Hadas Brandes
email: hadas.brandes@concordia.ca

Nadia Hardy

From single to multi-variable Calculus: a transition?

Keywords: transition to and across university mathematics, assessment practices in university mathematics education, teaching and learning of analysis and calculus, Anthropological Theory of the Didactic, praxeology

We recently used the notion of praxeology from the Anthropological Theory of the Didactic to model the knowledge that is necessary for students to learn in order to succeed in an undergraduate multivariable Calculus course. We considered the presence and absence of elements of the knowledge to be taught, as proposed by curricular documents, in the knowledge to be learned, as indicated by final exams. Our results indicate that the mathematical activities expected of students at this level align with the activities observed in differential and integral Calculus, where exercise-driven assessments set students' work mainly in the recognition of types of tasks and recollection of appropriate techniques.

INTRODUCTION

So far, research on the teaching and learning of Calculus has focused on singlevariable Calculus. Cognitive and epistemological obstacles have been illustrated against students' learning of Calculus [START_REF] Tall | Concept image and concept definition in mathematics with particular reference to limits and continuity[END_REF][START_REF] Sierpinska | Understanding in Mathematics[END_REF] and an institutional perspective has also been taken to study the influence of institutional practices on students' learning of Calculus [START_REF] Barbé | Didactic Restrictions on the Teacher's Practice: The Case of Limits of Functions in Spanish High Schools[END_REF][START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF]. There's a pattern that indicates Calculus students mostly engage in procedural work that requires only a superficial grasp of the underlying concepts [START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF][START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF][START_REF] Selden | Do Calculus students eventually learn to solve non-routine problems?[END_REF].

We recently undertook a study that shifts the focus to multivariable Calculus courses [START_REF] Brandes | The bare necessities for doing undergraduate multivariable Calculus[END_REF]. Our goal was to determine the knowledge that is essential for students to learn in order to provide acceptable solutions on the final exam of an undergraduate multivariable Calculus course. To this end, we used the notion of praxeology from the Anthropological Theory of the Didactic [START_REF] Chevallard | Organiser l'étude 1. Structures & fonctions[END_REF] to model the knowledge students are expected to learn and the knowledge to be taught. We present our operationalization of this concept in the first part of this paper. In second stage, we discuss a partial result of our study that places this multivariable Calculus course along the transitions that university mathematics students undergo in their engagement with mathematics (Winsløw, Barquero, de Vleeschouwer, & Hardy, 2014).

THE EDUCATIONAL SYSTEM

We studied a 'Multivariable Calculus I' course offered to students in two mathematics programs at a large North-American university. One of the programs is for those who plan to join the workforce after graduation; the other aims to prepare students for graduate studies in mathematics. Students in either stream will have completed one-variable differential and integral Calculus and an introductory Linear Algebra course on matrix and vector algebra. The multivariable Calculus course and its sequel ('Multivariable Calculus II') are prerequisite to most of the courses in the program geared towards graduate studies. Students usually complete Multivariable Calculus I and II within the first year of their degree.

In any given term, the course is split into two sections per program, with about 70 students per section. The course is heavily coordinated across sections and terms through a strict curriculum, course examiner, and common assessments. The course outline specifies what to teach every week along with exercises from the textbook. The course examiner writes common assessments for students in all sections. A student's grade is obtained from the highest of the following: 10% assignments, 30% midterm, 60% final exam, or 10% assignments and 90% final exam. Finals exams are therefore the crux of a student's performance; in turn, the exams are consistent from term to term in both format and content. Past exams are readily available to students, and concern with their reactions prevents changes being made to the final exams.

ROUTINE PROBLEMS IN SINGLE-VARIABLE CALCULUS

We are interested in the mathematical activities with which students of a multivariable Calculus course are expected to engage. We focus on the types of problems that typify the learning of multivariable Calculus; a wealth of studies do so for single-variable Calculus [START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF][START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF][START_REF] Selden | Do Calculus students eventually learn to solve non-routine problems?[END_REF]. These studies emphasize the exercise-driven quality of the course assessments, in the sense of [START_REF] Selden | Do Calculus students eventually learn to solve non-routine problems?[END_REF] routine problems, which "mimic sample problems found in the text or lectures, except for minor changes in wording, notation, coefficients, constants, or functions" and "can be solved by well-practiced methods" (p.18).

The exercise-driven quality of the course assessments extends to elements of the curricula [START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF]. Calculus textbooks traditionally adhere to a definitiontheorem-example-exercise format, wherein the exercises repeat the problematics of the examples and algorithms outlined in the text. [START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF] measured the extent to which intrinsic mathematical properties play a role in the minimal reasoning required to solve routine tasks in traditional Calculus textbooks. Lithner's classification of reasoning types runs along a scale of how big a role is played by the mathematical properties intrinsic to the problem versus the reapplication of known algorithms; this scale runs parallel to [START_REF] Selden | Do Calculus students eventually learn to solve non-routine problems?[END_REF] spectrum of problems from very routine to very non-routine, which vary based on how familiar the solver is with the given problem. The more routine the problem, the less interaction is required of the solver with the mathematics specific to that problem.

Assessments in North-American Calculus courses are largely drawn from the course textbook, which [START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF] showed to be steeped in routine problems. Accordingly, he found students' strategies to be anchored in what they recall superficially rather than in the mathematics specific to a problem. This correlates with Calculus students' failure to complete non-routine problems [START_REF] Selden | Do Calculus students eventually learn to solve non-routine problems?[END_REF][START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF]. If textbook exercises can mostly be solved by identifying superficial similarities with a known example [START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF], then students' nonreliance on intrinsic mathematical properties and over-reliance on the recall of algorithms may have roots in their learning environment. We follow this view by framing our study within the Anthropological Theory of the Didactic and focusing on elements of students' learning environment: curricular and assessment documents.

ANTHROPOLOGICAL THEORY OF THE DIDACTIC (ATD)

Framework

From the perspective of the ATD, knowledge does not exist in a vacuum, rather, it is bound to the institution in which it is shared and somehow connected to the knowledge shared in related institutions; such connection is called transposition and is of a didactic nature in the context of educational institutions [START_REF] Chevallard | La transposition didactique, du savoir savant au savoir enseigné[END_REF]. Didactic transpositions take place along a spectrum of knowledge in which scholarly mathematics (the knowledge developed, shared, and used by the experts -the mathematicians) is transposed into knowledge to be taught in a given institution, up to a transposition into knowledge actually learned by the students. This transformation of mathematical knowledge takes form in several stages: scholarly knowledge, knowledge to be taught, knowledge actually taught, knowledge to be learned, and knowledge actually learned.

An essential feature of the ATD is an epistemological model called praxeology. It allows the researcher to model knowledge at any stage of a didactic transposition. The notion of praxeology is based in the assumption that any human activity consists of a practical block (praxis) and a theoretical block (logos). The praxis is made up of tasks to be accomplished and techniques with which to accomplish them; the logos is the discourse that produces, justifies, and explains the techniques in the practical block. [START_REF] Chevallard | L'analyse des pratiques enseignantes en théorie anthropologique du didactique [The analysis of teaching practice in the anthropological theory of the didactic[END_REF] specifies two components of a theoretical block: technology , the discourse that produces and justifies the techniques in the practical block, and theory that justifies the technology.

In light of these theoretical considerations, and given our goal of finding the minimal core of knowledge that students must learn in order to succeed in their multivariable Calculus course, we treated three instances of didactic transposition. We created a model of the knowledge to be learned, as determined by the final examinations; to this end, we needed a model of the knowledge to be taught, as indicated by the curricular documents. In order to familiarize ourselves with the mathematics prior to these two instances of didactic transposition, we also created a reference model based on the scholarly multivariable Calculus knowledge to be transposed. Before we present our praxeological models of the knowledge to be taught and to be learned, we review some of the literature about mathematics students' praxeologies.

Transitions in students' praxeologies [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF] explain that students, at the pre-university level and in some cases at the university level, tend to have a praxeology defined mostly by practice. This is especially the case in differential and integral Calculus courses where assessment is concerned mostly with the practical block and does not address the ways in which the theoretical maintains the practical. This may have a precedent in the way knowledge is taught in the classroom, as teachers may not have time to justify tasks and techniques, given often-hefty curricula to deliver. Students, for their part, tacitly accept the existence of a theoretical discourse supporting the practical without concerning themselves with it [START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF][START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF]. Their work is mainly in recognizing types of tasks and identifying a suitable technique [START_REF] Hardy | Students' perceptions of institutional practices: The case of limits of functions in college level Calculus courses[END_REF][START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF], much as in Lithner's identification of similarities reasoning ( 2004) and Selden et al.'s routine problems (1999).

As students progress in university mathematics, they undergo two transitions. Where once they might have ignored theoretical blocks and worked exclusively within the practical block of a praxeology, they increasingly have to engage with theory and technology in their completion of tasks. [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF] call the transition from praxeologies that are purely practical to praxeologies that include a theoretical and a practical block a first transition of university mathematical praxeologies (p.101). For example, prior to the first transition, students complete tasks such as using derivative rules to find the derivative of a function. Here, differentiability is an always-met condition of the functions upon which students act in the tasks they do. Prior to the first transition, it is sufficient for students to attend only to the practical block of the mathematical knowledge; at the other end of this transition, students are required to acknowledge the theoretical block as the justification for the techniques they use for accomplishing a task. For instance, students may have to address the differentiability condition of a function before engaging in finding its derivative.

A second transition occurs when students reach courses whose curricula and assessment prioritize what once may have been the theoretical block of a praxeology; as students transition into proof-making and validating, theoretical blocks of the past become their practical blocks. For instance, the second transition will have occurred in a student who knows to use the definition or theorems about continuity to prove that, if a function is continuous, then some property of that function is true. The characteristics of a second transition are that students explicitly acknowledge and use the theoretical block to generate a technique for achieving a task.

KNOWLEDGE TO BE TAUGHT

The textbook of the multivariable Calculus course is typical of those used in North-American Calculus courses and follows the usual definition-theorem-exampleexercise format. The course outline lists the textbook sections to be covered each week and a choice of end-of-section exercises. By knowledge to be taught (KT) we mean the mathematical knowledge in the sections and exercises listed on the outline. To model the KT, we identified the praxeologies of which it consists.

In the case of the knowledge to be delivered in this course, we found that technology and theory can be taken as one. There is no clear distinction between the two in the textbook; the discourse throughout is set in the geometry and algebra of threedimensional space organized in the Cartesian system, and at times in Euclidean metric spaces. However, the theory is not made explicit and tends to be woven into the technology. Further, we found that the focus of the KT is mainly in the practical blocks. For the purpose of this study, then, it was sufficient to compile a list of items (definitions, theorems, etc.) that form the theoretical blocks of the praxeologies of KT without distinguishing theory from technology. This tended to the theoretical block of the praxeology that modelled each section of the textbook on the course outline. To identify the tasks to be accomplished, we considered the examples and the end-of-section exercises listed in the outline. To describe the associated techniques, we consulted the examples and discussion portions (theorems, explanations) of the text. To account for the build-up of knowledge between sections (e.g. the notion of derivative of a vector function is defined in one section and reused in later sections), we cross-referenced across theoretical blocks and across and within practical blocks.

KNOWLEDGE TO BE LEARNED

In an operational sense, we define knowledge to be learned (KL) as the subset of the KT which students need to know in order to provide solutions on final exams. This operationalization was necessary from a methodological perspective: while the questions in the final exams indicate the tasks to be accomplished, in most cases there is no indication as to the expected technique or theoretical justifications. The model of KT was therefore necessary to identify these elements of a mathematical activity. In this sense, the main purpose of the model of KT was to model the KL. Our operationalization, although useful to describe and characterize the KL, does not properly reflect the fact that a transposition takes place and that some of the praxeological elements (likely, elements of the theoretical block) are more likely illdefined than well-defined subsets of the praxeological elements of the KT. While the KL may borrow elements of the KT praxeologies, the discourse that unifies the two blocks of a praxeology might be distorted in the transposition.

Our model of KL is based on twelve final exams given recently within a span of three years. We described the solution to each exam question in terms of KT task-technique pairs that occur in the solution. Here is an instance of this work. Consider the following item from one of the exams:

Find the tangent plane that touches at , where the surface is given by

We recorded this as "to find the tangent plane to a surface at a point." This task corresponds identically to task from the KT model; in turn, the technique for this task requires the completion of : to find the value of the partial derivative of a function at a point. Thus, we associated to this task the KT sequence This particular exam task corresponded identically to a KT task; this was not always the case. Nevertheless, apart from a handful of cases, we were able to identify sequences of task-technique pairs that would form complete solutions to the exam questions; this methodological affordance may attest to the routine quality [START_REF] Selden | Do Calculus students eventually learn to solve non-routine problems?[END_REF] of the tasks students are expected to accomplish.

Next, we grouped tasks of the same type so as to reflect praxeologies that occur in the KT. For example, the following tasks come up in solutions to exam questions:

To find the first partial derivatives of a function To find the first partial derivatives of a two-variable function defined implicitly To verify that a two-variable function satisfies a partial differential equation This cluster of tasks is drawn from the praxeology of KT specific to partial derivatives. Altogether, we partitioned the model that captures KL about partial derivatives and surfaces into groups of tasks that match up with these praxeologies of KT: the above cluster specific to partial derivatives, along with tasks that draw from KT praxeologies specific to functions of several variables, the chain rule, tangent planes and linear approximations, directional derivatives and the gradient vector, extreme values, and Lagrange multipliers. Organizing the model of KL in parallel to the model of KT facilitated our analysis of the structure of the KL.

STRUCTURE OF THE KNOWLEDGE TO BE LEARNED

The KL has to do with partial derivatives and surfaces; space curves and vector functions; equations of lines and planes and distance in ; limits of rational functions; polar curves; and Taylor Series. Let's call 'ideal student' one who has the requisite knowledge to write acceptable solutions in a final exam. How might we characterize the praxeologies of an ideal student in this course? Below, we consider which parts of the KT praxeologies are to be learned and characterize them in the language of [START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF] and [START_REF] Selden | Do Calculus students eventually learn to solve non-routine problems?[END_REF].

Knowledge from all KT praxeologies occurs as knowledge to be learned. Thus, the KL is not necessarily a subset of KT in the sense that some praxeologies are to be learned while others are not. Rather, we found that the KL is a subset in the sense of what's left of the KT praxeologies after the didactic transposition of KT into KL.

First, the practical blocks of the KT praxeologies are downsized in this transposition. For instance, consider the praxeology of KT about polar coordinates. The ideal student can convert polar equations into Cartesian equations and sketch the curvegiven the following curves (up to a change in constants and functions sine or cosine):

;

The algebraic manipulations specific to converting these types of polar equations into Cartesian equations are in examples from the textbook, as is the technique for sketching them. The ideal student's topos ('action space') [START_REF] Chevallard | Organiser l'étude 1. Structures & fonctions[END_REF] does not need to extend beyond the point praxeology (a praxeology of knowledge that is particular to a single type of task) specific to these functions. We found many of the practical blocks of KT praxeologies to be reduced in this way to point praxeologies.

Most of the praxeologies of KT are downsized in another sense: their theoretical block is removed following the transposition from KT to KL. For instance, consider the praxeologies that constitute the knowledge to be taught about partial derivatives. We found that the practical blocks are reduced to computational tasks where the ideal student needs to apply the appropriate differentiation algorithm; the geometric interpretation of partial derivatives as slopes is unneeded and the ideal student does not need to know any of the theory or technology at the backbone of the procedures. The ideal student does not need to know the limit-based definition of partial derivative nor the definitions and roles of limits, continuity, and differentiability in the concepts of derivative, gradient, and extrema of a function. The theoretical blocks of these praxeologies vanish in the transposition from KT to KL. In general, it seems that the ideal student needs to be fluent in the algorithms prescribed by praxeologies of the KT but doesn't need to justify or explain them.

The absence of theoretical blocks in the ideal student's praxeology is manifested in several ways: first, the student needn't justify the validity or choice of technique (e.g. by verifying or stating that the chain rule is applicable, since the functions in the exams are always differentiable); second, the exam questions do not require students to interpret any results (e.g. by making a sketch of a surface near a point where some geometric properties of the surface were computed); and finally, it suffices to have a superficial grasp of the concepts in the theoretical blocks in order to accomplish the types of tasks in the final exams. We expand on this point.

In general, the ideal student can recognize task types and identify the appropriate technique, in reasoning similar to [START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF] identification of similarities (IS), whereby a strategy for tackling a problem is chosen based on the similarities of certain surface properties between the new problem and a known problem (e.g. given a limit-finding problem, note whether the limit is taken at a numerical value or infinity and identify the type of function involved). For instance, the exam questions specific to limits of multivariable functions are as follows: find the limit of a function at the origin, if it exists, or show that it does not exist. The function given in the exams is either an odd rational function with no limit at the origin or it involves a trigonometric component which could be rid of to reduce to a rational function in the process of an argument (in these cases, the exam functions invariably have limit ). This task occurs in examples in the textbook and exercises in these students' assignments. In general, tasks required by the exam questions were similar to those in the KT, so that students could rely on IS reasoning rather than on the underlying mathematics in order to choose the appropriate technique. This course is therefore in line with students' pre-university mathematics, where much of their responsibility is in recognizing types of tasks and choosing an appropriate known technique (Winsløw et al., 2004).

IS reasoning is characterized as requiring little reflection on the intrinsic mathematical properties of the problem at hand [START_REF] Lithner | Mathematical reasoning in Calculus textbook exercises[END_REF]. To successfully implement IS, the ideal student needs to recognize terms in the question statements (arc length, curvature, normal plane, binormal vector…) and the formulas for deriving them. But the ideal student is not tested on the meaning of these quantities and geometric properties as they relate to a curve at a point (e.g. a student might need to find the equation for an osculating plane, but does not need to explain what the osculating plane describes). The irrelevance of intrinsic mathematical properties to the tasks students need to achieve suggests that the theoretical block of KT praxeologies need not be present in the ideal student of either course.

Theoretical blocks are missing from the ideal student's topos in a few senses: the student is not required to justify or explain the techniques chosen to complete a task, and at times is even told which technique to use (e.g. via instructions to 'use Lagrange Multipliers' or 'use the chain rule'). The ideal student is not required to interpret the numerical or algebraic results of their calculus in any way; and it suffices to learn the components of the theoretical blocks only superficially. In all, this multivariable Calculus course seems to follow in the pre-university mathematics tradition whereby students need not link the practical and theoretical blocks of a praxeology [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF]. Further, the components of the practical blocks themselves are discrete, as the ideal student does not need to combine tasks in any way -for instance, the ideal student must know how to find invariant quantities of a curve, but needn't provide a local description of a curve based on its invariant quantities. This may be called the "compartmentalization of knowledge in calculus courses" described by Winsløw et al. (2014, p.104).

On the whole, it appears that only a surface version of the KT theoretical blocks is essential for the ideal student to learn: they need to know terms and associated formulas, in some cases have some intuitive image of certain concepts, and be fluent in the algorithms described by the technologies. This surface acquisition of the theoretical block serves to recognize routine tasks and identify a suitable technique.

In light of the absence of theoretical blocks in the minimal core of knowledge that is essential for students to learn in order to provide solutions to exam questions, we conclude that the KL cannot be described by actual praxeologies (made up of a praxis and a logos). Rather, the KL is an amalgamation of practical blocks. This places this university-level multivariable Calculus course in the stage prior to the first transition in university mathematics education previously discussed: where refers to the practical block of a praxeology and to its theoretical block. [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF] explain that this first transition occurs when students no longer work strictly within the practical block of a praxeology and begin to incorporate a theoretical block, perhaps by using it to justify or produce a technique; a second transition occurs when students' past blocks turn into their current practical blocks, as when they start making and validating proofs in Analysis.

CONCLUSIONS

The aim of our study was to determine the minimal core of knowledge that is necessary for students to learn in a multivariable Calculus course in order to provide acceptable solutions on their final exam. We found that the exercise-driven quality of the course assessments makes it essential for students to recognize certain types of tasks and to identify the appropriate technique, but does not require students to learn the theoretical block that maintains these tasks and techniques.

Historically, the studied educational system introduced the multivariable Calculus course as a prerequisite to Analysis in an effort to help students adapt to university mathematics in the first year of their studies. It seems, however, that the mathematical activities expected of students in this bridge between pre-university and university courses are of the type expected in past Calculus courses: students' action space is fully within the practical block of the praxeologies that model the knowledge to be taught. As a result, students are no more required to engage with the theoretical in this Calculus course than they previously were. Meanwhile, the mathematical activities in Analysis courses are two steps ahead, after the second transition described by [START_REF] Winsløw | An institutional approach to university mathematics education: from dual vector spaces to questioning the world[END_REF], where students must work within what once were the theoretical blocks that backed the practical of Calculus. The question therefore remains: what course would make it essential for students to incorporate the theoretical blocks of a praxeology into the work they do in a practical block?

Figure 1 .

 1 Figure 1. Transitions in university mathematics education (Winsløw et al., 2014, p.101)


Marc Lalaude-Labayle
email: marc.lalaude-labayle@univ-pau.fr

Patrick Gibel²
email: patrick.gibel@u-bordeaux.fr

Isabelle Bloch²
email: isabelle.bloch@u-bordeaux.fr

Laurent Lévi

A TDS analytical framework to study students' mathematical activity An example: linear transformations at University

Keywords: linear algebra, semiotics, theory of didactical situations, transition, tertiary level

come

A TDS analytical framework to study students' mathematical activity An example: linear transformations at University

INTRODUCTION

Our work deals with a double object: linear transformations as a structuring concept of the teaching of linear algebra and a particular institution at the undergraduate level, the Classes Préparatoires aux Grandes Écoles. Grounded on didactical motivations, our epistemological analysis allows us to exemplify the crucial role of linear transformations for the emergence of linear algebra concepts. This epistemological part of our research, mainly based on the works of Dorier and Moore, leads us to the use of Peirce's semiotic and to enrich the analytical model of [START_REF] Bloch | Un modèle d'analyse des raisonnements dans les situations didactiques: étude des niveaux de preuves dans une situation d'enseignement de la notion de limite, A model to analyse reasoning processes in didactic situations: a study about levels of proofs in a situation concerning the teaching of limits Recherches en Didactique des Mathématiques[END_REF] which is rooted in the Theory of Didactical Situations (TDS). The purpose of this paper is twofold:

 give some responses to the following question: which reasoning forms are actually produced by a student during the different stages of a situation of oral evaluation?

 show the utility of our framework to analyse the signs and arguments produced and thus take part to the development and enrichment of this model.

Within the French didactic tradition, we remind briefly the theoretical tools used in the elaboration of the framework. Then, we expose our model, using the terminology of Bloch and Gibel. Equipped with these tools, we use the model to analyse the arguments produced by students. But, at first, we succinctly introduce the institution of Classes Préparatoires aux Grandes Écoles by highlighting the differences with the University, especially regarding the transition phenomenon with secondary level.

I. THE CLASSES PRÉPARATOIRES AUX GRANDES ÉCOLES: A FRENCH POST-SECONDARY LEVEL INSTITUTION

The Classes Préparatoires aux Grandes Écoles, which we can translate as Higher School Preparatory Classes, are part of the French tertiary education system for over two centuries. They consist in two really intensive years which act as a preparatory course to train undergraduate students for their further enrolment in one of the French graduate schools called Grandes Écoles, such as École Polytechnique, École Normale Supérieure, École des Hautes Études Commerciales, also known as HEC School of Management … The enrolment in one of these Grandes Écoles depends on the rating obtained in national competitive and demanding examinations.

We summarize the main differences between the Classes Préparatoires and the University in the following table. Thus, we highlight some facts that Winslow (2007) showed to think about the study of the transition phenomena from the secondary to the post-secondary level.

Classes Préparatoires University

Full- As Winslow (2007), [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, Note de synthèse présentée en vue de l'[END_REF] and more recently [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] wrote, these differences have deep didactic implications, relative to the theoretical knowledge and praxeologies, to the problem solving approaches, to the evaluations and to the personal homework just to name a few. In our experimental work, we studied some arguments produced by second year students from one of these Classes Préparatoires during an oral evaluation on linear algebra.

As noted by [START_REF] Bloch | A model to analyse the complexity of calculus knowledge at the beginning of University course. Two examples: parametric curves and differential equations[END_REF] for calculus, in order to recognize and analyse the reasoning forms actually produced by a student,

II. A MODEL TO ANALYSE STUDENTS' REASONING

We then need a tool to modelize the students' productions, more precisely to identify, while in a mathematical situation, which is the knowledge they rely on during their activity; we want to identify the structure and functions of students' reasoning processes in this situation. Briefly, this model should allow us to seize the complexity of the reasoning processes a student has to cope with during the resolution of mathematical problems. By reasoning process, we mean valid or erroneous ones according to the work of [START_REF] Gibel | Mise en oeuvre d'un modèle d'analyse des raisonnements en classe de mathématiques à l'école primaire. A model to analyze reasoning processes at primary school[END_REF] about reasoning.

This model takes its origins in the Theory of Didactical Situations (Gonzalez-Martin, Bloch, Durand-Guerrier, and Maschietto 2014) and in the semiotics of Peirce. We briefly recall the main theoretical elements of TDS and semiotic used in the elaboration of the model.

The TDS theory

TDS relies on a two basic premises concerning the mathematical activity and the learning of mathematical knowledge. For TDS, the mathematical activity consists of distinct stages: a situation of action, followed by a situation of formulation and then a situation of validation phase. To take the learning activity into account, TDS adds two more stages: the phase of devolution and the phase of institutionalization. TDS defines three fields to construct and analyse such situations. The theoretical field which the domain of elaboration of fundamental situations relative to a knowledge. The a priori experimental field which envisages a situation at a specific level of teaching, taking into account the didactic repertory as defined by Gibel [START_REF] Gibel | Fonctions et statuts des différentes formes de raisonnement dans la relation didactique en classe de mathématiques[END_REF]. The a priori analysis of the situation, which checks if the conditions of devolvement are fullfilled, takes place at this second level. The third field is the field of eventuality where the situation is actually implemented. In short, TDS is a didactical framework which tries to implement situations with adidactical parts and offers tools to analyse the teaching and learning activities. These adidactical stages allow students to face a heuristic phase of research and then, through a confrontation to the elements of an adequate milieu, to test, validate or invalidate their conjectures.

The notion of milieu appears to have a central role. TDS organizes situations with up to seven logically successive phases, but in our work we will mainly work with three of them: a heuristic one, grounded in a problem, and a formulation and validation one, composing the adidactical moments of the situation, and then the institutionalization by the teacher or with his/her help, which is the didactical moment. The dynamic of the nesting of the situations with the paired levels of milieu illustrates the dynamic of the learning processes involved. The following chart [START_REF] Bloch | Quelques apports de la théorie des situations a la didactique des mathématiques dans l'enseignement secondaire et supérieur[END_REF][START_REF] Bloch | A model to analyse the complexity of calculus knowledge at the beginning of University course. Two examples: parametric curves and differential equations[END_REF]

Table 2: Structuration of the didactical milieu

The levels M-2 and M-1 are the ones that will allow us to identify, describe and analyze the elements (signs, processes …) associated to the emergence of an argumentation within the proof process. More precisely, as [START_REF] Bloch | A model to analyse the complexity of calculus knowledge at the beginning of University course. Two examples: parametric curves and differential equations[END_REF] write

The place where we hope to see the expected reasoning processes appear and develop is located at the articulation between the heuristic milieu and the reference milieu. (ibidem, p.46) Thus, the TDS theoretical framework allows us to consider not only the reasoning processes produced, but their functions within the situation and the levels of argumentation they rely on: it already gives us a glimpse of the multidimensionality of our model.

The semiotic tools

To take into account the semantic dimension, i.e. the meaning of the signs and arguments produced, with certain accuracy, we need semiotic tools. The signs a student produce during an adidactical situation, whether formal or informal, are the only observable phenomena that can sustain our semiotic analysis: roughly speaking, the signs produced (syntactic aspect) are in relation to an object (semantic aspect) creating a instantiated sign (pragmatic aspect) relative to the milieu, the didactical repertoire and the repertoire of representations as defined by [START_REF] Gibel | Mise en oeuvre d'un modèle d'analyse des raisonnements en classe de mathématiques à l'école primaire. A model to analyze reasoning processes at primary school[END_REF]. This triadic relation, linking the sign, the object and the student's instantiated sign, led us to use Peirce's semiotics. In this paper, we only analyse the relation between a sign and its object, its content. But in his thesis, one the authors (Lalaude-Labayle 2016) conducted a semiotic study, relying on the full Peirce's triadic classification of signs as 'put in algebra' by [START_REF] Marty | L'Algèbre des signes, Essai de sémiotique scientifique d[END_REF]. Applying the universal categories to the relation between a sign and its object refers to the notions of icon, index and symbol, describing the abstraction level of this relation. An icon is a sign which stands for an object because of its physical resemblance: a drawing, a triangular matrix represented with triangles … An index is a sign physically connected to its object: the columns of the pivots in a reduced echelon form of a matrix M is an index that points to the basis of Im(f), where M is canonically associated to f … A symbol is a sign that refers to its object by virtue of a law: ker is a symbol whose object is the concept of kernel of a linear transformation.

The logical inferences

In our work we understand the term 'reasoning' in it broadest sense. More precisely, by reasoning we mean a sequence of representations, following some intern and potentially explainable rules that lead to reach some explicit goal. Postulating, as it is the case within the TDS theory, some rationality of the students, we need to define these rules or inferences. Peirce saw mathematics as the science of drawing necessary conclusions, studying what is and what is not logically possible. But, since one does not think about logical propositions but about and with signs, Peirce broadened the notion of inference. He then distinguishes three kinds of rational inferences: abduction, induction and deduction. Deduction, or necessary reasoning, deduces a proposition B from a proposition A, where B is a formal logical consequence of A. Induction goes from the particular to the general; it allows inferring B from A. Abduction allows inferring A as a probable explanation of B. So, deduction proves that something must be, induction shows how something effectively operates and abduction suggests that something could be.

The framework to analyse students' productions

To analyse students' processes of reasoning, [START_REF] Bloch | Un modèle d'analyse des raisonnements dans les situations didactiques: étude des niveaux de preuves dans une situation d'enseignement de la notion de limite, A model to analyse reasoning processes in didactic situations: a study about levels of proofs in a situation concerning the teaching of limits Recherches en Didactique des Mathématiques[END_REF] develop a multidimensional model. They focus their didactical analysis on three main axes. The first axis is related to the level of milieu and so to the phase of the situation in which the student produces his/her reasoning (cf. Table 2). The second axis of the model is linked to the notions of didactical repertoire, of organizational system and of a repertoire of representations. It studies the functions of the reasoning produced and is in close relation to the first axis. As [START_REF] Bloch | A model to analyse the complexity of calculus knowledge at the beginning of University course. Two examples: parametric curves and differential equations[END_REF] state it, they aim at linking these two axes, showing how the reasoning functions are linked specifically to the levels of milieu and how these functions also manifest these levels of milieu. (ibidem, p.47) Semiotic analysis of observable signs constitutes the third axis of the model. Marc Lalaude-Labayle enriched the model by adding a fourth axis about the forms of inference applied. This logical axis links the second and third axis by setting out and clarifying the organization of the reasoning signs and their functions. This fourth axis helps to 'make visible' the organization within the system of representations and its actualization. We sum up this model in the following table

Milieu M-2 Heuristic level

Milieu M-1 Formulation, validation

Milieu M0 Institutionalization

Nature and functions of reasoning

Forms of reasoning

R4.1 -deductive -inductive -abductive R4.2 -deductive -inductive R4.3 -deductive

Table 3 -A model to analyse situations

SEM signifies that the formulations are made on a semantic mode whereas SYNT is for syntactic mode. This model emphasizes the fact that the mathematical activity, with its reasoning processes, appears in the heuristic and reference milieu (cf. Table 2). These two milieus, and the articulation between them, will thus be of particular interest for our work, even if the situation is an ordinary one.

Let us insist on the fact that the use of this model relies on a precise a priori mathematical analysis of the situation and of its components, e.g. the problem to be solved. Within the TDS this step appears to be necessary to clarify the didactical a priori analysis.

III. A PRACTICAL USE OF THE MODEL IN A LINEAR ALGEBRA ORAL EXAMINATION

We analyse the productions of a second year student of Classes Préparatoires in the context of an oral examination. The students of the Classes Préparatoires, by group of three, pass such an oral exam once every two weeks. It lasts one hour during which the teacher asks to the three students to solve different mathematical problems. They work simultaneously and individually on a large blackboard on the problem they just discovered, in front of the teacher. The teacher helps the students by giving advices or clues. Taking into account the writings noted on the chalkboard, she/he can ask for some explanation or clarification. This can give rise to an oral or written answer and possibly to some discussion to deepen and enrich the repertoire of the representations of the student. The oral exam we analyse deals with linear algebra, and more specifically linear transformations and matrices.

The student is confronted to the following problem:

Let n be a integer, greater than or equal to 2 and let φ defined on R n [X] by: for all polynomial P from R n [X], φ(P)=P(X+1)-P(X).

1. Show that φ is an endomorphism of R n [X].
2. Is φ injective? Surjective? Bijective?

In France, R n [X] symbolises the vector space of real polynomials of degree at most n.

As briefly explained earlier, our model provides a framework for investigating mathematical and didactic activities in terms of milieu, focusing on the reasoning processes, signs and their dynamics and on the conditions that enable their development during the situation, be it ordinary or not. As is done within TDS, our didactic analysis is divided in several stages: a detailed and strucured a priori mathematical and didactic analysis, enriched with a specific a priori analysis of the reasoning involved; follow then an a posteriori analysis organized in our model.

A priori analysis

From the mathematical point of view, showing that φ is an endomorphism of R n [X] can be approached in different ways, engaging several frames: indeed the stability of R n [X] under φ can be done in a purely algebraic frame using the degree and the composition rule, in a functional frame using the decomposition P= ∑ a i X i or in an algebraic frame using the linearity and showing that forall integer i between 0 and n, φ(X i )∈R n [X]. To study the injectivity and surjectivity of φ, the student can again choose between several frames and several registers of semiotic representation [START_REF] Duval | Understanding the Mathematical Way of Thinking -The Registers of Semiotic Representations[END_REF]: she/he can use an algebraic frame with an example φ (1)= ⃗ 0 and then applying the theorem linking rank and kernel dimension; but she/he can also try to find precisely Ker φ, that is find a basis. To do this, she/he can use the functional decomposition of polynomials, she/he can solve a linear system or she/he can determine the matrix of φ in the canonical basis of R n [X].

From a didactic point of view, the situation studied here is said to contain an adidactic dimension. Most of the actions, of the frames and of registers of semiotic representation are devolved to the student. So, a first difficulty that occurs for the student can be the control she/he has on the objects involved: a circular application of the definition of linearity of φ to prove its linearity, the complexity of formulas to write down φ(P) for a general P, the non operability of φ with wrong calculations of φ(1), φ(X), φ(X²) for example. For the injectivity of φ, the student can "forget" the structure of the space R n [X] he is working on and try to check whether φ(P)=φ(Q) implies that P=Q. All the reasoning processes and objects involved are part of the didactic repertoire of the class the student belongs to.

A posteriori analysis: analysis of a student's production

In the following, we extract some translated excerpts of the student's answer, the whole solution can be found in Lalaude-Labayle (2016).

To solve the first part of the problem, which is an ordinary situation in undergraduate level, the student is mainly confronted to the milieu of reference, articulating objects and processes involved in his repertoire of representations. Here the heuristic milieu is not really requested. He starts by showing the stability R n [X] under φ and that writes changing X with X+1 doesn't change the degree of P. He then proves the linearity in an algebraic frame. Doing so, he makes a formalization of proofs within the required theory and thus reaches the level R1.3. The semiotic analysis shows that he uses generic arguments (R2.2) and more formal one (R2.3). These arguments and signs don't give any hint to how φ operates on R n [X]. Its argumentation validates its use of the didactic repertoire, and reveals some implicit assumptions: φ(P) is a polynomial is here implicit, as is its use linking degree and composition of polynomials. He uses mainly hypothetical-deductive inferences. But, as an introduction to his argumentation, the student asks himself whether deg(φ(P)≤n: he formalizes here the start of an abductive reasoning.

To study the injectivity of φ, the student applies a transformation of register of semiotic representations to formalize the link with ker φ, starts within an algebraic frame then uses the decomposition of P to study ker φ, but without success: he cannot make φ operate on P and says not to have any clue to study the kernel. During this phase, the student tried to use some deductive reasoning involving objects from the reference milieu: the lack of articulation between the heuristic milieu and the reference milieu confirm the difficulties encountered in R1.3 with an aimless organisation of the signs. The semiotic analysis underlines the lack of quantifiers which leads to a incomplete apprehension of the objects (R2.2) and reinforces the feeling of lack of goal in the reasoning. The teacher asks then the student to consider the tools he has got in his repertoire to "calculate" objects in linear algebra. With no answer, he asks the student to consider the matrix of φ in the canonical basis. Doing so, he tries to force the student to face the heuristic milieu and tries to maintain an adidactic dimension in the situation. The calculations produced confirm the fact that the student doesn't know how to compute φ: he obtains the identity matrix. This matrix doesn't appear to be an index to control the reasoning (articulation R1.2 and R1.3). With a new oral intervention of the teacher, the student writes the right matrix. Some misinterpretations are following: R n instead of R n [X], surjectivity is meant instead of injectivity (R3.1), φ(1)=0 is used as an symbol for ker φ=vect(1) instead of a simple index of it (R2.1). The student uses the theorem of the rank (R3.2), relying then on a deductive form of reasoning. But some of his deductions rely on the preceding explicit calculations (articulation R4.2 R4.3).

CONCLUSION

As stated in the introduction, the purpose of this paper is twofold: try to determine the reasoning produced by a student in a specific mathematical situation and show the utility of our framework to analyse the signs and arguments produced in this situation.

Regarding the first question, our analysis allows us to say that this particular student has difficulties to reach fully the institutionalization milieu (level R3): the reasoning and the articulation of the objects involved do not ease his control over his arguments and eventually lead him to aimless computations. Moreover he seems to get stuck in the reference milieu (level R2) and to hypothetical-deductive inferences. The student does not rely on reasoning made in a heuristic milieu (level R1) that would be appropriate to linear transformations and polynomials. The problem we analyze in this work contains an adidactic dimension but fails in asking the student to make effectively operate φ on R n [X]. In semiotic terminology, we can postulate that the at least incomplete pragmatic dimension in the reasoning leads to some confusion and lack of pertinent association between the syntactic and semantic dimensions [START_REF] Bloch | A model to analyse the complexity of calculus knowledge at the beginning of University course. Two examples: parametric curves and differential equations[END_REF].

Regarding the second question, our work seems to confirm that the model used within TDS constitutes an efficient framework, as stated in Bloch andGibel (2011, 2016). It helps specifying the reasoning and signs on which it relies both for the a priori and a posteriori analysis and highlights the obstacles contained within the mathematical notions. Our future work should be working more specifically on the abductive reasoning and on situations encouraging students to adopt a heuristic approach.

  sums up the levels of milieu paired to the different phases of situations corresponding to the experimental situation M1 Didactical milieu E1: reflexive subject P1: P. planner S1: situation of project M0 Learning milieu : institutionalization E0: generic student P0: professor teaching

Table 1 : Comparison of CPGE and University

 1

it is necessary to classify the objects, signs and reasoning processes they have to cope with during resolution of calculus problems.(Bloch & Gibel, 2016, p. 44)