse, not only ws the theorem proved in specil sitution, but lso more importntly, the wys of proving the theorem in the generl cse cme up progressively. B. Generlizing: generl right-ngled tringle To prove the theorem in the generl cse, the techer orgnized nother group ctivity: fitting figures tht include squre by using four congruent generl rightngled tringles. Bsed on the ctivity, the students were sked to present their digrms, s shown in Figures 4(1) nd 4(), nd provide the relevnt explntions. As soon s the bove digrms were presented on the screen, the students were encourged to clculte the res in different wys (the djcent sides re nd b, nd the hypotenuse is c). Figure 4(1) Figure 4(1) Regrding Figure 4(1), one method is: S lrge squre =(+b) The other is S smll squres + 4S tringles = c b; Since the two methods re used to clculte the re of the sme squre, then S lrge squre =S smll squres + 4S tringles 1 Nmely, ( + b) = c + 4 b + b = c Anlyses Simplifying, we get Similrly, with the help of Figure 4(), nother proof ws introduced. Discussion With respect to the justifiction of the theorem, diverse pictures evolved. The Shnghi lesson seemed to emphsize multiple mthemticl proofs. Regrding the justifiction in the Czech lesson nd the Hong Kong lesson, the two techers tended to verify the theorem through physicl nd visul ctivities. In distinguishing them from ech other ccording to whether the justifiction is bounded up with mthemticl symbols, the Hong Kong lesson is closer to the Shnghi one, since no mthemticl symbols were used in the Czech lesson t this stge. It seems tht the Shnghi techer stressed deductive mthemticl resoning, while the Hong Kong techer nd the Czech techer preferred visul resoning nd verifiction Appliction of the theorem Giving problems for students to prctice is very importnt for students to lern the theorem. The kinds of problems techers use lso reflect techers interprettions of the objectives of teching the theorem Distribution of problems According to Gu (1994), problems cn be conceived s comprising three bsic elements. The initil sttus A is the given conditions of the problem. The process of solving the problem B is the trnsition of pproching the conclusion bsed on the existing knowledge, experience nd given conditions. The finl stge C is the conclusion. A problem is considered s prototype if it consists of n obvious set of conditions, conclusion nd solving process fmilir to the lerners. The prototype cn be trnsformed into closed vrition or n open vrition by removing or obscuring one or two of the three components respectively. Furthermore, Gu merges Blooms six teching objectives: knowledge, clcultion, interprettion, ppliction, nlysis nd synthesis, into three ctegories. They re memoriztion (including knowledge nd clcultion), interprettion (including interprettion nd ppliction) nd explortion (nlysis nd synthesis). It is found tht the three types of teching objectives cn be chieved by providing students with three kinds of vrieties of problems respectively. Regrding Pythgors theorem, these ctegories cn be illustrted in Tble 3. 71

5 Anlyses ZDM 00 Vol. 34 (6) Ctegory of problem Specifiction to Pythgors theorem Memoriztion [M] Clculting the power nd squre root; Nming hypotenuse in different right-ngled tringles in terms of their positions nd their lbels; Writing up the formule corresponding to different right-ngled tringles in terms of their positions nd their lbels; Interprettion [I] Clculting the length of the third side of right-ngled tringle when two sides re given explicitly; Judging whether n ngle is right ngle when the lengths of the three sides re given explicitly. Clculting the length of the third side of right-ngled tringle when the lengths of two sides re given explicitly in dily life context; Explortion [E] Clculting the length of the third side of right-ngled tringle when the lengths of two sides re given implicitly; Applying the theorem by constructing right-ngled tringles; Ill-structured problems (tril nd error, discussion with different situtions) Tble 3: Ctegories of problems According to this clssifiction, the problems used in these three lessons cn be depicted in Figure 5. Figure 5. Distribution of problems in the lessons It is found tht in the Czech lesson, 46% of the problems re t memoriztion level while the others re t the interprettion level. However, in the Hong Kong lesson, ll the problems re t the interprettion level. In the Shnghi lesson, three-fourth of the problems re t the interprettion level nd ll others belong to the explortion level. Overll, the Shnghi techer provided more chllenging problems thn others, nd the Czech techer offered the esiest problems to her students. Exmples of problems will be given in the following section. Problems nd explntion In the Czech lesson, the following types of problems were used: Clculting the hypotenuse when the two sides djcent to the right ngle re given in different right-ngled tringles in terms of their positions nd size Nming the hypotenuse in different right-ngled tringles in terms of their positions nd their lbels; Writing down the formuls corresponding to different right-ngled tringles in terms of their positions nd their lbels; Determining if triplets such s [9,40,41], [9,1,13] etc. re Pythgoren triplets M I E CZ HK SH Aprt from the types of problems given bove, the techer orgnized the following ctivity for students: Three students were sked to stretch three line segments of lengths 3 meters, 4 meters, nd 5 meters, so s to crete right-ngled tringle in the clssroom (ech student stood t vertex). Students were to judge perpendiculr reltionship between the two djcent sides (CZ, Interprettion). After this ctivity, students were sked to give piece of dvice to their fthers t home if they do not know how to mrk out right ngle for exmple when constructing fence. In the Hong Kong lesson, the bsic type of problems is to find the third side when two sides re given in different right-ngled tringles in terms of their position nd size. In ddition, the techer presented dily life problem: An rmy of soldiers wnts to ttck cstle, which is seprted from them by river nd wll. The river is 15 m wide nd the wll is 0 m high. ) How cn the soldiers rech the top of the wll (if they cnnot fly over the wll by ny mens) 4? b) Wht is the shortest distnce to get to the top of the wll? In the Shnghi lesson, the following bsic problems were provided: Exercise 1: In the following right-ngled tringles, given the lengths of two sides, fill the length of the third side into the relevnt brckets: 4 In ncient Chin, common wy for soldiers to rech the top of wll in the enemy s city is to construct ldder for soldiers to climb over the river to rrive t the top of the wll.

6 ZDM 00 Vol. 34 (6) ( ) ( ) 3 13 In ddition, one problem ws mde up from figure fitted by students themselves s follows: Exercise : In the following figure, if the re of squre EFGH is 0 cm, nd the rtio of the lengths of the two djcent sides is 1:, cn you clculte the perimeter? From the problems used in these lessons, it ws found tht the Czech techer nd the Hong Kong techer emphsized more the dily life ppliction nd the connection between mthemtics nd society. In prticulr, the Czech techer stressed the history of discovering Pythgors theorem, the beuty of Pythgors triplets nd the rel context ppliction of Pythgors theorem such s judging perpendiculr reltionship when constructing fence. On the other hnd, the techer in Shnghi tended to provide vriety of problems with different mthemticl contexts, which require severl steps nd different concepts nd skills to solve. Vrition in the exercises In ddition to noting the types of problems provided in the lessons, it is lso interesting to identify the wys the problems vried. If the chnges from the prototype of problem (in which the lernt knowledge cn be pplied directly) to its vritions re identified visully nd concretely (such s vritions in number, positions of figure etc.), but the relevnt concepts nd strtegies cn still be pplied explicitly, then this kind of vrition is regrded s explicit (see Exercise 1 in the Shnghi lesson). On the other hnd, if the chnges from the prototype to its vritions hve to be discerned by nlysis bstrctly nd logiclly (such s vrition in prmeter, subtle chnge or omission of certin conditions, chnge of contexts, or reckoning on certin strtegies etc.) so tht the conditions or strtegies for pplying the relevnt knowledge re implicit, then this kind of vrition is chrcterized s implicit (see Exercise in the Shnghi lesson). According to this explicitimplicit distinction, it ws found tht both types of vrition often ppered in the Shnghi lesson, but A H D E B F G C ( ) Anlyses bsiclly only explicit vrition ppered in the Czech lesson nd the Hong Kong lesson. By using implicit vrition in the exercise, not only will the problems be more difficult, but they will lso be more open-ended s well. Discussion It seems tht the techers in ll three plces pid ttention to prctice t the interprettion level. Yet, in the Czech lesson, there were one qurter of the exercise t the memoriztion ctegory but there ws no such kind of exercise in the Hong Kong lesson nd the Shnghi lesson. Moreover, the Hong Kong techer provided problem with rel life context for students to prctice, nd the Shnghi techer ssigned severl problems with complicted mthemticl contexts for students to solve. It seems tht the Hong Kong techer nd the Shnghi techer gve more chllenging problems for their students to tckle Clssroom interction Questioning As mentioned bove, there ws no question of the fourth ctegory in ll the lessons. The totl numbers of questions re 10, 48 nd 5 in the Czech lesson, Hong Kong lesson nd Shnghi lesson respectively. The distribution of the questions ws shown in Figure 5. Question Distribution of question CZ HK SH Plce Figure 5. Distribution of questions YES/NO NAME EXPLANATI ON The bove tble shows tht the second ctegory of questions ws the most commonly used in the lessons. In ddition, it ws found tht the Czech techer nd the Hong Kong techer dopted similr pttern of questioning, with more thn 70% of the questions requesting simple yes or no response, nd less thn 15% of the questions in the other two types. It is interesting to tke note of the Shnghi techer s pttern: the Shnghi techer only sked less thn 5% of the questions to request simple yes or no response nd sked round hlf of the questions to elicit students explntions Pttern of clssroom interction The following section tries to chrcterize the fetures of clssroom interction in ech country or city by quoting some excerpts. Czech Republic. After discussing the homework, the 73

7 Anlyses ZDM 00 Vol. 34 (6) techer introduced proof. The exchnges were demonstrted s follows: 1. T I will show you one more. I hve it prepred here. Yeh. I hope it will not fll down from it. The mgnet needles re lredy wek.. T So, the bsic thing is the bord here [there re three congruent right-ngled tringle models in the bord]. We will strt from it. Here I hve right-ngled tringle. This is...? 3. Ss Right ngle. 4. T One leg. The second leg. 5. Ss The second leg. The hypotenuse. 6. T The hypotenuse. Well, nd now- ll these three tringles re congruent. The id is lredy old, so, don t look t those worn-out vertexes. They re congruent, becuse they Ss Coincide. 8. T They coincide if we plce them this wy [overlpping two models together]. Yeh. And now I plce them this wy. I hope I will do it correctly. So. They stick, yeh [putting digrms s shown in Figure 1(1)]. Wht do you see here? 9. Ss A squre. 10. T A squre bove... one leg. And this? 11. Ss A squre bove the second one. 1. T It is squre bove the second leg. You cn see it well, yeh? So, the blue prt is Ss A squre. 14. T The sum of squres bove both legs. Is it so? 15. Ss Yes. 16. T Tht is this side. Well, nd now I set it nother wy. I set it in such wy tht I will mke the squre bove the hypotenuse. I cn keep this here [putting up digrm s shown in figure 1()] 17. T The id is old, but I think it is very grphicl. Here we hve the hypotenuse nd the blue prt which is there Ss A squre. 19. T The squre bove the hypotenuse. And cn you deduce from this tht those two squres together nd this squre- tht they hve the sme size? 0. Ss Yes. 1. T Well, yeh or no?. Ss Yes. 3. T Well, it fills gin the whole bord nd there re the three tringles which were there before, too [demonstrting the previous digrm s shown in Figure 1(1) ). 4. T So the two squres which you sw before... hence the sum of the res of the squres bove both legs is the sme s the squre constructed bove the hypotenuse. 5. T Tht is nother wy how to prove Pythgors theorem. And there re mny other wys. And these wys re even interesting in such wy tht they cn be used s puzzles. (CZ,trnscrpt, p.5-6) Techer demonstrtion with some students input. During the process of demonstrting the proof, the techer often sked students simple but criticl questions to check if they understood wht the techer tried to demonstrte. The techer clerly demonstrted nd explined the components of the models (1~7) nd the digrms put up (8~18), nd the reltionship between the two digrms (19~4) with the help of frequent questioning (, 4, 6, 8, 10, 1,14, 16, 17, 19, 1). Hong Kong. The process of proving the theorem cn be divided into the following: Clculting the re of the digrm by using Figure 1(1); clculting the re of the digrm by using Figure 1(); explining the equivlence of the two methods; nd (3) deducing Pythgors theorem. The following excerpt mnifests the second wy of clculting the re of the squres. 1. T Oky, cn nyone tell me, ctully this figure - this figure A plus B - the re of this figure nd the re of this squre C, wht is the reltionship?. Ss A plus B Ss A T Wht did you sy? 5. Ss A plus B equl C. 6. T Equl. A student sid the res of these two figures re equl. Why? 7. T Why did you tell me tht they re equl? 8. Ss Becuse the four 9. T Yes. A student hs told me. He noticed wht I did before ws tht I took wy three purple right-ngled tringles, right? 10. T Tht mens I knew then tht the res of these three right-ngled tringles re equl. 11. T If I tke wy these four right-ngled tringles t the sme time we hve sid tht they re the sme, right? 1. T These two their length nd width re both plus b. Their res re equl. 13. T If I tke wy four right-ngled tringles, I m tking wy the sme things. So the resulting re should the re of A plus B equl to C? 14. Ss Yes. 15. T Yes. We get this is the result. 16. T Oky, this squre A this re, plus the re of squre B re equl to C, the re 74

8 ZDM 00 Vol. 34 (6) of this squre. Oky? 17. Ss Yes. Techer demonstrtion with little student input. Bsed on the techer s visul demonstrtion, the students were ble to find out the reltionship mong re A, B, C (5), but they could not offer cler explntion (8). Then the techer demonstrted nd explined the re reltionship (9-17) in ccordnce with the principle tht tking wy the sme thing from the sme re results in the sme reminder. Shnghi. The conjecture mde by the students ws proved by the following steps: exmining specil cse nd then exploring the proof in the generl sitution. The discourse of introducing the first proof ws illustrted s follows. 1. T: + =c. Then is there such reltionship in generl right-ngled tringles? Let s go on putting up the figures. Plese chnge them into four congruent generl rightngled tringles. How mny wys of putting it up do you hve? Let s hve try. [The students put up squre in groups of four. The techer gives the students dvice when mking n inspection. The students who hve finished putting up squre on the projector]. T: Well, let s stop to hve look. The group of Chen Tingting hs put up squre like this just now. Well, Chen Tingting stnd up to nswer. Why is it squre tht you put up? 3. S: They re four congruent tringles. The djcent sides of two groups re equl. The sum of them is equl, too 4. T: The sum of them is equl 5. S: The squre cn be put up. 6. T: The squre cn be put up. 7. T: Sit down, plese. There ws nother method to put up squre just now. Let s hve look. 8. T: (Becuse of the light color on the projector) Deepen the color little. Well, the student who put up the squre just now stnds up to nswer. You put up squre. Why? Plese, Shen Lingjun? 9. S: Becuse they re congruent tringles. Their hypotenuses re equl. 10. T: hypotenuses re equl. 11. S: So wht I put up is squre. 1. T: Well, sit down. The students who put up this squre put your hnds up (Figure 6.7(4)). (The students rising their hnds. It is found tht the mjority of the students put up their hnds). Put them down! Is there nyone who put (the squre) like Shen Lingjun (did)? Yeh. So did your group. Well, put down your hnds. Wht cn we get through putting Anlyses up the picture? First of ll let s look t the squre tht Chen Tingting put up just now. If the short djcent side is, the long djcent side is b, nd the hypotenuse is c, wht is the re of the big squre? (The techer repeting) Well, Zhng Ling, plese. 13. S: It should be (+b) 14. T: Wht does (+b) indicte in the squre? 15. S: (b+) is (b+). 16. T: (b+) is (b+) 17. T: Any other wy? Any other wy? Well, Zhou Yu, plese S: 4 b+ c. 19. T: Wht re is it? 0. S: It s the re of the four smll tringles dded to c. 1. T: Wht re is this [pointing to the centrl smll figure]?. S: It s the re of the smll squre. 3. T: it s the re of the smll squre. 4. T: Wht reltionship cn you get? 5. S: (+b) =b+c. 6. T: Plese put it in order. 7. S: +b+b =b+c ; +b =c 8. T: Sit down, plese. We cn get wht +b is equl to through putting up (the picture) just now. 9. Ss: c. (SH09, trnscript, p.9) Techer control with much student input. In the bove excerpt, the techer tended to encourge students to present their digrms on the screen, nd give their resons (, 8). Moreover, the students were sked to tell the whole procedure of their clcultion (1, 17, 19, 4, 6). Discussion The scenrios described here seemed to be contrdictory to the stereotype tht the clssroom in Est Asi is knowledge trnsmitting nd techer-centered. While the Czech techer nd the Hong Kong techer preferred to demonstrte visully the process of proving with some elicittion of student s involvement, the Shnghi techer seemed to put much emphsis on how to help students mke their own conjectures nd proofs, nd encourge students to express their ides by orgnizing deliberte ctivities. This observtion my be due to the specificity of the prticulr content nd the idiosyncrsy of the prticulr techer, but t lest it should point to rethink bout cler-cut distinction between studentcenteredness nd techer-centeredness. 4 Discussion nd Conclusion Bsed on this study, the following observtions cn be mde: 4.1 Mthemticl proof versus visul verifiction The Shnghi techer seemed to put much stress on the 75

9 Anlyses ZDM 00 Vol. 34 (6) process of discovery nd the proof of Pythgors theorem. Not only were students sked to mke conjectures bsed on drwing, mesuring nd clculting, but lso multiple mthemticl proofs were explored by engging in certin ctivities. Moreover, the techer ws dept t providing scffoldings for students to develop the proofs by themselves. It seems tht the Shnghi techer tended to justify the theorem mthemticlly nd logiclly. On the other hnd, the Czech techer nd the Hong Kong techer tended to introduce multiple proofs by solving puzzles or demonstrting digrms visully. It seems tht the Czech techer nd the Hong Kong techer preferred to justify the theorem visully. Recently, the role of proof nd the wys of treting proof hve been re-vitlized in the new mthemtics curriculum interntionlly. For exmple, Principles nd Stndrds for School Mthemtics (NCTM, 000) suggests tht the mthemtics eduction of prekindergrten through grde 1 students should enble ll students to recognize resoning nd proof s fundmentl spects of mthemtics, mke nd investigte mthemticl conjectures, develop nd evlute mthemticl rguments nd proofs, nd select nd use vrious types of resoning nd methods of proof (p. 56). It seems tht the techers in this study ll meet this reformed orienttion to certin degree. However, the question is wht kind of proofs should be introduced? The mthemticl proofs like those in the Shnghi lesson, or the visul proofs like those in the Czech lesson nd the Hong Kong lesson? The nswer depends on wht kind of mthemtics re students expected to lern. 4. Knowledge constructing or trnsmitting It is quite surprising to find tht the Shnghi techer pid much ttention to the process of re-discovering the theorem: from mking conjecture to justifying the conjecture through series of well-designed ctivities. The techer in the Czech Republic nd the techer in Hong Kong put emphsis on introducing nd verifying the theorem by using visul ids. Moreover, it ws found tht students in the Shnghi lesson were quite involved in the process of lerning such s putting up nd presenting digrms, nd explining their understnding (lthough the techer guided these ctivities hevily). This is quite contrry to other observtions tht Est Asin students re pssive lerners (Pine, 1990; Morris, et l., 1996). The techer in the Czech Republic nd the techer in Hong Kong tended to demonstrte nd explin the process of knowledge construction nd try to elicit student s responses to certin extent. This finding seems to chllenge the stereotypes presented in much of the literture. 4.3 Explicit vrition versus implicit vrition All techers in the study emphsized clssroom exercise. It is importnt to tke note of the wys of vrying the problems in clssroom teching, which my ffect the qulity of doing exercise. This study seems to suggest tht the techer in Shnghi ws dept t providing both explicit nd implicit vritions of problems for students to explore, while the other two techers preferred using only explicit vrition of problems for students to prctice. Moreover, implicit vrition of problems my provide students with more chllenging nd open-ended problems to tckle, which my be helpful for developing students bility of problem solving. 4.4 Concluding remrks As Clrke (00b) rgued, n interntionl comprtive study should be undertken in nticiption of insights into the novel, interesting nd dptble prctices employed in other school systems of whtever culturl persusion, nd of insights into the strnge, invisible, nd unquestioned routines nd rituls of our own school system nd our own mthemtics (p.15). This study ims t describing mthemtics clssroom scenes vividly in different cultures for reders to reflect upon their own prctices. Moreover, this study employed methodology tht proves to be useful in cpturing the novel, interesting nd dptble prctices employed in the three plces concerned. In prticulr, the methodology supports the notion tht keeping the content invrint when compring mthemtics clssrooms in different cultures will mke study more profitble (Alexndersson, 00). Acknowledgements The dt collection ws supported by trvel grnt from the Hong Kong Culture nd Society Postgrdute Progrmme 000 of the University of Hong Kong. We would lso like to thnk the Czech coordintor in the TIMSS-R Video Study for permission to use their Videos in this pper. Our thnks re lso given to Miss Angel Chui nd Miss Ellen Tseng for their help in collecting the Czech nd Hong Kong Videos for this study. References Alexndersson, M., Hung, R.J., Leung, F., & Mrton, F.(00, April). Why the content should be kept invrint when compring teching in the sme subject in different clsses, in different countries. Pper presented in Americn Eductionl Reserch Assocition Annul Meeting, New Orlens, USA. An, S., Kulm, G, Wu, Z.(00). The impct of culturl difference on middle school mthemtics techers beliefs in the U.S nd Chin. Pre-conference proceedings of the ICMI comprtive conference (pp ). 0 th -5 th, October, 00. Fculty of Eduction, the University of Hong Kong, Hong Kong SAR, Chin. Bishop, A. J.(00). Plenry lecture entitled Wht comes fter this comprtive study: more competition or collbortion t the ICMI comprtive study conference 0 th -5 th, October, 003, Hong Kong, Chin. Clrke, D. J. (00, April). The lerner s perspective study: Exploiting the potentil for complementry nlyses. Pper presented t Americn Eduction Reserch Assocition Annul Meeting, New Orlens, USA. Clrke, D. J. (00b). Developments in interntionl comprtive reserch in mthemtics eduction: Problemtising culturl explntions. Pre-conference proceedings of the ICMI comprtive conference (pp.7-15). 0 th -5 th, October, 00. Fculty of Eduction, the University 76

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