Lecture notes on the Theory of Didactic Situations in mathematics

Lecture notes on the Theory of Didactic Situations in mathematics by Anna Sierpinska Concordia University, Montreal, Canada prepared for the course MATH 645 Topics in Mathematics Education Research given in year 2003 in the frame of the program Master in the Teaching of Mathematics at Concordia University

© 2003 Anna Sierpinska

MATH 645: Theory of Situations/ Lecture 1

Instructor: Anna Sierpinska

LECTURE 1 INTRODUCTION TO THE CONCEPT OF DIDACTIC SITUATION AND THE DISTINCTION BETWEEN SITUATIONS OF ACTION, FORMULATION AND VALIDATION

Note: All references to page numbers in brackets refer to the textbook. Other references are given in the footnotes. 1. WHAT IS A DIDACTIC SITUATION? The notion of didactic situation is grounded in certain assumptions about learning and teaching. These assumptions can be formulated using the metaphor of game (p. 40). * the teacher is a player faced with a system composed of a student and a didactic milieu * the student is himself a player in a game of him/herself with a didactic milieu * in the student’s game with the didactic milieu, knowledge is the means of understanding the ground rules and strategies and later, the means of elaborating winning strategies * the teacher’s aim is to engage the student in such a game; aiming at a particular mathematical knowledge, the teacher will try to set the student-milieu system so that, indeed, this knowledge would appear as the best means available for the understanding of the rules of the game and elaborating the winning strategy

What is a milieu? ‘Milieu’ should perhaps be understood in an ecological sense, as in ‘water is the natural milieu of fish’. Thus the ‘didactic milieu is the natural milieu of students’. A person, a human being, normally lives in several different milieus and plays different roles in them. In a family milieu one can be a child, a mother, a father, etc. In a sports milieu, one can be the player, the coach, etc. Other possible roles could be played in a workplace milieu, social milieu, etc. In the school milieu, one can be a student, a teacher or an administrator. In each course, the student has to cope with a specific milieu, and there are even more specific milieus for each class in a course. To ‘survive’ (‘to win’) in a milieu one has to get to know the ‘rules of the game’ and develop strategies for winning the game.

Difference between this approach to teaching and learning and the traditional point of view Learning is not reduced to the result of a transmission of information from teacher to students. Learning is understood more as sense making of situations in a milieu, and developing ways of 1



coping with them. Teaching of a knowledge K consists in organizing the didactic milieu in such a way that knowledge K becomes necessary for the student to survive in it. If the situations in a mathematics class are such that a certain type of social behavior is sufficient for survival in them, without any use of mathematical knowledge, then it is the social behavior, not the mathematical knowledge that the students will learn. If the teacher solves the problems for the students and only asks them to reproduce the solutions, they will learn how to reproduce teacher’s solutions, not how to solve problems. In this sense, the kind of game the student has to play with the milieu, to survive in it, determines the kind of knowledge that he or she will acquire. Thus, in the theory of situations, ‘knowledge is [understood as] the outcome of the interactions between the student and a specific milieu organized by the teacher in the framework of a didactic situation’ (Balacheff, 19931, p. 133).

The context of the didactic situation A didactic situation (DS) does not exist in a void. The persons who are the main actors in the DS, playing the roles of the teacher and the student, may look at the DS from the outside, not as actors but those who plan an action in view of some far away goals. The student will see the DS as a means to reach personal life goals (e.g. becoming an engineer, obtaining a high school diploma in order to get a better paid job, etc.). The teacher will look at the DS as an educational designer, or even as a researcher, and will design the DS in view of certain professional objectives, e.g. curricular objectives, assessment objectives, or research objectives. Each person takes the existence and characteristics of the other into account in the planning. In the diagram on page 248, the situation in which the persons who are only planning to act the roles of the teacher and the student (P1, S1) is called a ‘meta-didactic situation’. When these persons act as teacher and student (P2, S2), and interact about the learning situation of the student (e.g. the teacher gives a problem to the student and the student inquires if he or she understands well the conditions of the problem) then they are in a ‘didactic situation’. When the teacher withdraws from the scene, and the student engages in solving the problem for the sake of learning something, she is in a learning situation (S3). As the student endorses the problem as her own, she acts as a problem-solver (S4). In the problem, there may be a real or an imagined story with a material milieu with which some persons have to deal (S5) (e.g. persons purchasing some

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Balacheff, N. (1993): Artificial Intelligence and Real Teaching. In C. Keitel and K. Ruthven (Eds.), Learning from

Computers: Mathematics Education and Technology. Berlin: Springer-Verlag (pp. 131-158)

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goods and wanting to get a good deal). Sometimes the problem-solver identifies herself with these persons and solves their problem. 2. SITUATIONS OF ACTION, FORMULATION, VALIDATION AND INSTITUTIONALIZATION There are different types of didactic situations, depending on the kind of ‘game’ that the teacher plays with the student-milieu system (p. 161-2). For the description of the types of situations, I decided to go from situations in which the teacher is the most authoritarian figure (the most common traditional classroom situations) to situations in which there are almost no teacher interventions. Situation of institutionalization. The teacher plays the role of a representative of the official curriculum, the official mathematics as represented by the school institution, the textbooks officially approved by the ministry, and the official culture. He informs the students about the officially accepted terminology, definitions, theorems considered important from the institution’s points of view. For the students, the milieu thus obtains the explicit features of an institution, with clear assumptions and rules. Knowledge acquires the features of a law, rather than of an answer to scientific inquiry: it is validated and justified through the authority of the institution rather than through criteria such as internal, logical consistency and relevance for the solution of scientific or technological problems. Situation of validation. The teacher takes on the role of the theoretician evaluating the productions of other theoreticians, whose role, in the classroom, is played by the students. The students try to explain some phenomenon, or to verify a theoretical conjecture. The teacher acts as the chair of a scientific debate: s/he intervenes only to put some order in the debate among students, draw their attention to possible inconsistencies, and encourage them to be more precise and systematic in the use of concepts. For the students, the milieu resembles that of an academic seminar rather than that of a lecture room. Knowledge has the dynamic features of a theory in the making, not of a finished, institutionalized theory. Situation of formulation. The milieu for the students is developed on the basis of some previously shared experience or activity: The students exchange and compare observations between themselves. They may not have the language to formulate their observations, so their main effort in this situation goes into creating such a language and agree on some common meanings. The teacher chairs the exchanges (in order to avoid chaos) and highlights (repeats louder, writes on the board) some formulations of the students, in case they may not have been heard by other students. Knowledge, in this situation, appears as a result of a personal

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experience, which needs to be communicated, and thus slightly de-personalized and decontextualized, in order to be understood by others. Situation of action. The teacher organizes a milieu for the students to engage with but then completely withdraws from the scene. The milieu for the students is that of a problem so chosen and formulated that (a) the students are willing to adopt it as their own, and are interested in solving it to satisfy their own curiosity or ambition; (b) the students have the means to construct the solution by themselves, either by inventing a new procedure or choosing one among those they know, without, however, the teacher suggesting which one to choose. In this situation, knowledge appears as a means for solving a problem or a class of problems. In many mathematics classrooms, and certainly in most university lecture rooms, the institutionalization situations enjoy an absolute reign. Other kinds of situations do not appear, or they appear in degenerate forms. A degenerate form of a situation of validation is one where the students solve the ‘proof’ problems, e.g. ‘Prove that, in a right angled triangle, if one the acute angles is 45° then the triangle is isoceles’, without the statement having been formulated by the students as their own conjecture, and where the style of making the validating argument is prescribed by the teacher. A degenerate from of a situation of formulation can be one where the teacher asks the students to formulate definitions and theorems and sanctions the formulations with approval or disapproval (‘correct’, ‘incorrect’). A degenerate form of a situation of action could be one in which the teacher gives the students a problem to solve but then constantly gives them hints and suggestions about what to do and which method to use. But scientific knowledge does not grow that way: everything starts with a problem, tentative solutions, communication of the results, their justification, revision of the results in the wake of the criticisms and queries from the scientific community. Thus the ‘natural order’ of the growth of scientific knowledge is from action, through formulation and validation, to institutionalization. But in school we are not inventing new knowledge; we are teaching institutionalized knowledge most of the time. So we think that it is a waste of time to reproduce, in the classroom, the tortuous path of its production, and we find that it is more economical to teach directly the results. But, in actual fact, we do not gain anything, because, by teaching only the results of scientific inquiry, we are teaching not scientific knowledge but law. So we are completely missing the point of our teaching. Of course, we have to be realistic and admit that we cannot afford the time for teaching every single bit of the curriculum in this ‘genetic’ way. Therefore, we have to reflect on the knowledge we aim to teach. What parts of this knowledge can be and which cannot be sacrificed to direct institutionalization? Does it make sense to have the students re-invent decimal notation, 4



long division rules, algebraic notation? Is it possible to create situations in the classroom through which the students, in one term, would have re-invented the differential calculus? Probably not. What is possible, then? This is one of the questions posed by the theory of situations. 3. EXAMPLE OF A SEQUENCE OF CLASSROOM SITUATIONS: THE RACE TO 20 In order to get a better sense of the different types of situations, let us engage in a series of classroom activities that will exemplify these situations. These activities have been experimented by a team of researchers conducted by Brousseau in 1970s and described in a paper published in 1978 (pp. 4-18). The lesson (in grade 5 or 6) was divided into several phases. Phase 1: The teacher introduces a game to the students (5 mn) Teacher: Today we’ll play a game with numbers. It is called ‘A race to 20’. Two players play the game. One player says ‘1’ or ‘2’. The other can add 1 or 2 to the number of the opponent and says the result. He who first says ‘20’ is the winner. Let me play this game with one of you. Anyone wants to volunteer? The teacher starts playing the game with a student on the board. Both she and the student write their numbers on the board. After a few steps she relinquishes her place to another student. The record of the game could be, for example: T/S2

S1

2

3

5

7

8

9

11

13

14

16

17

19

20 Phase 2: The students play the game in pairs (4 rounds, 10 mn) Teacher: Now, sit in pairs and play up to 4 rounds of the game, keeping a record of the game on paper. It is expected that students will find that saying numbers at random is not the best strategy. Some will find that saying ‘17’ is a sure guarantee of winning the game. Phase 3: The students play the game in teams (6-8 rounds, 20 mn).

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Teacher: Now divide in two groups and play the game in teams. For each round, one student should be chosen as a representative of the team to play at the board. The teams can discuss their strategies between rounds. But do not interfere with the representative at the board. The teacher keeps the record of the results of the teams on the board. Phase 4: Game of discovery: Formulation of propositions Teacher: Now, I am inviting each team to formulate the strategies that they think allowed them to win. The other team then verifies the statement. If the statement turns out to be true, the team wins a point. If the statement turns out to be false, the team that proved it false receives 3 points. If the game of discovery grinds to a halt, the teams can return to playing the game. It is expected that the students will discover that playing 2, 5, 8, 11, 14, 17, 20 leads to winning the game, and that they will prove this statement by playing the game from each number on (action proofs). Now, in the case of a class of MTM students, one should expect some more sophisticated proofs. For example, the students might notice that the winning numbers are of the form 20-3k, k=0, 1, 2, 3, 4, 5, 6. This could be noticed right away, or after a phase in which the teacher would propose a change of the rules of the game: adding by steps of not 1 or 2 but of 1 or 2 or 3, and racing to, say, 25. The discovery game could then be generalized: the students would be expected to formulate propositions about the winning numbers in the general case of a race to n and steps 1 through m. The students would be expected to find out and prove that if the race is to n and the steps are 1, 2, ..., m, (m
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4. ANALYSIS OF THE EXAMPLE Let us reflect on the following questions: •

What type of situation does each of the phases represent?

•

What is the knowledge learned in each phase?

•

What is the milieu in each phase?

•

What are the rules of the game between the student and the milieu in each phase? What is the relation of the teacher to these rules?

Phase 1: In this phase we are in a didactic situation but not in a learning situation. The teacher and the students are in the roles T2 and S2, with the teacher explicitly setting up a milieu for the students and explaining to the students the rules of the game with this milieu. The milieu is an actual game - a game with numbers. The students, as students, know that they will be playing that game not just for fun but also to learn some mathematics; they do not know what mathematics they will be learning, but they promise themselves to be able to soon find out. This phase could be classified as part of a situation of institutionalization, which is spread through the whole activity, in periods when the teacher comes back and gives more instructions. The students obey the instructions because they accept the teacher’s authority as a representative of the school institution: ‘I don’t know why I should play that game, but I trust the teacher is doing it for some purpose which is useful for me, so I’ll play it’. When they think that way, the students are in the role of S3, and they see themselves in a learning situation within a didactic situation. Phase 2: In this phase the students are in the role of S4; they are problem-solvers, coping with the game, wanting to win, and they forget, for a moment, that they are students. The milieu, in that phase, is not didactic; it is the milieu of playing games and wanting to win. The students arrive at some intuitions about the winning strategies; for example, they may find out that it is good to say ‘17’ because they won several times by saying that number. This is a situation of action: the teacher is out of the scene, the students have endorsed the problem as their own; it is the situation itself that provides feedback and allows them to keep in control of the validity of their solutions. This way the students start developing some personal, still implicit, knowledge. Phase 3: The students are still in the roles of problem-solvers and the milieu is still that of a game, but the situation gets slowly transformed into a situation of formulation. By having to communicate with other members of the team and thinking out some common strategies, the intuitive personal knowledge gets de-personalized and de-contextualized. There may not be 7



much of justification at that moment, because things are happening too fast, and social rather than mathematical factors may impact on whether a suggestion of a strategy is taken or not in a team. The students may find out more winning numbers, and even describe, in general terms, their pattern (an arithmetic sequence starting from 20 with a difference of -3). Phase 4: The students are now in the roles of S3: looking at their problem solving actions from the outside and judging them, as well as submitting them for judgment by other students. With the teacher only chairing the session, this situation classifies as a situation of validation. The knowledge developed in this phase can be that of argumentation. This argumentation may not necessarily have the features of mathematical proof. The argumentation may be of a social nature, with some students using persuasion to convince the others, rather than some kind of objective and unemotional reasoning. The class ends in discussing the question: What kind of intervention should the teacher make in closing the activity, in a situation of institutionalization? What mathematical terminology and theory would she introduce? What was the mathematical knowledge that was aimed at in the activity?

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LECTURE 2 THEORY OF SITUATIONS AS A MEANS TO OVERCOME THE ‘PROCEDURES VS UNDERSTANDING’ DILEMMA IN MATHEMATICS TEACHING

In the ‘Introduction’ to Brousseau’s book, the descriptions and explanations of the situations of action, formulation and validation are based on the notions of ‘feedback’ and ‘dialectic’, not the metaphor of ‘game’ which I used last week to introduce the notion of ‘didactic situation’. In fact, the metaphor of game appeared in Brousseau’s theory only around 1985-6, while the text of the ‘introduction’ was based on a research done in the early seventies when the theory was still in diapers. It is nevertheless worthwhile trying to understand this older version of the theory because it contains seeds of many later developments, and gives us a better sense of where the theory comes from. I’ll start, therefore, by trying to explain the meaning of the notions of ‘feedback’ and ‘dialectic’ in the context of the theory of situations. This discussion will lead me to showing how the theory could be seen as an emergent of the dialectic between the fatalism of those claiming that nothing but procedures can be realistically taught in mathematics classes and the optimism of those who claim that teaching understanding of mathematics is just a matter of good will of teachers and of good teaching materials. I’ll conclude by stating that one of the main claims of the theory is that ‘mathematical meaning’ - this object of all our didactic desires - is not something given or absolute: it has to be researched, studied. The study of the meanings of the mathematical contents of our teaching is an absolutely fundamental element of research in mathematics education or ‘didactics of mathematics’, which permeates all and every single question in this domain. Questions related to the teaching and learning of mathematics which can be resolved without the study of the mathematical content of this teaching and/or learning are simply not part of ‘didactics of mathematics’; they could be legitimate questions in pedagogy, psychology, sociology, anthropology, communication studies, educational technology, etc. This class will close with an activity in which you will be invited to experience a phase of such ‘study of meaning’. The mathematical concept under scrutiny will be the operation of division.

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PART I: THE NOTIONS OF ‘FEEDBACK’ AND ‘DIALECTIC’ IN THE EARLY VERSIONS OF THE THEORY OF SITUATIONS

1. Feedback The term ‘feedback’ comes from cybernetics, which is defined as the study of communication and manipulation of information in service of the control and guidance of systems (biological, physical, chemical, cognitive, etc.). Suppose we have two systems, A and B, and system A makes an action aimed at system B. This action results in a feed forward of information from A to B. System B may re-act to this information by sending a feedback of information to system A. On the basis of this information, the system A may change its way of acting on system B. If the system B does not send feedback to A or A does not react to any information that may come from B, then the A-B super system is called an ‘open-loop’ system) In the opposite case, the system A-B is called a ‘closed-loop’ system. (An example of a closed-loop system is placing a heater in a closed room with a thermostat). If the system A acts on B in order to attain some goal, and reacts to its action by attempting to minimize the difference between the goal and the output of B, then the feedback is called negative feedback. If the feedback amplifies the difference, the feedback is called positive feedback.

Example 1 When I present a project of a research paper at a conference, I feed forward some information to the potential readership of the finished paper if published. During the discussion period after my talk I receive feedback from the audience. In the course of the discussion, the differences between my intentions and the audience’s interpretations are minimized. This makes me revise my paper so as to better match the interests and ways of understanding of the readers of the paper. In this example, the systems A and B were both cognitive systems, and the feedback was negative feedback. If, in the course of the discussion the differences between my intentions and the audience’s interpretations are amplified, the feedback was positive, and I produce a paper which is even less understandable than its previous version. Let us take some examples from the first phase of the class on ‘Race to 20’.

Example 2 Let A be the cognitive system of the high school teacher (Hteacher). Let B be the system composed of the students and the didactic milieu of the moment. The central element of this milieu is the game ‘race to 20’. The system A feeds forward the information about the rules of 2



the game ‘race to 20’ in a verbal form. In the aim of controlling the accuracy of the transmission of meaning, A ‘puts B in motion’, i.e. the Hteacher makes a student play the game. By acting, system B sends feedback to system A. This feedback carries information back to A on whether the rules of the game have been understood in the intended way. System A re-acts by more verbal feed forward in case B does not act in the intended way (negative feedback). Da capo.

Example 3 If we now enter inside the student-milieu system, we may look at the cognitive system of the student as one system, S, and the system of the game played by two players as the other system, G. G can be seen as a set of all possible two-column tables obtained in playing the game, some of them classified as ‘left player wins’ and other by ‘right player wins’. By entering the game as a player, say, a ‘left player’, S feeds forward some information into G, and obtains a feedback in the form of the ‘wins’ or ‘loses’ verdict. S’s goal is to always get a ‘wins’ verdict, and thus aims at controlling the system B so that it always produces such a verdict. The feedback, in this case, is a negative feedback.

Example 4 Let us now use the notions of positive and negative feedback to compare the effects of traditional and constructivist styles of communication between teacher and the student-milieu systems. In the traditional style of communication, the feedback of the teacher (direct corrections, pointing out of errors, hints) is a negative feedback, aiming at minimizing the differences between the expected output of the student-milieu system and the actual output. In the constructivist style of communication, the teacher’s feedback is a positive feedback: in an attempt to understand the student’s way of thinking the teacher will make the student-milieu system focus on the development of a knowledge that may have little to do with the knowledge intended by the teacher: the difference will be amplified.

2. What does the term ‘dialectic’ refer to? 2.1 Meanings of ‘dialectic’ in the history of philosophy In the history of philosophy, the term ‘dialectic’ has had many meanings. For example, in Plato’s Republic, ‘dialectics’ was synonymous with what we call ‘philosophy’ today, i.e. a systematized intellectual reflection on the nature and genesis of being (i.e. what is and what is not and how can we distinguish between one and the other). In the time of Plato, ‘philosophy’ had a much larger meaning of ‘all knowledge’. In the Middle Ages, the term referred to logic, in the context of a

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classification of the so called Liberal Arts into trivium and quadrivium. Trivium contained three domains of knowledge about language: grammar, rhetoric, and logic, called ‘dialectic’ at that time. All three ‘sciences’ provided a technical knowledge necessary for conducting debates. Grammar was the basis for constructing correct sentences and statements; rhetoric served the purpose of persuading the potential opponent that a given statement is true; dialectic or logic was meant to guide the opponents in examining their statements for consistency and truth. In modern times, the term ‘dialectic’ in philosophy has been associated with Hegel’s dialectic method of discussing and solving the various ‘dualisms’ such as the mind/body, freedom/determinism, universal/particular, the state / the individual dualisms. He claimed that these apparent oppositions can be proved as compatible with each other if seen from the perspective of a third, more general concept. In fact, he saw reality in its evolution as a continuous fight between opposing tendencies which are resolved through a more general tendency, of which the basic two can be thought of as particular cases. In formulating a philosophical argument aiming at resolving a duality he would use the pattern of THESISANTITHESIS-SYNTHESIS. In the ‘thesis’ he would argue in positive terms for one of the points of view. In the ‘antithesis’ he would argue in favor of the opposite point of view, stressing the contradictions with the first. In the ‘synthesis’ he would propose a point of view which would bring the former two together as complementary in the frame of a more general conceptual framework. For example, everyone can see the contradictions between morality from an individual’s point of view and morality from the society’s point of view. What is good and pleasant for an individual is not necessarily good and pleasant for the rest of the society. Doing only what is good for the society and not what we would really like to do limits our free will, to which we think we are entitled if we are not slaves. For Hegel, these contradictions are overcome by the concept of ethical life, which refers to modern institutions such as the family, the civil society and the state. These institutions are a realization of our individual free will. As a consequence, abiding by their rules does not constrain our free will. Philosophers have criticized this resolution of the morality dualisms, but this fact does not abolish Hegel’s theory of a dialectic character of the evolution of ideas. According to Hegel, there is no stop after a synthesis has been proposed. A synthesis becomes a thesis which can be subject to a critique leading to an antithesis. The resolution of the opposition gives rise to a new synthesis etc.

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2.2 The meaning of the term ‘dialectic’ in the distinctions between the types of didactic situations In the theory of situations, the term ‘dialectic’ refers to the method used by a cognitive system (teacher, student) to manage the contradictions between its expectations concerning the output from the system it attempts to control (the student-milieu system, the milieu, resp.) and the feedback. Feedback is just communication of information. The process of dialectic turns this information into knowledge: out of the contradiction, something positive is coming out, that explains the contradiction and generates ways of avoiding it in the future. In the situation of action, a student may expect to win by playing 7 on the basis of her belief that 7 is a lucky number. If she loses, she may resolve the contradiction between her expectations and the outcome of the game by concluding that 7 may be a lucky number but not in this game, and starting to notice the properties of the numbers 1-20, specific to the game. She may notice, for example, that when she played 12 she won, so next time around she’ll try to play 12. But, with a smart player, she’ll lose, and this will be again a contradiction with her previous strategy or ‘theory’. Overcoming the contradiction by means of a new ‘theory’, and continuing this ‘dialogue’ with the situation, the student will teach herself a method for playing the game so that she wins or is able to predict the outcome before the end. However, in the situation of action, there is no need for these ‘theories’ or ‘rules for action’ to be verbalized; they may thus remain largely implicit and unconscious for the student. The situation changes dramatically in the situation of formulation, where the necessity to communicate forces the students to bring the ‘theories’ or ‘rules for action’ to the level of consciousness. An explicitation of a strategy by a student may enter in contradiction with the milieu in two ways: - via feedback with respect to the form: other students may consider the formulation as unclear; - via feedback with respect to the validity: the strategy may prove to be ineffective in a game, or may be rejected by an argument of another students. The resolution of the contradiction in each case brings about some positive new knowledge about the situations: a better way of expressing one’s ideas or an improved strategy. Concerning the validity of the statements of the students, a situation of formulation does not force them to distinguish between validity of a statement and the efficacy of a strategy, nor between a convincing, or authoritative, or forceful statement and its truth value. The minds of the disputants are geared towards action and effective action in the situation of the game actually played or to be played in the future; all arguments are subordinate to this goal.

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The situation of validation changes the milieu with which the students play in that respect. The objects manipulated now are no longer moves in the game, but statements about the moves in the game. A student’s ‘theory’ may fall into a contradiction with another student’s ‘theory’. In the situation of validation the students will work on deciding which theory is ‘true’, but the outcome of this work may be a third theory, a clearer, more precise statement. If, in a didactic situation, the object of the students’ attention is the validity of certain mathematical statements, then it is considered a ‘situation of validation’, even if the arguments used to prove or disprove the validity of these statement are not, properly speaking, mathematical proofs. They can be mathematical arguments like ‘If I play 17, I win because my opponent can only play 18 or 19 according to the rules of the game, and in each case I can then play 20’. But they can also be empirical arguments like ‘If I say 15 I lose because each time I played 15 I lost’, or ‘Each time I play 14 I win; proof, let’s do it!’. In the situation of institutionalization, the students have to overcome the contradictions or just differences between their own ways of playing the game, speaking about it and justifying their strategies and the teacher’s ways of doing those things. In the ideal case, a student is able to resolve the contradictions and bring his or her own knowledge to a higher level of generality.

2.3 Theory of situations as the result of applying the ‘dialectic method’ to resolve the duality between ‘procedural teaching’ and ‘teaching for understanding’ in mathematics education. Theory of situations was first created as a synthesis aimed at overcoming the opposition, in the traditional teaching of mathematics, between procedural and explicit verbal knowledge (PK) on the one hand, and meaning and understanding (MK), on the other (pp. 128-1311). In the traditional teaching of mathematics only the former was the object of the teacher’s didactic concern and action. The latter was left to happen by itself, as a function of students’ intelligence and practice in the application of the procedures, definitions and theorems in solving exercises and problems. The two types of knowledge were opposed by several features: - PK is taught, MK is not; - yet MK is necessary for the acquisition of PK; students who do not understand fail at the examinations;

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see also: Brousseau, G. (1988): Représentation et didactique du sens de la division. In: G. Vergnaud, G. Brousseau,

M. Hulin (Eds.), Didactique et Acquisition des Connaissances Scientifiques. Actes du Colloque de Sèvres, mai 1987. Grenoble: La Pensée Sauvage éditions, pp. 47-64.

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- but they are not taught MK, so they are not to be blamed for their failure in PK; - teaching methods and curricula are to be blamed for students’ failure; - hence, it is necessary to change the curricula and teaching methods: let’s reform the system! This opposition, unresolved, produces ever new reforms. The slogans of ‘teaching for understanding’ are at the start of almost every reform, but, somehow, inevitably, the institutionalized teaching of mathematics converges towards the traditional teaching of PK and abandoning of the MK to the students’ own devices. A question for the research in mathematics education is therefore: What are the objective causes of this convergence? Is it the laziness of the teachers that is to blame? Or, rather, the fact that MK requires more time for preparing classes, less manageable classroom situations, a lot more reading of more voluminous students’ work? The workload of the teacher may increase exponentially with respect to the traditional teaching to the point where it becomes unmanageable. Moreover, statistically, the results on the official final examinations of the students subjected to the teaching of MK on top of PK do not normally show a spectacular improvement with respect to those taught only PK. Actually, a teaching focused on MK, with its openness to all kinds of interpretations and understandings, may leave some students with conceptions contradictory with the more official meanings of terms and lead to errors. Not only that, but some of these conceptions may become so entrenched that they become real mental obstacles to understanding new knowledge. Therefore, teachers lose motivation to put more work: they see it does not pay off proportionally to their efforts. Is it the teachers’ lack of mathematical and didactic knowledge that is to blame? Or rather the lack of such knowledge in the society and culture? This knowledge needs systematic research, invention and experimentation, and its development should not be left to teachers who have other matters to attend to. There exists a body of mathematics education knowledge, but it does not always translate easily or well into practical knowledge for teachers and their classrooms. Is it the students’ lack of interest in mathematics or lack of intelligence that is to blame? Or is it rather that the existing teaching methods and curricula disregard completely the basic laws of human learning? These are the naïve speculations and common answers to the traditional opposition between procedures and understanding. These answers are rather fatalistic: they lead to accepting the necessity of the predominance of the PK in teaching mathematics. 7



Theory of situations is an attempt to find a synthesis via a dialectic between the opposing terms instead of resigning to having to reject one of the terms. It uses the dialectic method on two levels - the development of the didactic theory (research on mathematics teaching and learning) as a synthesis out of the opposition between the teaching methods focused on PK and those focused on MK, and - the development of mathematical knowledge in students as a synthesis of oppositions between - implicit expectations of the results of an action on a milieu and the actual feedback from the milieu, - formulation of these expectations and the feedback from the objective and the social milieus, - the expectations and the validity of these expectations viewed as mathematical statements - the students’ own strategies, interpretations, formulations and arguments and the official mathematical algorithms, definitions, terminology, notation and proof methods. The didactic theory tries to identify and explain the phenomena of teaching and learning of mathematics; in particular those that are responsible for the convergence of the didactic system towards the PK focused teaching. It tries to capture these phenomena using concepts such as ‘didactic contract’, ‘epistemological obstacle’, ‘didactic obstacle’, ‘the Dienes effect’, ‘the Jourdain effect’ or ‘the Topaze effect’, ‘the metacognitive shift’, ‘didactic memory’. Attempts are also made to apply this theory to ‘engineer’ didactic milieus which would be less likely to degenerate into PK focused teaching. We shall be seeing both the theoretical and the engineering aspects of the theory in this course. PART II: AN EXERCISE IN THE STUDY OF MEANING OF A MATHEMATICAL CONCEPT: THE OPERATION OF DIVISION IN GRADE 6

In order to teach not only the procedures but also the meaning of mathematical concepts to the students one has to study this meaning. The meaning of mathematical concepts is not something absolute and given once for all. It changes in time, and it changes in function of the contexts in which it is used, and the purposes for which it is used. A concept like, for example, function, may have a different meaning for an algebraist, for a geometer, for an engineer and for a teacher. Thus the meaning of mathematical concepts is not a given in research in mathematics education: it is, rather, a problem.

8



Thus, studies of the meaning (or meanings, aspects, etc.) of mathematical concepts are an important part of the theory of didactic situations. Let us engage in such a study, taking, for example, the notion of division. We can pose the following questions: (1). What is the meaning of the operation of division for a research mathematician or for a university teacher? (2). What is the meaning of the operation of division for an elementary school teacher?

1. Meaning of the operation of division for a research mathematician For a research mathematician, the operation of division appears in the context not of arithmetic but the study of number and algebraic structures. The question, for them, is not How to divide?, but Is it possible to divide in this particular structure? Division is not discussed without the operation of multiplication. The operation of multiplication is thought of as any binary operation defined in a set of elements that need not necessarily be numbers. So one must first have a set of elements with an operation defined in it, called ‘multiplication’, or something else, it does not matter (you can call it ‘operation star’). In order to be able to speak about division, one must have the notion of the identity element, or an element e such that a*e = e*a = a for any element a in the set. Then one needs the notion of inverse: if a is any element, then it has an inverse if there exists a unique element b such that a*b=e. The inverse of a, if it exists, is denoted by a-1. Having all these notions we can now define division: to divide a by b means to multiply a by the inverse of b; or a/b = a*b-1. This means that division a/b is defined only if b has an inverse. In the ordinary field of real numbers, the identity is the number 1, and all numbers except for zero have inverses. In the ring of integers only 1 and -1 have inverses. In the algebra of matrices, multiplication is defined in the well known way (i’th row times j’th column), but there is no unique identity matrix for all matrices. In fact one has to take only square matrices of a given dimension in order to be able to speak about the identity matrix. Only very special matrices have inverses (those with non-zero determinants) and then to divide a matrix by another matrix is to multiply it by the inverse of that matrix. As another non-numerical example, let us take the set of four elements, denoted {0, 1, x, 1+x} and let’s define the operation * in it by the following table:

9

MATH 645: Theory of Situations/ Lecture 2 *

0

1

x

1+x

0

0

0

0

0

1

0

1

x

1+x

x

0

x

1+x

1

1+x

0

1+x

1

x


From this table we can see that all elements except for 0 have inverses. The inverse of 1 is 1, the inverse of x is 1+x, the inverse of 1+x is x. Dividing x by 1+x we have to multiply x by the inverse of 1+ x, which is x. But x * x = 1+x so, if we denote division by #, x # (1+x) = 1+x.

2. Meaning of the operation of division for a grade 6 teacher in 1936 and 1998. With respect to the question of the meaning of division for an elementary teacher, one could start by looking at textbooks, old and modern, classifying the school ‘problems on division’ into some types, according to some features that would have to be established. In this class we shall look at samples of division problems from two textbooks, one American textbook from 1936, and one Polish textbook from 1998, both addressed to 6-graders. The activity will proceed in several phases: Phase 1: Individual work. Each student received the two samples and classifies each sample according to some criteria chosen by him- or herself. Phase 2: Small group work. The class is divided into 4 small groups. Each group agrees on a common classification and writes down the classification criteria explicitly on a sheet of paper. Phase 3: Presentations: Each small group presents their criteria of classification and gives examples of problems from each category. Every next group stresses what is different in their classification with respect to the previous group. Phase 4: Whole class discussion on (a) the criteria of classification (b) the differences between the classifications obtained for the two textbooks (c) the differences between the meanings of division between the two textbooks (d) the differences between the meanings of division in the 6th grade and the meaning of division in academic mathematics. The text that follows will be available to the students only after class.

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2.1 Information about the textbooks 2.1.1 Categorization of division problems used by Knight, F.B., Studebaker, J.W., Ruch, G.M. (1936): Study Arithmetics. Grade Six. Chicago: Scott, Foresman and Co. In this textbook the classification of problems on division is done along one main variable: the kind of entities being divided: numbers, such as fractions, mixed numbers, decimals, or magnitudes or measures. Within each category, subcategories are defined: e.g. in dividing fractions, both elements can be fractions, one of the elements can be a whole number, the divisor can be a fraction with numerator equal to 1. Titles of chapters and sections related to division Chapter 3 - Dividing with proper fractions (Some sections are meant to introduce a concept, to explain; other - to practice a concept, yet other - to apply the concept to ‘real life’ situations; we arrange the tittles of the sections in indents, like this: [Explanatory section] [Practice section] [Application section]) Meaning of division by fractions Knowing what the divisor is Fraction divided by fraction Using division of fractions Mixed numbers in answers Halloween races Roman numerals Divisors with numerator 1 The state fair Whole number divided by fraction A Thanksgiving dinner Fraction divided by whole number Oyster farming

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Chapter 4 - Multiplying and dividing with mixed numbers ... Dividing measures Making Valentines Chapter 8 - Multiplying and dividing with decimals ... Decimal divided by a whole number Using division of decimals Remainders in division Elsie and Bill learn about bees A new idea in division Changing fractions to decimals Dividing a decimal by a decimal Dividing by .1, .01, and .001 Further work in dividing with decimals A week in a logging town Whole number divided by a decimal Problems using decimals Final work in dividing decimals Examples of problems: 1. Alice, Ruth and Mary were the Pop-corn Committee for the Pearson School Halloween party. The girls bought 3/4 of a quart of popcorn and divided it equally among themselves to pop. Each girl took what fraction of a quart of corn to pop? 2. Tom and Jimmy were to make a box for a game to be played at the Halloween party. They needed 4 boards each 3/4 ft. long. The janitor gave the boys a board 3 ft. Long. How many boards each 3/4 ft. Long could they have cut from the 3-foot board? 3. Henry brought 3/4 of a bushel of walnuts to the party. He divided the nuts into 50 equal shares. Each share was what fraction of a bushel? 4. The children had a peanut relay race. Each team ran 7/8 of a block, and each pupil on the team ran 1/8 of a block. How many pupils were on each team?

12



5. Each of the girls on the Refreshment Committee served 1/2 of a pumpkin pie at the party. The pies had been cut so that each piece was 1/8 of a whole pie. Into how many pieces was each 1/2 pie cut? 6. 9/10 ÷ 15/16 7. 5/8 ÷ 15/16 8. The cookie recipe that Mrs. White planned to use called for 3/8 cup of chocolate. She had only 1/4 cup of chocolate. What fraction of the full recipe could she have made with that amount of chocolate? 9. Divide and put your answer in simplest form: 9/10 ÷ 3/5. 10. On Halloween the Pine Hill School had some Hard Luck races. The route for the races was in three laps. The first lap was from the school to Five Corners: 1/4 mile. The second lap was from Five Corners to Orr’s Sawmill: 7/8 mile. The third lap was from Orr’s Sawmill to the school: 3/4 mile. Helen said that the second lap of the route was 31/2 times as long as the first. Jane said that it was 33/8 times as long. Which girl was correct? 11. Divide: (a) 76 ) 912

(b) 431 ) 35351

12. Woods family went to the State Fair. Father and Andy drove to the fair in the truck, taking some cattle to be entered for prizes. Mother and Ruth drove the family car. On the way to and from the Fair, Father used a total of 24 gallons of gasoline and 5 quarters of oil for the truck. The gasoline cost 18 cents per gallon, and the oil cost 30 cents per quart. Father drove the truck 107 7/10 miles in going to the fair and 108 3/10 mile in returning. Besides the cost of the gasoline and oil, the expenses for the truck were $1.00 for repairing a tire. To the nearest cent, what was the cost per mile for the truck for the round trip? 13. Divide 3/4 by 5/9. 14. Nancy earned her Christmas money making Christmas cards. She bought 2 sheets of cardboard at 5 cents each, a bottle of drawing ink for 25 cents, and some watercolors for 25 cents. (A) How much did all these things cost? (B) The cardboard sheets were 22 inches by 28 inches in size. She cut each sheet into strips 22 inches long and 5 1/2 inches wide. How many of the 5 1/2 inch strips did she cut from the 2 sheets? How many pieces were too narrow for her to use? 15. Sally and Ruth decided to make some valentines, which would be different from those they could buy in the stores. They bought a sheet of red paper 22 inches by 28 inches. Each girl took 1/2 of it. How many hearts could each girl have cut from her share, if each heart used up to 1 square inch of paper? 15. 2 )155.8

13



16. 32 ) 5.12 17. 6 ) .828 18. During 8 hours on Tuesday there was .96 inch of rainfall. This was an average of what decimal fraction of an inch per hour. 19. Mr. Burns and his family drove their car and trailer to Arrow Head camp to spend a few days. They drove 297.5 miles in 8.5 hours in traveling to the camp. How many miles per hour did they average? 20. 7.8 ) 7581.6 21. Mr. Mills told Ned and Alice that they could sell vegetables during the summer and keep half of the profits. Mr. Mills helped Ned build a stand. To make the boards below the shelf, they sawed up some 14-foot boards. How many boards 3.5 ft. Long could they have sawed from each 14-foot board? 22. 1.25 ) 3

2.1.2 Categorization of division problems used by Zawadowski, W. et al. (1998): Matematyka 2001. Podrecznik do klasy 6 szkoly podstawowej. Warszawa: WSiP. The categorization in this textbook goes still along the same variable: ‘type of numbers divided’, but it is a lot less detailed, and less explicit for the student. A new type of numbers appears: ‘rational numbers’. Decimals are just the positive rational numbers written in decimal notation. Rational numbers are all numbers which can be written in the form p/q where p and q are integers and q ≠ 0. The information about a section being related to division (or some other curricular topic) appears only in the table of contents as an additional information addressed mainly to the teacher. The curriculum is, in a sense, hidden in the textbook. This may have been done to avoid compartmentalization of knowledge in students into ‘rubrics’ such as ‘multiplication’, ‘division’, etc. Titles of sections related to division 5. Instead of dividing…

— division of ordinary fractions

… 12. About three princes who shared their gold

— division of decimal numbers

13. Minus times minus

— multiplication and division of rational numbers

14



… 15. A difficult choice

— operations in rational numbers

… 25. Which way is the best?


… 33. As close as possible!


Examples of problems 1. Look at the two series of operations. How do the divisors and results change? Can you find the missing results? 8:8=1

3/16 : 8 = 3/128

8:4=2

3/16 : 4 = 3/64

8:2=4

3/16 : 2 = 3/32

8:1=8

3/16 : 2 = 3/16

8 : 1/2 = ?

3/16 : 1/2 = ?

8 : 1/4 = ?

3/16 : 1/4 = ?

8 : 1/8 = ?

3/16 : 1/8 = ?

•

Add two further operations to each column.

•

By what number should 3/16 be multiplied in order to obtain 3/128?

•

By what number should it be multiplied to obtain 3/64?

•

What operations could replace each of these divisions? Do you see a rule?

•

Write a similar series of operations and give the results.

2. Mom said to Jack: I bought 6 liters of honey. We’ll pour it into 1/2 liter jars. Bring the jars from the cellar. (A) How many jars should Jack bring? (B) How many jars of 1/4 l would he have to bring? And how many jars of 3/4 liter? 3. The quotient is equal to the divisor and it is 4 times larger than the dividend. What is the dividend? 4. Find a number which is 4 times larger than the quotient of the numbers 3 1/2 and 2 4/5 increased by 1. 5. 2 1/3 + 3/4 : 1/2 6. -12,8 x (-0,2) 7. 3 1/3 : (-5/6) : (-2) 15



8. Decide which product is less expensive (a) Margarine sold in cups of 250 g for 1,32 zl vs margarine sold in cups of 500 g for 2,49 zl. (b) Yogurt sold in cups of 150 g for 0,93 zl vs yogurt sold in cups of 500 g for 2,60 zl.

2.2 Brousseau’s criteria of classification of division problems In his already mentioned paper on the ‘didactics of the meaning of division’2, Brousseau identified two sets of criteria for the classification of division problems: one related to contextual variables of the problems, and one related to the concepts involved in the solution of the problems. (1) Contextual variables Group 1: type of numbers involved in the division natural numbers, decimal numbers, rational numbers, real numbers, etc. representation of numbers (fractions, decimals) size value of the numbers (<1, >1, small numbers, big numbers) the mathematical function of the numbers (cardinal numbers, measures, scalars, linear transformations) Group 2: type of magnitudes physical magnitudes dimensions definition mode: magnitudes defined as products of magnitudes, quotients of magnitudes, etc. Group 3: type of didactic situation Group 4: previously taught techniques of calculation sharing manipulations repeated subtraction factoring systematic approximations from above and from below

2

Brousseau (1988), ibid.

16



reduction to operations in natural numbers ways of presenting the computations (2) Conceptions of division Group 1: Sharing (finding the number of parts, finding the value of a part) Group 2: Finding the unknown term of a product Group 3: Fractioning (fractioning of a unit, commensuration, decimal approximation of a fraction) Group 4: Linear transformation (finding the value corresponding to 1, ratio of measures, a function ‘divide by’, etc.) Group 4: Composition of linear transformations

17

LECTURE 3

1999 MATH 645 CONCORDIA UNIVERSITY LECTURE NOTES ON THE THEORY OF DIDACTIC SITUATIONS – ANNA SIERPINSKA

LECTURE 3 THE NOTION OF ‘DIDACTIC CONTRACT’ PART I - IDENTIFICATION OF A PHENOMENON: THE ‘DIDACTIC CONTRACT’, AND ITS IMPACT ON THE MEANING OF MATHEMATICAL CONCEPTS TAUGHT AT SCHOOL

We have assumed that the didactic situation can be described as a game between a (person in the role or position of the) teacher and the student-milieu system. Every game has its rules and strategies. The rules and strategies of the game between the teacher and the student-milieu system, which are specific of the knowledge taught, are called the ‘didactic contract’ (p. 41). The assumption that the rules taken into account are pertinent from the point of view of the knowledge taught or aimed at is essential for the definition of the didactic contract. If the rules taken into account have nothing or little to do with this knowledge, but are relevant from the point of view of, say, classroom management, or political correctness, or the general culture, then we might speak of other types of contract, maybe (e.g. pedagogical contract) but not of a didactic contract. It is assumed in the Theory of Didactic Situations that ‘didactics’ refers to the problems of teaching and learning of a particular knowledge, not any knowledge in general. Contrary to games such as chess or bridge, the rules of the didactic contract are not explicit and they can be slightly different from classroom to classroom, culture to culture, and they can even change in the history of a single classroom with the same teacher and the same students. The fact remains, nevertheless, that in every didactic situation there is a didactic contract, and that across different cultures, classrooms and time some rules remain constant, such as, for example, that the teacher is expected to perform teaching actions such as giving the students tasks that are specific to the knowledge he or she aims at, and the student is expected to attend to the tasks given by the teacher. The rules of the didactic contract are implicit: the teacher and the students do not sign a chart of ‘rights and obligations’ upon entering into a didactic relation. But they are there and we know that they are there when they are broken. Suppose that one day, in the middle of a class, I suddenly sit down, open a newspaper and start to read. What would you say? That I am not doing my job! I was supposed to teach you something, start speaking to you, assign tasks, etc., and here I am, sitting, and reading a newspaper! You would immediately report on my strange behavior to the chair of the department. 11/4/03

1

LECTURE 3 Let us look at another situation: The teacher gives the students the task to write the numbers 38, 24, 49, 46, 51 in an ascending order. The students are left to solve the problem individually, and then the teacher writes on the board the solution to this exercise. Then she turns to the class and asks: ‘Why did we put 46 and then 49?’ A student answers: ‘Because if we only had the other numbers, it would have been too easy’. The teacher answers with some anger in her voice: ‘This is not what you are asked for!’ (‘Ce n’est pas ce qu’on te demande!’). Another student says: ‘Because 46 is smaller than 49’. Teacher: ‘Yes, very good’ (and she writes the symbol ‘<‘ between the two numbers). We can see that the first student, in her intervention, stepped out of her role of student: she expressed her view on the didactic reasons in designing the task, thus entering the role reserved for the teacher. The teacher felt angry with the ‘usurper’.

The historico-epistemological context of the emergence of the notion of didactic contract in the theory of didactic situations The intellectual background of the idea of ‘didactic contract’ has been identified and analyzed by Bernard Sarrazy in (Sarrazy, 19951). ‘The concept of didactic contract has been introduced by Brousseau in 1978, as a possible cause of the so-called ‘elective failure’ in mathematics (some students appear to have difficulty only in mathematics at school, while succeeding reasonably well in other subjects). He used it to explain the case of a young boy, Gaël, who was repeating his first grade because of difficulties in mathematics. The observation of the behavior of the boy during the remedial sessions with a tutor showed that, for Gaël, knowing something meant only to be able to repeat certain ritualized actions, modeled by the teacher. He was putting so much effort into finding out what actions the teacher expected him to perform, or into ‘uncovering the implicit contract’, that he was not able to engage with unraveling the meaning of the mathematical knowledge involved in the tasks posed to him. When asked questions like ‘why did you add these two numbers?’ he would invariably answer: ‘because this is what the teacher said that we have to do’, ‘this is how I was taught’, etc. The boy was not retarded or unintelligent but simply had a notion of the didactic contract that did not allow him to learn any mathematics. The solution of the boy’s problems came with changing his notion of the contract by creating didactic situations for him in which the contract was obviously not the same as with his schoolteacher. The boy was no longer told ‘Do as I told you!’. Instead the tutor would play a game with Gaël, betting on the possible outcomes of a calculation, and then finding out who was right.

1

Sarrazy, B. (1995): Le contrat didactique. Revue Française de Pédagogie 112, 85-118.

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2

LECTURE 3 Describing the problem of Gaël in the sociological and cultural terms of ‘didactic contract’ rather than in psychological terms of intelligence quotient, or personality, was quite in tune with the spirit of the times. In the second half of the 70s, there was, in the explanations of the school failure, a shift from the macro-sociological theories which would put the blame on the socio-cultural background of the students (e.g. the theories of Basil Bernstein regarding the school failure of working class children), and on the workings of the educational system as a whole (e.g. Bernstein’s theory of ‘transmission of culture’ stressing the conservative character of educational systems), towards micro sociological theories which focused on classroom interactions. This ‘interactionist trend’ in educational sociology had a well-developed theoretical background in ‘symbolic interactionism’ and ‘ethnomethodology’, a research paradigm that emerged in USA in 1930s (the so-called ‘Chicago school’ of sociology: W.J. Thomas, F. Znaniecki, R. Park, H. Mead, G.H. Blumer). The papers and books of Ernst Goffmann that appeared in mid-70s became very popular in intellectual circles and had a strong impact on the way people started thinking about even their everyday interactions with other people and institutions. Today, the ideas of Goffmann and others working in the interactionist paradigm of sociology have become part of textbook knowledge and any Cegep student enrolled in the Social Sciences program has to learn about them. It is Goffmann who started identifying the various ‘contracts’ that bind our interactions with other people in everyday and professional lives. He called them ‘frames’ (kind of scenarios) that can be played in different ‘keys’, (e.g. as comedies or tragedies). Applied to the classroom context, one could speak of several identifiable ‘frames’, such as ‘lecturing’, ‘questioning’, ‘reprimand’, ‘praise’, etc. The frame of ‘school questioning’ is very different from the frame of ‘asking a question’ in a non-didactic situation: in the latter, the person who asks normally does not know the answer! A child who comes to school first time may be quite astonished that the teacher is asking questions she certainly knows the answers for! When she accepts this strange situation as normal, she has already understood the frame and became aware of the existence of a definite didactic contract that binds her own and teacher’s behavior.

Paradoxes of the didactic contract The student expects the teacher to teach, and for some students, this simply means to tell the student how to solve the assigned problems and what answers to give. But if the teacher complies with this expectation, the student will not learn anything, because she will not have had to make a choice of one strategy among other possible strategies, and of one interpretation among other possible interpretations (p. 41). This is analogous to the information value of a message: if the

11/4/03

3

LECTURE 3 probability of a message is 1, then the amount of information it carries is 0. Solving a problem one knows how to solve adds nothing to our knowledge. So the didactic contract puts the teacher in front of a paradox: everything that the teacher undertakes to make the student produce expected behavior tends to deprive the student of the necessary conditions for understanding and learning of the notion she aims at; if the teacher tells the student what she wants, she can no longer obtain it’ (p. 41). So how can a teacher convey new knowledge to the students? How can a student learn new knowledge? The Theory of Didactic Situations assumes that learning in a school situation is an adaptation to a milieu. The task of the teacher is then to organize the milieu in such a way that the adaptation will result in the student developing the target knowledge. This is, however, easier said than done. In the second part of the class we shall be analyzing the ‘division problems’ of last week, from the point of view of their ability to produce, in the students, an adaptation resulting in an understanding of the operation of division. PART II - EXAMPLES OF THE IMPACT OF THE DIDACTIC CONTRACT ON THE MEANING OF DIVISION IN TWO GRADE

6 TEXTBOOKS

Results of the activity of classification of ‘division problems’ in Week 2 Each student received a copy of two sets of ‘division problems’, one taken from a 1936 textbook (label this set ‘I’), and the other - from a 1998 textbook (label this set ‘II’). Students worked in four groups of three: G1, G2, G3, G4, for about 20 minutes. Their task was to classify the two sets of ‘division problems’ according to some criteria of their choice. The last 15 minutes of the class were devoted to a presentation, by each group, of the criteria they chose. A priori, many different criteria could be chosen. For the 1936 textbook, one criterion could be ‘the seasons’: there were problems for Halloween, problems for Thanksgiving, problems for Christmas, for Valentines, for spring rains, for summer camping, for summer jobs, etc. Another criterion could be ‘types of magnitudes’: pure numbers, physical magnitudes (measures and ratios of measures), money. Yet another could be ‘kind of numbers involved’: whole numbers, fractions, decimals, or natural numbers, integers, rational numbers. One could also classify the problems into one-step problems, two-step problems, etc. In class, students chose different criteria: Here are these criteria: Group G1: classification according to an aspect of the operation of division Division as the inverse operation with respect to multiplication (e.g. I.6) 11/4/03

4

LECTURE 3 Long (synthetic) division … of whole numbers (e.g. I.11) … division convertible to division by a whole number (e.g. I.15) … division convertible to division of whole number by a whole number (e.g. I.20) Number of parts in a whole

(e.g. I.1, I.2, I.4)

Repeated subtraction (e.g. I.4) Groups G2 and G4 Procedural problems Disguised procedural problems Conceptual problems Group G3 Enigma type problems (the problem can be solved backwards) (e.g. II.3,4) Problems involving judgment (II.8) Only group G1 used criteria specific to the notion of the operation of division. Other groups used more general criteria that could be applied to problems related to any mathematical notion. These are important criteria but they are not helpful in the study of the meanings of the operation of division aimed by the textbooks.

Study of the meanings of division conveyed by two grade 6 textbooks If our aim is to study how the meanings of division conveyed by the problems, it may be a good idea to find out - What mathematical notions and techniques are sufficient to solve each problem, as compared to - The mathematical notions and techniques expected to be used by students by virtue of the implicit ‘didactic contract’, whose rules are conveyed by the context of the problem.

Analysis of the 1936 textbook problems 1. Alice, Ruth and Mary were the Pop-corn Committee for the Pearson School Halloween party. The girls bought 3/4 of a quart of popcorn and divided it equally among themselves to pop. Each girl took what fraction of a quart of corn to pop?

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5

LECTURE 3 The problem can be solved by dividing one whole number (3) by another whole number (3): 3 quarters of a whole to be shared by 3 girls. Here, division is an answer to the question: A quantity Q is divided into n parts: how much of the quantity in one part? The division does not have to be done numerically. It is enough to visualize 3 things and their distribution among three people: 2. Tom and Jimmy were to make a box for a game to be played at the Halloween party. They needed 4 boards each 3/4 ft long. The janitor gave the boys a board 3 ft long. How many boards each 3/4 ft long could they have cut from the 3 ft board?

The problem is not a ‘division problem’; it is an ‘addition-in-disguise problem’. It can be solved by adding fractions, or even whole numbers. Four boards each 3 quarters of a foot add up to 12 quarters of a foot board, which is a 3 feet board. So the janitor gave Tom and Jimmy enough board to make their box: they could cut their 4 boards from the 3 feet board. 3. Henry brought 3/4 of a bushel of walnuts to the party. He divided the nuts into 50 equal shares. Each share was what fraction of a bushel?

One could solve the problem by dividing 3/4 by 50 (a fraction by a whole number), but this is not the only way. Actually, one can do with the notion of sharing a quantity Q (= one quarter of a bushel of nuts) among n (=50) people, and continue with the operation of addition. To answer the question in the problem one needs to understand fractions as ‘p parts out of a whole made of q parts’. Here is how the reasoning could go: Henry had 3 quarters of a bushel of walnuts. He divided each one of these quarters into 50 equal parts. The whole bushel would thus be composed of 200 such parts (a bushel is composed of 4 quarters of a bushel, and 50 + 50 + 50 + 50 = 200). He took one such part from each quarter to make a share (1+1+1): so he took 3 out of 200 parts, or 3/200 of a bushel. 4. Children had a peanut relay race. Each team ran 7/8 of a block, and each pupil on the team ran 1/8 of a block. How many pupils were on each team?

This problem can be solved by dividing one whole number (7) by another whole number (1). The problem is a division problem of the type: There are m things and these things are divided into groups of n things: how many groups can be formed? The division can be performed by repeated subtraction: 7 - 1 - 1 - 1 - 1 - 1 - 1 - 1 = 0, so 7 ones in 7. It can also be solved by counting, after having visualized 7/8 as 7 segments of equal length. It is completely irrelevant that eighths of a block are considered. 5. Each of the girls on the Refreshment Committee served 1/2 of a pumpkin pie at the party. The pies had been cut so that each piece was 1/8 of a whole pie. Into how many pieces was each 1/2 pie cut?

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LECTURE 3 The textbook probably expected the students to write 1/2 : 1/8 in answer to this problem, but the problem can be solved by multiplying a number by 1/2: there are 8 parts in each pie, so half of the pie has half the number of parts, i.e. 4. Here the fraction 1/2 is seen as an operator. Again, visualizing and counting could be enough to find the answer, without writing any arithmetic operations. 6. 9/10 ˆ 15/16

The text probably expects the students to use the procedure for the division of fractions: ‘When the divisor is a fraction, we must rewrite the example, making two changes: 1. Change the division sign to a multiplication sign 2. Invert the divisor (p. 69)’ introduced in the same section of the book. This procedure uses the algebraic notion of division as multiplication by the inverse. This notion is in sharp contrast with the arithmetic notions of sharing, partitioning or counting that were sufficient for solving the introductory word problems, in which numbers did not appear as pure numbers but as quantities of something. In fact, these ‘introductory problems’ did not build the expected scaffolding for the notion of division now officially introduced, and no conceptual link can conceivably be formed on the basis of just the problems such as the above. Any simple arithmetic problem about concrete quantities can be solved with some arithmetic notion of division. The notion of division as multiplication by the inverse is necessary in the problem of creating a unified theory of number systems, and this is certainly not a task accessible to 6 grade children. If the formal rule is given at this point, the students will never be given a chance to make this link, because the work needed to make it is far too costly in terms of conceptual effort in comparison with the mechanical application of the rule. Let us try to understand what would be involved in solving the present problem without knowing the ‘rule’. I claim that treating this problem as a ‘real problem’ and giving it to the students prior to giving them the rule, in the form of: ‘What would it mean to divide 9/10 by 15/16?’ could actually create a situation in which the students would have the chance to finally going beyond their whole number conceptions of division and reorganizing it into a new conception, namely the ratio conception of division. This notion is still an arithmetic notion, albeit a quite elaborate one. The link with the previous conceptions would thus be made, and the leap into the abyss of empty formalism would be avoided. Let us speculate how a student, who was not given the ‘rule’ before, could solve this problem. Suppose the student understands this question as: "how many 15/16 in 9/10?". By representing these two fractions as parts of a whole (e.g. a strip of 10 squares), the student may 11/4/03

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LECTURE 3 realize that 15/16 is greater than 9/10 by a little and it does not fit into 9/10 a whole number of times. He must, therefore, understand the question differently and thus change his notion of division: not ‘how many parts worth 15/16 of the whole in a part worth 9/10 of the whole’, because ‘how many’ presupposes a whole number as an answer, but ‘what is the ratio between a part worth 9/10 of the whole to the part worth 15/16 of the whole’? He could then see the whole as composed of, say, 160 little parts; 9/10 of the whole is composed of 9*16 = 144 such little parts, and 15/16 of the whole is composed of 150 such little parts. So the ratio of the 9/10 part to the 15/16 part is the same as 144 to 150, which is the same as 24 to 25. We can see that in this reasoning, the hypothetical student went from viewing division as partitioning or grouping (how many groups of so many elements) to viewing it as the operation of estimating the ratio of two magnitudes. Let us notice also that the notion of division as multiplication by the inverse is not necessary for the solution of the problem. Indeed, a generalization of the solution given above could lead to the following algorithm of the division of a fraction by a fraction: to divide a/b by c/d , represent the whole as bd little parts. Then a/b is ad of these parts, and c/d is cb of these parts. Then a/b divided by c/d represents the ratio of ad little parts to cb little parts. Symbolically: a/c : c/d = ad : cb. Of course, the ratio can sometimes be simplified, like in fractions. 7. 5/8 ˆ 15/16

The method elaborated in the previous problem allows us to treat this question as a simple application of the method: we can see the whole as composed of 128 little parts, 5/8 as 80 of these parts and 15/16 as 120 parts; the answer is the ratio of 80 to 120, which can be represented by the reduced fraction 2/3. 8. The cookie recipe that Mrs. White planned to use called for 3/8 cup of chocolate. She had only 1/4 cup of chocolate. What fraction of the full recipe could she have made with that amount of chocolate?

The problem can be solved using the notion of division as representing a ratio of quantities and a notion of the equivalence of one quarter and two eighths: one quarter of a cup is the same as 2 eighths of the cup. The amount of chocolate that Mrs. White had was thus to the full amount needed as 2 to 3. So she could only make two thirds of the recipe. This is a kind of proportional reasoning that does not require the writing of any arithmetic operations, and certainly not the operation implicitly expected by the authors of the textbook, namely 1/4 : 3/8 = 1/4 * 8/3 = 1/1 * 2/3 = 2/3. 9. Divide and put your answer in simplest form: 9/10 ˆ 3/5.

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LECTURE 3 Division as ratio can be used here again; 3 fifths is the same as 6 tenths, so 9 tenths is to 3 fifths as 9 tenths to 6 tenths, i.e. as 9 to 6 or as 3 to 2. This appears to be quite simple, but this is not what the text expects the students to do: the ‘simple form’ means the ‘mixed number’ form, i.e. 3/2 is expected to be represented as 1 1/2. But this representation requires to understand division not arithmetically, as ratio, but algebraically, as an operation on numbers, yielding a number, and not some new kind of entity. Seeing ratios as numbers and extending the four arithmetic operations from whole numbers to fractions is epistemologically quite difficult, because, from the former domain to the latter, these operations change their meaning. One can no longer think of addition as ‘bringing together, or of division as sharing, or partitioning a quantity. In the history of mathematics, a unified notion of number has been, de facto, an achievement of the nineteenth century. The solution of the problem, as it stands, without the implicit ‘didactic contract’ conveyed by the kind of worked out examples preceding this problem, does not require the above sophisticated notion of division as an arithmetic operation on numbers. The notion of ratio of whole numbers is sufficient. If a student solves the problem with the notion of division as ratio, and is punished by the teacher because she did not transform 3/2 into 1 1/2, this is a sign for her that the rules of the game are not what she taught they were. She does not know what is wrong with her thinking and she is unable to yet understand the notion of division of fractions in an algebraic way. Different scenarios are possible from this point on for the student, some ending in frustration and failure, some - in the student suspending her ‘situational’ understanding of fractions and happily engaging in the formal game on fractional expressions, changing division into multiplication, inverting, simplifying, extracting the whole part, etc. We can see now that, in the textbook, the ‘algebraic notion of division’ does not emerge as a ‘conceptual necessity’ for solving problems, but is enforced by a ‘didactic contract’ in a situation of institutionalization, as if it were a law and not a mathematical concept. 10. On Halloween the Pine Hill School had some Hard Luck races. The route for the races was in three laps. The first lap was from the school to Five Corners: 1/4 mile. The second lap was from Five Corners to Orr’s Sawmill: 7/8 mile. The third lap was from Orr’s Sawmill to the school: 3/4 mile. Helen said that the second lap of the route was 3 1/2 times as long as the first. Jane said that it was 3 3/8 times as long. Which girl was correct?

The problem does not require division, but multiplication of fractions and comparison of fractions. A possible line of reasoning could be: If Helen is right then the second lap would have to be 3 1/2 times 1/4 mile. This is equal to three quarters of a mile plus a half of one quarter, i.e. six eighths of a mile plus one eighth, i.e. seven eighths of a mile, so she is right. If Jane were also right then the second lap would have to be 3 3/8 times 1/4 mile. Here the multiplication becomes more complicated, so a mental calculation may be risky and it is better to write down the 11/4/03

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LECTURE 3 operations: 3/4 mile + 3/8 * 1/4 mile = (3*8)/(4*8) mile + 3/(8*4) mile = (3*9)/(4*8) mile = 27/32 mile. Now, 7/8 mile is the same as 28/32 mile, so Jane’s estimation is inaccurate by 1/32 mile. If a student chose to solve the problem by dividing 7/8 by 1/4 and checking whether it is equal to Helen’s proposal or Jane’s proposal, then a new conceptualization of division would be required; ‘new’ with respect to the conceptualizations needed in the previous problems. To solve the problem with division, the student must first understand the mutually inverse character of division and multiplication: if y is a times x then a is y divided by x. In the problem, y and x are magnitudes - distances, in miles; a is a factor, a ‘pure number’, so to say. Division could be understood as a ratio, but the ratio must be treated as a number, otherwise the expression ‘a times as long as’ wouldn’t make sense. The ‘must’ is a constraint of the problem and the chosen strategy, not of a didactic contract, as in the previous problem. Here is a possible line of reasoning: 1/4 of a mile is the same as 2/8 of a mile, so the second lap is to the first as 7 eighths to 2 eighths. So the ratio is 7/2 or 3 1/2. Therefore the second lap is three and a half times longer than the first, which is what Helen said, and not what Jane said. So Helen is right. 11. Divide: (a) 912 / 76 (b) 35351 / 431 Question (a) could be solved by repeated subtraction and counting how many times 76 fit into 912 (12 times). But applying the same strategy to (b) would be a bit tedious: 35351 = 431*82 + 9. One could, of course, use some informed guessing: there are 80 four hundreds in 320 hundreds, so let’s try 431*80: this is 34480. Adding 431 twice to the result yields 35342 and there is 9 left. In this reasoning, numbers are whole numbers and the division is the ‘Euclidean division’, which identifies the integer number of times a number fits into another number and computes the remainder. The textbook expects the students to use the long division algorithm and represent the result in the ‘mixed number’ form: 82 9/431. 12. Woods family went to the State Fair. Father and Andy drove to the fair in the truck, taking some cattle to be entered for prizes. Mother and Ruth drove the family car. On the way to and from the Fair, Father used a total of 24 gallons of gasoline and 5 quarters of oil for the truck. The gasoline cost 18 cents per gallon, and the oil cost 30 cents per quart. Father drove the truck 107 7/10 miles in going to the fair and 108 3/10 mile in returning. Besides the cost of the gasoline and oil, the expenses for the truck were $1.00 for repairing a tire. To the nearest cent, what was the cost per mile for the truck for the round trip?

This is ‘multiple step’ problem with several calculations to be made before the answer can be given, but the ultimate operation is division. (24 gallons of gas * .18 $/gallon + 5 quarters of oil * .30 $/quarter + 1.00 $) : (107 7/10 + 108 3/10) = 3 cents per mile 11/4/03

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LECTURE 3 Division has to be understood as a ratio here, but the ratio is not a pure number, it is a magnitude, because magnitudes of different kinds are compared: cost per mile. This is again something new with respect to the conceptions needed in the previous exercises, where ratios of homogeneous magnitudes were considered only. In the history of mathematics and physics, the notion of ratio of magnitudes of different kinds was very slow to appear. In Euclid’s ‘Elements’, Book V devoted to the ‘theory of proportions’, starts with several definitions regarding ‘magnitudes’ and ‘ratio of magnitudes’. Definition 3 states: ‘A ratio is a sort of relation in respect of size between two magnitudes of the same kind’. For Euclid, area and length, weight and volume were not magnitudes ‘of the same kind’. Also areas of circles and areas of squares were not of the same kind, because figures bound by straight lines and figures bound by curves were regarded as totally distinct entities. Thus, one of the theorems states: Circles are to each other as the squares built on their diameters. In today’s terminology and way of thinking about numbers we would say: the ratio of the area of a circle to the area of the square built on its diameter is constant and equal to the number Pi. But Euclid refused to measure the area of the circle with the area of a square and thus he could not obtain, in his system, the number Pi. The requirement of comparing magnitudes of the same kind only has been an obstacle to the development of mathematics for many centuries, but when it was finally overcome and mathematicians started to allow themselves to use ratios of non-homogeneous magnitudes and think of ratios as numbers which could equal to each other and not be just ‘proportional’, they could invent the notions of velocity (distance per time), acceleration (increase or decrease of velocity per time), and the differential calculus. The point of all these historical remarks is to stress the non-trivial character of the new conceptualization involved in solving the present problem using division. 13. Divide 3/4 by 5/9.

The problem can be reduced to finding a ratio of two whole numbers, 27 to 20: 3/4 is the same as 27/36 and 5/9 is the same as 20/36, so 3/4 to 5/9 is the same as 27 to 20. 14. Nancy earned her Christmas money making Christmas cards. She bought 2 sheets of cardboard at 5 cents each, a bottle of drawing ink for 25 cents, and some watercolors for 25 cents. (A) How much did all these things cost? (B) The cardboard sheets were 22 inches by 28 inches in size. She cut each sheet into strips 22 inches long and 5 1/2 inches wide. How many of the 5 1/2 inch strips did she cut from the 2 sheets? How many pieces were too narrow for her to use?

Both questions (a) and (b) can be solved by addition. A. 5 + 5 + 25 + 25 = 60 (cents) B. Suppose Nancy cuts across the longer side. One strip of 5 1/2 and another of 5 1/2 make 11 inches; with another pair of 5 1/2 inch strips she has already disposed of 22 inches, so she could

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LECTURE 3 cut yet another strip, and thus 5 strips come up to 27 and a half inches, and a strip of one half inch wide and 22 inches long is left over from one sheet of cardboard. So, from the two cardboards she could cut 10 5 1/2 inch strips and had two pieces two narrow for her to use.

15. Sally and Ruth decided to make some valentines which would be different from those they could buy in the stores. They bought a sheet of red paper 22 inches by 28 inches. Each girl took 1/2 of it. How many hearts could each girl have cut from her share, if each heart used up to 1 square inch of paper?

The only division that has to be done in this problem is 22:2 or 28:2 and the only interesting question that the students may have about this problem is whether it matters how the paper is cut in two: across the longer side of across the shorter side. Otherwise the problem can be solved by multiplication understood as repeated division, or even by drawing and counting: 15. 155.8 / 2 16. 512 / 32 17. .828 / 6

The textbook regards the above three exercises as practice in applying the algorithm of the long division of decimals. But problem # 15 could be done mentally, by representing it as sharing the amount of 155 and 8 tenths of something between two people. Splitting 155 into 2 gives 77 and a half. Splitting 8 tenths in half gives 4 tenths. So altogether each person gets 77+(1/2 + 4/10) = 77 + 9/10 = 77.9 Problems # 16 and 17 differ from # 15 in that the divisor is greater than the dividend, so division has to be seen as a ratio. In # 16 the dividend is still a number greater than 1, in # 17 it is less than 1, but this may not make a big difference. How small is 5.12 compared to 32? Changing the unit one can say that 5.12 is to 32 as 512 is to 3200, which can be simplified to 128 to 800 (dividing by 4), and then to 16 to 100 (dividing by 8). The ratio 16 to 100 can be written in decimals as .16. Problem #17 could be interpreted as an exercise in division of a whole number by a whole number, in its conception as sharing or partitioning: changing the units one could think of .828 as representing 828 grams of something (e.g. chocolate), and then partition or share this amount into 6, obtaining 138 grams. This can then be represented as 0.138 of the kilo. In each case, as we could see, the long division algorithm for decimals could be avoided at almost no cost. 18. During 8 hours on Tuesday there was .96 inch of rainfall. This was an average of what decimal fraction of an inch per hour?

Like in the problems above, long division could be avoided. We could think of .96 of an inch as being 96 ‘centi-inches’, and partition these 96 units into 8 hours (by repeated subtraction, or 11/4/03

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LECTURE 3 repeated addition or a combination of multiplication and addition). This gives 12 units per hour. The 12 ‘centi-inches’ can be written as .12 of an inch. 19. Mr. Burns and his family drove their car and trailer to Arrow Head camp to spend a few days. They drove 297.5 miles in 8.5 hours in travelling to the camp. How many miles per hour did they average?

In this problem division has to be thought of as a ratio resulting in a new magnitude and not a pure number. Average velocity in miles per hour has to be calculated: 297.5 / 8.5 [mi/h]. Long division algorithm for decimals can be avoided by a process of rough estimation, multiplication and addition. For example, let’s first look at how many times the 8 hours could fit into 297 miles. If our guess is 30, then we calculate 8*30 = 240 and we add 30 halves, which is 15. So 8.5 * 30 = 255. This falls short of 297.5 miles by 42.5. Now 8*5 is 40 and 5 times a half is 2 and a half, so 8.5 * 5 = 42.5 which is exactly the missing remainder of the road. Hence 8.5 * 35 = 297.5 and so the average velocity was 35 mi/h. 20. 7581.6 / 7.8

This problem can be reduced to division of whole numbers, sharing or partitioning 75816 into 78. The result is a whole number, 972. 21. Mr. Mills told Ned and Alice that they could sell vegetables during the summer and keep half of the profits. Mr. Mills helped Ned build a stand. To make the boards below the shelf, they sawed up some 14-foot boards. How many boards 3.5 ft long could they have sawed from each 14-foot board?

The problem could be solved by addition: adding 3.5 + 3.5 + .... until something close to 14 is obtained. In this case, adding 3.5 four times gives exactly 14, so the answer is 4. 22. 3 / 1.25

This problem can be reduced to finding the ratio of 3 to 1.25. Making the unit hundred times smaller, we can think of the problem as finding the ratio of 300 to 125. This ratio can be simplified to 12/5. Representing this ratio in mixed number form we get 2 and 2/5. The ratio 2/5 can be represented as 4/10, so the result is 2.4 in decimal notation.

Analysis of the 1998 textbook problems 1. Look at the two series of operations. How do the divisors and results change? Can you find the missing results?

8:8=1 8:4=2 8:2=4 8:1=8 8 : 1/2 = ? 8 : 1/4 = ? 8 : 1/8 = ?

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3/16 : 8 = 3/128 3/16 : 4 = 3/64 3/16 : 2 = 3/32 3/16 : 2 = 3/16 3/16 : 1/2 = ? 3/16 : 1/4 = ? 3/16 : 1/8 = ?

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LECTURE 3 (a) Add two more operations to each column (b) By what number should 3/16 be multiplied in order to obtain 3/128? (c) By what number should it be multiplied to obtain 3/64? (d) What operations could replace each of these divisions? Do you see a rule? (e) Write a similar series of operations and give the results.

The notion of division that appears to be aimed at by this series of exercises is the formal mathematical one: to divide by a number means to multiply by its inverse. Question (a) of this problem could be solved without even looking at the left hand sides of the equalities, and the implicit didactic goal of the exercise could be missed. Questions (b) and (c) appear to aim at re-focusing the students’ attention on the left hand sides of the equalities. Then, perhaps, looking for a pattern in the first column the student might come to a generalization: when you divide a whole number a by a fraction 1/c (c being a whole number), the result is the whole number a*c. Looking for a pattern in the second column, the student might come up with two ‘rules’: (1) in dividing a fraction a/b by a whole number c, the result is a/(b*c), (2) in dividing a fraction a/b by a fraction 1/c , the result is a times the ratio of c to b. To answer question (b), the first rule could be reformulated as, ‘a/b : c = a/b * 1/c’. However, the rules obtained by the students might be a lot more specific than the target general notion of division as multiplication by the inverse. 2. Mom said to Jacek: I bought 6 liters of honey. We’ll pour it into 1/2 liter jars. Bring the jars from the cellar. (a) How many jars should Jacek bring? (b) How many jars of 1/4 l would he have to bring? And how many jars of 3/4 liter would he have to bring?

The notion of division as multiplication by the inverse is not necessary to solve the problem. One can solve it by drawing a diagram and counting: (a) The diagram could be made of 6 pairs of squares, each square representing 1/2 l. So 12 jars are needed for 6 liters of honey. (b) The diagram could be made of 6 groups of 4 squares, each square representing 1/4 l. So 24 such jars are needed for 6 liters of honey. (c) The diagramcould be made of 24 squares; partioning these into groups of 3 gives 8 (jars). 3. A quotient is equal to the divisor and it is 4 times larger than the dividend. What is the dividend?

To solve this problem it is not necessary to understand division as multiplication by the inverse. Division as ratio, with ratio understood as number, is enough. But it is necessary to understand the transitive property of equality (if A = B and A = C then B = C), and it is quite useful to be able to represent unknowns by letters. This is more an exercise in logical and algebraic thinking than in division of fractions. Here is how the reasoning might go: Let the dividend be a and the divisor b. We are told that a : b = b and a : b = 4a. This implies that b = 4a (by transitivity of equality). Then a : b = a : 4a = 1/4 (a whole to four times a whole is as 1 to four). 11/4/03

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LECTURE 3 But a : b = b, so b = 1/4 (transitivity of equality). But b is 4 times as large as a, so a is a quarter of b; but a quarter of a quarter is 1/16, so a = 1/16. However, the notion of division as a ratio is not given a chance to develop in the textbook, so the students can be left with the formal notion of division, and if they are not too good at formal algebraic and logical thinking, they have no chance of solving the problem unless they treat it as a riddle to be solved by guessing or experimenting on their calculators. 4. Find a number which is 4 times as large as the quotient of the numbers 3 1/2 and 2 4/5 increased by 1.

Division as ratio is sufficient to solve the problem: 3 and a half can be seen as 7 halves; 2 4/5 is 14 fifths; finding a common measure (one tenth) leads to seeing 7 halves as 35 tenths and 14 fifths as 28 tenths; so the ratio is 35 to 28 which is the same as 5 to 4. Now the unknown number is 4 times 5/4 + 1, so it is equal to 5 + 4, i.e. the number is 9. The problem is mainly an exercise in translating verbal representations of relations between numbers into arithmetic operations. It is also a preparation for algebraic thinking. 5. 2 1/3 + 3/4 : 1/2

Students guided by this textbook will use the rules of arithmetic on fractions, and the algebraic notion of division as multiplication by the inverse. One can, however, expect a lot of mistakes, because the students have no means to verify, by themselves, if their solution is correct, and the meaningless rules are easily forgotten. If the students could create for themselves a model of the expression, then they could perhaps judge of the validity of their result. 6. -12,8 x (-0,2)

Early introduction of operations on rational numbers (and not just decimals) is another symptom of the algebraic character of the notions of operations on numbers favored by the textbook. 7. 3 1/3 : (-5/6) : (-2)

The main problem here for the students is the order of operations, which is based on pure convention, and not on any rational base. 8. Decide which product is less expensive (a) Margarine sold in cups of 250 g for 1,32 zl vs margarine sold in cups of 500 g for 2,49 zl. (b) Yogurt sold in cups of 150 g for 0,93 zl vs yogurt sold in cups of 500 g for 2,60 zl.

Problem (a) can be solved by proportional thinking: since 500 is the double of 250, it is enough to find the cost of two cups of the margarine sold in cups of 250 g for 1,32 zl each: that would be 2,64 zl, which is more than 2,49. So the margarine sold in cups of 500g is less expensive. Also problem (b) can be solved avoiding division, and using only proportional reasoning: 3 cups of the first product is only 450 grams, but it would cost 3*0,93 zl = 2,79 zl, which is 11/4/03

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LECTURE 3 already more than the 500 grams of the second product. So it is less expensive to buy yogurt in the larger cups.

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1999 MATH 645 CONCORDIA UNIVERSITY LECTURE NOTES ON THE THEORY OF DIDACTIC SITUATIONS – ANNA SIERPINSKA

LECTURE 4 THE TOPAZE, THE JOURDAIN AND THE DIENES EFFECTS IN THE PROCESS OF MATHEMATICS TEACHING PART I - PROBLEMS OF TEACHING DIVISION - CONTINUED In analyzing the two sets of division problems in Lecture 3, we found the following conceptions of division:

Elementary arithmetic conceptions: Sharing: A whole number N of objects is shared among a whole number of persons P (or is put in boxes, in piles, etc.). N / P represents the number of objects that each person gets (there might be a remainder). Partitioning: A quantity Q is distributed in packages of q; Q / q represents the number of packages. (Q and q can be any numbers)

A geometric conception: Ratio as a comparison of two magnitudes of the same kind: if A and B are amounts of two magnitudes of the same kind then one can sometimes express the relation between these amounts as a proportion, ‘A is to B as p is to q’, where p and q are whole numbers, and write the ratio as a fraction p/q, without necessarily understanding it as a number in the same sense as whole numbers are numbers; in this conception, a ratio could still be understood as a pair of whole numbers.

A physical conception: Ratio as a compound magnitude: if A and B are amounts of two magnitudes, not necessarily of the same kind, then the ratio of A to B expresses the amount of the first magnitude per a unit of the second magnitude.

An advanced arithmetic conception: Ratio as pure number: in the conception of ratio as a compound magnitude, the ratio of the amounts A to B is abstracted from the magnitudes, and treated as a pure number.

An algebraic conception: Multiplication by the inverse: a/b = a*b-1. These conceptions form a hierarchical list in the sense that every next conception (except perhaps the first two) can be construed on the basis of the previous ones, through generalization, abstraction, and extension of the meaning of terms. But every next conception has to be construed ‘against’ the previous ones because it requires a change of perspective, a certain reorganization of knowledge and even a rejection of beliefs related to the previous ways of understanding division. In both textbooks, division was defined algebraically as the multiplication by the inverse but the students were not given a chance to construct this conception on the basis of and against the previous conceptions. The word problems given to the students could all be solved either with the elementary arithmetic conceptions or were so formulated that the use of the division operation was almost explicitly suggested. Having been given the procedure(s) for performing division the students were able to solve the problems without having to develop the geometric and physical notions of division. But without these conceptions, they had no grounds for developing a meaning for the procedures, and therefore could only learn the procedures but not the algebraic conception of division. The huge gap between the arithmetic conceptions of division and the algebraic conception has not been filled by an appropriate choice of problems and didactic contracts. Everything would be fine if the authors of the textbooks said: our aim is only to teach the students some algorithms of dividing ordinary and decimal fractions or rational numbers. However, if asked, they might claim that they teach the students the notion of division, and support their claims by arguments such as: (1) Aren’t the students choosing the operation spontaneously in solving problems when appropriate? (2) Don’t the students actually correctly perform the operation of division? Our answer would be: Yes, the students were choosing division in solving problems but this operation was strongly suggested by the text or the context of these problems1; the hint was 1

Most problems in the 1936 textbook were included in sections whose titles explicitly mentioned division, like

‘Problems using division of fractions’ (p. 71), and many had verbal clues within the text of the problem, like ‘He divided the nuts into 50 equal shares’, in Problem # 3, or ‘what fraction of the full recipe…’ in Problem #8 , or ‘what was the cost per mile…’ in Problem # 12.

so strong that the students did not have to choose among other possibilities - they knew, by virtue of the implicit contract, that division was what they were expected to do. This phenomenon of giving the answer in the question has been called by Brousseau: ‘the Topaze effect’ (p. 25). Concerning argument (2), what can be called ‘performing the operation of division’ by a mathematician, is not that at all from the point of the student, who is just, for example, changing the sign of “/” to “*”, inverting the divisor, and using the multiplication table. This second phenomenon, of giving a scientific name to a trivial activity has been called ‘the Jourdain effect’ (p. 25-26).

Exercise Find a didactic situation through which the student would be led to construct one of the ratio conceptions of division without being coerced into using this concept by an explicit hint from the teacher. Here is what I think might work with students (but I would need to experiment it to verify my conjecture) with respect to the ‘physical conception of ratio’. I would give the students a project titled: ‘How much of your own weight do you eat?’ Students could investigate this problem concerning themselves, members of their family, their pets (situation of action). They would come back with some data to class, and present the data to the class, perhaps in the form of tables. The problem would then arise of comparing, who eats more than someone else, not in an absolute way, but relative to one’s weight (situation of formulation). For example, children may find out that a cat weighing 5 kg eats 200g of food daily and a dog weighing 20 kg eats 1 kg of food daily. Ways of making such comparisons would then be proposed and defended by different students or groups of students (situation of validation). For example, it could be proposed to compare the ‘food weight per body weight’ quantity. In the case of comparing the absorption of food by the cat and the dog, the question would be reduced to comparing the ratios 200 g to 5000 g and 1 kg to 20 kg. The main problem would then be: how can one compare such ratios? Maybe some students would propose to represent the ratios by fractions: 200/5000 and 1/20, and use the techniques of comparing fractions. If not, then the teacher could suggest this and negotiate the sense of using this representation with the students (institutionalization). Once the students are convinced about this representation, the teacher would pose a question in which the weights themselves would be represented by fractions of some whole and then the ratio would be a ratio of fractions, thus leading to considering the division of fractions as a ratio. For example, the teacher could propose to compare the food weight/body weight ratio of a chickadee and a sparrow, saying that a chickadee eats 9/10 of a small bird feeder daily and weighs 15/16 of the

weight of the feeder, while for the sparrow the ratio is 21/25. Who eats more of his own weight and how much more? The students would then be left on their own to figure out the answer (action), then formulate their strategies (formulation), and compare their strategies for validity (validation). The teacher would then ask the students if it would be possible for them to generalize their results and say how to compare such data as in the situation with the cat and the chickadee. With some hints from the teacher (institutionalization) a formulation, verbal or using letters, of some rule of dividing fractions in the sense of finding the ratio between two fractional quantities could be written. The reason why not just ratios are considered in this sequence of problems but comparison of ratios is that division is necessary only when a comparison has to be made. Otherwise one would be satisfied with saying that, for example, for the chickadee, the food weight / body weight ratio is as 9/10 to 15/16, and no further processing of the data (representing the ratio in the form of one number) would be necessary. PART II - THREE DIDACTIC PHENOMENA: THE ‘TOPAZE EFFECT’, THE ‘JOURDAIN EFFECT’, AND THE ‘DIENES EFFECT’.

1. The ‘Topaze effect’ The name of the ‘Topaze effect’, which has been described above as ‘giving away the answer in the question’ in teaching, comes from a play by Marcel Pagnol, written in 1928, in Paris. Pagnol, born in 1895 in Aubagne, near Marseilles, had been a teacher and taught English in various lycées in Southern France and then in Paris. He abandoned this profession in 1922 and devoted himself entirely to writing for the theater and then for the cinema. The play ‘Topaze’ is set in a private boarding school. Topaze is a teacher in that school. The first scene of the play shows Topaze giving a dictation to a pupil during the recess. The boy, as described by Topaze in Scene III, ‘is a conscientious worker but he had some trouble keeping up with the class because no one seems to have taken an interest in him until now’, and Topaze decided to help him a little in his free time. Here is how the first scene starts: As the curtain rises, Mr. Topaze is giving a dictation to a pupil.… The pupil is a 12-year-old boy. He is turning his back to the public. One can see his ears that stick out and his thin bird-like neck. Topaze is dictating and from time to time bends over the shoulder of the little boy to see what he is writing. TOPAZE: (He dictates while he walks up and down). “Some… lambs… Some lambs… were safe… in a park; in a park. (He bends over the shoulder of the pupil and continues) Some lambs… lambz (The pupil looks at him, bewildered). Now, child, make an effort. I am saying lambz. Were (he repeats very distinctly) we-re. That shows that there was not only one lamb. There were several lambz”. (The pupil looks at him, dazed).

(Marcel Pagnol, Topaze, Translation and Introduction by Renée Waldinger. Great Neck, N.Y.: Barron’s Educational Series, Inc., 1958, p. 2).

Topaze wants the student to succeed; after all, part of the didactic contract is the obligation, for the teacher, to do all he or she can to help the student succeed. But the way he is going about it, is not leading to the student’s learning, but to the student’s producing a correct answer in spite of not having learned anything. I found myself caught into the trap of this phenomenon in my linear algebra class. After having introduced the notion of elementary matrices and discussed and illustrated the fact that an elementary row operation on a matrix A has the same effect as multiplication of A, on the left, by an elementary matrix, I gave the students a problem to solve. In the first question of the problem, a 2x2 matrix was given and the task was to find two elementary matrices E2 and E1 such that E2 E1 A = U, where U is the 2x2 identity matrix. The problem, I said, is a typical final examination problem; this immediately arose the students’ interest. The students were working on the problem in their exercise books, and I was walking around, looking above the shoulders of the students. A few of them were row reducing the matrix, but many were slowly just copying the matrix, and obviously did not know what to do. After 7 or 8 minutes some 6 students out of the 32 in the class had solved the problem. I let the other students work for another 2 minutes, and then got to the board, and said: ‘You have to row reduce the matrix and keep track of the elementary operations; then you just translate the operations into the elementary matrices’. So I told the students how to solve the problem, but did nothing to help them make the conceptual link between elementary matrices and row reduction, or, in other words, to help them understand the theorem. I thought I had explained the theorem sufficiently before, and I simply ignored the fact that, most of the time, explanation does not automatically cause understanding.

2. The ‘Jourdain effect’ The name of this phenomenon of teaching alludes to the play by Molière, ‘The Cit turned Gentleman’ (Le bourgeois gentilhomme), which was first acted at Chambord in October 1670, during the reign of Louis XIV. The play is about the sin of vanity in men who endeavor to appear above of what they actually are. In the play, Mr. Jourdain, a good but simple and not too highly educated citizen with no relations to aristocracy, aspires to becoming part of ‘la noblesse’, if not by blood then at least by manners and education. So he hires a music master, a dancing master, a fencing master, and a philosophy master. The name of ‘Jourdain effect’ refers to Jourdain’s lesson with the philosophy master (Act I, Scene 6), where Mr. Jourdain learns that he had been speaking prose all his life without knowing it:

Mr. Jourdain: … I must commit a secret to you. I’m in love with a person of great quality, and I should be glad you would help me to write something to her in a short billet-doux, which I’ll drop at her feet. … Philosophy-Master: Is it verse that you would write to her? Mr. Jourdain: No, no, none of your verse. Philosophy-Master: You would only have prose? Mr. Jourdain: No, I would neither have verse or prose. Philosophy-Master: It must one or t’other. Mr. Jourdain: Why so? Philosophy-Master: Because, sir, there’s nothing to express one’s self by, but prose or verse. Mr. Jourdain: Is there nothing then but prose, or verse? Philosophy-Master: No, sir, whatever is not prose, is verse, and whatever is not verse, is prose. Mr. Jourdain: And when one talks, what may that be then? Philosophy-Master: Prose. Mr. Jourdain: How? When I say, Nicola, bring me my slippers, and give me my nightcap, is that prose? Philosophy-Master: Yes, sir. Mr. Jourdain: On my conscience, I have spoken prose above these forty years, without knowing anything of the matter; and I have all the obligations in the world for informing me about this.

We have to do with the ‘Jourdain effect’ each time we describe the productions of our students in mathematical terms, which presuppose an elaborate conceptual activity, while having no evidence of such an activity. We say that a student has divided a fraction by another fraction, while it would have been more appropriate to say that the student has changed the sign of division into a sign of multiplication, put the second fraction upside down and used his memory of the multiplication table to produce two numbers separated by a horizontal little line. We also say that ‘a student has solved an equation’ to refer to an activity of transforming an expression containing numbers and letters into another such expression, according to certain rules. Most students have no notion of equation, just as Mr. Jourdain had no notion of ‘prose’. In a recent research made by an M.T.M. student, Sonia Manago2, a pair of 16 years-old students were given a set of algebraic expressions to classify according to criteria of their own choice, and their classification proves quite clearly that ‘solving an equation’ does not mean, for these students, finding those values of the variables for which the equality condition expressed by the equation is satisfied. The girls came up with the following classification, and names for the categories:

2

Manago, S. (1999): Two Students’ Thinking Process on the Notions of Equation and Solution. Research Intership

Project, presented in the Master in the Teaching of Mathematics programme, Concordia University, Montreal.

Equal equations: 2=2, x+1=x+1, 3=7-4, x2=9, x/5 = 0.2 x Not exactly equal equations: 0.99 = 1, 8=9, 3/4 = 5/6 Formulas: A = bh/2, P = xy/2, h = 2A/b, y= 2p/x, Ax+B = Cx+D Equations that equal to zero: 2X = 0, 2a - 5 = 0, 2x - 5 = 0, X = 0, X + 5 = X Simple equations: y = 2x, y = ax+c, y = x+1, y= x2 + 1, x2 + 1 Complicated equation: 2x - x Y or X equals a fraction: y = 5/2h, x = 5/2, y = 5/2 x In the interview with Sonia, one of the students expressed her doubts with accepting x=0 as an equation; she said, ‘I don’t really think it’s anything. I think it’s an error!’. Asked, ‘what is a solution to an equation’, one of the students said, ‘The answer to an equation’, and the other ‘It’s the answer to the equation you are trying to figure out. So if you have a long equation, you do all the steps to get to the answer… It’s the final answer’. For the latter student, the solution to x+5 = x was ‘5=0’, because this was ‘the final answer’.

2.1 An extreme case of the Jourdain effect: the mathematical program of Zoltan Dienes. Extreme cases of the ‘Jourdain effect’ could be observed during the wave of the so-called New Math reforms in 1960s and 70s. Children were given all sorts of ‘manipulatives’ to play with, toys, dolls, and blocks. When children were sorting toys, their activities were called using the terms of set theory such as ‘finding the intersection of two sets’. The case of claiming that some kindergarten or first grade children manipulating cups of yogurt ‘constructed the group of Klein’ became legendary; it was then used as a joke to ridicule the New Math approaches and unjustified ambitions (p. 139). A side explanation: the group of Klein

Here is an explanation of what the group of Klein is and what it may have to do with the manipulation of cups of yogurt, for those of you who never heard of it before. A ‘group’ in mathematics refers to an algebraic structure of a special kind. An algebraic structure is a set closed under some operations defined on the elements of that set, and satisfying some conditions (‘axioms’). From this point of view, the set of all real numbers with the operations of addition and multiplication, is an algebraic structure; because of the properties of these operations (commutativity, associativity, distributivity, existence of a neutral element for

each operation, namely 0 and 1, existence of the additive inverses and multiplicative inverses) this structure is called a ‘field’. Now, a ‘group’ is any non-empty set, say G, closed under one operation, say *, which satisfies the following conditions: (1) the operation is associative (2) there exists, in G, a neutral element for that operation (i.e. an element e such that for element x in G e*x = x*e = x) (2) for every element x in G there exists an element y in G such that x*y = e (i.e. every element in G has an inverse in G). ‘Group’ is, obviously, a theoretical model of such familiar structures as the set of all integers with respect to addition, the set of all real numbers except zero with the operation of multiplication, the set of all real numbers with respect to addition, the set of all rotations of the plane around the same point, the set of all dilations of the plane with respect to the same point, etc. All these examples are examples of infinite groups, i.e. groups composed of an infinite number of elements. But there exist groups with a finite number of elements, and these were considered appropriate to be introduced to very young children at the time of the New Math reforms. The reformers were supporting their projects by reference to Jean Piaget’s psychological theories, and, in particular, to his thesis that the most primitive cognitive ‘operational structures’ in the child resemble the three most general mathematical structures, which he called ‘mother structures’, because all other can be derived from them: the algebraic structures, of which the structure of group of transformations is the prototype; the order structures for which the structure of lattice of relations is the prototype; the topological structure of objects invariant under continuous transformations (Piaget, 1969, p. 673). According to Piaget, child’s mental development follows the logical development of the theory, from the most general ideas to the most particular ideas, rather than the historical order of the development of mathematics, from knowledge of individual cases, through abstraction, to the general cases. The child is cognitively organizing her experience in transforming objects according to the possibility of reversing and combining these transformations in her mind if not in reality. The child’s world in organized in hierarchies of relations between things, events and people that can be modeled by the structure of lattice. The child’s geometry is closer to topology than the Euclidean metric geometry which is taught at school; in particular, any closed smooth curve is put in one class by children and called ‘a round’ - topologically, all such curves are indeed, continuous one-to-one images of a circle.

3

Piaget, J. (1969): Psychologie et pédagogie. Éditions Denoël.

From this ‘logical parallelism’ he concluded that teaching of mathematics should be focused, in early grades, on set theory and isomorphisms of structures. The New Math curricular movement fully endorsed this point of view. However, Piaget was warning the reformers against assuming that the child’s cognitive structures are some kind of conscious knowledge, and was even evoking Molière’s Mr. Jourdain to make his point. For him, the discovery of the ‘logical parallelism’ does not solve the pedagogical or didactic problem of teaching mathematics; the problem remains in the form of finding a way of helping the students pass from an unconscious use of the structures to a reflection and theorization of these structures in a symbolic language. … il se trouve que ces trois structures mères correspondent d’assez près aux structures opératoires fondamentales de la pensée. Dès les «opérations concrètes»… on trouve des structures algébriques dans les «groupements» logiques de classes, des structures d’ordre dans les «groupements» de relations et des structures topologiques dans la géométrie spontanée de l’enfant (qui est topologique bien avant d’atteindre les formes projectives ou la métrique euclidienne, conformément à l’ordre théorique et contrairement à l’ordre historique de la constitution des notions)… S’inspirant des tendances bourbakistes, la mathématique moderne met donc l’accent sur la théorie des ensembles et sur les isomorphismes structuraux plus que sur les compartimentages traditionnels, et tout un movement s’est dessiné qui vise à introduire de telles notions le plus tôt possible dans l’enseignement. Or, une telle tendance se justifie pleinement, puisque précisement les opérations de réunions ou d’intersections d’ensembles, les mises en correspondance sources des isomorphismes, etc., sont des opérations que l’intelligence construit et utilise spontanément dès 7 ou 8 ans et bien plus encore dès 11-12 ans… Seulement l’intelligence élabore et utilise ces structures sans en prendre conscience sous une forme réfléchie, non pas comme M. Jourdain faisait de la prose sans le savoir, mais plus encore comme n’importe quel adulte non logicien manipule des implications, des disjonctions, etc., sans avoir la moindre indée de la manière dont la logique symbolique ou algébrique parvient à mettre ces opérations en formules abstraites et algébriques. Le problème pédagogique subsiste donc entièrement, malgré le progrès de principe réalisé par le retour aux racines naturelles des structures opératoires, de trouver des méthodes les plus adéquates pour passer de ces structures naturelles mais non réfléchies à la réflexion sur de telles structures et à leur mise en théorie (Piaget, ibid. P. 67-68).

But let us come back to our explanation of the notion of ‘group of Klein’. Felix Klein was a German mathematician of the second half of the 19th century, who contributed to the theory of groups, among others. The group of Klein is a four-element group, whose all elements are inverses of themselves. More precisely, any set {e, a, b, c} with an operation * such that the following ‘multiplication table’ holds, *

e

a

b

c

e

e

a

b

c

a

a

e

c

b

b

b

c

e

a

c

c

b

a

e

One can construct many different models of this group. One is, for example, the set of all plane symmetries that transform a figure made of two parallel segments of equal length, such as the equal sign "=", into itself. There are four such symmetries: the identity transformation, a vertical flip, a horizontal flip and a central symmetry. It is easily verified that each of these symmetries, when repeated, brings the figure back to the initial position and that the combination of any two non-trivial out of them gives the third. Another example of a model of this group can be constructed using some transformations of the position of a cup of yogurt, which has some inscription or decoration on its side. Suppose at the beginning the cup is standing upright with its decoration in the front. Four transformations of the cup’s position are considered: the cup is left untouched (e); the cup is put upside down so that the decoration goes to the other side (a); the cup is put upside down so that the decoration stays on the same side (b); the cup is given a half turn (c). It is easy to verify that these transformations form a group of Klein. As children were playing with the cups of yogurt, they were performing such transformations and their combinations and were, maybe, noticing, that doing the same transformation twice would put the cup back in the original position, and that combining any two of the transformations a, b and c would give the third one, and from this, the ‘structurally minded’ observers were concluding that the children ‘have constructed’ the group of Klein. But what the children were actually doing had nothing to do with the identification of the group structure in their manipulations. They were even more in the dark than Mr. Jourdain, because they would not have been able to identify the part of their activity called ‘construction of the group of Klein’, as Mr. Jourdain could identify when he was speaking prose and when he was not; and they would not be able to produce, as Mr. Jourdain could, further examples of their activity, now sanctified by a scientific term.

3. The ‘Dienes effect’ The Dienes effect has more to do with the work of researchers and innovators in mathematics education than with the classroom practices of teachers, like the previous two phenomena. But it is important for us to understand because we are all, in this class, at least part time researchers and innovators in mathematics education. It is important also for our understanding of the theory of didactic situations, because this theory was born, in part, as a result of Brousseau’s understanding of what was wrong with Dienes’ theory of mathematical instruction. We start with presenting, briefly, the person of Zoltan Paul Dienes and his theory.

3.1 Zoltan P. Dienes During the period of the New Math reforms Zoltan P. Dienes, a Hungarian by origin, but fluent in English, French, German and Italian, was a professor at the University of Sherbrooke, in Québec (Servais & Varga, 1971, p. 394). He became well-known to teachers and parents of elementary school children around the world for his blocks (ibid., p. 38, 108) designed for the teaching of position systems of writing numbers in various bases, as well as blocks for the teaching of logic (the set, available commercially in some countries, was often called ‘Dienes blocks’). For mathematics educators he became known for his theory of the ‘psychodynamic process’ of teaching and learning mathematics. When the New Math reforms started being criticized and withdrawn from schools, Dienes left Sherbrooke, and now lives in Wolfville, Nova Scotia, Canada. He still writes books. His latest is "I'll tell you algebra stories you have never heard before" (2002; Upfront publishing, Leicestershire). [the last three sentences were written in 2003].

3.2 The concerns of Zoltan Dienes in mathematics education; the definition of the ‘Dienes effect’ Dienes’ main problem with mathematics education was no different than that of Brousseau: the discrepancy between the stress, put in mathematics education, on teaching of techniques and the attention awarded to the teaching of the understanding of mathematical ideas (Dienes, 1960, p. 45). But, unlike Brousseau, he did not seek to find the causes of the phenomenon and ways of shifting the pedagogical scales more towards understanding of the actions of the teacher and her interactions with the students and the mathematical content. He ‘rule[d] out bad teaching as a regular contributory cause to the present state of affairs’ (ibid., p. 5), and set to devise an instructional theory and exemplary teaching materials that would be, in a sense, ‘teacher-proof’: if administered according to precise instructions of the teacher’s guide, they should work with any teacher. But they didn’t; Brousseau has tried them himself and observed them when used by other teachers and discovered that they only ‘work’ with a committed teacher who, wanting to prove that they do work, would make special interventions and modifications, in order to give meaning to the students’ actions and help the students become aware of this meaning. The materials did

4

Servais, W. & Varga, T (eds). (1971): Teaching School Mathematics. A UNESCO Source Book. Middlesex,

England: Penguin Books. 5

Dienes, Z.P. (1960): Building Up Mathematics. London: Hutchinson International.

not induce understanding in students who would work with a teacher who would only just hand out the worksheets, and encourage the students to continue. This belief in the existence of some kind of infallible ‘artificial genesis of mathematical knowledge’ that would be independent of the teacher’s personal investment in the learning process has been called the ‘Dienes effect’ (p. 37).

3.3 Dienes’ instructional theory The central concept of both Dienes’ theory and Brousseau’s theory is ‘game’. But for Dienes, a game is a special type of play: it is defined as a rule-bound play (Dienes, 1963, p. 24) 6, and it is not assumed that, in a game, one must have winners and losers, and that there are some strategies for winning the game, as in Brousseau’s theory. In the theory of didactic situations (TDS), a game is more of a problem situation; the problem is to find a way or a strategy to win the game. The discovery of the finding of this strategy is done through the construction of some new knowledge or a new understanding of some old knowledge, and it is this knowledge that is the true outcome of the game for the winner: she has learned something new. In TDS knowledge is the outcome of the game but the game does not resemble this knowledge, and the players are not playing with that knowledge; they are using or expected to use that knowledge to win the game. But ‘playing with knowledge’ or ‘playing the target mathematical knowledge’ is exactly what the games are all about in Dienes’ theory. In order to teach the students a mathematical notion, Dienes proposed to give students sets of manipulatives so structured as to faithfully represent this notion. The students would thus be given some physical representations of the mathematical notion. They would first be allowed to freely play with one representation (Brousseau, TDS, p. 139). Then the play would become more structured: the students’ attention would be directed to some properties or actions that would be specific of the target mathematical notion. The students would then be given several other physical representations of the notion and would be directed to focus on the things that were common in all those different ‘games’ they were playing (the stage of isomorphic games and abstraction). Next, the students would be encouraged and helped to construct a schema of all these games, and formulate it in words or represent it by spontaneously drawn diagrams (the stage of schematization and formulation). The sequence would stop at that stage in earlier grades. In secondary school, the sequence could continue with a symbolic representation of the schema in some accepted mathematical notation, and the formalization of the properties of the target notion in the form of an axiomatic theory (the stages of symbolization, formalization and

6

Dienes, Z.P. (1963): An Experimental Study of Mathematics-Learning. London: Hutchinson.

axiomatization) (ibid.). The ‘notions’ that Dienes had mostly in mind were those closely related to Piaget’s ‘mother structures’ and logic: algebraic structures such as groups of transformations or vector spaces, combinatorial problems of ordering, sets and operations on sets, laws of predicate calculus.

3.4 Examples of Dienes’ ‘lessons’ Example 1: Vector spaces

During a sequence of activities supposedly leading to the identification, by the children, of the structure of vector space, children are given objects (or pictures of objects) of several kinds: two kinds, three kinds, four kinds. The objects are so chosen as to appear in two ‘opposite’ states. For example, in one activity, they are given cups, gloves and boxes. The cups can be right side up, or upside down. The gloves can be right side out and inside out. The boxes can be open or closed. The children are asked to count how many items of each kind they have, assuming that the rule of this game of counting is that opposite states cancel each other out, so that if a child has, in her set, an open box and a closed box, she puts the pair aside and does not count it (Dienes, 1963, p. 32-33). For example, if a child has 3 closed boxes, 5 open boxes, 2 gloves right side out and 1 glove inside out, 6 cups right side up and 3 cups upside down, her ‘book-keeping record’ would show: Boxes

Gloves

Cups

2o

2r

3u

The children would be then encouraged to put their and a neighbor’s collections together and count the items. A bookkeeping record of such a ‘merger’ of possessions could be: Boxes

Gloves

Cups

Jane

2o

2r

3u

Mary

5c

3r

1d

Together

3c

5r

2u

Children would also be asked question such as, ‘What would be the state of your record if you doubled/tripled, etc. your possessions?’, ‘What would happen if your neighbor had the same number of items of each kind but in the opposite state?’, etc. Playing this way with a variety of collections of objects, the children would be encouraged to speak about the common features of all such situations and notice some common

features, which could eventually be schematized in the form of, say, a set of book-keeping records regarding a number of kinds of things, each kind of thing appearing in two possible states; the totals of the records can be made by addition and subtraction of numbers in each rubric. From this kind of plays to the concept of vector space, claimed, by Dienes, to be the target mathematical notion, there is still a long way to go. First of all, in a vector space, one must be able to multiply vectors by any number, not just a whole number. In plays with discrete items such as boxes or gloves, it does not make sense to multiply even by fractions: what is 1/2 of a glove?! But Dienes was not claiming that the target had to obtained in a short sequence of sessions. The sessions could be scattered along one year, two years or more, and the ultimate stage of axiomatization would not be attained till the end of secondary school. Example 2: A lesson in logic

The description of this lesson, actually conducted by Dienes in a French-speaking classroom in Sherbrooke, comes from (Servais, Varga, ibid. p. 38-40). Children are given a set of the so-called ‘attribute blocks’ (also called Vygotsky’s logic blocks, or Dienes’ logic blocks). Each of the blocks can have one of the 4 kinds of attributes: Shape: round, triangular, square, rectangular (but not square) Color: yellow, blue, red Size: big, small Thickness: thick, thin The children work on various problems in groups. Not all groups work on the same problem. In one of the groups, the teacher suggests that the children pick out of the whole set four blocks. Suppose the children have picked the following four blocks: one triangular, red, small, thick one square, yellow, small, thin one square, blue, big, thin one square, blue, small, thick The teacher asks the children to formulate sentences of the form ‘If… then… ‘ about this set of blocks and then decide which are true and which are not. Children formulate sentences and write them down in two columns, one for true sentences and one for false sentences. Here is a sample noted by the observers of the lesson: True

False

If red then triangle

If red then square

If triangle then red

If triangle then blue

If blue then square

If square then blue

If thick then small

If small then thick

If large then blue

If blue then large

Now the teacher changes the assignment. He writes the last true sentence on a slip of paper and asks the children to add a fifth block, which would make the sentence false. A child proposes to add a large yellow and thin round block. Then the children are invited to produce sentences that are true about this new set. One of the sentences is: ‘If large then thin’. The teacher again proposes to add a block that would make that sentence false. And so it continues. When the children arrive at ten blocks it is not easy to find a simple sentence that is true. So they start the game over game, now with one of the children proposing a true statement about a few blocks, and another falsifying it by enlarging the set.



LECTURE 5 MORE PHENOMENA OF TEACHING: THE METACOGNITIVE SHIFT, THE METAMATHEMATICAL SHIFT, THE IMPLICIT SUGGESTION OF ANALOGY, THE PARADOX OF LEARNING BY ADAPTATION, THE PARADOX OF THE ACTOR

This week we continue discussing the phenomena of mathematics teaching as identified by Brousseau. 1. THE METACOGNITIVE SHIFT The phenomenon of the ‘metacognitive shift’ occurs when a teaching aid becomes an object of teaching itself (p. 26-7). Teachers use all kinds of teaching aids to convey the meaning of the abstract mathematical concepts. They can be material objects such as counters, sticks, blocks, or graphical representations, or orally communicated metaphors. However, the interpretation of actions on and with these objects as representations of the particular mathematical concepts that the teacher has in mind requires that the students focus their attention on certain features of the objects and not on others, and manipulate these objects in some appropriate ways for this particular goal. Otherwise, they may miss the concept completely. If, in order to avoid this, the teacher starts teaching the students the rules of interpreting and using things that were supposed only to help the students grasp the meaning of a mathematical concept, we have to do with the ‘metacognitive shift’ in teaching. Example 1: The Venn diagrams When, during the New Math reform period in the 1960s, elementary school children were taught operations on sets, they could not be expected to reason about them using the language of formal logic. Thus a less formal language was needed, and it was proposed to use the so-called Venn diagrams. A Venn diagram is composed of a number of circles, each representing a set, and the mutual position of the circles is intended to represent the relation between the sets. Thus, for example, the set A» (B« C) could be represented by the diagram:

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1



The set (A» B) « (A» C) can be represented by the diagram

Comparing the two diagrams the students would be expected to ‘discover’ the distributivity law: A» (B« C) = (A» B) « (A» C) . The representation seems to be quite simple and straightforward, as long as it serves not as a formal language but a heuristic tool; something one quickly draws such a diagram on scrap paper to convince oneself about the truth or falsity of a relation between sets, and then writes a proof in ordinary mathematical language (based on the definitions of set operations and the tautologies of the predicate calculus). But, in the teaching of set theory at an elementary level the ‘ordinary mathematical language of set theory’ was not available. So there was no alternative language to communicate and test the validity of statements: there was only the ‘language of

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strings’1 or ‘Venn diagrams’. To smooth out communication between the teacher and the students, and avoid misunderstandings, the ad hoc drawings for the personal use of mathematicians had to be developed into a language. And this is indeed what we can observe in the abundant literature for teachers produced at the time of the reforms. For example - in the ‘Unified Modern Mathematics’ series (1972) produced for the use of Teachers College, Columbia University by an impressive international board of mathematics educators called ‘Secondary School Mathematics Curriculum Improvement Study’ (including, beside the American teacher educators, such ‘big names’ as Gustave Choquet from France, Lennart Råde from Sweden, and Hans-Georg Steiner from Germany). The Volume 2.1 of this series contains 6 pages of all kinds of rules concerning the interpretation of Venn diagrams. In particular, it is proposed to write the symbol ø in a region of a Venn diagram to mark that it represents an empty set and the symbol ‘x’ to mark that the region is not empty (ibid., p. 14)2. In this approach, the students were given exercises just for the practice of the conventions of the representation (ibid., p. 19). At the time of the New Math reforms, Venn diagrams were used everywhere, but in some places this representation received more attention than in others. In Europe the most famous advocate of the New Math reforms who contributed a lot to the spreading of the use of the ‘language’ of Venn diagrams and arrow diagrams (for relations and functions) was certainly the Belgian mathematician Papy. He added color coding to his Venn diagrams and represented empty sets not by the symbol ø but by hatching the regions of the diagram3. When presenting the phenomenon of the ‘metacognitive shift’, Brousseau gives the example of the abuse of the Venn diagrams used in teaching set algebra and arrow diagrams in teaching about relations and functions, but can we think of other examples, not necessarily coming from the history of the New Math reforms but from the present day mathematics teaching? I have thought of two examples; can you come up with more?

1

In some American textbooks for teachers, the expressions ‘the language of strings’ rather thah ‘Venn diagrams’ was used, e.g. Comprehensive School Mathematics Program 1978, CSMP Mathematics for Intermediate Grades. Part III: The Languages of Strings and Arrows. CEMREL, Inc. (USA). Experimental Version 5 -25401. 2 See Appendix 1. 3 Papy (with the collaboration of Frédérique Papy), 1968: Modern Mathematics, Volume 1. London: Collier-Macmillan Ltd. (Papy signed his books with his last name only).

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Example 2: The metaphor of scales in teaching equations The metaphor of ‘scales’ and ‘balancing of scales’ has been used extensively for the purpose of giving meaning to operations on equations such as, e.g. adding or subtracting the same number to or from both sides of the equation4. For the metaphor to work, one must think not of the modern scales, with just one plate, which automatically displays the weight (and sometimes the price) when something is put on the plate, but of the old-fashioned two-plate scales with the product put on one plate and weights on the other. Such scales are, presently, museum objects, and most high school children have never seen them. The drawings of scales in textbooks thus become representations of objects just as abstract as the concept of equation. The understanding of the metaphor requires therefore the teaching of the rules of interpretation of the sequences of drawings of the scales with different objects on both sides and the focusing of the students’ attention on whether the plates are drawn on the same or different level - they have no notion of ‘balancing the scales’. A teacher might, of course, just ignore the metaphor and teach the operations on equations directly, but some teachers do work with the metaphor for a long time with the students, and produce a whole ‘language of scales’ to reduce the ambiguity of communication, just as their predecessors were producing the ‘language of Venn diagrams’. The risk of ambiguity and unintended interpretations of the scales metaphor is real; there are accounts of this happening in published research papers56.. Example 3: The use of technology in mathematics teaching: can it favor the ‘metacognitive shift’ phenomenon? An important part of the controversy about using or not using calculators and mathematical computer software is concerned with exactly the risk of teaching the teaching aid rather than the mathematics it is supposed to help understanding. In my own practice of teaching linear algebra with Maple, I remember that, at the beginning, I used to spend a lot of time teaching my students the commands of Maple and the quite awkward syntax of the software. Students were spending a lot of time trying to figure out why a command didn’t work they way they expected (just to find out, for example, that they had forgotten to put the semi-colon at the end of the command line). 4

See Appendix 3: two pages from a Polish tetxbook for Grade 7 students: Zawadowski, W. et al., 1996: Matematyka 2001. Podrecznik dla klasy 7. Warszawa: Wydawnictwa Szkolne i Pedagogiczne. English translation of the text is provided. 5 Pirie, S.E.B. (1998): Crossing the Gulf between Thought and Symbol: Language as (Slippery) SteppingStones. In H. Steinbring, M.G. Bartolini-Bussi, A. Sierpinska (eds.), Language and Communication in the Mathematics Classroom. Reston, VA: National Council of Teachers of Mathematics, pp. 7-29. 6 MacGregor, M. (1998): How Students Interpret Equations: Intuition versus Taught Procedures. In H. Steinbring, M.G. Bartolini-Bussi, A. Sierpinska (eds.), Language and Communication in the Mathematics Classroom. Reston, VA: National Council of Teachers of Mathematics, pp. 262-270.

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At the end, they were not sure what they are learning in the course: linear algebra or the Maple language. Today, I no longer demand that the students do their homework assignments using Maple; I allow them to do so, if they want. Each student in the class has the right to one-hour tutorial on the use of Maple, and I provide help during my office hours and via e-mail. I am using Maple for teaching: my lectures are written in a Maple worksheet. The worksheet is active during the class; it is projected on a large screen and student generated examples and conjectures are tested using the software, thus avoiding tedious calculations by hand. But, in my lectures, I do not discuss the syntax of the language nor do I make any suggestions of the type, ‘Look, here is a useful command’. However, in printing the class notes for the students I do not delete the commands; I was explicitly asked by some students to leave them on, so that they can use them when working with Maple on their own. It is not clear, however, if technology can be classified as a ‘teaching aid’, aimed at overcoming the difficulties in the weak student and enhance understanding in the stronger student. In the mathematician’s hands technology is an instrument, useful or sometimes even indispensable in the execution of certain tasks. And then, the teaching of how to use technology in doing mathematics is not a symptom of meta-cognitive shift. It is the teaching of the craft of using a tool in an intelligent way. Some research which has already been done on the teaching and learning of mathematics with technology7, shows that for many students the calculator is the ultimate reference and not a ‘mathematical instrument’ in solving problems. This research also points to the need of preparing the students to correctly interpret and use computer’s outputs. For example, if I ask Maple to solve the equation x2 - (1+√2)x + √2 = 0, I obtain 1, and √2. But when I ask Maple to verify if √2 satisfies the equation, the response is ‘false’ (see the bordered figure below). One has to understand that for Maple, √2 does not exist; ‘√2’ is only a symbol for a decimal number, an approximation of √2. But no approximation of √2 satisfies the above equation.

7

see, for example, Guin, D. & Trouche, L. (1999): The Complex Process of Converting Tools into Mathematical Instruments: the Case of Calculators. International Journal of Computers in Mathematical Learning 3, 195-227.

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> f:=x->x^2-(1+sqrt(2))*x+sqrt(2); 2 f := x -> x

- (1 + sqrt(2)) x + sqrt(2)

> solve(f(x)=0,x); 1, √2 > evalb(f(1)=0); true > evalb(f(sqrt(2))=0); false

If I ask the calculator TI-92 to solve the same equation in the exact mode, the response I obtain is ‘x = -(√(3-2√2) - √2 - 1)/2 or x = (√(3-2√2) + √2 + 1)/2’. Of course, 3 - 2√2 = (1 - √2)2 and the same numbers 1 and √2 are obtained, but the calculator will not spontaneously simplify an expression. Besides, one has to know that the calculator is simplifying an expression upon the command ‘expand’ and it’s no use looking for a ‘simplify’ command, because it does not exist.

2. THE META-MATHEMATICAL SHIFT This phenomenon, only briefly mentioned by Brousseau (p. 39), consists in the ‘replacing [by the teacher] of a mathematical problem by a discussion of the logic of its solution and attributing all sources of error [in its solution] to [a misunderstanding of this logic]’. An example of this could be the situation where the teacher, in the aim of improving the students’ performance in solving equations and understanding what they are doing, teaches the students a theory of equations: gives a definition of an equation, and before that, a definition of a variable, an algebraic expression, the logical axioms of equality (reflexivity, symmetry and transitivity) and the algebraic properties of equality in number systems (e.g. if a = b and c = d then a + c = b + d), etc. Such knowledge belongs to the so-called ‘meta-mathematics’, i.e. a theory of the language of mathematics, which makes abstraction from the intuitive meanings of particular mathematical statements or objects and occupies itself only with their general form. The New Math reforms were fraught with proposals amounting to no more and no less than teaching meta-mathematics to school-children; after all, logic and theory of relations can be regarded as parts of meta-mathematics. Today, the shift is less felt, but some elements of it still

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exist, I believe, in the introduction of equations in secondary schools via the notions of ‘open’ and ‘closed sentences’. 3. THE IMPLICIT SUGGESTION OF ANALOGY Most of the phenomena of teaching mathematics we have reviewed so far represent the different ways in which teachers try to (a) maintain the fiction that learning does, indeed, take place, and that, therefore, they are doing what is expected of them as teachers (Topaze, Jourdain), or, (b) genuinely help the students learn better, but the method chosen does not and cannot bring about the expected results (Dienes, the meta-cognitive and the meta-mathematical shifts). The phenomenon discussed in this section (p. 27) belongs to the category (a) of these phenomena. The teacher gives the students problems formulated so as to highlight their analogy with problems previously solved by the teacher on the board or discussed in class. The teacher does not want to explicitly say, ‘solve this problem just like we solved problem number so and so’ but she gives a hint, sometimes just by asking the question in exactly the same form. The students are expected, in this game, to get the hint. The teacher feels miserable when they don’t because this forces him or her to explicitly point to the analogy and the falsehood of her game is revealed. 4. THE PARADOX OF LEARNING BY ADAPTATION The remarks that Brousseau is making in the section ‘Paradoxes of learning by adaptation’ (pp. 44-47) could be understood as a criticism of the constructivist epistemology and psychology of learning and a promotion of the interactionist stance in these matters. Constructivism as a psychology and epistemology made its way into mathematics education in the late 70s and, in the USA, it generated a lot of basic research into children’s processes of acquiring a notion of natural number, fractions, arithmetic operations, and also more mature students’ processes of learning such notions as the exponential function8. It developed into a certain ‘ideal’ of teaching mathematics, where there would be no lecturing, no drill exercises, but individual children ‘constructing their own knowledge’ (a well known constructivist slogan) by solving problems, with the teacher’s role reduced to that of an interviewer (‘Tell me how you solved this problem’). By adapting to a stimulating environment, with interesting problems, children’s ‘cognitive structures’ would grow and evolve in a natural 8

See, for example, the volume: Steffe, L.P. & Gale, J. (Eds.), 1995: Constructivism in Education. Hillsdale, New Jersey: Lawrence Erlbaum Associates, Publishers.

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way. The metaphor here is that of a biological organism changing through adaptation to its environment. From the constructivist perspective, as Piaget stressed, knowing is an adaptive activity. This means that one should think of knowledge as a kind of compendium of concepts and actions that one has found to be successful, given the purposes one had in mind. This notion is analogous to the notion of adaptation in evolutionary biology, expanded to include, beyond the goal of survival, the goal of a coherent conceptual organization of the world as we experience it. (Von Glasersfeld, 1995, p. 7)9.

Constructivism has developed in opposition to Behaviorism, a theory of learning quite prominent in the USA, based on the fundamental assumption that a rewarded response is the action that will be repeated, ‘reinforced’. Behaviorism led to a theory of instruction based on drill and practice and a system of rewards and punishments. Constructivism attracted mathematics educators interested not only in early childhood education, but also those working with university students. For example, the so-called APOS10 theory of learning mathematics advanced by Ed Dubinsky and his collaborators, is firmly based in the constructivist epistemology. Here is how Dubinsky defines mathematical knowledge: Mathematical knowledge is an individual’s tendency to respond, in a social context, to a perceived problem situation by constructing, re-constructing and organizing, in her or his mind, mathematical actions, processes, objects and schemas with which to deal with the situation (Dubinsky, 1997, p. 95)11.

Thus, for a constructivist, knowledge is a psychological entity: an individual’s network of cognitive structures, schemas, constructed through the individual’s experience in solving all kinds of problems, practical and theoretical. From this perspective, objectivist notions such as ‘truth’ and ‘validity’ of knowledge which refer to a ‘correct representation of reality’, do not make sense; they are replaced, in constructivism, by the notion of ‘viability’. The very concepts of ‘correct’ and ‘reality’ are questioned by constructivism. To the biologist, a living organism is viable as long as it manages to survive in its environment. To the constructivist, concepts, models, theories, and so on are viable if they prove adequate in the contexts in which they were created. Viability - quite unlike truth - is relative to a context of goals and purposes. But these goals and purposes are not limited to the concrete or material. In science, for instance, there is, beyond the goal of solving specific problems, the goal of constructing as coherent a model as possible of the experiential world (Von Glasersfeld, ibid.).

9

von Glasersfeld, E. (1995): A Constructivist Approach to Teaching. In L.P. Steffe & J. Gale (eds.), Constructivism in Education. Hillsdale, New Jersey: Lawrence Erlbaum Associates, Publishers, pp. 3-15. 10 ‘APOS’ stands for Action - Process - Object - Schema. 11 Dubinsky, E. (1997): Some Thoughts on a First Course in Linear Algebra at the College Level. In D. Carlson, C.R. Johnson, D.C. Lay, A. Duane Porter, A. Watkins, W. Watkins (eds.), Resources for Teaching Linear Algebra. The Mathematical Association of America, MAA Notes, Volume 42, pp. 85-105.

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Brousseau holds a very different view of learning and knowledge. He claims that the notion of learning by adaptation is inconsistent (p. 44-45). ‘Adaptation - he claims - contradicts the idea of new knowledge’. If a person solves a problem different from all the problems she had solved so far, with some adaptation of the knowledge she already had, why would she think she has invented some new knowledge? In a similar situation she could go, if necessary, through the same process, from scratch, and therefore there is no need to identify the process as a new ‘method’ or ‘new knowledge’. But if this person shows her way of solving the problem to some other persons who were also trying to solve it but were no able to, then their interest in it, eagerness to understand it, and their appraisal of its more general and not just local value, will indicate to her that some new knowledge has actually been invented. This is what appears to be meant by the statement: ‘knowledge is almost the cultural recognition that direct knowing is impotent to solve some situations naturally (by adaptation)’ (p. 45). This view of knowledge is strongly reminiscent of the position taken by interactionism in social psychology. Interactionism in social psychology has its roots in the pragmatic positions of Peirce, James, Dewey (USA, end of the 19th and beginning of 20th century) and the sociological research of the so-called Chicago school of sociology12. Its epistemology stresses the roles of experience, the common sense, and the existence of multiple interpretations. All one knows is experience; but this experience is not a sequence of isolated sensations but a culturally shared world which we take for granted in the everyday life. Experience is thus the common sense, where the ‘common’ means the ‘shared’. And ‘shared’ means ‘objective’. We do not want to have ‘private knowledge’ or ‘subjective knowledge’ - we do not value it. When we notice that our knowledge is different from the shared knowledge, because, for example, we are unable to achieve goals that others achieve, we treat this knowledge as something subjective and we reconstruct it until it allows us to achieve the goals. Then we treat it as objective again (Hammersley, 1989, p. 56). The basic unit of the interactionist psychology is a goal directed action; it is assumed that our actions are social, even if we perform them alone, because we are able to view ourselves as objects. In undertaking an action we imagine the effects that it will produce on others and how they could react (ibid., p. 59): It is this socially generated ability to view oneself as an object, and to interpret the world in alternative ways, that allows people to modify their interpretations and to choose different courses of action. As a result, human action… is constructed through a reflexive process which takes the form of a person making

12

See an interesting account of interactionism in Hammersley, M. 1989: The Dilemma of Quantitative Method. Herbert Blumer and the Chicago Tradition. London: Routledge.

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indications to himself, that is to say, noting things and determining their significance for his line of action. (Blumer, cited in Hammersley, ibid., p. 130).

Applying this theory to learning, one might perhaps say that, from the interactionist perspective, new knowledge coincides with choosing different courses of action, and different interpretations of a class of situations, and this can happen only when the learner sees himself or herself as an object - a member of a society and judges his or her actions from that external point of view. In a sense, one can learn new knowledge if one is, at once, a learner, and one’s own teacher. In a sense, also, one knows only if one ‘acts’ one who knows - ‘if one acts as a knowledgeable person’, according to what the society takes for granted as the behavior of a knowledgeable person. Only ‘shared’ or ‘public’, or culturally identified as such knowledge is considered to be knowledge. 5. THE PARADOX OF THE TEACHER AS LEARNER For constructivists, the ideal teaching situation is one where the teacher does not know the solution of the problem given in a didactic situation. The problem could have been invented by the student, as a result of being in some more general problem situation, and the teacher and the student work together on it as partners-in-mathematics. Brousseau compares this situation to one in a theater, where the actor would not just act a feeling (e.g. joy or anger) but actually experience this feeling. Referring to Diderot’s analysis of acting, Brousseau claims that this would result in quite poor performance: such acting might not be very convincing: ‘the more the actor feels emotions he wants to play, the less he is able to allow the audience to share this feeling’ (p. 46). Being on stage, visible for the audience, the actor could become ashamed of his private feelings and try to conceal their perceptible symptoms rather than amplify them, which is what he has to do if he wants the audience to understand what is going on. The point Brousseau is trying to make, I presume, is that if a teacher finds it useful to act as if she did not know how to solve the problem, this should only be good acting and not the actual state of the teacher’s mind. Not only should the teacher know the knowledge that she intends to teach but she should use all the means in her repertoire of ‘didactic tricks’ to put on stage and devolve (p. 31) to the students a problem situation they would consider it their own responsibility to solve and which would lead them to develop or use the knowledge in question.

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MATH 645: Theory of Situations/ Lecture 6 MATH 645: Theory of Situations/ Lecture 1

Instructor: Anna Sierpinska Instructor: Anna Sierpinska

LECTURE 6 DIDACTIC ENGINEERING 1. THE NOTION OF FUNDAMENTAL SITUATION.

1. DEFINITION OF THE METHODOLOGY OF DIDACTIC ENGINEERING Suppose you want to conceive and try a new way of teaching a piece of mathematical knowledge. Then you have, roughly speaking, two ways of going about that. Choice 1: The Comparative Study of Experimental and Control Groups You write down a scenario of classroom activities, with a precise description of the role and actions of the teacher and the expected reponses of the students. The scenario contains advice for the teacher in case the students commit errors and mistakes of various types. The decisions made in writing the scenario pertaining to the choice of the mathematical activities and problems could be justified by curriculum prescriptions, some theory of learning, some principles of teaching, knowledge of mathematics and personal teaching experience. But the evaluation of the teaching project will not be evaluated of the basis of this justification. In fact, this justification may not be even written down or otherwise made explicit in the final report of the project. The project will be evaluated by testing the scenario on a group of students. A control group will also be chosen. The control group will be taught the same mathematical content with traditional methods, and both groups will be administered identical pre-tests and post-tests. In the case of similar results on the pre-test, if the experimental group performs better on the post-test, then the teaching project will be evaluated as ‘effective’. Choice 2: The methodology of instructional development You reject the ‘comparative study’ methodology because you do not believe that it is possible to teach ‘the same mathematical content’ with two different sets of mathematical activities and diferent pedagogical approaches. You also don’t believe that one can assess what the students have learned by counting their scores on a standardized test. You start by writing a detailed scenario, just like someone who picked Choice 1, but you do make explicit the rationale behind all your decisions, based on your theory of learning, your instructional theory and your theory of what it means to know the particular mathematical content that you plan to teach the students, because this is going to be the ground with respect to which your project will be evaluated. You



make predictions concerning the knowledge that the students should construct as a result of participating in the planned activities. Then you try out your scenario in a class with someone else, not yourself, as a teacher. You are sitting in the classroom as an observer. You collect all possible documentation concerning the students’ mathematical work. You audio- and videotape the classes, and you collect all the students’ written work. Then you analyze this material with the question: Has the anticipated knowledge developed in the students? If not, then what knowledge has developed? What are the reasons behind the discrepancies between the anticipation and the actual outcome? Can these be explained in terms of the theoretical frameworks assumed a priori? Is there a need to search for an alternative theory, or for an amendment of the theory that has been used? How can the scenario be improved to decrease the difference between the anticipated and the actual knowledge produced by the scenario? On the basis of this analysis, you re-design your scenario and try it again in the same way. Etc. If you made Choice 2 and your theoretical framework is based on the Theory of Didactic Situations, then you can say that you are using the methodology of Didactic Engineering in developing a teaching project in mathematics. The term ‘engineering’ in the name of the methodology comes from an analogy of this kind of conception, design, and implementation work in mathematics education with the work of a civil engineer. An engineer does not validate his or her design of, say, a bridge, on the basis of a comparison with already constructed bridges (which have not fallen down), but on the basis of (a) predictions of its properties (stability, capacity, etc.) which can only be deduced from the mathematical and physical theory, and (b) the realization of the project, checking if the predictions were correct1.

2. A TOOL FOR DIDACTIC ENGINEERING: THE NOTION OF FUNDAMENTAL SITUATION If you plan to teach students some piece of mathematical knowledge so that they learn it, then the theory of situations suggests that you have to organize the didactic milieu and the game of the students with this milieu in such as way that this particular mathematical knowledge will appear as the best means available for the understanding of the rules of the game and elaborating a winning strategy (see Week 1 Class notes). You know from the theory, that there are many ways 1

An interesting analysis of the methodology of didactic engineering can be found in the article Artigue, M. & Perrin-Glorian, M.-J., 1991: Didactic Engineering, Research and Development Tool: some Theoretical Problems linked to this Duality. For the Learning of Mathematics 11.1, 13-18.



in which you may fail to reach this goal (Jourdain, Topaze, Dienes effects, and other phenomena), so you have to organize the milieu so as to avoid falling into the trap of wanting to preserve the fiction of ‘doing your job’ at all costs. This assumption is based on a view of mathematics, advanced by the theory of situations, according to which ‘mathematical knowledge cannot be apprehended otherwise than through the activities that they allow us to realize and therefore the problems that it makes it possible to solve. Mathematics is not simply a logically consistent conceptual system for the production of rigorous proofs; it is, first of all, an activity which is realized in a situation and against a milieu. The activity is a structured one, with distinguishable phases of action, formulation and validation, devolution and institutionalization’2. Theory of situations and this particular conception of mathematical knowledge can be used both to identify what mathematical knowledge is being constructed by the teacher and his or her students in an actual lesson and to ‘engineer’ situations aimed at the construction of a certain knowledge by the teacher and the students. In this class we are interested in the latter activity, and our questions is: How do we go about finding out what kind of situation would generate a given mathematical knowledge? We have to start by analyzing the knowledge K that we aim to teach. We will want to define this knowledge by the general characteristics of a problem-situation to which it could be considered as an optimal solution strategy. A problem-situation is more than just a problem (and it is not a school exercise); it is characterized by what is at stake in solving or not solving it, the possible states of the system it which it has appeared, rules of action, aims of solving the problem. By listing those factors (variables) of the problem-situation which are pertinent from the point of view of the knowledge K associated with it, one obtains a model called The Fundamental Situation associated with K. Symbolically, FS(K)=[C1, … , Cn]. Normally, a piece of knowledge does not appear in its full generality, but in a variety of meanings or concepts depending on the domain of its use or application. We can assume that each such ‘conception’ of K is associated with certain ranges of values of the variables Ci. Thus, the FS(K) can be thought of as a generator of specific situations (SS) associated with the different conceptions of K. We have to decide which conception of K we want the students to construct.

2

Translated from the article Bosch, M. & Chevallard, Y., La sensibilité de l’activité mathématique aux ostensifs. Objet d’étude et problématique. Recherches en Didactique des Mathématiques 19.1, 77-123, p. 81.



Example of an analysis of a piece of knowledge in terms of variables of situations in which it appears as an optimal solution - Part 1: FS and SS Knowledge K = Number FS(K) := [size of set (small, big; finite, infinite), order type of set (continuous/discrete), context of use (comparison of size of sets (which can be close to each other or far away), counting, measuring, coding elements in a set, marking rythms (chanting)), representation (numerals (in various systems: decimal, binary, roman, etc.; verbal, written), sets of physical objects, abaci, ...)] Conception K1 of K = everyday names of natural numbers in their counting function SS-K1 := [size of set (not very small, say 20-30; finite), order type of set (discrete), context of use (comparison of size of sets (which are far away), counting), representation (oral numerals) The next step is to conceive of a didactic situation in which the teacher would devolve to the students a problem-situation whose constraints would satisfy the values of the variables of the chosen Specific Situation. This didactic situation would be defined by some values of its pedagogical variables that would ensure the devolution of the problem situation and would not contradict the problem-situation. Example - Part 2: variables of a didactic situation DS(K1) := [motivation (K1 must be a means to solve the students’ problem, not a school problem); prerequisite knowledge (the students know the names of numbers in proper order: they can chant, ‘one, two, three,…’ till at least 20); kind of problem (the comparison of two biggish sets of objects placed far apart is involved in the problem to be devolved to the students)]

Only after this preliminary analysis of the knowledge to be taught can one start thinking of a possible realization of the didactic situation in the classroom. There are, a priori, many ways in which such a situation could be materialized, theoretically conserving the property of leading to the construction of the same knowledge by the teacher and the students3. Example - Part 3: a classroom realization of a SS for the counting function of number The teacher in a kindergarten class of 5 years old children has arranged the following: In opposite corners of a large room there are two tables. On one table there are 23 pots of paint, on the other, there are about 30 brushes. The teacher says: ‘Anybody who brings, from that table over there, as many brushes as there are pots of paint on this table - I mean one brush for each pot of paint - wins a prize’. Children are motivated: they want to win a prize. They are five year olds

3

Think of the various classroom situations proposed by the class in Week 5 with respect to the ratio conception of division.



who know their number sequence till at least 30, so the prerequisites are there. There are too many pots of paint for the children to grasp their number visually - so they will have to use some coding. They will not be allowed to use marks on paper or beads or anything other than their voice to code the number. The most economical solution will be to count the pots of paint, remember the last numeral and then count the brushes till that numeral and take those counted.

Exercise 1 (a) Define the FS for the knowledge: operation of division (FS(Div)) (b) Define the SS for the physical ratio conception of division (SS-Div/r) (c) Define a didactic situation for SS-Div/r

Hints for a solution (a) The Fundamental Situation for the operation of division could be defined by the following variables:

•

level of generality (an operation in abstract algebraic structures understood as the multiplication by the inverse, an operation in a concrete number structure or measure space)

•

kind of entities involved (numbers only, a number and another entity (e.g. a physical magnitude, a transformation, a matrix, etc.), non-numbers only)

•

kind of numbers involved, if at all (natural numbers, integers, rational numbers, real numbers, complex numbers, etc.)

•

kind of representation of numbers used, if numbers used (exact, decimal approximations, decimal system, another position system)

• • •

size of numbers, if numbers used (big, small, less/greater than 1) relative size of the dividend and the divisor, if numbers used (in a/b, a < b or a > b) context of application (sharing, partitioning, geometric ratio, physical ratio, multiplying/combining by/with the inverse of an element in an algebraic structure)

(b) A Specific Situation related to the Physical Ratio conception of division could be defined by the following values of the above variables: SS-Div/r is concerned with an operation in a concrete measure space, involving two physical magnitudes, expressed by rational but not integer numbers, represented in either a fractional or



decimal form, and involved in a problem of comparison of ratios (otherwise the ratio would not have to be understood as a number).

Exercise 2 (a) Read Brousseau’s description and analysis of a classroom situation: ‘The thickness of a sheet of paper’ (pp. 195-212) (b) Determine the mathematical knowledge K and its specific conception K1 that this situation is aiming at and describe K and K1 in terms of their defining variables (c) Describe the didactic variables of the classroom situation corresponding to K1. (d) In what way the classroom situation realizes SS-K1? (e) By what kind of behavior does the teacher avoid (or not) the trap of the phenomena of Topaze, Jourdain, the metacognitive shift, the metamathematical shift, the implicit suggestion of analogy? For a better ‘feel’ of the above situation, I propose the following small activity in the class: I’ll put five books on the table: 1. ‘Thinking mathematically’, 2. ‘Elementary Linear Algebra’, 3. Sheakespeare, 4. Matematyka-7, 5. Robert Dictionary. I’ll ask you to evaluate the thickness of the paper used in each of the books and order the books from the one with the least thick paper to the one with the thickest paper. All you will have is a ruler. How will you go about it? When I tried to evaluate the thicknesses myself, I was measuring the thickness of all inside pages of the book and counted the number of pages. Here are the results I got: width

# sheets

ratio

1

12 mm

120

0.1

2

20 mm

271

0.074

3

13 mm

120

0.108

4

21 mm

231

0.909

5

57 mm

525

0.108

When I then calculated the ratios width/# sheets, it turned out that the book (5) has thicker paper than (1) which was obviously not true; I could feel it. Then I decided to measure the width of the same number of pages in each book; I chose 240 pages or 120 sheets. I then obtained the following table: Book

width

# of sheets

1

12 mm

120



2

8 mm

120

3

13 mm

120

4

11 mm

120

5

7 mm

120

Now I did not have to calculate the ratios, I knew that the order of the books from the book with the most thin pages to that with the thickest pages is: 5, 2, 4, 1, 3. It has become clear for me in this exercise that had I chosen to take the same number of pages from each book to start with, I would not have to calculate the ratios. So if the aim of the situation has something to do with ratios, division or rational numbers, then assigning an exercise like this one for each individual child to do on his or her own, would not help achieving it. Now I understood why the situation as described in Brousseau’s book was a lot more complicated, with children not just measuring and calculating individually but in groups and having to communicate and compare their results. It is rather unlikely that all groups would have chosen the same number of sheets to measure.



LECTURE 7 DIDACTIC ENGINEERING 2: CAUSES OF DIFFICULTIES OF REALIZATION OF DIDACTIC ENGINEERING PROJECTS.

It is often the case that a teaching project which, a priori, appears to create the appropriate conditions for the construction, by the students, of a given mathematical knowledge, is disappointing when it is experimented. Students either do not learn anything new or learn something different from the target knowledge. What could be the objective1 causes of this failure? We shall discuss three types of causes, labeled ‘didactic transposition’, ‘teacher’s epistemology’, and ‘obstacles’ of various origins. It may also happen that a teaching experiment ‘works’ once, but the success is not repeated when it is run for a second time. Why? This is the problem of reproducibility of the products of didactic engineering.

1. DIDACTIC TRANSPOSITION The didactic engineering methodology presupposes that, at least in the phases of action, formulation and validation, the students will act not as students whose sole aim is to satisfy the teacher and pass the course, but as learners whose aim is to gain some new mathematical knowledge by solving a problem. It is implicitly expected that, at some point, the students will behave as mathematicians. This expectation can be justified only if the target knowledge is, indeed, of the same nature as mathematicians’ knowledge. But this need not be the case. In general, school mathematics knowledge differs quite substantially from mathematicians’ knowledge, not only quantitatively: school mathematics knowledge is not a small subset of the mathematician’s knowledge. The school institution has developed mathematics as a teaching subject which may have parts not included in the mathematicians’ knowledge. One could say that, in general, mathematical knowledge (or any kind of knowledge for that matter) is a function of the institution in which it ‘lives’. The mathematics developed and 1

‘Objective’ here means: independent from the personal characteristics of the teacher and the students (e.g. poor class management of the former and the incurable laziness of the latter).



used in an Arts and Science Department is different from the mathematics developed and used in the Engineering Department, which is yet different from the mathematics as it serves the medical and biology research centers. There is Actuarial Mathematics and Financial Mathematics. There is not one ‘La Mathématique’, as it was held in the 1960s, during the New Math reforms. Of course, all these various institutions are not hermetically closed and there is a continual flow of ideas between them. When an institution picks up some body of knowledge from another one and adapts it, changes it to fit her own goals and tasks, then we speak of an ‘institutional transposition of knowledge’2. When the transposition goes from an institution which produces knowledge to an institution which teaches it to students, then we speak of ‘didactic transposition’3. The process of didactic transposition is inevitable when it comes to teaching mathematics at school; however teachers and curriculum developers must beware of producing knowledge that would have only some kind of ‘internal’ value for the functioning of the school as an institution, but not outside of it. We must constantly remind ourselves that the aim of the school is to prepare the children for life and professions outside of the school. The risk is real; we have seen monstruous examples of such unnecessary ‘didactic creativity’ in mathematics teaching, such as the ‘language of strings and arrows’, in the 1960s New Math reforms. There are less monstruous examples such as the ‘Big cosine’, Cos and the ‘Small cosine’, cos. Exercise 1 Can you give other examples of school mathematical concepts which do not exist in the academic mathematics? Aside from creating new ‘objects of knowledge’, school mathematics differs from ‘research mathematics’ also in other ways. A part of some general knowledge can be isolated and given an important status of a ‘topic’ with a special name and place in the curriculum. Example: ‘les identités remarquables’ in the French curriculum. This refers to algebraic properties of operations on real numbers which are, in the theory, simple consequences of the properties of the real number system

2

Chevallard, Y. 1992: Fundamental concepts in didactics: perspectives provided by an anthropological approach. In R. Douady & A. Mercier (eds.), Research in Didactique of Mathematics. Selected Papers. Grenoble: La Pensée Sauvage éditions, pp. 131-167, p. 165. 3 Actually, in participating in the present course, you are witness of a didactic transposition of a body of knowledge that has first evolved as a university research domain, the ‘Theory of Didactic Situations’. I shall not dwell on the differences between this academic body of knowledge and the knowledge that we are constructing together in the course, but I am fully aware of the existence of such differences.



(commutativity and associativity of addition and multiplication, distributivity of addition with respect to multiplication, etc.), e.g. (a-b)(a+b)=a2 - b2.4 The curriculum devotes several class periods for this topic and students are given many exercises for their application; a special test is usually designed for the assessment of the students’ mastery of these identities. A technique that mathematicians are using as an instrument and are not studying per se becomes an object of teaching: this is the case of ‘paramathematical notions’ such as equation, parameter or proof5. A mathematician would not say: ‘today, I’ve been doing proofs’, or ‘I’ve been solving equations with parameters’ because he or she is doing proofs every day or every time he or she is doing mathematics, and in all his problems there are constants, variables and parameters. But a teacher can be ‘doing proofs’, or teaching the ‘notion of equation’, or ‘the notion of parameter’ to his or her students today and consider the topic finished and done with tomorrow6. Perhaps the most important difference between research mathematics and school mathematics is the aim of the mathematical activity. The mathematician wants to know and in the aim of knowing he uses his or her mathematicial competences; the student has to demonstrate that he or she knows what he or she is expected to know and possesses the competences aimed at by the curriculum. For example, a mathematician will transform an expression like 4x2 - 36x into 4x(x-9) or (2x-6√x)(2x+6√x) depending on the assumptions and purposes of the problem in which this expression appeared. But the student is quite likely to receive this expression isolated from any mathematical problem, in an exercise like ‘Factor 4x2 - 36x’ and he or she will know to be expected to answer with 4x(x-9) or (2x-6√x)(2x+6√x) depending on the didactic, not mathematical context of this exercise, i.e. on what the teacher had been teaching before giving this exercise. So, in choosing what to write the student is not so much solving a mathematical problem as deciphering the rules of the didactic contract7. A thorough analysis of the distinction between school and research mathematics in general is available in Chevallard (1991) (see footnotes). An in-depth study of the didactic

4

Chevallard, Y. 1991: La transposition didactique du savoir savant au savoir enseigné. Grenoble: La Pensée Sauvage éditions, p. 43. 5 Proviso: the ‘notion of proof’ is an object of study for logicians working in the so-called ‘foundations of mathematics’. But even for them mathematical proof is an everyday tool; they justify their assertions by means of proofs. 6 Chevallard, 1991, p. 49-51. 7 ibid., p. 52.



transposition of a particular mathematical concept, the concept of distance, can be found in Chevallard & Johsua (1991)8. 2. TEACHER’S EPISTEMOLOGY Knowledge functions in the school institution in a very different way than in a research institution. It may therefore not be very realistic to expect that, even through the best - a priori didactic engineering, students will develop mathematical ways of thinking and concepts similar to those of research mathematicians. The way mathematical knowledge functions at school is influenced by the teacher’s knowledge relative to the acquisition of mathematical knowledge. It is not necessary that the teacher be aware of that knowledge, that he be able to verbalize it and discuss it against other possible views on the acquisition of mathematical knowledge. The existence of this knowledge manifests itself in the teacher’s practices. Brousseau calls this knowledge ‘teacher’s epistemology’ (p. 35)9. Here are some examples of such manifestations. 1. A teacher will classify some students’ errors as ‘une étourderie’ (a result of absentmindedness), especially if they come as an answer to an ‘easy’ question that most students already are familiar with. They will classify some other errors as ‘conceptual’ or simply ‘serious’. The former will not entail any didactic action on their part; the latter will - the teacher will engage in ‘revisions’, ‘remedial activities’, etc., with respect to students whom they suspect of having ‘conceptual problems’. This is a symptom of some kind of ‘spontaneous psychology’ that the teacher seems to profess, which has little to do with research in cognitive psychology, where responses to cognitive problems are not classified into ‘serious’ and ‘not serious’. 2. The teacher expects the students to produce their answers according to a schema which they consider as ‘correct’; they expect the students to produce their answers in an ‘intelligent’ way, acceptable within the mathematical culture they believe the school should represent. Some teachers thus confuse the laws of the production of knowledge with the systematization and organization of knowledge. But even those who know that, in the domain of real mathematical research, solutions to problems are not found this way or that, in general, human cognition does not work in such a systematic, orderly and logical fashion, will still hold such expectations and

8

Chevallard, Y. & Johsua, M.-A. 1991: Un exemple de transposition didactique. Grenoble: La Pensée Sauvage. 9 The explanations of the phenomenon of ‘teachers’ epistemology’ that you will find in this section are a free translation and adaptation of a part of Brousseau’s clarifications which I obtained from him in private e-mail correspondence.



requirements with respect to the students’ answers: this is a necessary implication of the didactic contract. The teacher cannot transform his or her class in a psychology lab, and he or she cannot straighten up all the individual or even collective trajectories of learning because this would be too time- and labor consuming and perhaps technically impossible. Thus, in his or her practice the teacher uses certain ‘concepts’ or ‘laws’ related to epistemological questions such as ‘what does it mean to investigate?’, ‘to learn?’, ‘to understand?’ which define the students’ field for action and justify the teacher’s decisions. This spontaneous epistemology is accompanied by a whole mythology of metaphors and symbols thus forming a system - a praxis of the teacher - which makes his work appear possible and legitimate. Here are some more examples of teachers’ decisions that reflect this spontaneous epistemology and show how distinct it can be from an epistemology actually governing the work of mathematicians. In mathematics it does not matter very much how a result has been established: whether by way of a witty reasoning or by way of a laborious calculation and verification of all possible cases. What matters is to show that the solution solves the problem. But if a student solves an equation by trying some numbers at random or even systematically, his solution will not be considered correct by the teacher. It is worthless from the point of view of the teacher’s epistemology, because it has no ‘positive didactic value’: the student has not demonstrated he or she had learned the method. The aim, at school, is not to solve the problem but to demonstrate one has learned what one has been taught10. Teachers’ spontaneous epistemology of mathematics is often at odds with the foundations of the theory of situations. Since it is this theory that underlies teaching projects conducted according to the methodology of didactic engineering, this discrepancy may explain the difficulties of the classroom realization of these projects. Exercise 2. In an article published in English in the journal For the Learning of Mathematics in 1991, Michèle Artigue and Marie-Jeanne Perrin-Glorian11 report about a didactic engineering experience in a middle school (10-12 years old children) with students who had difficulties in mathematics (and other subjects). It turned out to be practically impossible to realize the didactic engineering projects, and the researchers started to study the reasons for this failure. They have studied, in particular, the teachers’ conceptions related to the teaching and learning of mathematics. 10

This is the end of Brousseau’s explanations. Artigue, M. & Perrin-Glorian, M.-J. 1991: Didactic Engineering, Research and Development Tool: Some Theoretical Problems Linked to this Duality. For the Learning of Mathematics 11.1, 13-18. 11



Based on your reading of this part of the article, explain in what way could these conceptions contribute to the failure of the didactic engineering? In what way are these conceptions incompatible with the theory of situations? 3. REPRODUCIBILITY OF DIDACTIC SITUATIONS It has proved extremely hard for teachers to reproduce a didactic situation so that the mathematical meaning of the students actions is conserved. This phenomenon is called, by Brousseau, the ‘obsolescence of didactic situations’ (p. 193). By obsolescence we mean the following phenomenon: from one year to another, teachers have more and more trouble reproducing the conditions likely to lead their students to create, perhaps through different reactions, the same understanding of the notion taught. Instead of reproducing conditions which, while producing the same result leave the trajectories free, they reproduce, on the contrary, a “history”, a development similar to that of previous years, by means of interventions that, even if discrete, completely change the nature of the didactic conditions guaranteeing a correct meaning for the students’ reactions; the obtained behavior is apparently the same but the conditions under which it was obtained modify the meaning (p. 193).

The failure of even the first realization of a didactic engineering project is related to this difficulty in reproducing the mathematical meanings. In writing up a scenario for a didactic situation, it is rather easy to describe the physical environment, the tasks, the verbal interventions and the actions of the teacher, the expected reactions of the students. It is much more difficult to enumerate and highlight those features of the milieu which are indeed responsible for the emergence of the target mathematical meaning. Sometimes it can even be hard to tell what these features are, and it may well be that the dynamics of a didactic situation represent a chaotic rather than a stable system, meaning that a tiny alteration of its conditions (e.g. an apparently unimportant remark or a hint from the teacher, an expression of approval or disapproval on his or her face) may cause huge changes in the knowledge that it produces12. In order to increase the chances of reproducibility, the didactic engineering methodology is very demanding with regard to the a priori analysis of the target knowledge and the work of identifying the critical variables of the didactic situations that are supposed to lead to it. Example. A group of researchers13 have studied the problems of the reproducibility of the products of didactic engineering in the following setting: The same, a priori, didactic situation has been implemented by two different teachers in two different classes of 13-14 years old students. The

12

ibid., p. 14. Arsac, G., Balacheff, N., Mante, M. 1992: Teacher’s Role and Reproducibility of Didactic Situations. Educational Studies in Mathematics 23, 5-29. 13



mathematical content of the situation was the activity of proving. The situation was composed of two main phases: (a) the research phase, in which the students are given a problem and write their solutions on a poster; the problem is so chosen that there are many solutions possible, and it is likely that the students will disagree on many points; (b) the debate phase: the students’ solutions are written on large sheets of paper and are then displayed as posters on the walls of the classroom. Students are divided into teams which analyze the solutions. Each team then delegates one spokesperson who presents the result of the analysis to the whole class. A whole class debate is then engaged. The situation was planned to last over two one and a half hour periods. In Class I, the teacher was a member of the research team. In Class II, the teacher was not a researcher but the scenario was thoroughly presented and discussed with her. In Class I the problem given to the students was the following:

‘Write for other students a message allowing them to come to know the perimeter of any triangle a piece of which is missing. To do so, your colleagues will have at their disposal only the paper on which is drawn a triangle and the same instruments as you have (rulers, etc.).’ In Class II the problem given to the students was the following:

M



‘The two lines intersect outside a page. Write a method allowing anybody to draw the line through M and the point of intersection without going out of the page.’ The teacher was supposed not to interfere at the mathematical level during the whole activity. The students had to feel that it is entirely their reponsibility to decide which solutions are valid and which are not. Otherwise, the researchers claimed, the real activity of mathematical proving will not emerge, and the students will only want to produce statements acceptable by the teacher and the teacher’s authority will be the ultimate criterion of validity of solutions. The teacher was allowed to intervene only at the level of the presentation of the problem and the organization and chairmanship of the debate. However, in the implementation in Class I, the teacher intervened in ways that did affect the students’ attitude towards the mathematical activity and, in actual fact, relieved them from the responsibility for the mathematical validity of their solutions. Here are some examples of this kind of interventions, as reported by the researchers: - in order to guarantee that the research phase be not too long, the teacher invited the students to propose a solution as soon as she thought that what they obtained was sufficiently developed, and not when she was sure that the students thought so; - the teacher was drawing the students’ attention to some crucial words in the formulation of the problem (in particular, the word ‘any’ in Class I); - the teacher in Class I kept continuous contact with the students, making about one intervention every minute over an 80-minute period (from ‘Are you okay?’, to ‘Have you read the problem carefully?’); - when she considered that a discussion is irrelevant from the mathematical point of view and leads nowhere, she would urge the students to dismiss it and go further; she would also focus the students’ attention on solutions and ideas that she considered relevant, and sum them up for the students. As a result of these interventions the students ‘got confused and were no longer committed to any real discovery of the solution’. In a repetition of the experiment in another class, the teacher was asked explicitly not to intervene at all, and stay at her desk during the whole phase of discovery. She did. She only told the students what is the problem that they have to solve and told them that they have so much time for it. But this did not make the students engage in the activity of proving, either. The students compared the solutions with respect to their simplicity, clarity, usability, but not their validity. When, after one solution was accepted by the whole class, the teacher asked if the authors of this solution are sure of it. They responded: ‘Yes, because we have done it in a lot of



cases’. So it was not even enough for the teacher to directly ask for a proof, to make the students produce a proof, although, as it was found out, they were perfectly able to prove that their solution was correct, in a mathematical way. In Class II, the teacher intervened in a stronger way even than the teacher in Class I: she asked leading questions (‘Do you all agree? It’s not proved’), she did not transcribe some solutions on the big sheets of paper, she intervened directly on the content of the debate, she reinforced some students’ ways of proving (‘Now you make a drawing in each group’. ‘It works, but only if M is on the bisector’, ‘Look at your drawing and come to an agreement’). The researchers concluded by saying that as soon as the teacher (a) tells the students that they have a limited time to solve a problem, and (b) shows the students that she endorses the ‘epistemological responsibility’ for the mathematics produced in class (e.g. when she refuses to write a false statement on the board), the students have no reason for entering a genuine activity of proving in the mathematical sense. It turns out that it is very difficult to implement a teaching design in a way which conserves the intended mathematical meaning of the activities. The meaning of the activity may change from one implementation case to another, due to the teacher’s interventions. The teacher’s interventions are caused by her beliefs about her role and her duties as a teacher, and when she acts on the basis of these beliefs, she may not support the development of the students’ development as autonomous thinkers. Exercise 3. Find a few critical features of the didactic situation ‘Race to 20’ (pp. 3-18) that, if altered, would lead to its failure in attaining such objectives as: discovery and proof, by the children, of a sequence of theorems related to the notion of Euclidean division in integers (‘division with a remainder’). 4. COGNITIVE, DIDACTIC AND EPISTEMOLOGICAL OBSTACLES When students start upon learning a new mathematical concept, their minds are not blank slates, they are filled with all kinds of knowledge, beliefs, experience. New knowledge is not simply added on, it must be merged with the old knowledge. But new knowledge can sometimes contradict the old knowledge, and then the old knowledge functions as an obstacle to learning the new one. The contradictions are inevitable because whatever knowledge we have, is an answer to a domain of questions and problems. This is usually a limited domain, but as long as we have not transgressed its limitations, we believe that there are no limits and our knowledge is universal. So



when we find ourselves in front of new questions and problems we want to solve them with the knowledge we have and this knowledge may not be capable of solving these new problems. We need to reject parts of our old knowledge, or re-organize it, or generalize it and recognize that it had a limited domain of application. This process feels as if there was something blocking our mind - an obstacle. It is not always possible to know in what ways the students’ old knowledge can function as an obstacle in learning the knowledge aimed at by a didactic engineering project. But this old knowledge may bias the students’ interpretation of the tasks given to them and they may thus completely miss the mathematical point of the activities. According to Brousseau, obstacles that appear in the teaching of mathematics can be of various origins: ontogenic, didactic, epistemological, cultural (p. 86). Ontogenic obstacles are ways of knowing whose limitations are due to the stage of the mental development of the child. A 6 year old child cannot be expected to understand the principles of an axiomatic theory. Didactic obstacles are ways of knowing whose limitations stem from a certain way of teaching. For example, in elementary courses in physics at high school, vector magnitudes such as force or velocity are taught in a context of problems that minimizes the complexity of mathematical computations. If the students have not studied the cosine theorem in mathematics but did study the Pythagorean theorem, composition of magnitudes problems will deal with orthogonal vectors only. The students risk to develop a conception that vector addition applies to orthogonal vectors only, which will function as an obstacle in their study of vector geometry and physics later on. Moreover, school problems related to vector magnitudes in physics tend to concentrate on the measures of the magnitudes; the direction of action of a force or motion is given either implicitly in a diagram accompanying the problem, or in non-mathematical terms such as ‘North-West’ in the text. Direction is not something that is calculated and it is normally not the object of the question in the problem. So it becomes unimportant. This may explain why in vector geometry, many students believe that a vector can be completely defined by its length14. Epistemological obstacles are those whose limitations are related to the very meaning of the mathematical concepts. A mathematical concept has many levels of generality and abstraction, and many aspects, developed during its history and depending on the context of its use. Each level and aspect has its limitations, and if one thinks of a concept in a meaning that is not appropriate for the given context or problem, then this way of thinking functions as an 14

Knight, R.D. 1995: The Vector Knowledge of Beginning Physics Students. The Physics Teacher 33, pp. 74-78.



obstacle and one makes mistakes or cannot solve the problem. For example, one can think of real numbers as measuring numbers, or as elements of an algebraic structure called a well ordered commutative field. The latter understanding is useless in the context of problems related to the measurement of lengths, areas and volumes of geometric figures. The former is useless when one deals with questions such as: could we put all real numbers in a sequence?15

15

More about epistemological obstacles can be found in: Sierpinska, A. 1994: Understanding in Mathematics. London: Palmer Press, pp. 112-137. Sierpinska, A. 1992: On Understanding the Notion of Function. In E. Dubinsky and G. Harel, The Concept of Function, Aspects of Epistemology and Pedagogy. MAA Notes, Vol. 25, pp. 23-58. Sierpinska, A. 1991: Some Remarks on Understanding in Mathematics. For the Learning of Mathematics 10(3), 24-26.



LECTURE 8 DISCOVERY TEACHING/LEARNING AND THE THEORY OF DIDACTIC SITUATIONS.

1. HOW ‘GOOD’ OR ‘BAD’ IS THE THEORY OF SITUATIONS? IS THIS A GOOD QUESTION TO ASK ABOUT THE THEORY? Lessons or lesson plans cannot be said to ‘conform’ or ‘not conform’ with the theory of situations or didactic engineering. The theory of situations provides us with a perspective from which to better understand what is going on in the classroom in terms of the students’ learning, the mathematical meanings that the students construct, and the role, in this process, of the teacher. When we design a lesson with the theory of situations as a guide then we are not trying to conform to some set of rules or norms but we are trying to foresee the effects of our decisions and choices with the tools of the theory. Thus, we may come to the conclusion, that in organizing the classroom activities in such and such way, all we’ll obtain is an expected behavior in the students (for example, they will produce an expected equation) but not the expected understanding. Then we may try to change the lesson plan so that the expected understanding does develop in the students. However, no matter how hard we try, we may still be wrong in our expectations, and when experimented, the lesson will not produce the understanding and knowledge we aimed at. If this happens, it is not the theory that is to blame but our design in which we may have failed to take into account certain variables. Here is what Brousseau wrote in respect to the status of the theory of didactic situations in an e-mail message sent to me in the last week of November 1999. ‘The theory of situations is aimed to serve both the study and the creation of all kinds of learning and teaching situations, whether they are “spontaneous”, or the product of an experience or of a special engineering project, and whether they are efficient or not. It is not a method of teaching. The theory can provide some methods of teaching, it can justify some methods and disqualify some other methods, as the case may be. The theory contains models that may support certain plans of action aiming at making the students (re)discover some mathematics. This way the theory can make suggestions for engineering. The didactic situations of “rediscovery” can thus be, in general, linked to these models’ (my translation).

The practical significance of a didactic theory can be defined, partly, by its relationship with a methodology of instructional design. In the case of the theory of didactic situations, the corresponding methodology is that of didactic engineering. This is how Brousseau explains the relationship between the two:

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‘Theory of situations has been elaborated to provide a framework for the study and means of description of any situation in which there is an intention of teaching someone some precise knowledge, whether it succeeds in it or not. It can take into account all the forms of learning identified in all kinds of research. The theory does not pretend to present all aspects of teaching situations and replace all the approaches: psychological, psychoanalytical, linguistic, statistical, etc.. But it tends to put the contribution of these approaches in the perspective of their function and their generality in the description of “didactic phenomena”. One could use here an analogy with economics and its relation with commerce. Economics does not deal with the psychology of the vendor or of the buyer but it can take into account their impact on the macroeconomical phenomena. Theory of situations is neither an ideology nor particular didactic method. In this sense, it has no technical alternative. It does not directly recommend this or that particular didactic procedure. Its theoretical concepts only allow one, for reasons of consistency, to predict the role of certain factors in some circumstances. This way it puts limitations on what it is possible to do or change in teaching, just as thermodynamics discards the possibility of building a perpetuum mobile but does not give precise guidelines for the construction of an ideal engine. On the other hand the theory manages well the contingency or the possibility that some of its theoretical predictions be rejected as false. For example, at one time I claimed that “a situation of formulation related to a precise piece of knowledge cannot function if a previous situation of action has not allowed the students to develop an implicit model of this knowledge”. But, later, I found counter-examples and I had to restrict the generality of the claim that, previously, made a lot of sense to me. Another example: I claimed that “An implicit model of action that is not, rather quickly, supported by a formulation, is lost as fast as it is learned”. I had to soften this declaration by provisions such as “in most cases”, “if the situation of action is evolving”, etc.. In this sense, the relevance of the theory lies in its ability to raise questions and classify and order the answers. (One of the difficulties of mathematics education is the multiplicity and diversity of the research work, classified according to criteria which appear to have nothing to do with each other. The important thing is thus to be able to judge of the relevance of the questions and the validity of answers). Yet, if I have developed the theory of situations it is because I needed it for didactic engineering. It is a product of my efforts to classify questions that are raised when one wants to organize, by whatever means, the learning, by anyone, of some knowledge. This explains why one finds, in the description of the types of situations, arrangements corresponding to the possibility of the construction of knwledge by the students in a non didactic situation. This is what resembles most, it seems, to the settings in which one can detect the “discoveries” of the students and imagine their role in the learning processes. There remains the question of interpretation, in the theory of situations, of the conclusions and claims of the observers of “discovery learning”. This way, the theory of situations restricts the meaning that one can attach to “didactic engineering”. The point is to distinguish between didactic “innovations” or “inventions” made for the educational market, and those products of the didactic work that are based on proved techniques, described by explicit and sensible technologies, which are, in turn, founded on falsifiable theories and verifiable experiences. The distinction is not easy because, in any work of didactic engineering, inventions play a very important part, as do empirical considerations which can be extremely complex and possibly judicious. Moreover the success of these products at differents levels (students, teachers, society) is not decisively determined by their belonging to this or that category. I am presently trying to fill the gap that I have left between the theory which presents the wide basic concepts and general didactic phenomena, and the different products of the didactics proper (the teaching of natural numbers, decimal numbers, rationnal numbers, measure, statistics and probability, reasoning and logic, space and geometry, elementary arithmetic and algebra) whose practice I have organized and observed. I am dreaming of a big treatise on didactic engineering which I consider as the didactics proper.’ (My translation)

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DISCOVERY LEARNING APPROACH I have judged one of the lessons presented in the class by students (related to Pythagoras theorem) as an example of the ‘discovery learning’ approach. I did not mean to say that it is not ‘conform with didactic engineering’. I only meant to say that, if one analyzed this lesson from the theory of situations perspective prior to actually conducting the lesson then one would either change the plan of the lesson or interpret the outcome of the lesson differently: the evaluation of what the students have actually learned in that lesson would be different. What I understand by ‘discovery learning approach’ is an organization of the students’ activity in a sequence of exercises so that, at the end, some representation of the target knowledge is produced by the students (e.g. a formula is written, a term is said, a drawing is produced), so that, in the eyes of one knowing the target knowledge, the manifestation of this knowledge in the lesson is obvious. In the approach presented by one of you, in the activity phase, the students were already manipulating representations of the target knowledge; the formulation phase was strongly guided by the teacher (Topaze effect), and the validation phase occurred after the ‘discovery’ (Jourdain effect): the students were then asked to prove a theorem which they already knew to be true. I asked Brousseau, by e-mail, for an explanation of the relations between the discovery learning approach and the theory of situations and he kindly responded (last week of November 1999, same message as above). I partly translate and partly summarize, below, his answer. ‘The theory of situations makes it possible to identify the circumstances of the discovery by the students (or anybody, for that matter) of some knowledge but, within the theory, the notion of “learning through discovery” appears contradictory. On the other hand, the theory can recognise the notion of “teaching through discovery” as referring to the organization of non didactic situations, and to the identification, the interpretation and the institutionalization of the knowledge produced by the students in these situations. The “teaching through discovery” approach creates situations based on “an epistemological and didactic fiction” which favors an auto-didactic reading of these situations by the students. Briefly speaking, the “discovery learning” appears to be a myth, a psychological extrapolation invented to justify and conceal a “discovery teaching”: a method of rediscovery. The problem solving method is based on heuristic interpretations of the mathematical activity, as opposed to the classical axiomatic and purely deductive interpretations. It leads to epistemological commentaries and “original” terminology and methods. Heuristics start to be taught as ordinary objects of teaching. These heuristics are conjectures about the learning of mathematics that are false in all their generality but useful or valid in an important number of cases. But they are indeterminate, otherwise they would become theorems. There is no reason why these heuristics would work better than the “real theorems”; in that case one could look for second order heuristics to produce and guide the first order heuristics. There is no doubt that heuristics, like, for example, analogies, are useful for the researcher who engages them under his or her own responsibility. But when they are proposed to the students as methods, in the frame of a didactic contract, they are nothing but some of the invisible means of the teacher to conceal his or her didactic strategies and produce the Topaze or Jourdain effects. This being said, there is no reason why heuristics should not be used when they work. The theory of situations does not make a mystery out of the theatrical component of any didactic situation. To play at

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discovery is not as realistic as to play at a competition of theorems but it can be as exciting and maybe more flattering, more gratifying for the ego of the student as well as of the teacher; briefly, it can be, locally, just as efficient for the student’s learning. From a strictly scientific point of view, it is not necessary to invent false psychological and cultural mechanisms to justify this didactic approach. But in pre-service teacher training, if didactic situations are presented for what they are, then perhaps the future teachers will not fall for it. It might be that they need to “believe in the reality of movie characters”. To be successful in teaching one needs such a will or conviction that perhaps the easiest thing to do would still be to justify this professional “faith” by appropriate beliefs?’ (my translation)

It seems to me that the debate should focus on three points: - The relations between the theory of situations and the didactic engineering; - The relations between the ‘discovery learning’ approach and the theory of situations; - The possible difference between the meaning of the word “theory” in Educational Sciences, especially anglo-saxon, and its interpretation in natural sciences, and the difference of perspective on the nature of didactic phenomena where the make believe and the masquerade are the standard practice.

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LECTURE 9 DIDACTIC VARIABLES. WHAT TO OBSERVE IN A CLASSROOM? HOW TO PRESENT A TEACHING PROJECT?

1. DIDACTIC VARIABLES Let us ask the question: What is a teaching situation a function of? To answer this question we must first decide what we shall understand by a teaching situation. In the theory of situations, the teaching situation is a game between the teacher and the system composed of the student and the milieu:

r

STUDENT

MILIEU

R TEACHER

Thus, the teaching situation depends on - the personal characteristics of the TEACHER; in particular his or her personality, beliefs about what it means to teach and to learn at school (i.e. his or her ‘epistemology’), his or her mathematical knowledge, cognitive style,… - the personal characteristics of the STUDENTS; in particular the stage of their cognitive development, their cognitive style, their personalities (affective, motivational),… - the composition and arrangement of the MILIEU; in particular, the task proposed to the students (exercise, open-ended problem, project, …), the resources and tools put at their disposal such as texts, game boards, geometer’s kits, calculators, computers, pens and pencils, crayons, scissors, paper, exercise books, desks and chairs, the arrangement of the desks and chairs in the classroom,…) - the RELATIONS r between the students and the milieu; in particular - the students’ relation to the task (school problem vs taken-as-own problem) - the students’ contact with the task (mental vs physical)

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- the students’ relations with the resources and tools (in particular, the students can be using a calculator as a computational device only, or as a heuristic tool) - the RELATIONS R between the teacher and the student - milieu system; in particular - proportion of teacher’s interventions with respect to students’ time on task - the subject of teacher’s interventions (organizational vs mathematical) - the kind of teacher’s interventions (presentation of task, hints, genuine questions (e.g. tell me how you solved this problem), rhetorical questions (i.e. questions to which the teacher knows the answer), questions meant to maintain the contact and students’ attention (e.g. ‘Are you with me here?’ and the teacher continues his or her lecture without waiting for a reply), exposition of theory, solving exemplary test problems, discussion with students, etc. I have thus listed a large number of variables on which a teaching situation appears to depend. But not all of them will be called ‘didactic variables’; only those that can be controlled by the teacher, not as a person, but as an element of the didactic system. From this point of view, the didactic variables will be those related to R, r and MILIEU. Variables related to the Teacher and the Student as persons are excluded, not because they are not important in the process of teaching but because they would qualify rather as psychological and sociological variables and not as didactic variables. The assignment of values to didactic variables in the design of a lesson must take into account the sociological and psychological variables but it has no control over them. 2. WHAT TO OBSERVE IN A CLASSROOM? HOW TO ANALYZE AN OBSERVED LESSON? When you are sitting in a classroom observing, take note of those facts that are pertinent from the point of view of the didactic variables. Make sure you are able to answer questions like: - What were the tasks proposed to the students? - What were the resources and tools put at the students’ disposal? - How was the classroom furniture arranged and how were the students and the teacher positioned? - What were the students’ relations to (a) the task (b) the tools and resources ? - What was the ratio of time of teacher talking to students talking or working on tasks? - What kinds of interventions did the teacher use? - What did the students do? When analyzing the data obtained through observation, ask yourself questions related to the mathematical content of the lesson: - What mathematics did the teacher appear to want to teach the students?

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- Were the didactic means he or she used likely to help him or her to meet this goal? (Why yes or why no?) - What were the students learning? - How would you change the lesson if you yourself were to teach the same mathematical topic?

3. HOW TO PRESENT A TEACHING PROJECT? In presenting your teaching project, first describe the mathematical knowledge that you expect your students to develop through participation in the proposed activities. Then describe the organization of the MILIEU and explain why do you think that this organization can help the students to develop the target knowledge. To describe the organization of the milieu means to assign values to the didactic variables, and thus this description should contain answers to questions related to the variables M, R and r: - What are the tasks proposed to the students? - What are the resources and tools put at the students’ disposal? - How is the classroom furniture arranged and how are the students and the teacher positioned? - What are the expected students’ relations to (a) the task (b) the tools and resources ? - What is the teacher’s role in the situation and what kinds of interventions is he or she expected to make? - What are the students expected to do? The justification of your choices in regard to these questions should be guided by your aim of teaching the students the assumed knowledge.

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Lecture notes on the Theory of Didactic Situations in mathematics

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