Mathematical Knowledge for Teaching and Didactical Bifurcation: analysis of a class episode

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1 Mathematical Knowledge for Teaching and Didactical Bifurcation: analysis of a class episode Stéphane Clivaz HEP Lausanne Equipe DIMAGE, University of Geneva Switzerland Context This text takes its place in an ongoing doctoral research, under the direction of Jean-Luc Dorier (University of Geneva), planned to be completed in 2011 about Mathematics for Teaching: Analysis of the Influence of Vaud s Teachers Mathematical Knowledge on their Teaching of Mathematics in Primary Schools. The object of this work is to describe the influence of primary school teachers mathematical knowledge on their didactic management of school mathematical tasks. The research questions are: 1. What is the knowledge of elementary mathematics of Vaud s teachers? 2. In mathematics and for primary school teachers, in what aspects of the teacher s work does that teacher s knowledge manifest itself? 3. What are the effects of the pertinence of teachers mathematical knowledge or lack of pertinence on their didactic management of mathematics lessons in primary school? Using semi-directed interviews, partly from Ma (1999), staging classroom situations and putting to use the teacher s mathematical knowledge, the results are compared to those pointed out by Ma for US and Chinese teachers. Using four sets of classroom observations about the teaching of multi-digit multiplication, the kind of Mathematical Knowledge for Teaching 1 (Ball, Thames, & Phelps, 2008) and Mathematical Pertinence of the Teacher (Bloch, 2009) are more finely analyzed through the Levels of Teacher s Activity (Margolinas, Coulange, & Bessot, 2005). This text will summarize the analysis of one class episode by the mean of these Levels. Theoretical frame and methodology Margolinas (Margolinas, et al., 2005) has developed a model of the teacher s milieu 2, based on Brousseau (1997) which she uses, in a somewhat weaker version, as a model of the teacher s activity. 1 Common Content Knowledge, Horizon Knowledge, Specialized Content Knowledge, Knowledge of Content and Teaching, Knowledge of Content and Students, Knowledge of Content and Curriculum. 2 Milieu is the usual translation for Brousseau s French term milieu, but, in French, it refers not only to the sociological milieu but it s also used in biology or in Piaget s work. A more accurate translation would be environment. 1

2 +3 Values and conceptions about learning and teaching - Educational project: educational values, conceptions of learning, conceptions of teaching +2 The global didactic project - The global didactic project, of which the planned sequence of lessons is a part: notions to study and knowledge to acquire +1 The local didactic project - The specific didactic project in the planned sequence of lessons: objectives, organization of work 0 Didactic action - Interactions with pupils, decisions during action -1 Observation of pupils' activity - Perception of pupils activity, regulation of pupils work Table 1: Levels of teacher s activity (Margolinas, et al., 2005, p. 207) At every level, the teacher has to deal not only with this level, but also, at least, with the upper component and the lower component. This tension makes the linear interpretation of teacher s work inaccurate (Margolinas, et al., 2005, p. 208). In fact, a more complete model, including the student (E), the teacher (P), the milieu (M), made at each level i by the lower E i-1, P i-1 and M i-1 component and the situation S i, made at each level i by E i, P i and M i. That can be written as S i = (M i ; E i ; P i ) and M i = S i-1, or more visually represented in an onion diagram (Figure 1) or more practically in a table (Table 2) where the teacher s levels range from +3 to -1 and the student s from -3 to +1. Figure 1: Milieu s structuring, level -2 to +2 M +3 : M-Construction P +3 : P-Ideological S +3 : Ideological situation M +2 : M-Project P +2 : P-Constructor S +2 : Construction situation M +1 : M-Didactical E +1 : E-Reflexive P +1 : P-Projector S +1 : Project situation M 0 : M-Learning E 0 : Student P 0 : Teacher S 0 : Didactical situation M -1 : M-Reference E -1 : E-Learner P -1 : P-Observer S -1 : Learning situation M -2 : M-Objective E -2 : E-Acting S -2 : Reference situation M -3 : M-Material E -3 : E-Objective S -3 : Objective situation Table 2: Milieu s structuring (Margolinas, 2002, p. 145), translation from Margolinas (2000), my translation for the italicized terms. 2

3 Therefore, S 0, the didactical situation, can be determined either from the teacher s point of view, by a downward analysis, or from the student s perspective, by an upward analysis. This latter may conduct to one or more didactical situations S 0 which may not be the same as the situation determined by the downward analysis. Margolinas calls this a didactic bifurcation (2004). The downward analysis uses mainly the audio-recorded interview we had with each teacher before the succession of lessons about the teaching of multidigit multiplication. The upward analysis is a priori in the sense that it doesn t depend on experimental or observational facts 3 (Margolinas, 1994, p. 30). Both are tools to analyze the classroom observations in an a posteriori analysis. The 27 minutes analyzed episode is taken from the series of seven lessons which Dominique, a 4 th grade 4 teacher, devoted to this algorithm in his class. It was transcribed and coded with Transana software (Fassnacht & Woods, ), accordingly to the categories of mathematical knowledge for teaching, the mathematical pertinence and the levels of teacher s activity. This extract can be situated in the series by means of a synopsis (Schneuwly, Dolz, & Ronveaux, 2006) and a macrostructure (Dolz & Toulou, 2008). Preliminary results The a priori (downward and upward) and a posteriori analysis can t be developed here. We will very briefly describe the situations S 0 these a priori analyses reveal and give a hint for the reasons of highlighted troubles, in terms of Mathematical Knowledge for Teaching (MKT) and mathematical pertinence. Dominique thinks that students must understand the algorithm, particularly the zero in the second row. So he decided to show previously a way to multiply by two-digit number in a table. This way of multiplying had been done in the previous year for multiplication by one-digit numbers (Figure 2, left). He planned then to show the usual way en colonnes 5 (EC), showing the parallelism between the result in the first table s row and the first EC row, writing the zero because when one works with tens, a zero must be added 6 (zero rule), computing the second row and showing that the result was the same as the table s, and then adding. This is the didactical situation from the teacher s perspective. Figure 2: Two algorithms for the multiplication 12x17, written by Dominique on a poster. The elaborate upward analysis, starting from M -3 material milieu, shows that the student can deal with M -3 in different ways about the parallelism between the table and the EC: one row with one row (same correspondence as teacher), partial product to partial product, just copy the results or do the two algorithms independently. He/she can also apply the zero rule in three ways: 3 «dans le sens qu elle ne dépend pas des faits d expérience ou d observation», my translation years old student 5 Literally in columns but the accurate English name would be long multiplication 6 «Quand on travaille avec les dizaines, on ajoute un zéro» 3

4 he/she can write a zero before beginning the second row with no further interrogation, he/she can link this zero to each zero in the table s second row, and he/she can apply literally the teacher s explanation, adding a zero each time he/she works with a ten. The combination of these two dimensions conducts to twelve situations S -2, from which six seem to be consistent and lead to six S 0 didactical situations, none of them being the teacher s one! The 27 minutes episode shows that most of the students considered the two algorithms independently or in a row-by-row correspondence, and wrote the zero without interrogating. These students are in two of the S 0 situations, the upward analysis determined. However, one student, EAr, repeatedly asked questions about why the teacher didn t add a zero each time he used tens: he is in another S 0 situation. Dominique, on the other hand, kept to his S 0 situation and was not able to even understand EAr s interrogations. The proliferation of didactical bifurcations, and the inability of the teacher to notice that the students S 0 radically differs from his, have their origins in teacher s choices made at +3 to 0 levels. And these choices may be understood as a consequence of teacher s MKT. The first choice was to use the table algorithm and particularly the correspondence between the rows sum in the table and in the EC s rows, but with no explicit correspondence between each partial product. Dominique s MKT about the table are accurate, as is revealed in the interview, but they are not pertinent, since they don t allow him to interact with the students on the mathematical parts of the situation (pertinence s first criteria according to Bloch (2009, p. 33) ). The reason for this discrepancy between knowledge and pertinence is the knowledge of multiplication itself. For Dominique, multiplication is only a shortcut for an iterated addition. He never considers it as a Cartesian product or as the area of a rectangle. Therefore EC and table algorithm are two ways to perform a multiplication, they are not linked to the multiplication itself and they are just linked to each other because they give the same result. The second choice is the recipe for zero rule. This rule is problematic in many ways: use of additive words (add a zero), lack of link with place value and above all, fallacy if literally applied. Dominique has a working Common Content Knowledge, but he can t unpack it and can t use the corresponding Specialized Content Knowledge. Conclusion Milieu s structuring allowed an elaborate description of a class episode and an underlining of diverging didactical situations for the teacher and for the students. The categories of MKT and the mathematical pertinence lead to a link between the mathematical knowledge of the teacher and his didactic management leading to this divergence. Other ways to bring these links to light will be found in other parts of our dissertation. 4

5 References Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), Bloch, I. (2009). Les interactions mathématiques entre professeurs et élèves. Comment travailler leur pertinence en formation? Petit x, 81, Brousseau, G. (1997). Theory of didactical situations in mathematics: didactique des mathématiques, : Kluwer Academic Publishers. Dolz, J., & Toulou, S. (2008). De la macrostructure de la séquence d enseignement du texte d opinion à l analyse des interactions didactiques. Travail et formation en éducation, (1). Retrieved from Fassnacht, C., & Woods, D. K. ( ). Transana (Version 2.41) [Mac]. Madison: University of Wisconsin. Ma, L. (1999). Knowing and Teaching elementary Mathematics : Teachers' Understanding of fundamental Mathematics in China and the United States. Mahwah, NJ: Lawrence Erlbaum Associates. Margolinas, C. (1994). Jeux de l'élève et du professeur dans une situation complexe Séminaire de didactique et technologies cognitives en mathématiques (pp ). Grenoble: LSDD- IMAG. Margolinas, C. (2000). Modelling the Teacher s Situation in the Classroom. Paper presented at the Proceedings of the ninth international congress on mathematical education. Margolinas, C. (2002). Situations milieux, connaissances: Analyse de l'activité du professeur. In J.- L. Dorier, M. Artaud, M. Artigue, R. Berthelot & R. Floris (Eds.), Actes de la 11e école d'été de didactique des mathématiques (pp ). Grenoble: La pensée sauvage. Margolinas, C. (2004). Points de vue de l'élève et du professeur. Essai de développement de la théorie des situations didactiques. Unpublished HDR, Université de Provence - Aix- Marseille I. Margolinas, C., Coulange, L., & Bessot, A. (2005). What Can the Teacher Learn in the Classroom? Educational Studies in Mathematics, 59, Schneuwly, B., Dolz, J., & Ronveaux, C. (2006). Le synopsis: un outil pour analyser les objets enseignés. In M.-J. Perrin-Glorian & Y. Reuter (Eds.), Les méthodes de recherche en didactiques : actes du premier séminaire international sur les méthodes de recherches en didactiques de juin 2005 (pp ). Villeneuve d'ascq: Presses univ. du Septentrion. 5

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