Second-Degree Equations as Object of Learning

Transcription

1 Pper presented t the EARLI SIG 9 Biennil Workshop on Phenomenogrphy nd Vrition Theory, Kristinstd, Sweden, My 22 24, Abstrct Second-Degree Equtions s Object of Lerning Constnt Oltenu, Ingemr Holgersson, Torgny Ottosson Kristinstd University College The purpose of this pper is to report spects focused by techers in clssroom prctice when teching the solving of second-degree equtions (x 2 + bx + c = 0 with, b nd c prmeters nd 0) by help of the formul p x 1,2 = ± 2 2 p 2 q (p, q prmeters) nd the students wys of discerning prticulr spects. The presenttion is bsed on dt collected in n upper secondry school in Sweden. Dt consists of video-recordings of lessons, individul sessions, interviews nd the techers /resercher s review of the individul sessions. Test results lso constitute n importnt prt of the dt. The study includes two techers nd 45 students in two different clsses. In the nlysis, concepts relting to vrition theory hve been used s nlyticl tools. Dt hve been nlysed with respect to spects focused on by the techers during the lessons, spects tht re ignored, nd ptterns of dimensions of vritions tht re constituted. Dt hve lso been nlysed with respect to the students focus when solving different problems in tests. The results show tht the techers focused on the prmeters nd the unknown quntity of n eqution in different wys nd this implicted tht the students hd the possibility to discern different spects referring to the reltion between seconddegree eqution nd the p-q-formul. Furthermore, some of these spects re identified s criticl spects in the students lerning. Bckground Vrious documents, such s the most recent course syllbi in mthemtics in upper secondry school in Sweden (Skolverket, 2000), specify tht students should be ble to solve seconddegree equtions nd pply this knowledge in problem solving. Despite this, severl investigtions t different universities point out tht big prt of the students re unble to solve this type of equtions (see, e.g., Högskoleverket, 1999; Pettersson, 2003, 2005; Thunberg & Filipsson, 2005). Although the solving of second-degree equtions is crucil for solving number of problems, it is surprising tht reserch into the teching nd lerning of this topic is so scrce (Oltenu, 2007; Viyvutjmi, 2004; Viyvutjmi & Clements, 2006). Reviews of reserch in lgebr eduction hve so fr generlly been silent bout the teching nd lerning of second-degree equtions (e.g., Kiern, 1992; Kiern & Chlouh, 1993; Wgner & Prker, 1993). The chpter on lgebr in the Interntionl Hndbook of Mthemtics Eduction (Filloy & Sutherlnd, 1996) does not refer to second-degree equtions. Viyvutjmi (2004) reported tht immeditely fter the students hd prticipted in lessons on second-degree equtions, 70 % of their responses to stndrd second-degree eqution tsks were incorrect. Zslvsky (1997) investigted misconceptions with respect to qudrtic functions in 25 different schools in Isrel. Zslvsky s reserch emphsis ws qudrtic functions however, nd her report touched only incidentlly on students responses to second-degree equtions. In the chpters on lgebr in the lst two four-yerly reserch

2 summry publictions of the Mthemtics Eduction Reserch Group of Austrlsi (Wrren, 2000; Wrren & Pierce, 2004), the word qudrtic ws used when Wrren nd Pierce (2004) referred to smll study by Gry nd Thoms (2001) bout the use of grphics clcultor nd multiple representtions to explore second-degree equtions. The results of Gry nd Thoms study indicted tht the students did not improve their bility to solve second-degree equtions. Hoch nd Dreyfus (2004) rgued tht wheres 30x 2 28x + 6, for exmple, is equl to (5x 3)(6x 2), students with poor sense of structure my not relise tht the qudrtic trinomil nd its fctorised equivlent re different interprettions of the sme structure (p. 51). Lithner (2006) rgues tht the finding of solutions to second-degree equtions builds on n lgorithmic resoning. This type of resoning depends on the steps tht should be effectuted when solving n eqution. If second-degree equtions re to remin n importnt component of mthemtics curricul round the world, reserch bout guiding techers to improve their students wys of understnding how to identify the solution for this type of eqution is needed. To do this, it is necessry to understnd where the problem is, nd in which wy the contents treted in lessons influence the students lerning. Reserchers point out tht there re few empiricl studies nlysing how techers tret mthemticl contents in the clssroom, both ntionlly nd interntionlly (Löwing, 2004; Oltenu, 2007; Runesson, 1999; Viyvutjmi, 2004). We do not know wht is presented to the students in teching sitution nd wht is criticl in their lerning of them. In this rticle, the emphsis is on how two techers tech students to solve second-degree equtions nd wht the students re lerning from it. A centrl question in this context is which spects the techers focus on when they tret the content, which spects re discerned by the students nd which of these spects tht re criticl for their lerning. Is it possible to find n explntion to this problem by incresing our knowledge bout the reltions between wht the students cn be wre of in content tht is presented in the clssroom nd how the students perform when they solve different equtions? The nswer to these questions cn give importnt informtion concerning how to improve students lerning to solve second-degree equtions. Theoreticl frmework The theoreticl perspective in this study is the vrition theory s it is described by Mrton nd Booth (1997), Bowden nd Mrton (1998) nd Mrton, Runesson nd Tsui (2004). One of the min points of the theory is tht lerning is wy of experiencing or coming to experience the world in certin wy, nd tht different wys of experiencing will led to different lerning outcomes. A wy of experiencing something cn be defined by the spects discerned. Tht is: An spect of thing corresponds to the wy in which tht thing might differ from, or be similr to, ny other thing, tht is, the wy it is perceived to be, or the wy tht it is experienced by someone s different from, or similr to something else. (Mrton et l., 2004, p. 9) If the thing is second-degree eqution (n eqution in the form x 2 + bx + c = 0, where, b nd c re prmeters, with 0), the spects re the unknown quntity x, the prmeters (, b, c), multipliction nd ddition s opertions nd the equlity between the two sides of the eqution. To discern certin spects of the eqution, vritions must be experienced in these spects. An spect is therefore vlue in dimension of vrition. The problem is to know in which spects to crete dimensions of vrition in order to constitute the mening of seconddegree equtions. These spects re clled criticl spects or criticl fetures. If we now think tht the mening of second-degree eqution is to find its solutions by help of the formul 2

3 2 p p x1,2 = ± q (clled p-q-formul in this pper), the presumptive criticl spects 2 2 could be the unknown x, the prmeters p nd q, the opertion between different prts of the formul (, ±, ) nd the opposite numbers of 2 p. The criticl spects prescribe tht the eqution nd/or the p-q-formul cn be seen in certin wy (for exmple s x 2 + px + q = 0) or in different wys (for exmple s x 2 + bx + c = 0,, b nd c prmeters nd 0). Mrton et l. (2004) specify tht the criticl spects must be found empiriclly. The totlity of criticl spects defines n object of lerning. Vrition theory tkes the object of lerning s the point of deprture nd clims tht the wy in which lerners experience the object of lerning depends on which spects of the object of lerning tht re focused upon nd discerned s criticl. The only wy to discern the criticl spects is to experience how they vry, either t the sme point in time, or by remembering erlier relted experiences. The concept of experience hs guided the nlysis of teching nd lerning in this study. Lerning, which in this perspective is sine qu non with the experience of vrition, hs been studied from two perspectives, nmely the encted nd lived object of lerning. The encted object of lerning is wht ppers in the clssroom nd the techer nd the students constitute it jointly. This is wht it is possible for the student to experience nd lern in specific setting from the point of view of wht is intended to be lerned. The wy in which students experience the object of lerning is the lived object of lerning. The nlysis of the empiricl dt hd s point of deprture tht to experience something is to discern prts nd the whole, spects nd reltions. To experience how to solve seconddegree equtions is to experience the mening nd the structure of them nd these two mutully constitute ech other. So neither structure nor mening cn be sid to precede or succeed the other. In structuring n experience, it is importnt for the techer to be ble to focus the students ttention on the criticl spects of tht experience, to distinguish this experience from ny other experience, nd to mke them understnd the reltionship between the criticl spects. The origins of this study were in the first uthor s (Oltenu, 2007) investigtion of the teching nd lerning of second-degree equtions nd functions. In this pper we investigte the lerning tht occurs in the clssroom when the students solve equtions in the form: x 2 + bx + c = 0 (, b nd c prmeters nd 0) To experience the p-q-formul in solving second-degree equtions, it is necessry for students to relte p nd q in the formul to given numbers, tht is, to see symbols s generlized numbers. In other words the students need to discern the prts in n eqution nd in the formul, relte them to ech other nd to the eqution nd formul s whole. If the eqution s x 2 -coefficient equls 1, this reltionship is direct. P nd q in the formul re then the sme s the x-coefficient nd constnt term of the eqution. If the eqution s x 2 -coefficient is different from 1, p nd q cn be identified by dividing the eqution with this coefficient (see equtions 2 nd 3 in Tble 1). This mens tht the use of the p-q-formul needs rewriting the eqution until the x 2 -coefficient equls 1 nd the other side of the eqution is zero. In the process of rewriting second-degree equtions, it is necessry for the students to discern tht the prts in n eqution cn be relted to ech other in different wys while the solutions remin invrint. In the presented eqution, the unknown quntity x is invrint. Tht mens tht x is invrint in the p-q-formul since the structure of this formul is invrint. In Figure 1, it cn be seen tht the prmeters vry through different rewritings or through dopting negtive nd/or positive numbers. It is the identifiction of the prmeters with p nd q in the formul tht mkes it possible to find out the unknown quntity. 3

4 x 2 + bx + c with, b, c R, 0 x 2 + bx + c = 0 x 2 + bx + c = d, d 0 x 2 + bx + c - d = 0 x 2 + b x + c = 0 x 2 + b c d x + = 0 b p = c q = b p = c d q = x 2 + px + q = 0 Figure 1. Vrition of prmeters. To nlyse the students lerning in reltion to the vritions opened in the clssroom, our point of deprture ws tht the students lern to solve second-degree equtions by pplying the p-q-formul nd tht this requires them to experience certin ptterns of vrition. This experienced vrition mkes it possible for them to discern some spects tht re necessry for developing bilities to solve this type of equtions. For this reson, in nlysing the dt the spects tht students focused on when they solved different equtions in test situtions were first identified nd therefter the spects tht re criticl in the students lerning. After tht, the opportunities tht the students were presented to experience in the clssroom when the techers were teching how to solve second-degree equtions were investigted. Method nd collection of dt The study ws performed in n upper secondry school in Sweden during the spring term During this period, two techers treting the sme topic in mthemtics were observed, together with the resulting lerning of the students. In both clsses, the sme textbook ws used. Students were selected from the Nturl Science Progrmme. A totl of 45 students (25 mle, 20 femle) nd two techers (Ann nd Mri) prticipted in the study. The present pper focuses how students in the two clsses experienced the solving of second-degree equtions. This mens tht the lessons in which the mentioned content ws treted in the clssroom hve been selected from the min study. 12 consecutive lessons in ech clss, eight individul sessions, eight interviews nd sequences where the techers nd the resercher looked t some video sequences together were selected from the min study for the nlysis. Also, the techer s plnning of the course contents, resercher field notes nd two tests tken by the students re included in the nlysis. Thus, combintion of dt ws used, the most importnt, however, were video-tped lessons nd the tests. Since both techers tught the sme content nd used the sme textbook, it ws possible to identify nd describe differences between the techers teching in reltionship to the content they hd tught. Before, fter nd during the observed lessons, ll students took two tests. Since the students solved the sme exercises in different tests, it ws possible to identify nd describe differences in their experiencing the contents. 4

5 Results The results presented in this pper re divided into two prts. The lived object of lerning is described in the first prt nd the encted object of lerning is described in the second prt. Description of the lived object of lerning The nlysis of the tests shows tht there re differences between the two clsses nd tht these re bsed on the wy in which the students discerned the equtions prmeters (the coefficients nd the constnt term) nd the unknown quntity. The coefficients nd constnt term refer to numbers tht cn be positive, zero or negtive, nd the unknown quntity refers to rel numbers tht stisfied the eqution. Tble 1. Vritions in the students wys of solving second-degree equtions. No. Second-degree equtions 1. x 2 + 6x + 5 = x x + 14 = = 8,5 + 9,8t 4,9t 2 Vritions in the students nswers Solve the eqution by using the formul Mri s clss Ann s clss 100% 55% Don t distinguish the constnt term 0% 20% Don t solve the eqution 0% 25% Identified p with 6 nd q with 7 91% 65% Identified p nd q with other numbers 0% 13% Identified p with 12 nd q with 14 0% 9% Don t solve the eqution 9% 13% Solve the eqution by using the formul 55% 35% Identified p with 2 50% 35% Identified p with 2 5% 0% Identified q with 1,5/4,9 45% 30% Identified q with 1,5/4,9 10% 5% Hve on side different from zero nd divide only the other side by 4,9 5% 10% Solve other equtions or exercises 15% 15% Only nswer 5% 15% Don t solve the eqution 20% 25% Tble 1 shows the percentges of the wys in which students solved different equtions nd the spects they discerned in this process. The equtions presented differ in subtle but mthemticlly importnt wys. The first difference refers to the unknown quntity, nmely it ppers twice in n eqution nd therefore it forms the eqution s whole with the help of ddition s opertion. The second difference refers to the prmeters (x 2 -coefficient, x-coefficient nd constnt term). The prmeters pper s explicit in the first eqution (p nd q in the formul cn bee identified directly) nd implicit in the second nd third equtions (p nd q in the formul cn be identified fter some rewriting of the eqution). The results show tht in both clsses lrge prt of the students preferred to pply the p-q-formul to find the solutions of second-degree equtions. Aprt from this there re students tht use the null product lw if b = 0, then = 0 or b = 0 to solve n eqution written s x 2 + px + q = 0. The two different wys in which students identify the unknown quntity show tht they discern some spects of the eqution through seprting the whole into different prts. 5

6 A closer look t the test results shows tht ll the students in Mri s clss could discern the prmeters in the first eqution, but only 55 % in. Furthermore, the students in developed n incorrect wy of solving n eqution in which the x 2 -coefficient equls 1 (se Eqution 1 in Tble 1). We cn see n exmple of this in the following figure: Figure 2. Håkn in. These students did not understnd how the null fctor lw should be pplied in the context of generl second-degree eqution. They did not think bout the fct tht the two vlues of x give the result tht 5 is equl to zero, which is not possible. The students did not discern the importnce of the constnt term to fctorise qudrtic trinomils, nd did not check whether ny of the solutions tht they obtined were correct. This wy of solving equtions could not be identified when the students solved second-degree equtions hving the x 2 -coefficient different from 1 (see Equtions 3 nd 4 in Tble 1). The second eqution in Tble 1 hs, for instnce, the coefficients 2 nd + 12, nd the constnt term is Since the x 2 -coefficient no longer equls 1, the use of the p-q-formul demnds rewriting of the eqution to n equivlent eqution. For this, the students hve to understnd tht the prts constituting the whole in the following equtions 2x x + 14 = 0 x 2 6x 7 = 0 cn be relted to ech other in different wys, but, despite this, the equtions hve the sme solutions. If the students do not understnd this reltionship, it leds to giving p the vlue + 12 nd q the vlue + 14 in the formul. This mens tht these students hve hrd time to pprehend in which wys the prmeters in n eqution of type x 2 + bx + c = 0 (with 0) relte to n eqution of type x 2 + px + q = 0 (normlised qudrtic equtions). In the normlised stte, the coefficient in the qudrtic term is 1 nd one side equls 0. In order to mke this reltion it is importnt for the students to discern the x 2 -coefficient. The results in Tble 1 show tht the students who understnd the reltion between the structures of the two equtions lso understnd the structure of the p-q-formul. If the x-coefficient nd the constnt term become negtive numbers fter rewriting, the ddition is used s n implicit opertion in the eqution x 2 6x 7 = 0 (the eqution is relly x 2 + ( 6) x + ( 7) = 0). In this cse, there re students tht discern wys in which the prts relte to ech other, tht is, these students discern tht the minus sign is used in order to highlight negtive number nd tht for instnce x 2 nd 6x re relted to ech other through ddition. The wy in which the prts re relted to ech other re discerned by ll students solving the equtions in Mri s clss, in comprison with only 65 % in. If the constnt term in second-degree eqution ppers on both sides of the equl sign (see Eqution 4 in Tble 1), tht is, the constnt ppers s two prts, there re students tht first discern the constnt term s whole. Therefter, these students discern the eqution s x 2 -coefficient nd divide the eqution by this coefficient. In this wy, it is possible for them to discern p nd q in the p-q-formul. The prt of students tht could identify p nd q in the formul with p nd q in the equtions hving their x 2 -coefficient different from 1 nd the constnt term on both sides of the equl sign, decreses in both clsses but more in Ann s clss. Only 45 % of the students in Mri s clss nd 35 % in could distinguish the eqution s prmeters nd relte them to the p-q-formul in the lst eqution presented in 6

7 Tble 1. Becuse the normlised form of the eqution is obtined from generl formultion in which the constnt term ppers on both sides of the eqution, it is necessry for the students to discern the eqution s constnt term. The results presented in Tble 1 show tht there were students in both clsses tht did not discern ll the prts of the constnt term. Also, there were students tht only divided by 4,9 on one side of the eqution, nd therefter used the formul s the following exmple shows: Figure 3. Ludvig in Mri s clss. This fr we cn estblish tht the prmeters forming the whole in second-degree equtions re criticl spects in students lerning. This is becuse the students hve difficulty in seeing the equtions in certin wy, nmely s x 2 + px +q = 0 nd therefter be ble to identify the prmeters of the equtions with p nd q in the p-q-formul. The encted object of lerning One possible wy to ccount for these differences in lerning is the structurl difference observed in the two techers wys of hndling the object of lerning. In order to develop the students bility to solve some second-degree equtions, severl equtions were chosen for presenttion in the clssroom nd in textbooks (see Tble 2). Tble 2. Different forms of second-degree equtions pplied in the clssroom nd in the textbook. No. Mri s clss Textbooks 1. x 2 = 144 x 2 4 = 0 x 2 = x 2 = 845-2x 2 = (x + 14) 2 = 4 (x 1) 2 9 = 0 (x + 3) 2 = x 2 6x + 9 = 0 x 2 4x 5 = 0-5. x x + 35 = 0 - x 2 + 5x + 6 = 0 6. x 2 + 2x 15 = 0 - x 2 + 6x 16 = 0 7. x 2 x 30 = x 3x 2 = 1 0,01x 2 + x + 2,3 = 0 4x 2 12x 7 = 0 The distinction between the selected equtions is tht the x 2 -coefficient in Equtions 1 nd 4-7 equls 1, while this coefficient is different from 1 in the Equtions 2 nd 8. In Mri s clss nd in the textbook, the x-coefficient nd constnt term re represented by lternting positive nd negtive vlues. This indictes vrition in the wy in which the prts of equtions re 7

8 relted to ech other. Furthermore, the unknown quntity ppers both s monom (Equtions 1 nd 2), nd binom (Eqution 2). Another distinction between the presented equtions in the two clsses is tht in Mri s clss, the constnt term ppers on the sme side s the unknown quntity in Equtions 4-7 nd on the opposite side of the unknown quntity in Equtions 1-3 nd 8. In, this phenomenon could not be identified. In the textbook, both these spects re focused. In the exposé of how to solve the first nd second eqution presented in Tble 2, Mri focused on two spects. The first spect refers to the squre root of positive numbers being both positive nd negtive numbers with the students erlier experiences s bckground, tht is, the students hve only experienced the positive squre root in the geometric context. At this opportunity the difference between first nd second-degree eqution is lifted to the front. The second spect refers to the eqution s x 2 -coefficient, nmely tht this coefficient must be 1 in order for it to be possible to extrct the squre root. This cn be seen in the following exmple: [1.2] Mri: If I then hve 5x 2, I cn t begin to tke the squre root of the right side directly, nd wht should I do first? [2.2] Leonrd: Divide. [3.2] Mri: Yes, I must divide wy the five becuse I must hve only x 2 before I, eh, tke the squre root of. (Lesson 24, ) This focused spect is relted to the wy in which it is possible to identify the unknown through focusing on the fct tht the eqution cn only be solved if the x 2 is on one side of the equlity sign nd the number is on the opposite side nd only if the x 2 -coefficient equls 1. In Ann s exposé, it could lso be identified tht she focused on two spects, but it ws not directly possible to discern the importnce of the x 2 -coefficient for drwing the squre root. This spect is importnt for the students when they solve eqution 8 in Tble 2. Therefter both the techers present tht the sme pttern cn be used to solve seconddegree eqution in which the unknown ppers in binom, nmely to drw the squre root of the number on the other side of the equl sign. This cn be seen in the following figures: Figure 4. Implicit x 2 nd explicit Figure 5. Implicit x 2 nd explicit constnt term (Mri, Lesson 24). constnt term (Ann, Lesson 25). From Figure 4 it is pprent tht Mri clerly present the two first-degree equtions tht result from the squre root drwing. In this wy she shows how the reltions between the eqution s prts chnge through squre root drwing. Furthermore, she points out tht the solutions of the first-degree equtions re the sme s the solutions of the second-degree eqution. Ann solves the eqution without clerly writing which the first-degree equtions re nd through only focusing on the nswer [10.2]. This mens tht Ann does not discern the wy in which the prts relte to ech other nd to the equtions s whole. [8.2] Ann: It wnts to sy [9.2] Lydi: Mm x minus one is equl to the squre root of nine, plus minus 8

9 [10.2] Ann: Plus, minus squre root of nine, tht mens x is 1 plus, minus, we cn write three insted of squre root of nine (Lesson 25, ) In order to solve second-degree equtions tht hve the x-coefficient nd the constnt term different from zero, Mri developed the wy in which the prts of n eqution cn be relted to ech other to obtin the squre of binom by introducing the completing squre. She does this in the following wys: Figure 6. Completing the squre (Mri, Lesson 25). In the numericl exmple Mri focused on vrying the reltionship between the eqution s prts severl times nd she points out tht it is possible to identify the eqution s unknown quntity in this wy. Therefter Mri uses generlised numbers (p nd q) s x-coefficient nd constnt term nd shows how the p-q-formul is obtined. In this wy Mri focuses on the reltion of one prt of the eqution (the unknown quntity) with the other prts of it (the x-coefficient nd constnt term), nd obtins the p-q-formul s n entity. In order to solve different second-degree equtions (see Tble 2), Mri constntly used the newly introduced formul. This signifies tht she ccentuted the mening of the reltion between the x-coefficient nd constnt term of second-degree equtions with p nd q in the p-q-formul. From the following dilogue, tking plce when Mri solves the eqution x 2 + 2x 15 = 0, it is cler tht it sometimes is not stright-forwrd for the students to do this identifiction: [1.10] Mri: Is it written in the form of x 2 + px + q = 0? [2.10] Emili: No [3.10] Mri: Is it not? Yes, it is minus, yes. But if we still compre it (the techer writes x 2 + px + q = 0 on top of the given eqution), I sy tht it is the sme, but wht is q equl to? [4.10] Emili: 15 [5.10] Mri: Yes, tht is minus 15, since the minus sign lies in q, q is equl to minus 15 nd p is equl to 2, nd I cn t hve ny coefficient in front of x 2, nd it should be equl to zero, so it hs tht form. (Lesson 25, ) From the trnscript, it is cler tht Mri first ssumed tht the students could identify p nd q with 2 nd 15 nd therefter use the p-q-formul [3.10]. But it shows tht there re students tht do not immeditely see tht 15 is the sme s +( 15), which is fundmentl convention in mthemtics. This leds to the fct tht these students cnnot see the reltion between the generl eqution x 2 + px + q = 0 nd the given eqution (see e.g. [2.10] in the bove extrct). Through pointing out tht the minus sign lies in q nd tht q in this cse is 15 nd p is 2 [5.10], Mri points out tht presumption in order to identify the solutions to given eqution of the second-degree with help of the p-q-formul is to discern the eqution s x-coefficient nd constnt term nd relte them to p nd q in the p-q-formul. Therefter Mri used positive vlue for the eqution s x-coefficient (p) to keep it invrint, while the constnt term (q) could hve both positive nd negtive vlues. This mens tht n opportunity to discern the vlue of q in the p-q-formul ws creted. In the next 9

10 step Mri kept q invrint with the help of negtive number nd vried the vlue of p, with both negtive nd positive vlues, which mde it simpler to discern p in the p-q-formul. After this, Mri vried the eqution s x 2 -coefficient nd constnt term. She mde this vrition through contrst, tht is, the x 2 -coefficient equls 1 or is different from 1 nd the constnt term ppers on the sme side s the unknown quntity or on both sides of the eqution. Mri generlised the importnce of the x 2 -coefficient when she for exmple solved the eqution 2x 3x 2 = 1. [1.12] Mri: Wht do you sy bout this then? Is it redy for the formul? Is it written in the form x 2 + px + q = 0? [2.12] Sune: No [3.12] Mri: No, it s not. Firstly, this is the x 2 term. It should be positive (points t x 2 in x 2 + px + q = 0) nd it's not. Then I ll begin by mking it positive nd I move it over (points t 3x 2 nd the right side) nd it becomes 3x 2, nd then it shll lwys be gthered on one side in order to mke it equl to zero [ ] [6.12] Mri: Is it redy for the formul now? [7.12] Josefin: No [8.12] Mri: No, it s not, for there cn t be 3 in front of x 2, only x 2 is llowed, so wht re we going to do with the 3? [9.12] Vivek: Divide. [10.12] Mri: Yes, divide, nd then I divide ll the terms. (Lesson 25, ) Mri rewrites the eqution in order to use the p-q-formul [1.12]. In her rewritings, she simultneously vries it one side from being different from zero to being equl to zero [3.12]. This mens tht she focused on the fct tht the constnt term must be on the sme side s the x 2 - nd x-coefficient. Therefter Mri simultneously vries the x 2 -coefficient from being different from 1 to equlling 1 by dividing the eqution with 3 [8.12]-[9.12]. She ccentuted tht ll terms must be divided with the x 2 -coefficient, which mens tht she lso implicitly focused on the vrition in the eqution s x-coefficient nd constnt term [10.12]. By doing this Mri shows tht the prts tht constitute these equtions cn be relted to ech other in different wys, but tht they despite this hve the sme solutions. In order to solve the lst eqution, Mri uses the p-q-formul by focusing on identifying the x-coefficient nd the constnt term in the eqution with p nd q in the p-q-formul. The dimensions of vrition ppering in Mri s clss nd referring to the eqution s x 2 -coefficient, x-coefficient nd constnt term open up strong reltionship between these nd p nd q in the p-q-formul. To constitute this reltionship, the focus ws on vrying the prts tht constitute the equtions whole, but to llow the structure of the p-q-formul still to remin invrint. This leds to the possibility of generlizing this pttern of solving seconddegree equtions nd the reltions tht pper in between in the use of the p-q-formul nd these equtions. This generliztion mkes it possible for the students to constitute confident mnipultion of different types of second-degree equtions, to lern the mening of these, relte the second-degree equtions coefficients nd constnt terms to the p-q-formul, nd use these reltions for new mnipultions. In Ann s exposé of the sme contents, the p-q-formul ws presented in symbolic form without connection to the students experiences. [37.4] Ann: And now you will lern little formul [ ] 10

11 [49.4] Ann: And so if you look into your formule book, nd you don t hve to, becuse I ll write it up for you, you will see tht there is written something like this: x 2 + px + q = 0 (the techer writes on the blckbord nd tlks out loud). [50.4] Ann: And then, under it, you ll see tht it is written like this (the techer writes on the blckbord: p x = ± 2 2 p 2 q ) (Lesson 26, ) From the trnscript, it is cler tht Ann begins to solve the new equtions, which the students now encounter for the first time, by formulting the p-q-formul. Furthermore, she presents this formul only in symbolic form, tht is, Ann does neither explin wht different symbols stnd for, nor in which wy these symbols relte to given second-degree eqution. Aprt from this she does not connect the introduced formul to the students erlier experiences of rewriting second-degree polynom. This leds to discussions nd commotions mong the students. Therefter the p-q-formul is used in numericl form, but Ann neither focuses on the fct tht p nd q cn be positive s well s negtive numbers, nor on the fct tht one side of the eqution cn be different from zero, which limits the students possibility to discern the eqution s x-coefficient nd constnt term in reltion to p nd q in the formul. The only vrition tht ppers in refers to the x 2 -coefficient tht vries from being equl to 1 to being different from 1. This is presented with the help of trnsprency in the following wy: [29.5] Ann: How do I get this? How do I get this step? The techer points t x 2 10x 2,3 = 0, s the following figure shows: Figure 7. Focused x 2 -coefficient (Ann, Lesson 31). [30.5] Eve: You multiplied by 10. [31.5] Ann: Why? And not just with [32.5] Eve: 10. [33.5] Ann: Why? [34.5] Eve: In order to get x 2. [35.5] Ann: This is very importnt informtion. (Lesson 31, ) From the trnscript, it is cler tht Ann focused on the fct tht n equivlent second-degree eqution cn be obtined through multiplying the eqution with 10, but she does not explin why this is very importnt informtion 1 [35.5]. In other words, Ann does not clerly ccentute tht the p-q-formul cn only be used if the x 2 -coefficient is 1 nd tht this leds to n ltertion in the eqution s x-coefficient nd constnt term. 1 In ddition, we cn notice tht the first nd the second step re not equivlent, tht is, the constnt term, despite the multipliction with 10, still is 2,3 (compre step 1 nd 2 in Figure 7). 11

12 Ann s exposé is chrcterized by not tying together the students erlier experiences with new experiences nd she minly leves the students to red the most importnt explntions in their textbook. In the textbook s exposé dimensions of vrition in the bove-mentioned spects re constituted, but it is ssumed tht the students understnd the messge of the textbooks. Tht this is not obvious for the students cn be seen in the following trnscript: Hnn sks wht she did wrong when she solved the eqution x 2 6x + 5 = 0. Ann looks t wht she wrote. [1] Ann: Hlf the coefficient (points t 6) with opposite sign, nd so it s simpler to do tht, hlf of tht (points t 6) with opposite sign. [2] Hnn: But I did it like it s written in the textbook [3] Ann: OK, then you hve to, sve yourself on frctions qurters (looks t wht the student wrote), nd then you forget tht there should be minus, opposite signs, tht s the only error tht you mke Hnn erses wht she hs written. [4] Hnn: Should the sign be reversed for everything or? [5] Ann: No, for it in the formul is written, it is written you do hve the formul (strts to look for the formul in the textbook). If you look here, it is written plus (points t p in x 2 + px + q = 0) nd minus is written here (points t 6), it mens in relity then, there is plus here (points t p in x 2 + px + q = 0) nd it becomes minus there (points t p/2 in the p-q-formul), if there is minus in front of this (points t p in x 2 + px + q = 0) it becomes plus (points t p/2 in p-q-formul), with opposite signs for everything, opposite, so nd then it will be simpler for you. (Lesson 27, ) If the students understnd the textbook s messge, nd it is not t ll obvious tht they do, it becomes possible for them to develop their bility to generlize the use of the p-q-formul. This mens tht the students in hve hd less possibility to experience pronounced vrition in the spects tht constitute the structure of second-degree equtions nd the reltions between these equtions nd the p-q-formul. Conclusion Two reltions could be identified in the encted nd lived object of lerning, nmely the reltion between the focused spects nd the reltion between the opened vritions. The first reltion refers to the identified spects tht the techers nd the textbook focus on nd the identified spects tht were shown to be criticl in the students lerning. The second reltion refers to the constituted ptterns of vrition in the encted nd lived object of lerning. These different wys in which the students discern the prmeters nd the unknown quntity of equtions re represented in Tble 3. The reltion between focused spects If we look bck t the wy in which the students in the two observed clsses experience second-degree equtions, we cn estblish tht lmost ll students in Mri s clss cn solve these equtions, in comprison with bout hlf of the students in. Furthermore, the mount of students tht solve these equtions if the x 2 -coefficient nd constnt term is written in different wys decreses in both clsses (see Eqution 3 in Tble 1), but gin, there is greter mount of students solving them in Mri s clss thn in. The differences in the students wy of solving second-degree equtions with the help of the p-q-formul minly depend, firstly, on the wy in which they discern the equtions prmeters (coefficients nd constnt term), nd secondly on the wy in which they relte 12

13 these prmeters to the prts of the formul. Furthermore, the wy in which the students discern the prmeters ffect the wy in which the students solve different equtions. This mens tht the students in Mri s clss hve developed greter bility to solve seconddegree equtions thn the students in. Wht is the reson for this? We cn see some explntions to this in the following tble: Tble 3. The encted nd lived object of lerning discerned spects nd ptterns of solving second-degree equtions. Second-degree equtions x 2 +px + q = 0 x 2 + bx + c = 0 ( 0, 1) x 2 + bx + c = d ( 0, 1) Aspects (encted object of lerning) Discerned Not discerned Discerned the unknown Mri s clss Mri s clss Mri s clss Mri s clss Mri s clss Mri s clss p > 0 q > 0 p < 0 q < 0 c nd d Mri s clss Mri s clss Ptterns of solving equtions (lived object of lerning) p-q-formul Discern the two solutions correctly (Mri s nd ) p-q-formul Discern the two solutions correctly (Mri s nd ) p-q-formul Discern the two solutions correctly (Mri s nd ) p-q-formul Discern the two solutions correctly (Mri s nd ) p-q-formul Discern the two solutions correctly (Mri s nd ) p-q-formul Discern the two solutions correctly (Mri s nd ) p-q-formul Discern the two solutions correctly (Mri s nd ) Did not discern the unknown Applied the null product lw wrong () Discern the two solutions wrong () Discern the two solutions wrong () Discern the two solutions wrong () Discern the two solutions wrong () Discern the two solutions wrong () Discern the two solutions wrong (Mri s nd Ann s clss) 13

14 In Mri s nd the textbook s exposé of the eqution x 2 + px + q = 0, the students hd the possibility to discern prmeters s generl numbers. The prmeters tht constitute this form of eqution, tht is, the x-coefficient nd constnt term, re generlised with the strting point in solving specific cses of equtions (see Equtions 1-4 in Tble 1). Therefter the number of prmeters with x 2 -coefficient increses. In this process the prts, nmely the prmeters, were discerned by letting the unknown quntity x remin invrint. These prts re discerned by successively vrying the prmeters. In, this systemtic vrition (generliztion) is not experienced. Ann presents only certin cses of second-degree equtions nd her students only hve the possibility to generlize these with the help of the textbook s exposé. The students lso hd the possibility to experience prmeters s vribles in different wys. The prmeters re exposed s vribles in Mri s nd the textbook s exposé through the fct tht tht the prmeters p nd q in n eqution were both positive nd negtive numbers. This vrition mkes it possible for the students to discern the prts tht constitute n eqution of the second degree nd relte them to the prts tht constitute the p-q-formul. In, these vritions re not experienced. Aprt from the focused spects mentioned, it ws lso estblished tht the techers prefer to develop the students bility to solve second-degree equtions, tht is, to find out the unknown quntity x, by using the p-q-formul. In this formul the prmeters p nd q pper s given numbers. These spects re experienced in both clsses nd in the textbooks. There re students in both clsses (see Tble 1) tht in order to decide these numbers simultneously discern the prmeters s generlized numbers (in the p-q-formul) nd s given numbers (in n eqution) nd therefter relte them to ech other. In other words the students simultneously discern the prts tht form n eqution of the second degree nd the prts in the p-q-formul. They lso discern the reltions between the structure of the equtions nd the formul. If there re students tht do not develop this bility, these students understnd the p-q-formul wrongly (see Tble 1). The second spect tht is focused on in the encted object of lerning is the unknown quntity. This spect refers to the fct tht the unknown quntity x cn pper in n eqution in explicit (see Equtions 1, 2 nd 4-8 in Tble 2) or implicit form (see Eqution 3 in Tble 2). This mens tht the unknown quntity cn be obtined either by using the p-q-formul or by reducing n eqution of the second degree to the first-degree equtions. The wy in which the unknown quntity ppers in second-degree equtions leds the techers nd the textbooks to prtly present this quntity s invrint, prtly to give it new mening by reducing second-degree equtions to first-degree equtions. This reduction is mde with the help of different lgebric rewritings tht refer to completing the squre (see Figure 6). Through these rewritings, the x-exponent chnged from being of the second degree to tht being of the first degree. The mening with the reduction of second-degree equtions is presented in Mri s clss nd in the textbook but not in. This leds the students in to elborte other wys of solving second-degree equtions (see Figure 2). This fr, we cn estblish tht there is strong reltionship between the spects tht hve been focused on in the object of lerning nd the wy in which the students discerned these spects. Furthermore, there is strong reltionship between the spects tht re focused on nd the students possibility to develop the bility to solve second-degree equtions. This cn be seen in Mri s clss when she does not focus on the importnce of the constnt term ppering s seprte prts in equtions nd the use of the p-q-formul (see Tble 1 nd Figure 3). This mkes it possible to rrive t n importnt conclusion, nmely tht the techers unwrily used dimensions of vrition in certin spects but tht crucil role in order for the students to develop their lerning is to crete vritions in the spects tht mke 14

15 it possible to understnd the reltions between the prts tht constitute second-degree eqution nd the p-q-formul, nd second-degree equtions nd the p-q-formul s whole. Reltion between the opened vritions The second reltion tht could be identified is clled reltion between the opened vritions nd is chrcterised by the vritions tht re opened up in some spects of the contents in the techers nd textbook s exposé of the object of lerning nd the displyed vritions when the students experience these spects. The techers nd the textbook opened up dimensions of vrition in two different wys. In the first wy, vritions were opened up through different exmples focusing on the sme spect. Therefter they were tied to the object of lerning s whole. This kind of vrition is clled convergent vrition. A convergent vrition mens tht different spects of the object of lerning refer the prts to the reltions between these prts nd therefter crete the whole in the object of lerning. It seems to be this vrition tht leds to positive development in the students lerning. This wy to open up dimensions of vrition cn be observed in Mri s clss nd in the textbook s exposé. In the second wy, the object of lerning is presented s whole nd therefter the prts tht constitute this object re vried without being discerned. This vrition is clled divergent vrition nd could be observed in Ann s clss. The nlysis of the students lerning indictes tht it could be this vrition tht mkes the students focus on finding lterntive solutions nd ides tht re not mthemticlly correct. If the mening of n spect is vried simultneously without first discerning this spect it seems to become problemtic for the students since these spects often refer to other mthemticl objects. These two wys of creting dimensions of vrition re represented in the following figures: x 2 -coefficient x-coefficient generlized prmeters in second-degree equtions constnt term the equtions nd p-q-formul s whole the reltionship between the eqution s prts generlized prmeters in p-q-formul the reltionship between the p-q-formul s prts Figure 8. Focused spects in Mri s clss nd in textbook. 15

16 x 2 -coefficient the equtions nd p-q-formul s whole generlized prmeters in p-q-formul x-coefficient constnt term Figure 9. Focused spects in nd in textbook. The wy in which vritions re opened up in the encted object of lerning seems to hve direct connection to the mening in which the students experience the spects. The reltionship between the spects focused on in the encted nd the lived objects of lerning disply tht severl focused spects in the encted object of lerning led to less vrition in the students experience of them, nd, on the contrry, if less spects re focused on, the students experience this object of lerning in more wys (see Tble 3). This mens tht the focused spects nd the wy in which these spects re focused mkes it possible for the students to discern the mening of the object of lerning or to develop own ides tht often differ from the mthemticl thinking (see Figures 2 nd 3). The wy in which vritions re opened up cn probbly contribute to n explntion of why so mny students think tht mthemtics is difficult. References Bowden, J., & Mrton, F. (1998). The university of lerning: Beyond qulity nd competence in higher eduction. London: Kogn Pge. Filloy, E., & Sutherlnd, R. (1996). Designing curricul for teching nd lerning lgebr. In A. J. Bishop, K. Clements, C. Keitel, J. Kilptrick & C. Lborde (Eds.), Interntionl hndbook of mthemtics eduction (pp ). Dordrecht, The Netherlnds: Kluwer Acdemic Publishers. Gry, R., & Thoms, M. O. J. (2001). Qudrtic eqution representtions nd grphic clcultors: Procedurl nd conceptul interctions. In J. Bobis, B. Perry & M. Mitchelmore (Eds.), Numercy nd beyond (Proceedings of the 24th Conference for the Mthemtics Eduction Reserch Group of Austrlsi, Sydney, pp ). Sydney: MERGA Kiern, C. (1992). The lerning nd teching of school lgebr. In D. A. Grouws (Ed.), Hndbook of reserch on mthemtics teching nd lerning (pp ). New York: Mcmilln. Kiern, C., & Chlouh, L. (1993). Prelgebr: The trnsition from rithmetic to lgebr. In P. S. Wilson (Ed.), Reserch ides for the clssroom: Middle grdes mthemtics (pp ). New York: Mcmilln. Hoch, M., & Dreyfus, T. (2004). Equtions structurl pproch. I M. J. Høines & A. B. Fuglestd (Red.), Proceedings of the 28th conference of the Interntionl Group for the Psychology of Mthemtics Eduction (Vol. 1, pp ). Bergen, Norge: Bergen University College. Högskoleverket. (1999). Räcker kunskpern i mtemtik? Stockholm: Högskoleverket. Lithner, J. (2006). A frmework for nlysing cretive nd imittive mthemticl resoning 16

17 (Deprtment of Mthemtics nd Mthemticl Sttistics, Umeå University, Reserch Reports in Mthemtics Eduction, no. 2). Umeå: Umeå universitet. Löwing, M. (2004). Mtemtikundervisningens konkret gestltning: En studie v kommuniktionen lärre elev och mtemtiklektionens didktisk rmr (Göteborg Studies in Eductionl Sciences 208). Göteborg: Act Universittis Gothoburgensis. Mrton, F., & Booth, S. (1997). Lerning nd wreness. Mhwh, NJ: Erlbum. Mrton, F., Runesson, U., & Tsui, A. B. M. (2004). The spce of lerning. I F. Mrton & A. B. M. Tsui (Red.), Clssroom discourse nd the spce of lerning (pp. 3 40). Mhwh, NJ: Erlbum. Oltenu, C. (2007). "Vd skulle x kunn vr?" Andrgrdsekvtion och ndrgrdsfunktion som objekt för lärnde. (Diss., Doktorsvhndlingr inom den Ntionell Forskrskoln i Pedgogiskt Arbete, 10, doktorsvhndlingr i Pedgogiskt rbete, 19 och Skrifter utgivn vid Högskoln Kristinstd, 10). Umeå: Umeå universitet, fkulteten för lärrutbildning, Ntionell Forskrskoln i Pedgogiskt Arbete. Petterson, R. (2003). Resultt v dignostisk prov i mtemtik för nyntgn teknologer vid civilingenjörslinjern Chlmers, Göteborg: Chlmers teknisk högskol. Pettersson, R. (2005). Resultt v dignostisk prov i mtemtik för nyntgn teknologer vid civilingenjörslinjern Chlmers Göteborg: Chlmers teknisk högskol. Runesson, U. (1999). Vritionens pedgogik: Skild sätt tt behndl ett mtemtiskt innehåll (Göteborg Studies in Eductionl Sciences 129). Göteborg: Act Universittis Gothoburgenesis. Skolverket. (2000). Nturvetenskpsprogrmmet, Gy2000: Progrmmål, kursplner, betygskriterier och kommentrer. Stockholm: Fritzes. Thunberg, H., & Filipsson, L. (2005). Gymnsielärres syn på KTHs introduktionskurs i mtemtik. Viyvutjmi, P. (2004). Correct nswers do not imply understnding: Anlyses of student solutions to qudrtic equtions. In I. P. Cheong, H. S. Dhinds, I. J. Kyeleve & O. Chukwu (Eds.), Globlistion trends in science, mthemtics nd technicl eduction (pp ). Gdong, Brunei Drusslm: University Brunei Drusslm. Viyvutjmi, P. Clements, M. A. (2006). Effects of Clssroom Instruction on Students' Understnding of Qudrtic Equtions. Mthemtics Eduction Reserch Journl, (VOL 18; pp ). Wgner, S., & Prker, S. (1993). Advncing lgebr. In P. S. Wilson (Eds.), Reserch ides for the clssroom: High school mthemtics (pp ). New York: Mcmilln. Wrren, E. (2000). Reserch in teching nd lerning lgebr. In K. Owens & J. Mousley (Eds.), Reserch in mthemtics eduction in Austrlsi (pp ). Sydney: Mthemtics Eduction Reserch Group of Austrlsi. Wrren, E., & Pierce, R. (2004). Lerning nd teching lgebr. In B. Perry, G. Anthony & C. Diezmnn (Eds.), Reserch in mthemtics eduction in Austrlsi (pp ). Flxton, Austrli: Mthemtics Eduction Reserch Group of Austrlsi. Zslvsky, O. (1997). Conceptul obstcles in the lerning of qudrtic functions. Focus on lerning problems in mthemtics, 19(1),