In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of .... Sociomathematical norms: Looking at the way the teacher poses questions, provides.
ON THE MATHEMATICAL KNOWLEDGE UNDER CONSTRUCTION IN THE CLASSROOM: A COMPARATIVE STUDY M. Kaldrimidou1, H. Sakonidis2 and M. Tzekaki3 1
University of Ioannina / 2Democritus University of Thrace / 3 Aristotle University of Thessaloniki, Greece
The present article reports on an attempt to identify the epistemological status of the mathematical knowledge interactively shaped in the classroom. To this purpose, three theoretical approaches are utilized in order to comparatively analyze a lesson provided by a well-experienced teacher on algebra, aiming at identifying the epistemological status of the knowledge under construction through the lenses offered by them. The results show that this parallel and complementary exploitation is especially valid for deepening our understanding of the mathematical knowledge under construction in the classroom. INTRODUCTION School mathematics and experts’ mathematics are two epistemologically distinct bodies of knowledge as they differ in form, context and use from one another (e.g., Sfard, 1998). However, the former draws from the latter, thus preserving certain connections with it, which are though rather blurred. As a result, one could hardly justify why a meaning, an activity or an outcome emerging in the school context can be characterized as ‘mathematical’. The research dealing with this issue is very limited and mathematics education does not still appear to have detailed criteria of whether what is personally or socially constructed in the classroom is or is not mathematical. The study of teaching and learning phenomena and, in particular, the study of the interaction in the mathematics classroom remotely focuses on the nature of the mathematical knowledge shaped in it, which is greatly determined by this interaction (Steinbring, 1998). In searching for criteria to analyze didactical phenomena within the perspective of the nature of the knowledge constructed in the classroom, the requirement of the underpinning fundamental and operational characteristics of mathematics, namely of its epistemological features, seems absolutely essential (e.g., Rouchier & Steinbring, 1989). To this direction, the focus of the present work is on the nature of the meaning emerging in the classroom characterized as ‘mathematics’ in connection with the classroom phenomena which determine this construction. In particular, in an attempt to identify the nature of the mathematical knowledge interactively constructed in the classroom contexts, we utilize the analytical tools offered by three relevant theoretical approaches, i.e., of socio-mathematical norms (Yackel & Cobb, 1996), of the epistemological triangle (Steinbring, 2005) and of the management of the epistemological features (Kaldrimidou, et al, 2000). The comparative reading of the 2007. In Woo, J. H., Lew, H. C., Park, K. S. & Seo, D. Y. (Eds.). Proceedings of the 31st Conference of the International Group for the Psychology of Mathematics Education, Vol. 3, pp. 89-96. Seoul: PME.
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Kaldrimidou, Sakonidis & Tzekaki same lessons through the lenses offered by these three approaches allows us to sharpen the analysis related to this nature. THE THEORETICAL APPROACHES Our purpose was to exploit the possibilities offered by each of the three approaches in order to identify the particular epistemological features of the subject matter knowledge they claim that is shaped in the classroom, as a consequence of the personal, social and epistemological constraints present. These approaches are briefly described below. a. Sociomathematical Norms: The notion of sociomathematical norms was conceived in order to analyze the mathematical aspects of teachers’ and students’ activity in the mathematics classroom (Cobb & Yackel, 1996). These norms are collective criteria of values with respect to mathematical activities, which are interactively constituted (Voigt, 1995), not predetermined, but continually regenerated and modified by the interactions taking place between the teacher and the pupils. The sociomathematical norms are established in all types of classrooms and they are context dependent. The most common sociomathematical norms reported in the literature are related to explanations, justifications and solutions. With respect to explanations and justifications, the main sociomathematical norm detected is related to ‘what counts as an acceptable mathematical explanation’ (Yackel & Cobb, 1996). Concerning solutions, the relevant sociomathematical norms refer to ‘what is valued mathematically; what a more sophisticated solution is; what is mathematically efficient and/or different’ (Yackel & Cobb, 1996). b. The epistemological triangle: Steinbring (1998), adopting the view that knowledge is represented by a specific way of constructing relations (Rouchier & Steinbring, 1989), advocates that the epistemological status of what is interactively constructed by the students in the classroom can be identified through a relational structure called ‘the epistemological triangle’. In particular, he argues that in the course of classroom interaction, students have to actively construct relationships between signs/symbols and reference contexts. This construction becomes ‘official’ in negotiations with the teacher and the other students. As a result, the analysis of the classroom production of mathematical meaning from an epistemological point of view needs to take into account the relationship between two interrelated dimensions: (a) the construction of meaningful relations for sign systems is regulated by the reference contexts exploited and b) the meaning construction processes are embedded and at the same time interfere with the social conditions at work in the instruction process. In the course of the interaction between the sign system and the reference context, the role of which can be exchanged, the production of mathematical meaning can be seen
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Kaldrimidou, Sakonidis & Tzekaki as a process of meaning transition from a rather familiar situation (the reference context) to a still unfamiliar sign system. c. Classroom management of the epistemological features of mathematics: Mathematics science functions with concepts, which are theoretical objects, with definitions as means of recognition and differentiation of objects, with theorems as means of presentation of attributes and relations and follows certain processes as means of management of objects, relationships and results. These aspects are not easily developed in students’ minds, but doing mathematics or acquiring a mathematical culture is unavoidably connected with functioning with the same "means" as mathematics does (Brousseau, 2006). Relying on the above, we claim that in order to identify the nature of the mathematical knowledge constructed in the classroom, we need to analyze classroom interaction on the basis of the management of these specific epistemological characteristics of mathematics by both teachers and students. Obviously, these elements are not always explicitly identified by the students. However, the teacher needs to control and handle them in ways that support students’ understanding with respect to the nature, the meaning and the role of these features in the mathematical activity. We argue that this aspect constitutes an important dimension of the teaching/learning process if students are to learn how to work mathematically. Hence, it is of great importance to look at how the teacher and the students deal with a concept, a definition or a theorem, how they function in solving, proving or validating procedures and, in general, if and in what degree these important characteristics of the scientific activity are valued in the classroom. To this purpose, there is a need to focus on each discursive contribution made by both teachers and pupils in the course of classroom interaction, examining the characteristics (a) assigned to it from a scientific mathematics point of view and (b) attributed to it in the context of the specific interaction. Collating these two aspects, we can identify congruencies or misrepresentations existing between the contribution made and the mathematical meaning or function underpinned, thus deepening our understanding about the nature of the knowledge shaped in the classroom. THE STUDY In order to study what emerges interactively in the everyday classroom as mathematical knowledge, we exploited the analytical tools suggested by the above three approaches. Our intention was to provide a comparative reading of the status of the mathematical knowledge under construction in the context of the interaction taking place in the classroom, through the lenses offered by these approaches. For the purposes of the present study, a videotaped and transcribed ‘regular’ lesson in algebra taught by a teacher with a university degree in Mathematics and more than fifteen years of teaching experience is exploited. The class consisted of 21 students of 15 years old pupils (third year of a gymnasium located in the northern part of Greece). PME31―2007
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Kaldrimidou, Sakonidis & Tzekaki The lesson is focused on solving quadratic equations, but the teacher begins by reminding to the students what an algebraic fraction is, a topic that they had discussed in the previous lesson (with linear expressions as denominators), with the intention of moving to algebraic fractions with quadratic expressions as denominators. Analyzing the data in the light of the above three perspectives, we followed an interpretive approach. Specifically, we focused on the classroom interaction, trying to identify episodes which could be discussed simultaneously from the point of view of the three theoretical perspectives. We then considered the nature of the knowledge emerging, claimed to be ‘mathematical’, by resorting to the epistemological features of the knowledge shaped. DATA ANALYSIS AND DISCUSSION The analysis that follows concentrates first on the notion of the sociomathematical norms, then on that of the epistemological triangle and, finally, on the management of the epistemological features of mathematics. Sociomathematical norms: Looking at the way the teacher poses questions, provides explanations or justifications and promotes ‘better’ solutions, it can be argued at first that the sociomathematical norms established in the classroom are mainly guided by her. While she proposes to the students to take initiatives and formulate their own ideas, she immediately corrects or rejects their contributions or provides the correct answer (e.g., lines 82-84 & 98). Her main concern ‘to avoid errors’ (“don’t lose any root”, lines 94 & 98) leads her to emphasizing procedural and morphological elements in her explanations (e.g. lines 86 and 94) and to suggesting approaches, even contradictory, to ensure ‘correct’ solutions (e.g., lines 71-72 and lines 81- 82). Thus, the fundamental norms about what is mathematical dominating in the classroom are either of descriptive character or concern procedures; explanations on objects are avoided. For example, in lines 71-72, the student proposes a procedure, but the teacher rejects it as ‘unsafe’. No exploration of the context within which the procedure could be utilized is carried out. Epistemological triangle: Analyzing the lesson from the perspective of the epistemological triangle, that is, in terms of the relationship between reference context and sing system, a change of the former is noticed through the lesson: from rational algebraic fractions (introductory part of the lesson) to rational numbers (lines 61- 62), then to operations with whole numbers (line 63) and finally to solving quadratic equations that can be factorized (the rest of the lesson). Similarly, the sign system exploited changes in the course of the lesson development, without this becoming clear. Thus, in some parts of the teaching the focus is on the left hand side of the equation (algebraic expressions), while in others on substituting values for x to find out whether the equation is true. It is apparent that the above 3-92
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Kaldrimidou, Sakonidis & Tzekaki changes make the relation between reference contexts and sign systems rather problematic. Furthermore, it should be noted that each time the teacher herself (line 84) or a student (lines 91, 95) make a contribution, which allows for the discussion to focus on relations of general or theoretical objects, the teacher avoids taking the opportunity of doing so by either rejecting the chance or by providing explanations and justifications of mainly procedural or morphological type (e.g., lines 83-86, 94, 96, 98). Management of epistemological features: Within this perspective, a dominance of procedural and practical directions as well the lack of complete justifications based on the nature and the attributes of mathematical objects can be identified. This results in different mathematical objects appearing in a homogeneous manner. For example, in lines 61- 65, three different mathematical objects (fractions, division, algebraic fractions) are implicitly connected to equations (looking for denominators ≠0). These objects are mainly presented in a morphological manner and without any connection to definitions or properties, which could support students’ identification of the new object under consideration (equation). This interplay between different mathematical objects, not clearly defined and vaguely interconnected cannot but lead to the distortion of the mathematical meaning of quadratic equations. Similarly, in lines 84–86, the incomplete reasoning utilized limits students’ thinking concerning the solution of quadratic equations and two different mathematical concepts are treated as one (in Pythagoras Theorem, the equations represented relations between lengths of line segments and not only line segments; thus, only the positive solutions had meaning). Moreover, the use of ‘rules’ (“here it needs ±√4”, line 86) instead of an argumentation based on properties results in the outcomes to appear as the result of statements. So, rules, properties and statements are treated in a homogeneous way, without any differentiation as for their nature or role. Finally, the emphasis on descriptive elements in various parts of the lesson results in the downgrading of the meaning of concepts (e.g., lines 92-94, “algebraic are the numbers which have + and –”). This, together with the dominating undifferentiated use of rules, properties and procedures prevents the attributes of the mathematical objects to function as frameworks for dealing with mathematical objects, as well as with mathematical relations. This is apparent in lines 97-98, where the student’s suggestion is rejected and not discussed on the basis of an argument which is based on the results of the strategy (“you lose a root”) and not with reference to the restrictions and the ways in which an algebraic expression can be simplified. DISCUSSION AND CONCLUSION It is widely accepted today that students’ learning of mathematics is greatly shaped by the meanings constructed through negotiations in the classroom. Thus, a systematic analysis of the interaction taking place within the mathematics classroom in relation to the mathematical meaning under construction is of particular interest. The three PME31―2007
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Kaldrimidou, Sakonidis & Tzekaki analyses presented above offer a different way of looking at this issue, highlighting different aspects of it. The first approach focuses on the processes adopted in the classroom, which shape what is appointed as ‘mathematical’ within it. On the basis of this approach, we can argue that, in the classroom under consideration, the rules are placed by the teacher, who accepts or rejects students’ contributions. The relevant criteria, often contradictory, remain ambiguous (e.g., the teacher first rejects and then accepts the student’s proposal “to separate known from unknown terms” with no justification, lines 71-82). As far as what ‘counts as mathematical’ in this classroom is concerned, it seems to be overtly determined by the teacher and we can only implicitly talk about its nature and whether it is or not mathematical. Steinbring’s approach allows us for an epistemological analysis of the mathematical knowledge interactively constructed in the classroom with reference to the nature and the character of the different objects involved in this interplay (whether or not this knowledge is relational and context-free). Using this approach to read the transcript of the lesson at hand, we can claim that the teacher pursues to arrive at a general idea (the solution of quadratic equations) via specific reference contexts. However, the way these contexts are handled as well as the different resolutions suggested (factorizing/separating terms) do not allow for a relational view of this idea to be developed. Finally, the third perspective explicitly focuses on the status that the knowledge shaped in the classroom acquires through the particular way it is managed, offering a lens to deciding whether what is developed in the classroom bears mathematical characteristics or involves students in genuine mathematical activity. On the basis of this analysis, we identified in the classroom under consideration the same homogeneous way in which mathematical objects, relations or procedures are treated in many other classrooms, as shown in earlier studies (Kaldrimidou, et al, 2000). This undifferentiated presentation of the various distinct objects, which are engaged in the interaction, as well as of their characteristics does not elevate properties and relations in a manner that would facilitate the management of new objects or relationships. Thus, in this particular case, the teacher places emphasis on morphological aspects or on earlier procedures, which are often used in different and even undefined ways in the new situation. This manner of dealing with mathematical objects and their properties distorts their nature and role, possibly leading students to difficulties in approaching the substance of the mathematical activity. The points raised above suggest that the parallel exploitation of the three approaches is especially valid. The first highlights the way in which the mathematical features are shaped in the classroom, the second focuses on whether the knowledge emerging is general or context-specific, while the third allows for the identification of the nature, the status and the function of the various ingredients of the mathematics shaped in the 3-94
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Kaldrimidou, Sakonidis & Tzekaki classroom. Moreover, the preceding analysis highlights the need to look closer at the particular epistemological features of the mathematical knowledge under construction in the classroom. The complexity of the didactical phenomena framing this construction imposes the need for a multiple approach to analyzing it, which will carefully incorporate the issues raised by all three perspectives. APPENDIX 61. T. Watch it, children. In order for the fraction to have meaning, what should the denominator always be? 62. Vasos. Different from zero. 63. T. That’s it! Because the division by zero is what? 64. Students. Impossible! 65. T. That is, before you simplify, you should place the denominator ≠ from 0... 71. George. To separate known terms from unknown terms. 72. T. You suggest we should separate. It cannot be done because both terms are unknown. Anyone else? …………… 81. Argyro. The 4 will not be moved to the other side and … 82. T. That’s right. Then, what do we have from here? x2=4. Let me hear now. What am I to write? x…, I am listening to the rest of you. What am I to write? 83. Argyro. x equals square root of 4. 84. T. Bravo! Be careful children! This is what we were saying up to last year. Because, when we learnt how to solve this type of equations, x was a line segment. We saw this in Pythagoras Theorem, do you remember? And line segments are always…? 85. Students. Positive 86. T. Positive! What did we put then? Simply square root of 4. Here needs ±√4. Then, what can be concluded from here? x=±2. Because x takes both values, -2 and +2. If you substitute x=+2 in the initial equation, is the equation true? 87. Students. Yes 88. T. It is confirmed. Thus, x=+2 is a solution. However, if we substitute x=-2, is the equation also true? 89. Students. Yes. 90. T. It is again true. That is, we should not lose solutions. We should write ±√4. 91. Kostas. When we say x2=4, isn’t it x2=22? Thus, since the two squares are equal, should their bases be also equal? 92. T. Well, look. You will learn in Lyceum that if av is equal to bv, then we can say that a=b only if a and b are positive numbers. Our problem is different. We need to solve an equation. And what do we notice in this equation? Both +2 and -2 give us 4. Thus, the equation is true for both these values. So, we should not lose -2. From now onwards, we should always write it this way. Last year, in geometry, PME31―2007
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93. 94.
95. 96. 97. 98.
we wrote √4. And what does ‘algebra’ mean? What numbers does algebra deal with? Algebraic numbers. And which numbers are algebraic? Theodora. The number which have plus and minus. T. That’s it! The ones which have plus and minus. We found this example last year, when working on Pythagoras Theorem, on line segments. There, it was not necessary to put both signs. It is here. Because we ended up to a square root. When we factorize the left hand side, it becomes clear which solution is which. Whereas, when we use this way, it is not clear. So, be careful! Don’t be carried away and lose a solution. That is, the negative root. Kostas? Kostas. In x2-2, if we write x.x = 2x? The x is cancelled and then x=2 T. Be careful! Which x’s is going? Priority of operations... We first multiply… Kostas. Madam, we will do x2=2x … x.x = 2x… T. But you have a root! It is forbidden! Ok? You lose a root. Don’t do this kind of cancellations, because you lose roots. All right? However, when we take out the common factor, we don’t lose the root.
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