domain: math shs

text:
Thomas Lecorre
email: thomas.lecorre@wanadoo.fr

Q² game used in a task design of the double quantification

Keywords: formalism, quantification, limit, scientific debate in class, game Q²

This article deals with student's interpretation of multiply-quantified statements. The efficient interpretation of such statements is useful, for example, for those who try to understand and use the formalism of the definition of limit of a function. Literature points out some students' difficulties in the interpretation of double quantified statements. I am using the scientific debate methodology to design and implement tasks by means of two fundamental steps. In the first step, I draw on the results of a questionnaire to identify students' difficulties related to the order of quantifiers in the interpretation of such statements. The aim of the second step is to design the targeted tasks by considering the results of the first step; these tasks are based on a game called Q². The implementation of the design shows that students understand that there are at least two kinds of interpretations, and that conventions of interpretations are needed.

Quantified statements problem

Mathematics formalism uses massively quantification and specifically multiply-quantification. Research shows that students have difficulties with the interpretation of multiply-quantified statements [START_REF] Dubinsky | On student understanding of AE and EA Quantification[END_REF][START_REF] Chellougui | L'utilisation des quantificateurs universel et existentiel en première année d'université, entre l'explicite et l'implicite[END_REF][START_REF] Piatek-Jimenez | Students' interpretations of mathematical statements involving quantification[END_REF]. EA statements corresponding to "Exists…for all…" sentences must be distinguished from AE ones corresponding to "For all…there exists…". [START_REF] Dubinsky | On student understanding of AE and EA Quantification[END_REF] study the impact of two main variables on the interpretation of double quantified statements. The first variable is the order of quantifiers (AE or EA) and the second one is the kind of statement, mathematical or non-mathematical. They show that the interpretation of non-mathematical statements is essentially correct but the interpretation of mathematical ones is difficult for students. For AE mathematical statements, the students' interpretation is very efficient, whereas their interpretation of EA ones seems to be done through an inversion: the EA statement is very often interpreted as the AE corresponding statement (the variables remain with their own quantifiers but the order of quantifiers are changed). Moreover, it is noticed that "the students did not appear to care of the syntax of a statement to analyze it […] the student did not appear to be aware of having engaged in interpreting the questionnaire statements." (Dubinsky & Yiparaki, 2000, p. 53). The inversion of interpretation of an EA statement is also noticed by [START_REF] Chellougui | L'utilisation des quantificateurs universel et existentiel en première année d'université, entre l'explicite et l'implicite[END_REF] when she asks students to define the upper bound M of a set A and when almost all of them answer by what she calls "a strange definition": « ∀ x ∈ A, ∃ M ∈ IR, x ≤ M ». Piatek-Jimenez (2010) confirms the asymmetric perception and interpretation of those two kinds of double quantification in the mathematical field. There seem to be two different problems: the use of "strange" conventions and the unawareness to interpret statements.

To overcome these difficulties [START_REF] Dubinsky | On student understanding of AE and EA Quantification[END_REF] presented a game based on the dialogical logic (explained below) and used it with students to make them aware that two kinds of interpretation can be used and that this depends on rules of interpretation linked to the places of quantifiers. Results have shown that the game seems to help sometimes students to understand such statements but in some cases, it does not, and in a few cases this has created more confusion.

I want to pursue this research by designing tasks to overcome the difficulties of the interpretation of double quantified statements using the methodology of scientific debate. This implies two related goals. The first one deals with the identification of students' difficulties related to the order of quantifiers in a double quantified statement. I specifically explore the role of some other potential variables: the set of quantification (familiar or not, finite or infinite), the semiotic representation of the quantified variables (formal or usual language) and the kind of relation involved in the predicate (familiar or not). The second goal is to design and to implement tasks by considering the results of the study related to the first goal. A new game called Q² based on the interpretation of double quantified statements to determine a winning strategy will play a fundamental role in this design.

The aim of this paper is to study the following question: is there a domain of values of variables where students take into account the order of quantifiers and if yes how to take advantage of this domain to enlarge it? Specifically, how to make students aware that they are interpreting these statements and that they need rules for that?

The methodology of scientific debate: A tool for designing tasks

The scientific debate [START_REF] Legrand | Scientific debate in mathematics courses[END_REF]) is a socio-constructivist approach to learning and teaching mathematics based on two main principles: 1) the need of new knowledge can be obtained by making one realize that his/her previous conceptions may lead to contradiction; 2) the organization of appropriate debates among students permits firstly to express and share previous conceptions about the targeted subject, secondly to encounter the limits of these conceptions, and thirdly to be able to understand the institutionalisation related to this knowledge that is made by the teacher. Three fundamental steps shape the design of tasks by means of the scientific debate methodology:

1) The first step consists on epistemological and cognitive studies of the targeted mathematical concept: the role of quantifiers in formal statements and the interpretation of double quantified statements that stands on the use of a convention of interpretation. For this interpretation, I have chosen the dialogical logic (Lorezen, 1967). In this paper, we mainly explore the cognitive aspects and we use a questionnaire for that purpose.

2) The second step concerns the design of the tasks, which is done by using mainly two kinds of questions to initiate the debate. The first kind of question concerns the truth of a conjecture: is it true or false? This conjecture can be given by the teacher or can come from the students after a call for conjecture. The second kind of question concerns the nature of an object for a conjecture or a property: is it an example, a counter example, or off topic (neither example nor counter-example) for this conjecture, and is it an example for a property? The example can come from the teacher or from a call for an example. A vote is made: do you think it is true, false or something else (an example, a counter example, off topic or something else)? The possibility of voting for something else is given to preserve the authenticity of the other votes (voting True or False must be a choice founded on convictions). Then a debate is organized by the teacher among those who have different view-points. The teacher never gives any opinion about what is debated but tries to maintain a level of interaction by emphasizing the contradictions between students.

3) The third step concerns the level of experimentation and its analysis: what actually happens is confronted to what was expected to happen. Specifically, this analysis leads to discuss the efficiency of the choices made in the two first steps.

The whole study is conducted according to the aforementioned three steps. In this paper, I show my findings from the study related to the first step and I give more details about the elaboration and the results of the two final steps. For the first step, I have chosen to use a questionnaire on a sample of 181 students in their last year of secondary school to identify students' difficulties with the interpretation of double quantified statements in formal and non-formal context. For the second step, the results of this questionnaire are used in a way that is aiming to make students aware that the lack of the rules of interpretation is not a problem in a certain domain but leads them to conflicts out of this domain. This design is based on a game, called Q², in which the interpretation of double quantified statements is crucial: the question that will initiate the debate concerns the way to win this game. For the third step, I experiment this design for students in their last year of secondary school in a scientific class composed of 29 students (experimentations at university are planned).

Task design

Background: The result of the questionnaire I have mainly studied five variables in the interpretation of double quantified statements: place of quantifiers, kind of statements (mathematical or not), the set of quantification (familiar or not, finite or infinite), the semiotic representation of the variables (formal or informal language) and the kind of relation involved (familiar or not). In the questionnaire, words (for all, exists) are used instead of symbols because symbols are introduced after the game Q² as a solution to the conflicts that appear about quantifiers and associated conventions. A questionnaire has been given to 182 students. We will only give the findings here (for more details see Lecorre, 2016a). The first finding is that the rule of "correct interpretation for AE statements and inversed one for EA statements" seems not so obvious: some EA statements are perfectly interpreted whereas some AE statements are interpreted through an inversion of the quantifiers and the variables. The second finding is that some other variables are correlated with difficulties in the interpretation (e.g. width of the quantified set, formalization of variables...). The third finding is that there exists a domain of correct interpretation for both EA and AE. This domain is made of non-mathematical statements quantified on a "small" finite set (less than ten values) without any formal variable and with a familiar relation. These findings are then used to design the tasks aiming at discussing the interpretation of double quantified statements.

The game Q²

The game presented by [START_REF] Dubinsky | On student understanding of AE and EA Quantification[END_REF] is based on the dialogical logic of [START_REF] Lorenzen | Métamathématiques[END_REF] which gives a way to decide on the truth of quantified statements using a codified dialog between a proponent and an opponent. In this game, for example, if the sentence is "for All x there Exists y such that R(x;y)", the A-player chooses x and the E-player has to find a y such that R(x;y) is verified. If he/she fails to find such y, the A-player wins, otherwise the A-player can give another x (same kind of rule for EA). I call this game a direct game: given a statement and sets of quantification, you have to decide the truth. The Q² game is an inverted game: given a statement and the truth, the players construct the set of quantification to make a statement true or false.

For the Q² game I choose values of the variables that make it an easy game to play: nonmathematical field, small set of quantification, familiar relation. This choice is made to permit students to get into the game and into the interpretation of associated statements. This is a two players' game. This game is given by four elements: a starting rule, a winning rule, a statement and a grid. For example, red player has to start (starting rule) and plays for the falsification (winning rule) of the statement S: "For all red letters, there exists the same black letter" and the given grid: So red has, first, to circle one letter with his red pencil. Then the black player circles one letter and so on until all the letters have been circled. With the given elements of this example (starting and winning rule, statement, and grid) and the coloration of letters, if the statement S is false then the red player wins, but if it's true the black one wins. This game has, of course, many variants, beginning with the filling of the grid and the starting and winning rules. This game is the heart of a design which aims precisely to enlarge the domain of good interpretation. I am going to show that a smart use of Q² in the design has the potential to reach such a goal by emphasizing the lack of the convention of interpretation.

"The Q² situation"

The principle of the Q² situation is to provoke conflicts of interpretations that will lead students to the need of the convention of interpretation. The situation Q² is divided into four periods:

-The first period aims an appropriation of the game Q² by playing.

-The second period deals with the concept of winning grid for the game Q².

-The third period targets a conflict of interpretation, in a way that students feel the need of conventions of interpretation. At that stage, the conventions are given.

-The fourth period is just an application of these conventions on the unsuccessful domain where the values of the variables lead students to difficulties of interpretation.

In the first period, a paper is given that contains eight games of Q² to play (each game is defined by a statement, a starting rule and a grid). These games are designed for the two players to have opportunities to win and to start to have ideas on how to play to have good chances of winning. In fact, with such a game, with "good players", the winner depends only on the statement (EA or AE), the starting rule, and the grid given.

The second period aims at the definition of winning grid. The students are asked to give the winning grids for a given rule, then a debate is organized about these propositions: are they winning grids, or not? The contradictory opinions about the propositions should lead students to identify the lack of a definition of winning grid. A winning grid is, in fact, quite difficult to define in a mathematical way for pre-university students (double recursive definition). Here, a definition such as "a grid is a winning grid for red if when red plays "cleverly", he is sure to win, even if black also plays cleverly" is largely sufficient for this design. When the students show a need for a definition, the above definition is given.

The third period aims to highlight the lack of convention of interpretation. Once again, for a given rule, the students are asked to give winning grids. There should be no more conflicts about what is a winning grid in general, but new conflicts should appear about the propositions: is this grid really a winning grid with this rule? This should happen because the winning grids depend on the interpretation of the rule which is a double quantified statement. And the need for a convention should appear with the impossibility to find a common agreement (is it a winning grid or not?). The didactical principle which is used here is the following: it seems very difficult to organize a direct confrontation of the different rules used by students to interpret a double quantified statement, because this problem, taken as a general one, is too theoretical and depends on too many variables (findings of the questionnaire). On the contrary, it is much easier to create a conflict on concrete consequences of the interpretation of such statements. Here, the conflict holds about the question "is the grid a winning one or not?" Then, trying to understand each other, and trying to convince their peers, students are going to explain their own interpretation. And then it will appear that the implicit conventions used by students are contradictory. Students can realize that without common conventions, no agreement is possible. Deciding if a grid is a winning grid or not is possibly more complicated than the logical principles of deciding who won, but it leads students to materialize their own conventions through these grids and permit to confront these conventions.

Then these conventions are given in the manner of the dialogic logic (Lorezen, 1967), which is described above with the game of [START_REF] Dubinsky | On student understanding of AE and EA Quantification[END_REF]. At this stage, the game Q² plays as a preparation to such rules by simulating a game between a proponent and an opponent. The design here aims much more the awareness of the need of convention than the "right rules". The aim is to make students aware of the necessity to check the validity of their interpretation relatively to the adopted conventions.

The fourth period consists in verifying, still using debates, that the given rules can lead to agreements, and even in the domain where students used to fail: the rules are helpful, efficient.

Results

The first period of the situation Q² (playing the game) shows a good appropriation of the game: the winning grids were almost always won by the one for who the grid was a winning one, which means that the students were playing "cleverly". Some strategies seemed to begin to be used. And, above all, almost all the decisions about who is the winner were correct. All this is coherent with the results from the questionnaire in terms of domain of interpretation.

The second period led to the question of the definition of a winning grid. The definition is given.

The third period begun with a question of the interpretation raised by students in a debate. I present an extract of the script of this debate to explain this unexpected acceleration. Student are asked to give winning grids for red for the game "There exists a red square such that all black squares have the same symbol" where red starts and plays for true. The given statement does not specify if black squares shall have the same symbol as the red one or not. So, after six winning grids had been given by the students in the second period, this lack of information was intensely discussed. The grid P1 was put into debate and everyone agreed that it was winning grid for red. Then P2 was put into debate (28 votes for a winning grid for red and 1 for "something else"). Loïc who had voted "something else" changed his mind and explained why, for him, it was a winning grid for red:

Loïc: Because red starts and as he plays to win he takes the square A and… But Quentin disagreed with this explanation:

Quentin:

And because red has it also (One B square)

But Hadrien and Louis did not agree with this addition:

Hadrien:

No, he has A squares and B squares.

Quentin: Yes! Louis: I think that I should just say that there exists a red square.

The sequence above shows that these students do not need to disagree on the fact that it is a winning grid or not to begin to explain their own interpretation: "the same symbol as the red square" (Quentin), or "the same symbol for all black squares" (Loïc, Hadrien). Then Fabio, in the same way, explained that nothing must be added contrary to the sayings of Quentin:

Fabio: I do think that what Quentin added is not necessary.

Then Quentin proposed a grid to strengthen the differences of interpretations: Leaving aside for a while the problem of the winning grid, the teacher asked whether the statement S1 was true or not, according to the grid of Quentin. Twelve students thought that the statement was true, ten false, while six students voted something else. Some conclusions were then raised :

Fabio:

There are some, like me, that can think that all the black squares have got the same symbol is enough and some other, like Sébastien, who are thinking that there must be a red square that has got exactly the same symbol as any black square. These two interpretations are not directly linked to an EA/AE inversion but to the interpretation of the predicate. It would be interesting to investigate the role that the vernacular language (the linguistic subtleties that may eventually vary in French or English) may have played in leading students to interpret the two variables in the predicate as bound variables ("the same as red squares"), as shall be done in AE statements, where as it isn't the case in such an EA statement. In any case, the discussion among students led them to work on these interpretations and to get aware of the complexity of having a unique interpretation. The teacher then gave the conventions of interpretation of EA and AE logical statements. These conventions were then used successfully for the applications of the fourth period. One month later, the same students had to face double quantified statements in a situation aiming at the definition of limit. They experienced, once again, the need of conventions when they encountered another conflict of interpretation of these statements, so they checked the conventions to decide on their own (Lecorre, 2016b).

Mickaël

Later, another experimentation was realized with some other students using a "complete" rule ("…such that all the black squares have the same symbol as the red square") and this led students, with a grid of eight squares filled with symbols appearing twice, to the predicted conflict (EA/AE interpretation). A new didactic variable is then identified, the constitution of quantified sets (how they are filled relatively to the predicate), which raises an unexpected question: is the students' choice of convention between EA and AE convention for the interpretation more guided by the constitution of the sets of quantification than by the statement itself?

Conclusion

The Q² situation, with the scientific debate methodology, were used to make the interpretation of the double quantified statement the main object of students' discussion. The contradictions that the lack of conventions of interpretation was bound to imply then emerged in that discussion. This lack which was invisible to students suddenly came into light with these contradictions.

More precisely, the questionnaire led to identify some variables playing a role in the interpretation of students. There is a domain of the value of those variables which gives a good interpretation.

This Q² situation led students to be ready to receive the conventions of interpretation of double quantified statements. Indeed, they have experienced the need of shared conventions. The described situation Q² mainly aims the recognition that there are two kinds of double quantifications that should be differenced according to the order of the quantifiers in the statement and the convention of interpretation of the predicate but the validation using variables defined in function of the other variable remains a difficulty.

Figure 1 :

 1 Figure 1: A grid for the game Q²

Figure 2 :

 2 Figure 2: The winning grids for red given by the students

Figure 3 :

 3 Figure 3: Quentin's grid (As in black and Bs in red)

  : I'm asking: if you who think that only black squares have the same symbol is enough, this red square exists such that what? This made Hadrien change his opinion, but Louis did not agree with this change:

	Louis:	There are two opposite opinions just because we're not thinking the same.
	Teacher:	Ok, you are not reading the same way…That is what Fabio said…
	Louis:	Exactly!
	Juliette and Maxime then explained why they voted something else:
	Maxime:	That is exactly why, from the beginning, I voted Other.
	Juliette:	So do I.

Proceedings of CERME10


text:
Margo Kondratieva

Carl Winsløw
email: winslow@ind.ku.dk

A praxeological approach to Klein's plan B: Cross-cutting from calculus to Fourier analysis

Keywords: Calculus, Fourier analysis, transition, praxeology

In our previous work on Calculus-Analysis transition we independently explored the reasons of students' difficulties with studying analysis and observed that the problem is related to the discontinuity of students' experiences leading to their inability to interpret the (formal and more rigorous) ideas learned in analysis courses in terms of (practical) knowledge acquired in calculus courses, and vice versa. In this paper we continue and combine our work with two new contributions: a theoretical formulation of Klein's idea of a "Plan B" for teaching mathematics, applied to the transition in question; and a concrete student activity attempting to give flesh to this "plan" for the special case of introductory Fourier Analysis.

Introduction

Calculus and Analysis appear as related, but distinct subdisciplines in many contemporary university programmes. Calculus courses specialise in mathematical themes indicated by course titles such as "Integral Calculus", "Functions of several variables" or "Ordinary differential equations". Analysis courses, on the other hand, treat theoretical perspectives on these same mathematical themes, gradually moving from course titles such as "Real Analysis", "Fourier Analysis" towards more abstract areas such as functional and harmonic analysis. In short, calculus courses can be roughly characterized as teaching students certain calculation practices related to real and vector valued functions, with little theoretical precision or justificationwhile analysis courses tend to present "formal theory with little practice". This division is a didactical construct which is related to historical and institutional conditions (see [START_REF] Klisinska | The fundamental theorem of calculus. A study into the didactic transposition of proof[END_REF], for an in-depth analysis of the case of the "fundamental theorem of calculus").

The main reason for the division seems to be that the two types of courses cater to different student populations. While calculus courses are studied by a large cohort of students in natural and social sciences, much fewer students get to study analysis (mainly students of pure mathematics, theoretical physics and mathematical statistics). For these and other reasons, it may be difficult to change the course structure.

The transition from Calculus to Analysis presents mathematics students with several challenges (for examples, see [START_REF] Winsløw | Klein's double discontinuity revisited: contemporary challenges for universities preparing teachers to teach calculus[END_REF]. Here is a typical student formulation of some of these (interview with a student of the first author, summer 2016):

In calculus courses we learn methods, but usually the why questions are not explained or proved. (...) However, analysis courses felt as separate. They were more theoretical than applied. I never grasped them as well as Calculus. It was often unclear, what it was leading to. I wish we had a better sense of connection between the theory we covered in pure math courses and the methods shown in applied math courses.

Thematic Working Group 14

Proceedings of CERME10

We have explored this perceived lack of "connection" in earlier papers [START_REF] Kondratieva | The promise of interconnecting problems for enriching students' experiences in mathematics[END_REF][START_REF] Kondratieva | On advanced mathematical methods and more elementary ideas met (or not) before[END_REF][START_REF] Winsløw | Les problèmes de transition dans l'enseignement de l'analyse et la complémentarité des approches diverses de la didactique[END_REF][START_REF] Winsløw | Angles, trigonometric functions, and university level Analysis[END_REF]. In the present paper, we use the notion of praxeology [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF] to represent the general "connection" problem in more precise terms, and -as a theoretical case studyto present a new proposal for "connecting" Calculus and one of the basic theorems in Fourier Analysis. Our research results are thus basically theoretical. [START_REF] Chevallard | Steps towards a new epistemology in mathematics education[END_REF] defines a praxeology as a pair ) , ( L P consisting of a praxis block P and a logos block L. A praxeology is a minimal element of human knowledge, P representing the practical partthe "know how" -and L the intellectual part, the "thinking and explaining"often called "know why". The two are interdependent:

Theoretical framework

…no human action can exist without being, at least partially, "explained", made "intelligible", "justified", "accounted for", in whatever style of "reasoning" such an explanation or justification may be cast. Praxis thus entails logos, which in turn backs up praxis. For praxis needs supportjust because, in the long run, no human doing goes unquestioned. (Chevallard, 2006, p. 23).

As we focus here on mathematical praxeologies taught and learnt at university, it is obvious that praxis (e.g. computing the Fourier series of a given function) is intimately connected to various forms of logos -from ad hoc explanations of standard techniques to theories involving general definitions, theorems and proofs. To compare and contrast the praxeologies developed in calculus and analysis courses, we consider that they represent various affinities with the praxeologies of present-day mathematicians, which we shall represent suggestively using Greek letters : the praxis blocks, including computational techniques, are identical to those used (for tasks of the same type) by professional mathematicians, while the logos blocks i L are limited to informal explanations of smaller collection of practice blocks (like the various techniques for determining whether a series is convergent or not). On the other hand, analysis courses then focus on the scientific form of logos blocks. The taught and learnt praxeologies in such courses are therefore of the form ) , ( i i P  where each i  constitutes a logos block consistent with the scientific model, while the praxis blocks i P are didactic "afterthoughts" constructed to consolidate the acquisition of i  . As mentioned in the introduction -such teaching practices often fail to motivate students for i  and to provide them with a coherent, autonomous relationship with ) , ( i i  

. Our research focuses on how this issue can be addressed.

Taken together, calculus and analysis courses in principle provide students with praxeologies ) , ( i i   which, taken individually, are close to the scientific model. For instance, convergence tests used in Calculus praxis on series are now supplied with a theory involving precise definitions and proofs of the "criteria" for convergence. However, because the number and technical complexity of these praxeologies is quite high and the i  were taught in other courses, typically years before, some effort and support may still be needed for students to "assemble" individual praxeologies ) , ( i i  

. We can say that working along these lines corresponds to establishing complete but separate praxeologies within different small areas of mathematics, which is what Klein called "Plan A" for teaching: "Plan A is based upon more particularistic conception of science which divides the total field into a series of mutually separated parts and attempts to develop each part by itself." (Klein 1908(Klein /1932, p. 77, p. 77, see also [START_REF] Winsløw | Angles, trigonometric functions, and university level Analysis[END_REF] Within this approach two praxeologies are related only through strict logical dependency at the theoretical level and only within strictly confined areas (which, in terms of what students actually acquire, may be surprisingly small).

However as explained by Klein, the scientific practice (historically as well as currently) involves more than isolated or strictly dependent praxeologies. Klein (1908Klein ( /1932, p. 78, p. 78) recommended that also elements of "Plan B" be included in mathematics teaching both in schools and at university:

The supporter of Plan B lays the chief stress upon the organic combination of the partial fields, and upon the stimulation which these exert one upon another. He prefers, therefore, the methods which open for him an understanding of several fields under a uniform point of view.

In terms of the praxeological model above, we may thus summarize the two "plans" or strategies for developing and connecting students' previous knowledge as follows:

Plan A. assemble elementary praxeologies ) , (

i i  
from calculus and analysis elements, by establishing firm relations of type

i i   
. In fact, this is sometimes a possible function of the "fingertip" exercises, which constitute i P in many courses and textbooks on analysis.

Plan B. develop cross-cutting relationships among praxeologies which could be of one of the following types (or combinations among them):

B1. Relating praxis blocks (

j i   
) or logos blocks (

j i    )
B2. Relating otherwise unrelated praxis and logos blocks (

j i    )
It may be more easy and common to develop relations of type B1, even if they certainly appear more often in "mathematician" praxeologies than in typical course teaching. We now present and analyse an example of student activity aiming at developing relations of the last type (B2): namely, that students connect a collection of praxis blocks i  (concerning trigonometry, integration and convergence) to a logos block ( L 0 ) from Fourier Analysis.

A logos block from Fourier Analysis

For a 2-periodic, piecewise continuous function f : ℝ → ℂ, the Fourier series of f is defined as

        1 1 0 2 1 sin cos n n n n nx b nx a a
, where

      dx nx x f a n cos ) ( 1 and       dx nx x f b n sin ) ( 1 .
In general the two infinite series may not converge at a point x. In 1829, Dirichlet gave one of the first sufficient conditions for pointwise convergence of a Fourier series. A version of this result which is usually formulated for piecewise continuous functions, is stated below in a special case to avoid technicalities. We refer to it as Dirichlet's theorem, although we don't use his original claim.

Theorem If f : ℝ → ℂ is a continuous 2-periodic function with piecewise continuous derivative, the Fourier series of f is pointwise convergent to ) (x f at every  x ℝ.
Here we outline the main steps of the proof that appears in a typical formal course on Fourier Analysis (see e.g. Folland, 1992, pp. 30-36 for the wealth of computational details omitted here):

1. First, it is shown that under weaker assumptions, such as f being square integrable and 2periodic, the coefficients n a and n b tend to zero as n tends to infinity. (In fact, one demonstrates this by showing that the series  2 n a and  2 n b are both convergent.)

2. Next, by direct computation, we rewrite the Nth partial sum given by

       N n n N n n N nx b nx a a x s 1 1 0 2 1 sin cos ) ( as:       dy y K y x f x s N N ) ( ) ( ) (
, where N K is the Dirichlet kernel defined by

     N n N nx x K 1 2 1 1 ). cos ( ) ( Clearly    / ) 2 / 1 ( ) 0 ( N K N and 1      dx x K N ) (
. By clever use of the addition formulae,

) / sin( ) / sin( ) ( 2 2 2 x x Nx x K N    for 0  x . 3. Finally, let D N (x) = s N (x) -f (x). Using 2.
, a straightforward set of computations yields (*)

D N (x) = 1 p g x (
f '(x + 0) ¹ f '(x - 0
). The functions g x (y)sin(y / 2) and g x (y)cos(y / 2), notwithstanding the possible discontinuity of the latter at y = 0, are bounded, and thus, square integrable. Formula (*) shows that D N (x) is simply the sum of Nth Fourier coefficients of these two functions. Applying now 1., we conclude that the infinite Fourier series s ¥ (x) converges to f (x) because its partial sum s N (x) can be written as s N (x)

= f (x)+ D N (x), where D N (x)vanishes as N ® ¥.
The key point of the proof is (*): to rewrite D N (x) as a sum of two Fourier coefficients, together with the fact that the coefficients tend to zero as

N ® ¥.
According to the distinction we made above, the general result (and certainly its proof) does not belong to the realm of Calculus. When students are presented with the theory in a somewhat more general form, -they may not realize that the proof almost entirely draws on the notion of series convergence and on techniques known from Calculus. To make them discover that is the aim of the design that we present in the next section, focusing on the following special case:

Example. Applying the above Theorem for

=       1 2 2 ) 1 ( 4 3 n n n so that        1 2 2 1 . 12 ) 1 ( n n n
The latter -striking -result can be derived by other means, as a variant of the famous Basel problem (see e.g. [START_REF] Kondratieva | Didactical implications of various methods to evaluate )[END_REF]. One such approach is at the root of the design presented below.

Outline and a priori analysis of Student Activity

In continuation of earlier work of the first author [START_REF] Kondratieva | Didactical implications of various methods to evaluate )[END_REF], we took the Example above as a point of departure for constructing a sequence of exercise-like activities that would lead Thematic Working Group 14 Proceedings of CERME10 students through two approaches to computing the infinite sum considered in the Example: part 1 consisting of a series of "calculus-like assignments" which, without saying so, go through the proof of Dirichlet's theorem in the special case where 2 ) ( x x f  ; part 2 in which the students work directly with the result, as in the Example; and a final reflection in which the students are supposed to realize that the proof (known from a Fourier Analysis logos block L 0 ) amounts to nothing more than a generalization of the sequence of calculus techniques drawn upon in part 1. We notice here that the numbering suggests that the praxis and logos blocks thus connected through the activity are not, prima facie, connected -and, thus, the connection established is really of type B.

Part 1 begins with presenting the problem of determining the value of

    2 1 / ) 1 ( n S n
. The praxis blocks acquired in calculus courses do not provide ready-made techniques to solve this problem; instead, students are invited to do so through "several preliminary problems":

1. Compute the integral     dx mx x cos 2
for any natural number m ( : 1  integration rules). , and : 4  differentiation from first principles). ( 1  ).

Show that

     m n x x m nx 1 2 1 ) 2 / sin( 2 ) 2 / 1 sin( cos ( : 2  trigonometric formulae). 3. Show that       0 0 0 , sin / ) ( 2 x x x x x u defines a continuous function on   2 / , 0  ( : 3  techniques to compute limits, including the special result 1 sin lim 1 0    x x x ).

Find u and show this function is bounded on

  2 / , 0  ( 3 

Show

               2 / 0 2 / 0 2 2 ) ( 1
6. Show that the integral in 5 converges to 0 as   m

( 1  and ) 3  .

7. Finally, combine the results above to find S ( 1  ).

The only slightly advanced praxis (technique) involved in the above appears in 6., where 1  is supposed to include something like

   f f -or, alternatively, 3  should include a rule which permits to conclude that 0     I m m f lim under appropriate conditions on ) ( m f .
Part 2 of the activity invites the students to compute the Fourier series of

2 ) ( x x f
 and engage in some concrete computations related to its convergence which are in fact very similar to 1.-6. above.

The final reflection is supposed to make them discover the close parallel between the two parts.

In case they do recall elements of the proof of Dirichlet's theorem, they will recognize in Problem 2. the simplification technique of the Dirichlet's kernel in step 2., and in Problems 3.-6.a way to get the convergence result of step 3. Meanwhile, step 1 appears more indirectly in the concrete case, where both the Fourier coefficients of f and the auxiliary coefficients, appearing in (*), can be computed or estimated directly. Indeed, many textbooks present Step 1 as a corollary of a more general theorem on orthogonal sets in Hilbert space. This, together with the technicalities related to the possible non-continuity of f, contributes to the impression that the proof is way beyond simple techniques from Calculus. Nevertheless, comparing the proof with the proposed activity, the students could realize that in the special case f (x) = x 2 the proof relies entirely on well-known praxis blocks ( i  , i = 1,2,3,4). Certainly, this could establish a strong relation L 0 « ( i  , i = 1-4) which might in fact be prepared by students' working Part 1 above prior to encountering L 0 .

Some experimental observations

To pilot and refine the above design before testing it with a larger group of students in a course on Fourier Analysis, we have done a preliminary study with five students who have completed at least 3 years of undergraduate mathematics program. These students were involved in summer research projects in mathematics at the Memorial University of Newfoundland. This involvement is an indicator of the students' high motivation and achievements in studying mathematics. The students volunteered to solve the problems from the activity (with no firm restrictions in time or access to any materials) and participate in a follow-up semi-structured interview. The students were asked whether they found the problems (a) familiar, (b) interesting, (c) and whether they saw any connections between praxis and logos of parts 1 and 2. All students regarded problems 1-6 as familiar from their calculus courses, and they found them easy. In words of one student, "I loved that stuff when I was in my calculus courses, so I found these problems pleasant… And they are not difficult, too." While problems 1-6 were familiar to the students, they clearly indicated that no projects of nature similar to problem 7 were present in their study: "I think it is a cool layout. Nothing of this format was in my calculus courses,when you need to use previous results to solve larger or more interesting problem." Students regarded the task of series evaluation as challenging but also most enjoyable: "The problems 1-6 were like baby steps… And they met together nicely in problem 7". So, at least these students were successful and appreciative of tasks in part 1. As for the accessibility of part 1 for an average student in a calculus course, we had overall a confirmative response: "I think it is accessible for a student who has done Integral Calculus.... if they are not confined to a very short period of time, then yes." Another student confirmed, "it could be a good exam sequence, more fun than just doing problems." However, a different perspective was also articulated: "…many students take this [Integral Calculus] course because it is a prerequisite for their programs, so maybe they would not be interested as much."

Among the five students only one had studied Fourier series in his courses, while others had heard the term but had very little familiarity with the subject. However, they all recognized the similarity in the technical praxis of parts 1 and 2, for example, that calculation of the Fourier series in part 2 resembles evaluation of integrals in problem 1 from part 1. Bridging the theory and connecting the idea of convergence of an individual series in part 1 and pointwise convergence of Fourier series was more challenging. This is where the role of an instructor might be critical: to help students to relate new theoretical constructs and ideas to familiar praxis. We realize that students' background makes a difference, however even learners previously unfamiliar with Fourier series seem to benefit from this activity. Students' responses based not on reproduction of known facts, but rather on reasoning related to their practical experiences, is an indication of establishing new mathematical relations. The following is an excerpt from an interview with students of the first author: M.K.:

Is it always possible to replace a function with its Fourier series in calculations? Student 1:

In my (applied) courses we were told that no (a function is not always equal to its Fourier series), but this was never proved. Now it kind of makes more sense.

M.K.: Do you think that familiarity with part 1 would help to exemplify general theory related to Fourier series and their convergence?

Student 1: Yes, definitely. I think it is more logical to go this way about discussing conditions of pointwise convergence of Fourier series. However, the experiences need to be close together in time, so that the second part occurs before students have forgotten the first portion.

The space available does not allow us to give the details of students' accomplishments and their impact on our design. We simply note that the sample students were by and large able to complete them and see the inner connections. Also, the students considered that building on the familiar computational tasks (1-6) on the one hand, and on new theoretical constructs (Example) on the other hand, organized around given problem (evaluation of the series S) was stimulating: "Suppose someone has a theoretical solution and I have a computational solution and they look completely different, but they give the same answer to the same problem so they have to be the same somehow… then I want to go back and find out why they are the same. I found it very interesting."

Conclusions

While calculus courses include praxis blocks i  compatible with those of professional mathematicians, their theoretical components are more informal and focused on algebraic computation. Moreover, these praxis blocks are often isolated from each other, as they occur within separate sections of textbooks and courses, and students typically don't get opportunities to apply them in combinations. When students meet Dirichlet's theorem, they are given a general and relatively complicated proof (in Analysis). In such courses, "simple applications" (such as the Example above) may be introduced as examples or exercises, to build an artificial practice block 0 P corresponding to the much richer logos block L 0 . The fact that the general proof ( L 0 ) is essentially linked to familiar praxis blocks from Calculus will then not appear. We propose that by replacing 0 P by a sequence of computational auxiliary tasks (1-7), similar to the steps 2 and 3 of the proof ( L 0 ), two goals can be achieved. First, students will see how different praxis blocks ) ,..., ( 4 1   from Calculus work together and combine to support 0 P by themselves. Secondly, this special case might help to prepare for the various general theorems on Fourier series convergence ( L 0 and beyond) by relating it to the concrete and familiar elements P 1 ,..., P 4 . This hypothesis will be investigated empirically. More generally, we hypothesize that situations which enable students to establish "cross cutting relations" above constitutes an essential complement to "Plan A" type courses. These constructions could emerge from detecting explicit links between different solutions of interconnecting problems [START_REF] Kondratieva | The promise of interconnecting problems for enriching students' experiences in mathematics[END_REF], as shown above. It will clearly necessitate a careful analysis of (central) theory blocks of more advanced courses, and resources found in reasonably well-established praxis blocks of previous courses. So, while the general hypothesis may look fairly simply, realizing it in concrete caseseven theoretically -represents a non-trivial didactical research programme.

Proceedings of CERME10


text:
Julie Jovignot
email: juliejovignot@gmail.com

Thomas Hausberger

Viviane Durand-Guerrier

Praxeological analysis: The case of ideals in Ring Theory 1

Keywords: Mathematical structuralism, structuralist praxeologies, ideals in Ring Theory, logic

. In this communication, we illustrate it on the case of ideals in Ring Theory. We provide a detailed study of a proof extracted from a textbook in Abstract Algebra showing that structuralist praxeologies involve interplay between intertwined algebraic, set-theoretic and logical praxeologies, revealing a hidden complexity.

Introduction

Hausberger (2016), with the introduction of the notion of structuralist praxeology, underlined the importance of praxeological analysis in the didactical study of phenomena related to the teaching and learning of Abstract Algebra at University level. His work is based on an epistemological investigation of algebraic structuralism that showed that mathematical practice in Abstract Algebra may be interpreted as an application of the axiomatic method, structures being used as tools by mathematicians in order to prove statements on objects. In the Anthropological Theory of the Didactics [START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic[END_REF], a method is a set of techniques. In fact, ATD poses the general model that every human activity may be described by quadruples [T,τ,θ,Θ], called praxeologies, which correspond to the organisations it sets up: these combine a praxis (a type of tasks T and a set of techniques τ) with a logos that include two levels of description and justification of the praxis: the technology θ and the theory Θ. [START_REF] Hausberger | A propos des praxéologies structuralistes en Algèbre Abstraite[END_REF] made the assumption that clarifying the structuralist techniques may illuminate practices in Abstract Algebra, make their rationale more visible and ground them as a coherent whole. [START_REF] Hausberger | A propos des praxéologies structuralistes en Algèbre Abstraite[END_REF] described common tasks and techniques in the arithmetic of abstract rings and studied the structuralist praxeologies developed by students on a mathematical forum online. By contrast, the empirical data presented here is an extract of the solution of an exercise on Noetherian rings proposed by teachers in a textbook. The central mathematical notion at stake is the notion of ideal. By a detailed study of this example, we will develop the argument that structuralist praxeologies involve interplays between algebraic, settheoretic and logical praxeologies, thus revealing a hidden complexity.

Structuralist praxeologies as intertwined algebraic, set-theoretic and logical ones

The notion of structuralist praxeology Structuralist techniques are the by-products of the complete rewriting of classical algebra operated by Noether's school in the 1920s [START_REF] Hausberger | On the concept of (homo)morphism: a key notion in the learning of Abstract Algebra[END_REF][START_REF] Hausberger | A propos des praxéologies structuralistes en Algèbre Abstraite[END_REF]. They are based on the now standard structuralist constructs: sub-structures, homomorphisms, isomorphism theorems, products or sums of structures, quotients, etc. [START_REF] Hausberger | A propos des praxéologies structuralistes en Algèbre Abstraite[END_REF] stressed that common tasks in Abstract Algebra may often be solved using elementary techniques. Whenever its logos block contains a theorem on structures, the praxeology may be called structuralist. Nevertheless, a gradation of its structuralist dimension (loc. cit.) may be observed. In fact, structuralist praxeologies reflect the concrete-abstract and particular-general dialectics that are at stake in Abstract Algebra: tasks involving concrete and particular objects are completed by using abstract and general considerations on structures. Examples will be given in the sequel. The particularity of structuralist praxeologies that will be investigated in this article is that they often involve sub-praxeologies of algebraic, set-theoretic or logical type.

Algebraic and set-theoretic praxeologies

Noether qualified her own work of "set-theoretic foundation for algebra" [START_REF] Hausberger | On the concept of (homo)morphism: a key notion in the learning of Abstract Algebra[END_REF], following Dedekind. On an epistemological point of view, it is characterised by the transition from thinking about operations on elements to thinking in terms of selected subsets and homomorphisms. The distinguished subsets are the kernels of homomorphisms, hence the normal subgroups in Group Theory and the ideals in Ring Theory. Noether uncovered the importance of the chain condition on ideals that led to the definition of Noetherian rings (see below). In other words, set-theoretic operations on ideals are connected to algebraic properties on elements. We will present below this connection by means of a "dictionary". It explains the intertwining of algebraic praxeologies (on the level of elements) and set-theoretic praxeologies (on the level of structures), but it leads also to the use of logical praxeologies, notably for the descent from the ideals toward the elements at stake.

Logical praxeologies

Many tasks in Abstract Algebra involve proof and proving, thus logical praxeologies. [START_REF] Durand-Guerrier | Truth versus validity in mathematical proof[END_REF] has enlightened that the natural deduction developed by [START_REF] Copi | Symbolic Logic[END_REF] provides a powerful tool to analyse and check mathematical proofs. In particular, it allows identifying those steps where mathematical arguments are silenced, supporting the claim that mathematics and logic are closely intertwined in proof. We will rely on Copi's natural deduction to describe logical praxeologies likely to appear in proof and proving: elimination and introduction of implication, universal quantifiers and existential quantifiers, restriction of the domain of quantification. The theory is the First order logic (Predicate calculus) and the technologies are logical theorems (i.e. statements true for every interpretation in any non-empty domain). In Copi's natural deduction, one deals with a generic non-empty universe, and some aspects need pragmatic control in order to ensure validity, as we will see below. The following table details common logical praxeologies that can be involved in a proof and hence in the study of structuralist praxeologies.

We provide triplets (type of tasks, technique, technology): No logical theorem. Need to control that the name of the element has not been used prior in the proof Using an AE statement ("For all, Exists") in a proof.

L7

Restriction of the domain of quantification Given a universal statement true in a domain A, assert it on a subdomain B of A.

[(x (A(x)  F(x))  (x (B(x)  A(x))]  [(x (B(x)  F(x))
Fitting the statement with the antecedent of a conditional statement L8

Transformatio n of a statement preserving its truth value Substitute an equivalent statement to a given statement

In the case of implication: [(x (P(x)  R(x)) (x (Q(x)  R(x))]  [(x (P(x)  Q(x))
Using the dictionary of properties elements/structures (cf. table 2)

Figure 1: List of a priori logical praxeologies according to Copi

The case of ideals in Ring Theory

The notion of ideal and its ecology An ideal I of a ring (𝐴, +,•) is, by definition, a subset of A with these properties: (i) I is a subgroup of the additive group (𝐴, +); (ii) if 𝑎 ∈ 𝐴 and 𝑥 ∈ 𝐼, then 𝑎 • 𝑥 ∈ 𝐼. As part of her Master's degree dissertation, the first author conducted an epistemological and didactic study of the concept of ideal in order to explore the ecology, including the habitats and niche [START_REF] Artaud | Introduction à l'approche écologique du didactique : l'écologie des organisations mathématiques et didactiques[END_REF] of this concept in French university education. This epistemological study started with the creation of ideal numbers by Kumer in 1847 and it enhanced the rise towards abstraction leading in the 1920s through the work of Noether to the concept we use today (Jovignot, to appear). As far as the ecology of the concept of ideal is concerned, the epistemological study allowed the identification of the following a priori main habitats: general Ring Theory (quotient rings and isomorphism theorems), arithmetic of abstract rings and elimination theory. Bearing on those results, Jovignot developed an analytical framework to identify habitats and niches of the notion of ideal in algebra textbooks addressed to undergraduates and Master's students. A first study of 3 textbooks has led to improve this grid, that was then applied to a sample of 7 French textbooks that were considered as representative of the ecology of the concept of ideal and of its use in the different post-secondary institutions in which this concept is taught in France. This study confirmed general Ring Theory and arithmetic of abstract rings as major habitats of the concept of ideal, but it also allowed the exhibition of habitats that had not been previously identified, such as the theory of modules and algebraic geometry, which suggests the importance of leading a complementary study in contemporary epistemology. Finally, elimination theory appeared, in our sample, only in the specialized computer algebra manual.

Ideals and structuralist praxeologies in the arithmetic of abstract rings

Arithmetic of abstract rings as a mathematical domain is characterized by a mathematical structure in "Russian dolls": it involves Euclidean, principal ideal domains (PID) and unique factorization domains (UFD), which generalize properties of the ring of integers, and mathematical theorems that state inclusions from the former class to the latter. Common tasks consist in proving that a given ring, for instance Gauss's ring of integers Z[i], belongs to a class or the other. More abstract tasks, such as the one that will be analyzed below, involve making new connections between such classes. The central notion is the notion of ideal. In fact, the class may be defined directly by a property on ideals (such as PIDs) or by properties on elements (such as UFDs) which may be related to properties of ideals by means of the following "dictionary" which was already mentioned above. This dictionary will be useful to understand used praxeologies in the task studied below. The task under study

In the next section, we will present the praxeological analysis of an exercise involving the concept of ideal which is extracted from a book addressed to Master's degree students preparing the French Agrégation1 : Francinou, S. & Gianella, H. (1994). This book is widely used by university students in France. The authors sampled classical exercises in Algebra and provided proofs. In the chosen exercise, students are requested to establish a connection between Noetherian integral domains endowed with an extra property and PIDs. We clarify that this praxeological analysis is not a tool for teaching but could help us later for the design of experimentation with students.

The exercise is the following (our translation):

Let A be a Noetherian integral domain. We assume that every maximal ideal of A is principal. 1) Show that A is a unique factorization domain.

2) Show that every non-zero prime ideal is maximal, principal and of the form (p) where p is irreducible. 3) Show that A is a principal ideal domain. (loc. cit. p.57) We will restrict our study to question 1. The authors introduce the following classical criteria, in which E designates the property of existence of a factorization and U the property of unicity:

A is a UFD if and only if: a) every increasing chain (a1) < (a2) < (a3) <… of principal ideals is stationary (equivalent to E) b) every irreducible element is prime (equivalent to U)

The proof provided by the authors is the following (our translation from French):

Since A is Noetherian, A satisfies (E). To establish that A is a unique factorization domain, it suffices to prove that if p is irreducible, the ideal (p) is prime. Let us consider a maximal ideal M containing (p). By hypothesis M is principal generated by a. Thus a divides p. Since a is not a unit (because M ≠ A), p and a are associates and (p) = M is maximal. In particular (p) is prime.

Praxeological analysis of the task

Supplementing the proof

Reading the proof of the authors, it appears that a lot of steps remain implicit. In order to be able to study the full set of praxeologies involved in the proof, either explicit or implicit, we have supplemented it. We consider that the proof is complete when all the statements are obtained by natural deduction from previously established results or standard theorems in Abstract Algebra. We do not examine in detail in this paper the question of which of these supplements should be taught, but we will provide hypothesis that will be studied in further steps of this research. The steps of the proof presented in the textbook are numbered, our supplements appear in italic and are designated by letters whenever several steps are involved. The supplemented proof reads as follows:

1. Since A is noetherian, A satisfies (E). a. Indeed, A is Noetherian so every increasing chain of ideals is stationary by definition. b. In particular, every increasing chain of principal ideals is stationary. c. So, thanks to the criteria, A satisfies(E). 2. To establish that A is a UFD it suffices to prove that if p is irreducible, the ideal (p) is prime. a. Indeed, we need to show that every irreducible element is prime (criteria, b) b. And "p is prime" is equivalent to "(p) is prime" c. In fact, we will show that (p) is maximal. It is enough since every maximal ideal is prime in a ring.

Let p be an irreducible element of A and M a maximal ideal containing (p). a. If there aren't any irreducible elements, we are done. In fact, irreducible elements exist since A is

Noetherian, except if A is a field. b. p is not an unit, so (p) is proper and M exists according to Krull's theorem. 4. By hypothesis M is principal. Let a be a generator of M.

Thus a divides p. a. Indeed, (p) is included in M and M = (a), so (p) is included in (a). b. And (p) is included in (a) if and only if a divides p.

6. Since a is not a unit (because M ≠ A), p and a are associates -indeed, a|p so there exists b in A such that p=ab; moreover, p is irreducible so, since a is not a unit, b must be a unit and p and a are associates -7. and (p) = M is maximal since two principal ideals are equal if and only if their generators are associates. 8. In particular (p) is prime.

Praxeological analysis

We present the praxeological analysis as a tabular; in the column labelled "steps", we are indicating in which steps of the proof the studied praxeology appears. Only tasks, techniques and technologies are mentioned; the theory in the sense of ATD is Ring Theory. We note S structuralist praxeologies and A algebraic ones.

Conclusions of our analysis

This praxeological study allows us to highlight significant characteristics of the praxeologies used by the authors that we summarize below.

The dictionary elements/structure is used along the proof; indeed the proof involves relationships between properties of elements of the ring (being an irreducible or a prime element) and properties of subsets (being a principal, maximal or prime ideal). Moreover, the algebraic notion of generator and the dictionary of properties elements/structures allow the replacement of common set-theoretic praxeologies (such as proving an equality of two sets by double inclusion) by more powerful algebraic praxeologies (involving A1). This cultural shift that is characteristic of structuralist algebra may be pointed out as a potential obstacle (previous praxeologies hindering the use of the new praxeologies to be acquired).

The structuralist praxeology S1 decomposes into several sub-praxeologies S2-S6, A1, L1, L7, L8. In the authors' proof, only structuralist steps of the proof are given; the steps involving algebraic and logical praxeologies are nearly systematically hidden. We may hypothesize that these authors see structuralist steps as the architecture of the proof and expect students to be able to reconstruct the missing elements by themselves. On the contrary, we will argue in favour of setting out the nonstructuralist praxeologies and elaborate on their role in connection with structuralist praxeologies.

The nearly systematic omission of logical praxeologies raises the following issues and comments. First of all, the proof deals with generic objects, which is due to the level of generality of the statement of the exercise. It is already explicit in the statement itself, therefore both rules of elimination (L3) and introduction (L4) of a universal quantifier on the ring are not needed. In the sequel, the ideal M is introduced (step 3) without justification of its existence (Krull's theorem). The introduction of the generator of M is allusive and could be misinterpreted, letting think that this element has already been introduced. In both cases, the elimination of the existential quantifier (L6) remains implicit, letting thus implicit the statements themselves. The generic element p that plays a central role in the proof is not introduced, while it is a delicate step. Indeed, a classical way to prove a conditional statement by generic element is to introduce an element satisfying the antecedent, under the implicit assumption that such element exist; indeed, if not, there is nothing to prove (step 3.a). In addition, letting silent the restriction of the quantification domain (L7, step 1) hides the fact that this rule does not apply for existential statements, which might not be clear for some students. Finally, the substitution rule (L8) is a key for using the dictionary elements/structure by substituting a property of elements for a property of structure and vice-versa.

Giving such a proof requires the availability of the praxeologies cited above and a suitable understanding of their interrelations, or enough experience on the structuralist methodology in order to apply these praxeologies en acte. We hypothesise that the textbook's proof does not permit the appropriation of the structuralist praxeologies at stake. A didactical strategy to reach this goal may include, for instance, a "meta-discourse" on the crucial role of the dictionary elements/structure, together with making explicit the logical praxeologies whose role has been underlined above.

The particular construction of the proof (related to the decomposition of S1 into S2-S6-A1) can be understood by analysing the interplay between the blocks of the praxis and that of the logos of the different praxeologies engaged in the proof. However, the technological elements are seldom present in the proof written by the authors. For example, the properties of the dictionary are used but barely cited. Even if the students own in their praxeological equipment those technologies, the proof doesn't offer them the opportunity to identify those technologies in the context of the proof and thus build the associated structuralist praxeologies in order to be able to use them by themselves in another proof situation.

General conclusion and perspectives

Our praxeological analysis has highlighted the complexity of the chosen exercise. This complexity comes, in particular, from the decomposition of structuralist praxeologies into several structuralist sub-praxeologies and their interrelation with logical and algebraic praxeologies. These are fundamental in order to make the structuralist technologies practically operative. A sketchy proof which restricts to the structuralist steps, although it is seen as a clear and synthetic account by mathematicians, may therefore appear quite inadequate for self-learning by students who are not familiar with the structuralist methodology. In other words, our study contributes to break the "illusion of transparency" behind proofs that may be found in Abstract Algebra textbooks.

We aim to record and analyse the work of students who attempt to reconstruct the proof as we did, or to write a proof from scratch. In this forthcoming empirical study, our praxeological analysis will serve as an a priori analysis. It may also be used as a starting point in order to prepare clues for the students and other types of didactical intervention, as well as to lead semi-structured interviews. Moreover, we intend to interview the authors of the book in order to get insights in their goals and motivations for the choices they made when writing down the proof. More generally, it is expected from these praxeological analyses, conducted on a larger scale, a deeper understanding of structuralist praxeologies with a view to setting up didactic engineerings dedicated to the teaching of structuralist concepts and in particular the ideal concept.

Figure 2 :

 2 Figure 2: dictionary of properties elements/structures

Figure 3 :

 3 Figure 3: praxeological analysis of the task

Proceedings of CERME10

Competitive exam for prospective teachers for secondary and tertiary education


text:
Lynn Farah
email: lgf00@mail.aub.edu

The organization of study in French business school preparatory classes

Keywords: Mathematics learning, preparatory classes (CPGE), organization of study, teaching devices, teachers' practices

This paper presents parts of a doctoral research pertaining to the study of mathematics in French business school preparatory classes. In what follows, we identify the main features of the institutional devices designed and implemented by two mathematics teachers in their respective classrooms in order to influence and transform the working habits of their students. To do so, we rely on qualitative analysis of data collected mainly through interviews and questionnaires. Our conceptual framework borrows constructs from the Anthropological Theory of Didactics as well as several works in the sociology of education field.

Context

Student failure in mathematics during the first university years is a widespread problem in many countries including France, but it does not seem to affect in the same way students of all French higher education institutions. In fact, there are in France alternative institutions such as the Classes Préparatoires aux Grandes Écoles (CPGE in what follows) students achieve much better results in mathematics than those enrolled in regular French universities, as is reported in official statistics provided by the ministry of national and higher education and research 1 . The CPGE prepare students over two academic years after obtaining the French baccalaureate to enter the Grandes Écoles, which are mainly business schools or engineering schools, by passing the concours, national competitive written and oral exams specific to each type of school which students take by the end of the second preparatory year. In the French educational systems, the two preparatory years at the CPGE are equivalent to the first two years of undergraduate study at university and do not lead to obtaining a degree. The CPGE have three streams, scientific (S), business and economics (EC) and literary (L), which each have different tracks.

Our study focuses on the CPGE in the continuation of the work of [START_REF] Castela | Des mathématiques à leurs utilisations, contribution à l'étude de la productivité praxéologique des institutions et de leurs sujets / Le travail personnel au coeur du développement praxéologique des élèves en tant qu'utilisateurs de mathématiques[END_REF]. These institutions differ widely from regular French universities in elements commonly considered as the main causes of student failure (Farah, 2015b, chap.II, section 4). They are known for their selectivity in recruiting students who have obtained exceedingly above-average results throughout high school and in the French baccalaureate, as well as their supportive culture, which fosters student collaboration and provides them with close follow-up, in a relatively rigid high-school-like system within stable moderate-sized classrooms. In fact, these institutions resemble more the European and North American universities than the French universities in terms of teaching methods and studentteacher relationships. Therefore, it is important to point out that although our study is conducted in a 1 Source: http://www.enseignementsup-recherche.gouv.fr/cid75181/reussite-et-echec-en-premier-cycle.html Thematic Working Group 14 Proceedings of CERME10 specific environment, its results help us to understand more general issues that concern other institutions in France and in other countries.

With the change of institution, from high school to CPGE, students face a significant rift with respect to the work they have to complete in order to succeed in mathematics. In fact, in the CPGE, students are expected to develop without teacher supervision and in addition to the tasks that are prescribed to them, significant autonomous personal work in mathematics that is not necessarily in the continuity of what ensured their prior success in high school. Moreover, assessment in the CPGE is entirely conditioned by the nature of the examinations of the concours. In mathematics, it is cumulative, covering the content of both preparatory years, which is never the case neither at university nor even in high school. In this paper, we are interested in the ways these institutions shore up their students and help them develop a new working mode in mathematics, geared towards the CPGE requirements, during the first year of scientific track business school preparatory classes (ECS). Therefore, we focus on the relationships that exist between institutional and students' personal organization of study in mathematics.

Conceptual framework

We are mainly interested in the institutional dimension and its impact on students' learning in mathematics. We claim that through their ways of functioning, the CPGE institutions help their subjects (the students) construct a new working mode in mathematics adapted to the CPGE requirements. We refer to the foundations of the Anthropological Theory of Didactics (ATD) to examine the weight and action of these institutions [START_REF] Chevallard | Approche anthropologique du rapport au savoir et didactique des mathématiques[END_REF]. We endorse [START_REF] Chevallard | Approche anthropologique du rapport au savoir et didactique des mathématiques[END_REF] description of an institution as a social system that allows and imposes on its subjects, that is people who occupy different positions within the institution, ways of doing and thinking. Subjects are hence submitted to collective constraints and expectations that regulate their actions and thus subjugate them (in French, assujettir). For our study, we consider first, at a global level, the CPGE institution within which individuals occupy the positions of student, teacher, and administrative staff… At a local level, we focus first on the teaching of mathematics in the broader subinstitutions, the EC stream then the ECS track. Next, we consider the teaching of mathematics in the school institution. Lastly, we examine the institution of the mathematics classroom of each teacher, with two main positions: teacher and student.

Regardless of the level of institution in question, it is important to bring forward the idea of organizational stability emphasized by [START_REF] Darmon | Classes Préparatoires. La fabrique d'une jeunesse dominante[END_REF] and [START_REF] Rauscher | Les professeurs des classes préparatoires aux grandes écoles[END_REF]. Darmon identifies institutional devices that are shared among CPGEs, which put students to work while supervising them. In accordance with the ATD hypotheses, Rauscher advances that subjects of the CPGE institutions occupying the teacher position (per discipline, hence in mathematics in particular) predominantly share common experiences and background traits. They thus form a distinctive social group, as a result of several interacting mechanisms, and take decisions as a team, or a tribe (tribu) as Chevallard (2003, p.89) would refer to it. The hypothesized continuity and stability within the CPGE, as to the norms of the teacher profession and the study organization created by each teacher, enables us to foresee the influence of the global CPGE institution on the students' work.

The work of [START_REF] Darmon | Classes Préparatoires. La fabrique d'une jeunesse dominante[END_REF] in sociology allows us to clarify certain important aspects of the functioning and role of the CPGE institution, through which it exerts its subjugation actions and the resourcing of students' personal work. Darmon defines a specific type of institution based on the socializing functions of the CPGE and examines it as an institution where a specific type of person is manufactured. According to her, these "enveloping institutions" (p.10) shape and transform the students through preparatory institutional socialization processes. Therefore, she analyzes the different daily functioning devices that make it possible for the institution to exert its effects on the students ("surveillance, sanction, examination and pressuring techniques" p.16). It appears that this subjugation process in undertaken by taking into consideration the individuals involved; this is far from common in higher education practices in France and sounds highly paradoxical. In fact, Darmon puts forward the fact that the CPGE strives to soften the preparatory violence. She describes the institution as being powerful but not totalitarian, violent but concerned about the wellbeing of its members, it operates by individualizing to the extreme rather than homogenizing, thus reinforcing its take over the individuals which are its members (2013, p.28). Her findings converge with those of [START_REF] Daverne | Les bons élèves, expériences et cadres de formation[END_REF] who put forth the regretted yet accepted pressure to which CPGE students are subjected as well as the personalized adaptation of teaching.

Furthermore, we sought to develop the institutional dimension of our research from the point of view of the teachers. We hence considered two levels: the first once pertains to the way teachers are subjected to the CPGE institution; the second one is related to the mathematics classroom of each teacher, the local institution s/he creates thanks to stable devices which we seek to identify. We believe that the subjugations to the CPGE generate an environment in which each teacher enjoys a given autonomy and can freely express his/her individuality within the boundaries of the common CPGE teacher culture highlighted by Rauscher. Using Darmon's work, we bring forward the CPGE institutional functioning analysis in order to explore how the socializing function is exerted.

Therefore, based on the different didactical and sociological elements of our conceptual framework, we address the following research question in this paper: which institutional devices lead to the transformation of the students' personal work mode in mathematics, at both levels of an institution, ranging from the global CPGE institution to the local teacher classroom institution?

Methodology

A first phase of our study, which is beyond the scope of this paper, was entirely centered on students' personal work in mathematics. Using questionnaires and interviews of first year students from two ECS track preparatory schools in Paris (see Farah, 2015a), we gathered data about the organization of mathematics courses, the teaching methods, the assessment tools and the resources provided by the teachers to put students at work and accompany them in the study of mathematics. Based on this, we sought to approach the practices of the teachers by examining the teaching devices they design and implement in their classrooms as well as their meta discourse (Robert and Robinet, 1993, p.1). We must clarify that the word "discourse" refers to verbal expression, i.e. the use of words to exchange thoughts and ideas. It is not a theoretical construct borrowed from a conceptual framework. As for the word "meta", we refer to Robert & Robinet's definition (1993) whereby a teacher's discourse contains meta elements, i.e. about mathematics and about the ways of doing and learning mathematics. The second phase of our research followed from this.

To answer the research question addressed in this paper, we relied on qualitative analysis of data collected from two mathematics teachers of the schools involved in our study. We started with data obtained through semi-structured interviews conducted with each of the two teachers about the devices instituted in their classrooms. Then, we designed and had each teacher complete two questionnaires. The first one, inspired from Rauscher's thesis ( 2010), is about their career path and their choices with respect to teaching in the CPGE, which we believe determine their position and impact their subjugation within the CPGE institution. The second one, inspired from Darmon's book (2013), is about the assessment and pressuring devices the teachers implement in their respective classrooms to put the students to work, as well as their ways of softening preparatory violence in terms of the support and comfort they bring to the students.

We used Qualitative Content Analysis of the interview transcriptions and questionnaire answers to put together a description of institutional devices implemented by each teacher (local) and those common across the different institutional levels (global). The narratives were analyzed thoroughly, manually, line by line, in a search for keywords and vocabulary terms constituting the teachers' discourse about the ways students should study mathematics, while focusing on anything pertaining to institutionalization, regularity, and insistence on specific actions by the students or the teachers. Our search was structured around the following themes that determined the analysis rubrics of our content analysis: taking notes in class, managing work and revisions overall, studying between two mathematics sessions, using resources, preparing for an exam, the colles2 , collaboration between students, student difficulties. We then resorted to triangulation to confirm the information obtained from the teacher-designed instruments by comparing it with what we had gathered in the first phase of our study through the students. We must clarify that, besides the things that converge with the information gathered from the students, we had very few elements that would allow us to determine the propinquity between the teachers' statements and what actually takes place in their classrooms. In fact, one could be surprised that, in an analysis of teaching practices, there have been few filed observations. This limit is due to practical constraints in terms the duration of a doctoral thesis. The final output of our analysis is presented in the form of a description of the different institutional devices that organize and shape students' personal work in mathematics.

Main findings

The findings show that the teachers seek to put their students to work and mold their study methods in mathematics through numerous collective devices instituted in their classrooms. In addition, they closely follow-up on each student's work in mathematics through customized individual devices. Thanks to the latter, the teachers develop and apply diverse pressuring techniques in order to ensure the students' intellectual training and their successful passing of the concours. We provide below a description of the main devices, which are either dictated by the global organization of mathematics study in the CPGE institution and thus revealing how the teachers are subjugated to their institution, or specific to one of the more local sub-institutions (for more details, see Farah, in press).

The teacher's course and the follow-up beyond

The mathematics course organization and progression are the first aspects of guidance to students' work. For both teachers, when they explain a mathematics lesson, their first priority is to retain the students' attention while encouraging them to actively participate by regularly asking questions. The lesson is completed and illustrated through examples and exercises, which are solved in class or at home, then corrected in class. During regular classroom sessions, if needed, teachers wrap up the work that they have previously started during practical solving sessions (called Travaux Dirigés or TD). Theses special sessions give students room to work on exercises in small groups, thus fostering discussions with the teacher as well as classmates.

Both professors involved in our study use a handout as the baseline for the lesson explanation; they distribute it to students either systematically or occasionally. Depending on the teacher, the class or the chapter, this handout can be exhaustive or having blanks to complete, and teachers modify it regularly in order to tailor its contents to the level of the students and their capabilities (concentration, understanding, note taking ability) and the course pace is slowed down or increased accordingly. The main objective behind the use of such a device is to save note-taking time in class and ensure that students don't make mistakes in copying key elements. A typical handout contains mathematical definitions and notations, propositions and theorems with occasional short proofs, lesson examples and application exercises. During the lesson, the teachers spend most of the time completing the missing proofs then provide additional examples. They explain to students that the proofs are the basis of mathematics, and repeatedly underline the practical and generic aspects to be extracted. On the contrary, little importance is given to statements of theorems. This is an example of a specificity of mathematics teaching in this CPGE stream as opposed to the insistence on academic knowledge in universities. In addition, the teachers formulate several remarks that are not solely about theoretical mathematical knowledge. In fact, in addition to the mathematical content, the teachers make comments related to practical knowledge. These are part of their meta discourse which contains technological elements [START_REF] Castela | Des mathématiques à leurs utilisations, contribution à l'étude de la productivité praxéologique des institutions et de leurs sujets / Le travail personnel au coeur du développement praxéologique des élèves en tant qu'utilisateurs de mathématiques[END_REF] used to bring forward the know-hows linked to the mathematical content, thus allowing them to accompany students in their study.

In addition to the time dedicated to lesson explanation, exercises solving and correction, both teachers ensure to always be available to assist the students in the learning of mathematics outside the classroom. They are willing to answer questions, provide explanations, recommend and even correct additional work despite believing that the workload they assign is already enough (regular exercise sheets and occasional extra exercise sheets with their correction for some chapters). They usually encourage students not to look for more resources (textbooks, online) and focus on what they provide due to time constraints. Moreover, the teachers hold weekly tutoring sessions to ensure that students are getting all the needed help within the institution. Through their extended availability and individualized follow-up, the teachers are ensuring that all their students are provided with the necessary assistance for their learning, while they control and organize their study. They are thus softening the preparatory violence through surveillance and examination. This is one important manifestation of the CPGE teacher culture that is absent in French universities.

The recurring discourse about the ways of studying

The teachers encourage their students to regularly study their mathematics lessons and solve the assigned exercises (for both the regular sessions and the TD) and they always explain to them how they should proceed to do so. The teachers emphasize the importance of reading a mathematics lesson actively and critically. The objective is learning the keys lesson elements while thinking about them and asking the right questions to first understand then memorize. According to both teachers, validation of the learning should be done by playing-back important lesson contents mentally, then preferably in writing.

In addition, they both stress the crucial role of decontextualizing in mathematics learning. To do so, they underline the significance of both the results brought through proof and the use of generic components of reasoning, in addition to the techniques used in standard exercises which students must be able to acquire and plough back in other situations. In fact, they preach strategic exercise solving whereby students are expected to identify standard problems and recognize methods, techniques and tricks that can be used to solve them, which can also be applied to other problem situations. Thus, according to both teachers, students should have a transfer-oriented approach to exercises rather than one that favors only practice or reproduction [START_REF] Castela | Des mathématiques à leurs utilisations, contribution à l'étude de la productivité praxéologique des institutions et de leurs sujets / Le travail personnel au coeur du développement praxéologique des élèves en tant qu'utilisateurs de mathématiques[END_REF], the latter are dominant among successful university students but are deemed ineffective in the CPGE. The teachers also insist on the necessity of doubling efforts until mastery is attained when facing difficulties in solving an exercise.

We can refer to the notion of constructive help proposed by the teachers to guide students in studying the lesson, solving exercises and decontextualizing proof and exercises, when working on a daily basis between two mathematics sessions, as well as when preparing for an exam. In fact, we have identified several features of help common to both teachers in their discourse, about expected ways of studying mathematics and practical knowledge pertaining to the techniques which could help students gain know-hows relating to the awaited tasks. These illustrate the convergence of learning methods regularly repeated by two different teachers of the same stream and track.

The assessment tools

In order to ensure that the students are completing the assigned work (lesson and exercises) and to identify their weaknesses and difficulties in mathematics before the graded exams, teachers use personalized informal evaluation techniques during classroom sessions (both regular and TD). They often resort to oral interrogations about the lesson notions by randomly calling on students or choosing those who are inattentive or fall behind. Also, while the students are solving exercises in class, the teachers go around to check what they have done, they assess their understanding and help when needed. Then, the teachers encourage the students to engage in discussions about the exercises' solutions before correcting them or asking a student to do so. One of the two teachers gives special care to exercises preparation by the students prior to class. In order to push students to maintain regular work, he periodically calls students to the board and collects notebooks without prior warning whenever he notices that the work has not been fully done, without necessary grading any. These are all examples of surveillance and sanction techniques that allow the institution to monitor and redirect the work of the students' work.

The teachers have several types of more formal assessment devices, institutionalized at the global CPGE level, which allow them to evaluate the degree of investment and understanding of their students. Firstly, they use all sorts of written evaluations. Teachers mainly resort to short quizzes focused on the mathematics lesson content (definitions, theorems…) at the beginning of the school years to push the students to study, however they state that they cannot maintain them throughout the year due to time constraints. They also have monthly exams (called Devoirs Surveillés or DS), and bi or triannual mock concours which are summative and are conducted in conditions similar to the official concours. One of the teachers quizzes his students about the correction of previous DS exams thus allowing the students to detect and address their weaknesses. In addition, teachers assign and grade homework sets (called Devoirs Maison or DM) on a monthly basis and they usually invite students to work on those in small groups. All of the above are examination and pressuring techniques used across the CPGE institution, with specificities of each sub-institution.

Last but not least, the colles are the most important assessment tool that teachers use to evaluate their students in a highly customized manner. We summarize the main perks they list about this institutional device, which are for most specific to the case of mathematics colles in the ECS track, since their organization and functioning changes across disciplines, tracks and streams. The colles impose on the students a work and study regularity, which is certainly stressful and tiring for some, but the pressure is eventually seen as beneficial for the majority. Mathematic colles sessions are described as similar to private tutoring sessions where students can discover their weaknesses, ask questions, obtain additional explanations and a new point of view, and practice by solving additional exercises. Further to these mathematical learning related aspects, the colles are characterized by their interpersonal feature and the know-hows and social skills they teach (stress management, oral presentation, self confidence) which go beyond the scope of the classroom or even the school. Therefore, the colles are viewed as a summary of the best things the CPGE have to offer in terms of learning environment for their students (Daverne and Dutercq, 2013, p.182). They are to many teachers the secret to students' success in CPGE (ibidem, p. 182), despite the difficulties and constraints they are subjected to.

Discussion and conclusions

On one hand, we can conclude that the teachers who took part in our study are heavily involved in their students' learning. To accommodate the needs and level of a "new population" (ibidem, p.7) of CPGE students, more diversified in terms of academic and social backgrounds, teachers redefine their teaching modalities and pedagogical devices and adjust the level of their expectations. Daverne and Dutercq state that if some young students have good working habits when they enroll in the CPGE, none yet have the general culture nor the confidence needed to face the concours, which requires from teachers a high level of commitment towards them and a constant care for their moral (ibidem, p.8). Hence, the teachers participate in the didactical and pedagogical organization of their students' autonomous study thanks to the advice they provide and the devices they institute and regularly adapt according to their needs and capabilities. They are therefore clearly dedicated to their students' success. This is also reflected through the closeness in the student/ teacher relationships, which we do not tackle in this paper (for more information, see [START_REF] Farah | Étude et mise à l'étude des mathématiques en classes préparatoires économiques et commerciales: point de vue des étudiants, point de vue des professeurs[END_REF].

On the other hand, although the use of the varied pressuring techniques in mathematics differs among teachers and depending on the students' dispositions, the techniques themselves remain redundant across teachers and classes. This brings forward their generality and continuity within the EC stream of the CPGE institution, of which they become a specificity. As a matter of fact, we find in the teachers' discourse common features underlining the coherence in the practices of teacher tribes per class as well as the stability of practices within each preparatory school, within the ECS track, and even within the entire EC stream. Regardless of the level of the institution, the devices used are specific to the teaching and learning of mathematics, even though we do not examine them with respect to a specific mathematical content in this paper. We conclude that the coherence of practices noted between the two teachers involved in our study concurs with what the sociological studies of [START_REF] Rauscher | Les professeurs des classes préparatoires aux grandes écoles[END_REF], [START_REF] Darmon | Classes Préparatoires. La fabrique d'une jeunesse dominante[END_REF], and [START_REF] Daverne | Les bons élèves, expériences et cadres de formation[END_REF] have identified.

Proceedings of CERME10

A colle is an assessment tool specific to preparatory classes. In mathematics, it classically takes the form of a one-hour bi-monthly oral examination, in groups of three students working individually but simultaneously, answering lesson questions and/or solving problems on the board, managed by a colleur who is present to supervise and grade the work.Thematic Working Group 14