THE APPEARANCE OF ALGORITHMS IN CURRICULA A NEW OPPORTUNITY TO DEAL WITH PROOF?

Modeste Simon, Institut Fourier, Université Joseph Fourier, Grenoble. Ouvrier-Buffet Cécile, UPEC & Laboratoire André Revuz, Université Paris 7.

We deal with the concept of algorithm which is taking importance in curricula in many countries. In particular, we develop an epistemological analysis of this concept and discuss its place in the mathematical science and the link it has with proof. This analysis is enriched by the study of "how researchers know the algorithm". We conclude with implications of the changes in curricula on proof learning.

Discrete mathematics represents a mathematical field which takes a growing importance in our society. In particular, the accessibility of the concepts of this field brings new tracks to teach and learn proof (e.g. [START_REF] Grenier | Discrete mathematics in relation to learning and teaching proof and modelling[END_REF]ZDM, 2004). This paper will deal with one of the concept of this field: the algorithm. Indeed, with the omnipresence of computers and technologies in our society, it seems that algorithm will take more and more importance in curricula and it raises questions that impacts on the teaching of mathematics. Moreover, new types of proofs involving computations appear, and with them, algorithmic proofs for instance and the philosophical and epistemological questions of the use of computers to create and/or validate proofs [START_REF] Hanna | The ongoing value of proof[END_REF]. The 1998 Yearbook of the National Council of Teachers of Mathematics was entirely dedicated to the questions of algorithms and was "an attempt to answer many of [the questions provoked about the place of algorithms in today curricula] and to stimulate other questions that all of us in mathematical education need to consider as we continually adapt school mathematics for the twenty-first century" (NCTM, 1998, p. vii). Recently, the studying of algorithms got into the class of mathematics in the French curriculum of the secondary. The appearance of the concept "algorithm" in the mathematical curriculum questions the role in mathematics of an object which seems, at first look, to belong more to computer science. Actually, algorithm is first of all, from historical and epistemological points of view, a mathematical concept.

The links between algorithm and proof are not easy to describe and have not been much studied epistemologically. The goals of our study (started in [START_REF] Modeste | Algorithmique et apprentissage de la preuve[END_REF]) are here twofold: from a mathematically-centered perspective, we want to bring an epistemological analysis of the algorithm which emphasizes its interplay with proof. The way researchers in mathematics and computer science know the concept of algorithm allows a validation of the epistemological model we develop in this paper. From a didactical perspective, we ask how the concept of algorithm can enhance the curriculum, focusing on links between algorithm and proof (avoiding the ubiquitous computer aspect). We ultimately want to build Research Situations for the Classroom (RSC) [START_REF] Grenier | Spécificités de la preuve et de la modélisation en mathématiques discrètes[END_REF][START_REF] Chabert | A History of Algorithms From the Pebble to the Microchip[END_REF][START_REF] Godot | Research situations for teaching : a modelization proposal and examples[END_REF] involving algorithm and proof processes. In particular, the design of such situations implies to rely on an epistemological analysis and a study of the practises of researchers (see [START_REF] Knoll | Discussing the Challenge of categorising mathematical knowledge in mathematics research situations[END_REF]. Then the concluding session highlights the links between the concept of algorithm and the proving process and brings new research tracks in order to design and to analyze situations for the classroom.

EPISTEMOLOGICAL ASPECTS OF THE CONCEPT OF ALGORITHM

A common definition

A usual definition of algorithm is presented by Knuth who takes algorithm as an object of study (and also questions the differences between mathematical thinking and algorithmic thinking): "(...) an algorithm is a set of rules or directions for getting a specific output [1] from a specific input. The distinguishing feature of an algorithm is that all vagueness must be eliminated; the rules must describe operations that are so simple and so well defined that they can be executed by a machine. Furthermore, an algorithm must always terminate after a finite number of steps." (Knuth, 1996, p. 59) This implies that an algorithm solves a specific problem P by returning in the output the answer corresponding to the instance of P given in the input. It is important to remember that the input and output information are coded and "algorithms deal primarily with the manipulation of symbols that need not represent numbers." (Knuth, 1996, p. 61) The previous definition also shows the effective aspect of algorithm: no ambiguity must exist in the instructions so that any operator -most of the time a computer, and in this case the algorithm can be described by a program -gets the same output with the same steps. The finiteness cannot be dissociated from the notion of algorithm, as Chabert notes it in his book on the history of the algorithm: "Today, principally because of the influence of computing, the idea of finiteness [1] has entered into the meaning of algorithm as an essential element, distinguishing it from vaguer notions such as process, method or technique. […] Here we have a finite number of operations, a finite number of input values, but also a finite number of solution procedures, that is that each step should be able to be carried out by a finite processsomething which is not possible, for example, in determining the quotient of two incommensurable real numbers. We also refer to an effective procedure, that is one that will effectively achieve a result (in a finite time)." (Chabert et al., 1999, p. 2-3) This finiteness raises questions regarding complexity: given an input, how many steps does the algorithm take to answer? How much space does it need to store the involved information? These questions respectively deal with time complexity and space complexity. This complexity depends on the size of the input and can be studied from two points of view: the worst-case complexity, the maximum complexity of the algorithm for an input of size n and the average-case complexity, the average of the complexity for an input of size n (usually, the inputs of size n are considered to have the same probability). Since [START_REF] Knuth | Selected Papers on Computer Science. Center for the Study of Language and Information -Lecture Notes[END_REF], complexity is a specific and fundamental aspect of the algorithm which is absent from the other fields of mathematics. He says (about Bishop's mathematics [2]) that "[it] is constructive, but it does not have all the ingredients of an algorithm because it ignores the "cost" of the constructions" (Knuth, 1996, p. 110).

Here, we have detailed three important aspects of algorithm: the link with problems, the effectivity and the complexity.

Ambiguity of this definition

As examples of algorithms, authors often give Euclid's algorithm for gcd, arithmetic operations on integers, algorithms for sorting or algorithms for shortest paths in a graph... Among all these examples, one strikes us: the method to find the roots of the quadratic equation using the discriminant .

With more details, the algorithm is the following:

Input: if then return , else if then return , else return
Judging by the definition we gave above, we could say that it is an algorithm. But we find it surprising that such a method was chosen as an illustration of the concept "algorithm". Indeed, in each case, the algorithm is just a formula. Moreover, from the point of view of the "complexity", such an algorithm is not interesting, as the complexity is independent of the size of the input (we are not speaking here about the complexity of the arithmetic operations involved in the discriminant, which are for us better examples of algorithms). This example raises the question of the border between algorithmic and non-algorithmic areas and the "usual" definition is ambiguous about this. For our study, for a didactical purpose, it would be useful to distinguish this kind of formula with "real" algorithms.

A more theoretical definition

In the beginning of the 20 th century, the quest of foundations for mathematics caused mathematicians to give a more theoretical definition of algorithm.

"The works of Gödel inspired the research of Alonso Church, Stephen Kleene, Alan Turing and Emil Post. These mathematicians attacked Hilbert's Entscheidungsproblem and showed that there were, indeed, undecidable problems, that is mathematical statements for which no procedure exists by which it can be decided if the statement is true or false. To do this, each of them defined a concept of computability, that is a concept of algorithm." (Chabert et al., 1999, p. 457) Two of these concepts should be quoted: the Turing machine and the recursive functions. This theoretical work leads to a classification of problems depending on whether there exists an algorithm to solve them or not (undecidable and decidable problems) and if they are "easy" or "hard" problems, which means if they can be solved by a polynomial algorithm or not (we refer here to P and NP-hard problems [3]). This point of view will constitute a fundamental aspect of algorithm, the theoretical models.

Algorithm and proof

Algorithm and proof interplay in many ways, it will be another important aspect for us. First, an algorithm has to be proved; more precisely, it is necessary to prove its correctness (i.e. it gives the expected answer) and its termination (i.e. it always stops after a finite number of steps). And, once it is proved, an algorithm can be used as a step in another proof. Actually, all the aspects raised previously have a link with proof. In particular, correctness and termination correspond respectively with problem solving and effectivity. The complexity aspect involves proof too, and studying the complexity of an algorithm often needs substantial mathematics. The same is true of theoretical models, which only make sense in a proof process. Just like any mathematical object, the algorithm raises questions involving proof. But some of them are specific and only the algorithm raise the mathematical questions previously mentioned. Moreover, the algorithm is not only linked with proof on that way. An algorithm can also be a tool for proving a property, and for a given problem, an algorithm will give a constructive proof of its resolution (e.g. Euclid's algorithm provides a proof of the existence of the gcd of two integers and an effective way to compute it). Conversely, an algorithm often lies under a constructive proof and it can be interesting to formulate this algorithm clearly. For instance, from any proof by induction follow a recursive algorithm.

Algorithms, seen as proofs, allow to deal with two kinds of problems: existence problems and testing a property.

Recently, a link has been pointed out between proof and algorithm, with the computer-assisted proofs, that is the use of algorithms to build proofs which are much too long to be verified by a human being. For instance the four-color theorem has been proved this way. However, the algorithm has to be proved in order to validate the mathematical result. This new kind of demonstration asks philosophical and epistemological questions about the nature of proof.

Tool-Object

The aspects of algorithms underlined above can be divided into two parts since they refer to algorithm as a tool or as an object. Looking at the algorithm as an object means studying questions of validity, of complexity and description of algorithms.

Looking at the algorithm as a tool is focusing on the use of algorithms to solve problems. Among the aspects discussed here, the effective aspect and problem aspect refer to the algorithm as a tool whereas the complexity aspect, the theoretical models and the link with proof refer to the algorithm as an object.

MATHEMATICIANS' POINT OF VIEW ON ALGORITHM

This analysis of the algorithm concept is mainly theoretical and it would be interesting to compare it with the reality of mathematics, that is the ongoing research.

-Are mathematicians' representations in accordance with our epistemological study of algorithm? -How do algorithms interplay in their practice of research? -Which aspects of algorithm are involved in mathematical research and which ones are not? -Do researchers refer mostly to the algorithm as a tool or as an object? -Do these questions depend on their field of research?

Actually, the main point which interests us is the following: validating our epistemological analysis when comparing the descriptions of mathematicians of coming to use and to know algorithm and our epistemological model (in the same way that Burton did). Here, the form of our interviews does not permit to describe the whole conceptions of the researchers in a specific theoretical model. Right now, the trends in our results are enough to use this analysis as a preliminary work in order to build Research Situations for the Classroom (RSC).

Interviewing researchers

To answer the previous questions, we chose to interview researchers both in applied and fundamental mathematics. We also interviewed researchers from fields at the intersection of mathematics and computer science, like operational research, combinatorics, computational geometry... These researchers have a mathematical activity too, that is a proof activity, but should have a rich and different vision on the algorithm, provided by the links they have with computer science. We interviewed 22 researchers. From their point of view, they belong to the following fields:

Fundamental Applied Mathematics 10 7

Computer Science 5 5

Table 1: Distribution of the researchers [4]

In fact, in order to study the practice and the representations of mathematicians, the tool of interview seemed to be the most convenient possibility (in the same way as [START_REF] Burton | Mathematicians as enquirers. Learning about learning mathematics[END_REF]. We have met the researchers face-to-face and the interviews were audio taped. The researchers received a brief questionnaire to provide personal information (name, gender, function, discipline (mathematics, computer science; fundamental and applied mathematics [5]), research subjects, teaching level at university), and an excerpt of the new French curriculum with the appearance of algorithms [6]. We followed a list of questions (see above) in conversational style. These interviews last between 20 and 30 minutes and the part which interest us here makes up around half of this time.

The interviews

In the interviews, we chose to bring up three points:

1) What is an algorithm, how can one define it and recognize it, and what examples can be given?

2) How is it used in mathematics and in the field of the researcher?

3) The place and the role of algorithms in the researchers' work.

The questions were the following:

Q1: How would you define what an algorithm is? Q2: How can one recognize an algorithm? Q3: Give examples of algorithms, of non-algorithms. Is the discriminant (presented like above) an algorithm? Where is the border between algorithmic and non-algorithmic areas? Q4: What are algorithms useful for? What are their roles in your field? Q5: Are there algorithms in your personal research? Where?

The first three questions aim at making the researcher talk about the definition(s) of algorithm. Giving examples, counterexamples and thinking about the discriminant should make the researchers question their own definition of "algorithm". The last two questions aim at making them evoke the role and place of algorithms and more particularly in their field. All the questions were reformulated if necessary, so that there was no misunderstanding. And as it was an open discussion, these points were not been necessarily mentioned in this order.

Analysis of the interviews

In order to study the transcriptions of the interviews, we built an analysis grid based on our epistemological study. The goal was to find in the interviews which aspects were present. The difficulty was to give the grid a good granularity: the purpose was to associate each idea of the researcher to one or more precise aspect of algorithm, but many aspects were often vaguely mentioned. For example, as far as complexity is concerned, most of the time, the researchers just spoke about its importance but did not give details about the different kinds of complexities or about the question of optimality. After different draft versions of an analysis grid, we finally decided on the following:

Table2: The analysis grid

Results

Validation of the epistemological analysis

In the interviews, all the aspects we expected researchers to speak about were mentioned by them. Obviously the researchers did not all talk about every aspect and they did not always give many details. But each aspect has been mentioned enough to confirm our study. We will discuss in details how each aspect has been brought up.

Domination of the algorithm as a tool

The "effective" aspect and the "problem" aspect were mentioned by all the researchers. That means that the "tool" aspect is very important in their representation of the algorithm. The "effectivity" and "problem" aspects mainly appear in the definitions of algorithm researchers gave. The "effectivity" aspect is often linked with the use of computer, and many researchers pointed out the importance of computers for algorithms (and some of them made the confusion between algorithms and programs).

Definition of the algorithm

As the definitions researchers gave very often involve the effective aspect of the algorithm, we can say that their definitions are very close to the "usual definition" we mentioned at the beginning. To illustrate this, here are some definitions of "algorithm" given by the researchers: "A sequence of instructions which enable from an input to produce an output."

"A finite chain of steps which can be described, and which allows to compute or find the solution of a given problem."

"An effective process which allows to achieve a calculus or an automatic deductive task."

"An automatic method to solve a problem which does not need any human intervention, and which is workable for a machine."

About the discriminant, most of the researchers answered that, according to their definition, it was an algorithm. Some others felt embarrassed that their definition encompassed the discriminant and explained that it was an algorithm but had no interest:

"It is not a very rich algorithm, not a good example."

"We can compute it explicitly. It is just applying a formula."

"We are between the "method" and the "algorithm". There is no real process. If any time we have a formula we consider it as an algorithm, it reduces the meaning of algorithm."

"The problem has a set size... for me, in the idea of algorithm, there is an aspect of variable size, there is the complexity behind..."

The problem of the ambiguity of the definition we underlined seems to be shared by some researchers. The "complexity" aspect seems to be closely related to this problem. Most of the researchers must not have noticed this because, as we will see below, the notion of complexity is not of a big importance for them.

Presence of the "proof" aspect

The "proof" aspect, which is for us the most important when looking at the algorithm from a mathematical point of view, has been brought up by about half of the researchers (12 among 22). There is a link with their field of research: indeed, most of the researchers who brought up the proof consider themselves as fundamental researchers (in maths or computer science) whereas the majority of the others consider themselves as applied mathematicians or computer scientists. We can say that the importance of proof activity has a link with fundamental questions.

Moreover, among all the parts of the "proof" aspect, the most quoted is that an algorithm is a tool of proof, that is to say that the algorithm is associated with the notion of constructive proof. The notion of proof of an algorithm (correctness or termination) has not been mentioned much, and more precisely it is always the correctness which was quoted.

The "complexity" aspect

The "complexity" aspect was not mentioned by all the researchers, only 11 of them spoke about it. In this case, it seems to be linked with the computer science field of research: among the 10 (self declared) computer scientists, 8 raised the questions regarding the complexity involved by the algorithm. We can infer that the complexity is not really important from the mathematicians' point of view (as the quotation of Knuth about Bishop's mathematics let us think).

The theoretical models

Theoretical models were mentioned by only 8 researchers, not only from fundamental research but mainly from fields at the intersection of mathematics and computer science (computational geometry and topology, operational research, combinatorial optimization, graph theory or cryptology). In fact, theoretical models for algorithms have a very important role in these fields. That must be the reason why those researchers mentioned them. However, it should be noted that very few researchers from other fields (only two) brought up those theoretical models. It seems that this "recent" aspect of the algorithm (but older than the link with computers!) is not known by the researchers or does not seem important to them. The first possibility is, according to us, the most plausible. As an interviewed mathematician underscored:

"As far as I'm concerned, I'm aware of these questions [about algorithm and theoretical models] because my husband is a computer scientist. But this is not in mathematicians' culture..."

CONCLUSION AND PERSPECTIVES

Our epistemological analysis has been validated by the interviews. Hence, we can ask ourselves what this study implies for the teaching of algorithm and the teaching of proof. These questions have their importance at the moment in France, but also widely impact on mathematical education. We saw that if one does not want to teach algorithm as a tool only, but also as an object, it cannot be separated from the "proof" or "complexity" aspects. Learning algorithm seems to be a good way to learn proof, judging by the connections there exists between these two concepts.

Learning from these interviews, it seems that researchers in mathematics and in fields which link mathematics and computer science, do not have a wide view of the algorithm concept. We can say that little is known about this concept. We could explain this by the recent development of the study of algorithm, but this still seems pretty worrying. We can assume that this lack of knowledge about algorithm is shared by teachers of mathematics (at least in France) and their training curriculum has to be questioned.

This study of the algorithm in mathematics should allow us to study curricula and textbooks of mathematics of the secondary in order to know if the "tool" and "object" aspects are involved.

We would also like to study how the algorithm can be handled as an object by pupils and how it can make them enter in a proof process. We already made experimentations about this, at the beginning of university and in the training of primary teachers, and we obtained promising results (the students and pre-service teacher training were able to build several algorithms and their proof, see [START_REF] Modeste | Algorithmique et apprentissage de la preuve[END_REF]. Our goal is to carry on these kinds of experiments in the secondary level. [START_REF] Schuster | About traveling salesmen and telephone networks combinatorial optimization problems at high school[END_REF] has studied combinatorial optimization problems in the secondary and has obtained very positive results about pupils' skills in manipulating algorithms and proving. We could work on the "Konigsberg's bridges problem" studied by [START_REF] Cartier | Learners' conceptions in different class situations around Königsberg's bridges problem[END_REF] or on other problems studied by the "Maths à Modeler" team, but from an algorithmic point of view. The analysis of the way one can deal with algorithms and proof in the classroom and the results of such experiments will be the object of a new article, based upon the epistemological model developed in this paper.

NOTES

1. Bold types added.

2. In this article, Knuth studied books from many mathematicians and notice that, for him, Bishop's mathematics where the most close to the algorithmic thinking. However, even in Bishop's mathematics, Knuth noted that the notion of complexity was absent.

3. The definitions of P and NP-hard problems are not exactly these ones. For more details about theoretical models, one can read: Hopcroft, J.E., Motwani, R., & Ulmann, J.D. (2007). Introduction to Automata Theory, Languages, and Computation. Pearson Education.

4. The total number is not 22 because some researchers consider that they have 2 fields of research.

5. The choice of the researcher is not necessary dichotomous. Indeed, the presentation for their choice of the discipline was the following:


Simon Modeste

RELATIONS BETWEEN MATHEMATICS AND COMPUTER SCIENCE IN THE FRENCH SECONDARY SCHOOL: A DEVELOPING CURRICULUM

In the last decade, French curricula for secondary school have included contents in Computer Science inside Mathematics curricula. We study this particular situation by questioning the history of the introduction of Computer Science in school in France, by analyzing the links between Mathematics and Computer Science in the curricula (based on epistemological considerations), and by enlightening the issues of this curriculum under development.

This specific situation questions the relations between Computer Science and Mathematics, as disciplines, but also inside the French curricula of secondary school. This issues clearly regards STEM education, with a special focus on the relation between Mathematics and Computer Science among other scientific topics. In this paper, we address the following issues we have been working on since 2009 [START_REF] Modeste | Algorithmique et apprentissage de la preuve[END_REF][START_REF] Modeste | The appearance of algorithms in curricula, a new opportunity to deal with proof[END_REF][START_REF] Modeste | Enseigner l'algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l'apprentissage de la preuve ?[END_REF][START_REF] Modeste | Impact of informatics on mathematics and its teaching. On the importance of epistemological analysis to feed didactical research[END_REF], Modeste & Rafalska, 2017): What do Mathematics and Computer Science share as scientific disciplines and what kind of interactions between them can be developed in secondary school? How does the French curricula deal with this issue and in which direction are they developing?

In this paper, we first present briefly the theoretical framework that shapes our analysis. Then, we explore briefly the history of the introduction of computer science in school in France and we analyze the relations between Mathematics and Computer Science in the current curricula, based on epistemological considerations about the nature of these relations. Finally, we examine the institutional situation in France, through the processes that lead the development of these curricula Modeste and the actors who influence it. This will enable us to enlighten the issues of developing coherent curricula in mathematics and Computer Science.

A framework for analysing the curricula: ATD

In this paper, we will use some elements from the Anthropological Theory of Didactics (ATD) [START_REF] Chevallard | Concepts fondamentaux de la didactique : perspectives apportées par une approche anthropologique[END_REF][START_REF] Artigue | International comparative studies on mathematics education: A viewpoint from the anthropological theory of didactics[END_REF][START_REF] Bosch | Introduction to the Anthropological Theory of the Didactic (ATD)[END_REF]. Indeed, ATD is relevant to analyze curriculum design and allows taking into account various levels of analysis such as society, pedagogy or discipline, as well as institutions involved in the process, which fits our purpose. We briefly describe the elements of ATD that are used below.

ATD considers unit blocks of activity called praxeologies. They permit to describe human activities and their organization in specific institutions. Here we are interested in Mathematics praxeologies and Computer Science praxeologies involved in secondary school curricula. Another concept linked with ATD is the didactical transposition [START_REF] Chevallard | La transposition didactique. Du savoir savant au savoir enseigné[END_REF][START_REF] Chevallard | Didactic Transposition in Mathematics Education[END_REF], which is the process through which knowledge (i.e. the praxeologies) is transposed from institutions where its is produced to institutions where it is taught. In the case of scientific knowledge (like Mathematics or Computer Science), the didactical transposition concerns the passage from scholarly knowledge to knowledge to be taught and then to the taught knowledge. This concept is closed to the notions of intended and implemented curricula but incorporates views on curricula as products and as processes. The didactical transposition of a piece of knowledge involves many actors and institutions, at a higher level than the one of the teachers, called noosphere. In the last part, we give example of it.

To understand how didactical transpositions occur or how curricula are designed and evolve, we will rely on the levels of didactic codetermination [START_REF] Chevallard | Organiser l'étude 3. Écologie & régulation[END_REF], which allow taking into account the context (in terms of conditions and constraints) in which praxeologies exist and develop. These levels, that interact with each other, are the following: Civilization (9)↔ Society (8) ↔ School (7) ↔ Pedagogy (6)↔ Discipline (5) ↔ Domain (4) ↔ Sector (3) ↔ Theme (2) ↔ Subject (1)

In this article, we are interested mainly in Society, School, Pedagogy and Disciplines as levels that influence curricula, in particular on Mathematics and Computer Science and the way they interact.

Current situation in France: organization and design of the curricula

Currently, French secondary school is divided in two institutions. Lower secondary school (called collège) is mandatory and starts at grade 6 and finishes at grade 9, with a national assessment called Brevet. The next 3 years (grades 10 to 12) concern upper secondary school (called lycée). It has different streams (vocational, technological and general), and we will focus here on the general stream, which ends with a national assessment called Baccalauréat. The two national assessments, Brevet and Baccalauréat, have a strong influence on the way the intended curriculum is implemented by teachers.

Historically, France has a national curriculum elaborated by specialized committees. In mathematics, the curriculum is defined by different official documents. One is the program itself, describing contents and expectations about them, and the others are accompanying resources for helping teachers to implement the program, offering details about specific topics (propositions of Modeste learning activities and related explanations). Until 2016, the programs for secondary school were presented for each grade. Since 2016, in primary school and lower secondary school, the curriculum for each subject is derived from a common core of knowledge, competencies and culture and organized by 3-years cycles (lower secondary school is concerned by cycle 4 -grades 7 to 9 -and by the last year of cycle 3 -grade 6). These changes are clearly related to international movement towards enhancing competencies. For now, curriculum for upper secondary school has not changed but very recently the Ministry of Education has launched a wide project for changing the Baccalauréat and the whole structure of upper high school 1 .

Historical overview on computer science in French Secondary School

This section is based on a previous collaborative work [START_REF] Gueudet | Curriculum in France. A national frame in transition[END_REF] in which details about French curriculum's structure and evolution are presented.

In the 1980s, after a few experiments, the teaching of computer science in upper secondary school was introduced in France for the first time with an optional teaching called "Informatique des Lycées" (Computer Science for upper secondary school). This teaching was done by specially trained teachers, essentially mathematics teachers, and was centered on algorithmics and programming; the Ministry of Education invested a lot of effort into teacher training for this option. At this time, international work is already developing about the interactions between Mathematics and Computer Science, as attested by the first ICMI Study: The influence of Computers and Informatics on Mathematics and its Teaching (Howson & Kahane, 1986). However in France there was no social consensus on the finality of this teaching [START_REF] Baron | L'informatique et son enseignement dans l'enseignement scolaire général français: enjeux de pouvoirs et de savoirs[END_REF], and Computer Science as a scholar discipline, disappeared in the 1990s, replaced in the curricula by teaching how to use computers as tools in every discipline.

In the 2000s, the CREM 2 [START_REF] Kahane | Enseignement des sciences mathématiques : Commission de réflexion sur l'enseignement des mathématiques : Rapport au ministre de l'éducation nationale[END_REF] recommended in its report to "introduce some Computer Science in the teaching of Mathematics and in teacher education" and defended the importance of interactions between Mathematics and Computer Science. The report addresses many arguments summarized as follows: Algorithmic thinking, implicit in the teaching of mathematics, could be developed and enlightened with the instruments of algorithmics; Programming promotes formalized reasoning; Questions about effectiveness of algorithms involve mathematics; Data processing and digital computations are common in other disciplines; and finally, Computer Science transforms Mathematics, bringing new points of view on objects, bringing new questions, creating new fields in Mathematics that are growing rapidly, and changing the mathematician's activity with new tools.

Just after this report was published, algorithmics was introduced in mathematics at grades 11 and 12, but only for literature series, and in optional mathematics courses in the last year of the economics series (in an introduction to graph theory) and the scientific series (in an introduction to 1 A mission has also been given to the Fields medal awarded French mathematician Cédric villani on the teaching of mathematics at primary and secondary levels (http://www.education.gouv.fr/cid126423/21-mesures-pour-lenseignement-des-mathematiques.html) but we will not analyze it here.

2 In 1999, the French education ministry appointed a commission, the CREM (National Commission for Reflection on the Teaching of Mathematics) for rethinking the teaching of mathematics for the new century headed by Jean-Pierre Kahane. Jean-Pierre Kahane is a famous mathematician, member of the Academy of sciences, and former president of ICMI (1983ICMI ( -1990)). In 2002, the CREM published an important report on the teaching of mathematics.

Modeste

A3. Computer assisted mathematics and experimental dimensions. Computer Science permits to foster the experimental dimension of mathematics, allowing systematic exploration of various cases, or algorithmic solving of mathematical problems. Computer Science can bring complementary outlook on mathematical objects. A4. Modeling, simulation and relation with other disciplines. Both disciplines offer means for formalization that permits to model and simulate (or compute) situations from other disciplines. Mathematical models can be computed and simulations often involve many mathematical models.

Based on these four aspects, we analyze below the curricula of French lower and upper secondary school in Mathematics and Computer Science. This provides an insight on the didactical transposition and informs us about the factors influencing the curricula and the levels of codetermination at which they intervene.

Computer Science and Mathematics in lower secondary school curriculum (French collège)

In lower secondary school, inside the Mathematics programs for cycle 4, one domain is Algorithmics and Programming. The main objective is developing pupils' skills on: "Write, elaborate, and execute a simple program". For this goal, the program enumerates various programming notions (loops, conditions) and programming paradigms (parallel, event-driven…) and suggests to use a programming language with specific properties, very close to the language Scratch4 , which is, in fact, used in the accompanying resource (and, thus, used by all the teachers). This follows an international trend for introducing Scratch and block programming in curricula, but the specificity, in France, is its inclusion inside the Mathematics class (Modeste & Rafalska, 2017). This leads to two different kinds of activity in class: on the one hand, algorithmics and programming are widely used to solve Mathematics problems, particularly in algebra and geometry (this is A3); on the other hand, they are also used in projects which can be very far from Mathematics. A4 could be developed at this occasion, but neither the program nor the accompanying resource develop this aspect. A1 is not present (proof and logic have a small place in this curriculum); it is the same for A2 (not any element of mathematics used in Computer Science has been introduced when Computer Science entered the Mathematics curriculum). The epistemological reference remains unclear about the status of these Computer Science contents.

Besides the teaching of Computer Science within Mathematics, contents related to networks and to programming the behavior of objects, machines or systems are taught within Technology. For now, very few interactions exist between teachers of the two subjects and few elements are given for that in the curricula and resources.

We can see various processes involved here, at different levels. At the civilization or society level, international trends seem to have a strong influence, but the transposition is influenced at many other levels like School (there are no Computer Science teachers in French schools); Pedagogy (the pedagogy of project, suggested in algorithmics and programming, is likely to collide with habits in Mathematics teaching in France, despite the efforts made since the 2000s to change them), and Disciplines (the potential links between the two disciplines are superficially taken into account). Another element affecting taught knowledge is the national exam at the end of lower secondary school (the Brevet). The contents in the final assessment strongly influence the type of tasks that are

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proposed by teachers and which domains, sectors and theme are emphasized. The recent introduction of algorithmics and programming tasks, oriented towards completing or interpreting programs or algorithms because of the paper-and-pencil nature of the assessment, is likely to impact teachers practices and hence the implemented curriculum.

Computer Science and Mathematics in upper secondary school curriculum (French lycée)

The current curriculum of Mathematics at upper high school is the one introduced between 2009 (grade 10) and 2012 (grade 12). Algorithmics has been introduced in Mathematics through these programs. In 2017, the Mathematics program for grade 10, and in particular the algorithmic part, has been adapted, as algorithmics is not a new subject anymore for pupils at this level, and in order to make of the transverse content of algorithmics an authentic chapter untitled "algorithmics and programming". The programming language Python has been introduced in replacement of Algobox, a language designed for French secondary school. This process will necessarily be followed by changes in grades 11 and 12 in the next years.

It is worthwhile to compare these two programs and the two corresponding accompanying resources, regarding the interaction of Mathematics contents with algorithmics. From 2009, algorithmics must be a tool for solving mathematics problems and contents (A3), but this role is often restrained to illustrate mathematical concepts and simulate random experiments [START_REF] Modeste | Enseigner l'algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l'apprentissage de la preuve ?[END_REF]. There is also a collapsing between the notions of algorithm and program resulting in a collapse between input/output and typing/printing, and raising didactical issues [START_REF] Modeste | Enseigner l'algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l'apprentissage de la preuve ?[END_REF]. In the 2017 program and accompanying resource, some of these issues have been settled, and a point of view on algorithms as functions is expected (in order to make this change effective, model subjects for the Baccalauréat have been published with an emphasis on the way algorithms must be structured). We interpret this change as a better consideration of some points of view of Computer Science in algorithmics, with a potential to develop also algorithmics for itself, but at the same time, the examples given in the new accompanying resource are strongly oriented toward application to calculus and numerical methods. This contradiction illustrates the tensions, at school, pedagogy and discipline levels, between teaching some elements of algorithmics for themselves and organizing the whole teaching of algorithmics around mathematics problems.

In 2009 as in 2017, A2 is not developed: although there is a strong potential to support interactions between mathematics or consistent algorithmic problem solving at secondary school (and namely the opportunity of being taught in the same class), very few contents shared by Mathematics and Computer Science are offered in the curriculum (for instance, arithmetic and combinatorics have remained negligible in the new curricula). For A1, although some contents of logic are present in upper high school Mathematics, there is no mention of their links with algorithmics. Finally, A4 is also very little present in these curricula.

Evolution of curricula: Actors, processes and debates. An insight of the noosphere

In the recent development of the curriculum of Mathematics and Computer Science, many actors have been involved or have influenced the situation. The academic societies in Computer Science have been, for a long time, lobbying for the introduction of Computer Science as a Scholar Discipline in France, associations like EPI5 have been campaigning for introduction of Computer

For more details about the French Mathematics curriculum, see[START_REF] Gueudet | Curriculum in France. A national frame in transition[END_REF] or the ministry website: http://eduscol.education.fr/cid66998/eduscol-the-portal-for-education-players.html

https://scratch.mit.edu/

Enseignement Public et Informatique : https://www.epi.asso.fr/, association founded in 1971.

Communication supported by French National Research Agency <ANR-16-CE38-0006-01>.

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number theory). Then, between 2009 and 2012 in new official programs, algorithmics was introduced as part of Mathematics into all series of the general stream (literature, economy, sciences) from grade 10 to 12 (French lycée).

Finally, in the 2010s, computer science as an autonomous discipline came back in the upper secondary school, together with algorithmics and programming as part of the contents in Mathematics. In 2012, a new optional teaching of computer science (called ISN, for Informatique et Sciences du Numérique) was proposed in grade 12 for students of the scientific series. These teachings are made by teachers from various scientific disciplines, including a lot of mathematics teachers. A similar option (Computer Science and Numerical Creation) has been created in grade 10 and tried in various schools since 2015. Since 2016, Computer Science is also taught in cycle 4 (grade 7 to 9), but divided between two disciplines: Mathematics and Technology.

Following this increasing need for teaching Computer Sciences in secondary school, a Computer Science option has been introduced since 2016 in the competitive exam for recruiting mathematics teachers for secondary school (CAPES), leading to the involvement of university scholars in Computer Science in the preparation of these.

We can already notice that the appearing of Computer Science in French curriculum is not a linear process, illustrating the tensions in the phenomena of didactical transposition, submitted to conditions and constraints that falls under different levels of codetemination: School, Pedagogy and Discipline, but also Society, inasmuch as the relationship with Computer Science, Mathematics and Sciences in the French society and French academic world clearly affects the curricula.

After this brief presentation of French curriculum structure and organization 3 and this short historical perspective, we examine relations between Mathematics and Computer Science in the current programs for grades 10 to 12 in the general stream and in the new programs for cycle 4 (grade 7 to 9) of lower secondary school.

ANALYZING SECONDARY SCHOOL CURRICULA

Epistemological considerations on the relation between Mathematics and Computer Science

In our previous work [START_REF] Modeste | Enseigner l'algorithme pour quoi ? Quelles nouvelles questions pour les mathématiques ? Quels apports pour l'apprentissage de la preuve ?[END_REF][START_REF] Modeste | Impact of informatics on mathematics and its teaching. On the importance of epistemological analysis to feed didactical research[END_REF], we have developed epistemological considerationsin the sense of [START_REF] Modeste Artigue | Épistémologie et Didactique[END_REF] -on the relations between Mathematics and Computer Science. We have identified four aspects that we summarized here.

A1. Foundations, logic and proof. Mathematics and Computer Science share common logic foundations (Computer Science is born, in part, from development of logic) which guide the role of language in both disciplines. Proof, as a mean of validation, plays an important role in Mathematics and Computer Science, and is an important component of their epistemologies.

A2. Continuity and interfaces.

The frontier between Mathematics and Computer Science is impossible to draw, and a lot of objects and fields live at their interface like cryptography, games theory, graph theory, combinatorics, etc. Many new Mathematical fields developed in response to Computer Science needs and many algorithmic and computing questions arose from Mathematics.

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Science in School and have been contributing to the refection on this for a very long time. In the 2010's, many opinion columns were published in French newspapers and the French Academy of Sciences (2013) published a report on the necessity to introduce Computer Science in school in France. Many of the actors of these institutions (mostly influential researchers) have also been invested in curriculum design (for ISN or the cycle 4 curriculum of Computer Science) but also in developing reflections on what should be and should contain a teaching of Computer Sciences, and in developing resources and textbooks.

The main debates about the future of teaching Computer Science are now about teachers and teachers' training, and the question of bringing together or not the different pieces of Computer Science present in the curriculum. The creation of a Computer Science option in the competitive exam for recruiting Mathematics teachers let think that we head towards a teaching of Computer Science by Mathematics teachers, or more likely Mathematics/Computer Science teachers. This debate is not closed at all and other hypotheses are also likely. In particular, some defend the close connection between Computer Science and Information Technologies [START_REF] Tort | Informatics education: beyond the opposition between information technology and computer science[END_REF] that would be likely to increase the distance between Mathematics and Computer Science. In any case, enhancing interaction between Mathematics and Computer Science in school will remain an important issue.

CONCLUSIONS AND PERSPECTIVES ON MATHEMATICS CURRICULUM REFORMS

Computer Science is still looking for its place in the curricula, and questions the territories of other scientific disciplines. As we have seen, the interactions with Mathematics are important in the scholarly knowledge. This leads to important curriculum design issues and questions the teachers' needs in order to handle the intended curricula. The perception of these changes in the curricula by Mathematics teachers could be enlightening for understanding their needs and representations of Computer Science, in order to understand the effects on the implemented curricula.

In the noosphere, many actors influence the didactical transposition of Computer Science which has a direct impact on Mathematics curriculum in the French educational context. In our view, an important issue is the place that a curriculum can let to the interactions between Mathematics and Computer Science (we have seen that these interactions are very limited at present). We have also seen that the French curriculum of Mathematics and Computer Science is influenced by two tendencies: international trends and French specificity.

We view the relation between Mathematics and Computer Science in French secondary curriculum as a prototypical example that illustrates the difficulties faced in STEM education to design curricula that take into account the disciplines themselves without neglecting their interactions (and reciprocally), and that address institutional constraints like the profile of the existing community of teachers.

To study the curriculum issues, we think that ATD and didactical transposition can be relevant.