Cécile Ouvrier-Buffet
email: cecile.ouvrier-buffet@univ-reims.fr

Antoine Meyer
email: antoine.meyer@u-pem.fr

Simon Modeste
email: simon.modeste@umontpellier.fr

Discrete mathematics at university level Interfacing mathematics, computer science and arithmetic

Keywords: teaching and learning of number theory and discrete mathematics, teaching and learning of mathematics in other fields, proof, algorithms

Discrete mathematics is a recent field linked with Computer Science. We discuss its place in university mathematics curricula and in the particular case of France, where it has difficulties to find its place. We make explicit the didactical challenges posed by discrete mathematics at university level, and present DEMIPS network and its plans to tackle them. Through two detailed examples we discuss the reasons for teaching Discrete Mathematics at university level, and illustrate our conclusions.

INTRODUCTION

This paper points out the current need for the construction of resources and debates regarding discrete mathematics at university level. We wish to emphasize the features of the French context, both from an educator's and researcher's point of view, at the intersection of didactics, mathematics and computer science. Indeed, teaching and learning discrete mathematics involves mathematical skills and heuristics (e.g. different kinds of proofs and reasoning, several ways of modelling etc.) 1 and also develops objects, concepts, methods and tools that are necessary for computer science. This link with computer science brings new types of questions to mathematics (for instance, regarding algorithmic complexity). Then, our aims are to design original situations for schools and at university level, and to construct appropriate introductory situations for computer science and maths majors.

We propose an overview of discrete mathematics in mathematics education and make a focus on the interface between discrete mathematics and computer science. Then, after presenting our research group in France, we analyse two kinds of situations.

DISCRETE MATHEMATICS IN AND FOR MATHEMATICS EDUCATION How to define discrete mathematics?

Several mathematical topics are often gathered under the blanket term discrete mathematics. A first step in contributing to a thorough didactical study of discrete mathematics is to provide a satisfactory definition, or at least delimitation, of what it refers to. Several definitions exist, which either attempt to provide a general common trait to the covered topics, such as "the mathematics of discrete sets", or resort to an enumeration of objects, concepts or techniques most often associated with discrete mathematics. Most of these definitions include or are followed by a discussion on some typical difficulties, such as the distinction between finite, discrete and continuous mathematics (e.g. [START_REF] Maurer | What is Discrete Mathematics? The many answers[END_REF]. To clarify the distinction between finite and discrete mathematics, the MAA (1992) places finite mathematics in the precalculus category and discrete mathematics in the same category as calculus.

We advocate that an interesting way to define discrete mathematics both for research and didactical perspectives (for the design of courses and of didactical engineering at university level) is to emphasize the features of the modes of reasoning that are common (or specific) to the various topics usually recognized as discrete mathematics, and the discrete nature of the structures they involve. Moreover, a classification of problems is required in order to structure a didactical analysis of the field of discrete mathematics. Furthermore, as the development of discrete mathematics has been strongly directed by the needs for computer science, the links with computer science must be explicitly explored.

In 1974, Knuth, a pioneer in computer science and its teaching made a similar analysis (Knuth, 1974, p. 329) :

"The most surprising thing to me, in my own experiences with applications of mathematics to computer science, has been the fact so much of the mathematics has been of a particular discrete type [...]. Such mathematics was almost entirely absent from my own training, although I had a reasonably good undergraduate and graduate education in mathematics. [...] I have naturally been wondering whether or not the traditional curriculum -the calculus courses, etc. -should be revised to include more of these discrete mathematical manipulations, or whether computer science is exceptional in its frequent application of them."

We consider that these questions are still topical, even at university level, and deserve a careful didactical analysis.

Where is discrete mathematics? What questions are relevant at university level?

It is often stated that discrete mathematics can be a tool for improving reasoning and problem-solving skills (see for instance Rosenstein, Franzblau & Roberts (1997) who advocated an introduction of discrete mathematics in curricula, asked didactical questions, and made propositions that were taken into account for Principles and Standards for School Mathematics NCTM, 2000 for instance). Moreover, discrete mathematics is an active modern branch of contemporary mathematics with a wide range of applications in society, which is a very legitimate reason to teach it at school, high school and college. In fact, discrete mathematics courses are relevant to a wide variety of majors at university level, including mathematics, number theory, computer science, and engineering: from an epistemological point of view, discrete mathematics has an interdisciplinary nature and can provide a mathematical foundation (with specific ways of reasoning and proving, and mathematical concepts) for computer science and engineering courses. By 1989, an MAA report [START_REF] Ralston | Discrete Mathematics in the First Two Years[END_REF]) from an ad-hoc committee consisting of mathematicians and computer scientists recommended that "discrete mathematics should be part of the first two years of the standard mathematics curriculum at all colleges and universities". This report also emphasizes the notions of proof, recursion, induction, modelling, and algorithmic thinking, as well as the benefits of using discrete mathematics in the secondary level to improve problem-solving skills with the transition to university level in mind [START_REF] Ralston | Discrete Mathematics in the First Two Years[END_REF]. Moreover, [START_REF] Epp | Discrete mathematics for computer science[END_REF] points out the strong necessity of abstract thinking for the course and its applications in computer science. She underscores that it is done in the frame of the current curricular recommendations, prepared by The Joint Task Force on Computing Curricula (2013) of the ACM and the IEEE Computer Society, which gives discrete mathematics as one of the two largest components in the "core body of knowledge" recommended for all computer science students. Besides, discrete mathematics is in close relationship with other mathematical areas: other fields of mathematics use its methods and results, and, are useful for solving some discrete mathematics problems.

What is currently the place of discrete mathematics and its links with other scientific fields at university level? In several countries (Hungary, USA, Germany for instance), its significance in university programs is well-established and acknowledged. That is not always the case in France where the status of discrete mathematics in the first three university years is unclear, at least in mathematical curricula. However, discrete mathematics appears sporadically in few parts of mathematics curricula as probability theory (in particular combinatorics for discrete probability theory) or arithmetic. It sometimes appears in courses dedicated to the learning of proving, mathematical reasoning and problem solving, but we question whether its specificity is emphasized. One is more likely to find courses where discrete mathematics is taught for itself in computer science or mathematics and computer science curricula, where there exists a kind of common basis shared between teachers and including classical contents of discrete mathematics as can be seen abroad. These reports and recommendations coming from academic societies and the above remarks underscore two key questions for mathematics education at university level, and more specifically in France:

 What are the place and role of discrete mathematics at university level? How to design curricula and didactical engineering for the first university years ?

 What links are there between discrete mathematics and other areas (mainly of mathematics and computer science) and how are they (or should they be) practised / worked in the first university years?

These questions are particularly crucial for countries where discrete mathematics does not have a well-established status is the first university years.

What do we know from a didactical point of view?

In mathematics education, various research regarding the teaching and learning of discrete mathematics exist, focusing mainly on the primary and the secondary levels (ZDM ( 2004), Hart & Sandefur (in press) propose overviews). This research meets general approval, and points out epistemological features of discrete mathematics such as: discrete problems bring out different ways of proving [START_REF] Grenier | Spécificités de la preuve et de la modélisation en mathématiques discrètes[END_REF]; discrete structures enable work on the construction of mathematical models, optimization, operational research and experimental mathematics (e.g. [START_REF] Grenier | Spécificités de la preuve et de la modélisation en mathématiques discrètes[END_REF][START_REF] Maurer | What is Discrete Mathematics? The many answers[END_REF]; discrete concepts are accessible and problems are easy to understand [START_REF] Grenier | Spécificités de la preuve et de la modélisation en mathématiques discrètes[END_REF]De Bellis & Rosenstein, 2004); discrete concepts have different kinds of definitions and representations [START_REF] Ouvrier-Buffet | Exploring mathematical definition construction processes[END_REF][START_REF] Ouvrier-Buffet | A mathematical experience involving defining processes: in-action definitions and zero-definitions[END_REF]; some discrete problems are real world problems developing and using techniques from mathematics and computer science [START_REF] Schuster | About Traveling Salesmen and Telephone Networks -Combinatorial Optimization Problems at High School[END_REF], etc. Discrete mathematics problems are also a frame for developing and teaching algorithms; conversely, the study of algorithms requires a lot of discrete mathematics, and studying algorithms and programming can be a good way to justify the introduction of discrete mathematics contents (e.g. [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]. In all this research, discrete mathematics seems to be a powerful source of problems for teaching and learning mathematical proofs and processes and engaging students in developing new ways of thinking (such as recursive thinking), heuristics and problem-solving skills from primary school to university. Besides, some researchers point out that its teaching provides opportunities to bypass some of the sources of commonlyoccurring negative affect in students (e.g. [START_REF] Goldin | Discrete mathematics and the affective dimension of mathematical learning and engagement[END_REF]).

It appears that the features of discrete mathematics clearly represent challenges for university mathematics, in particular in France.

THE "DEMIPS" NETWORK -A WAY TO FEDERATE DISCRETE MATHEMATICS EDUCATION

Presentation of the DEMIPS network

In the French framework of mathematics education, there is a need to federate (isolated) research in university mathematics. Following the INDRUM momentum, the national network DEMIPS 2 supports the development of new research programs. DEMIPS's research involves around 40 researchers in mathematics, mathematics education, physics education, computer science, and epistemology and history of mathematics and sciences, and is concerned with five main topics: three topics dealing with mathematical contents (analysis; linear algebra and abstract algebra; arithmetic, discrete mathematics and algorithmics) at the secondary -post secondary transition and at university level (the links with physics and computer science are questioned); a transversal topic (logic, language, reasoning, proofs -from both a mathematics and computer science point of view); and a specific topic dealing with the practices of teachers and teachers-researchers at university level (in mathematics, computer science and physics).

We (the authors of this paper) organize federative research in the fields of arithmetic, discrete mathematics and algorithms. The members of our group are mathematics educators (didacticians) with specific skills in teaching and learning at university level, mathematicians, and researchers in computer science. We choose to study the parts of mathematics which lie at the intersection of "classical" mathematics and theoretical computer science (for instance discrete mathematics, arithmetic, and algorithms), which interact and complement each other. As theoretical background we will follow Brousseau's theory of didactical situations [START_REF] Brousseau | Théorie des Situations Didactiques[END_REF] for its notion of didactical engineering, and the notion of scheme [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF] in order to structure our analysis of mathematical concepts. We organize our questions around key axes regarding the French university level:

 What are the epistemological features of concepts and reasoning in arithmetic, discrete mathematics and algorithms? How do they interact? (And then, how can these interactions be used to enrich the way these concepts are taught?)

 What kind of situations can one design in these mathematical areas for the university level and for pre-service and in-service teacher training? What for?

 What kind of curricula are there for this kind of mathematics at university level? What can be said about the design of these curricula?

Our research questions try to break down the barriers between scientific disciplines involving discrete mathematics. They also underline typical situations and questions common to mathematics and computer science, and try to put to use didactical analysis techniques to cast a new light on the way these questions are, or could be, tackled at university level. We develop below two examples to illustrate our work, and elaborateon the place and role of discrete mathematics at university level.

SITUATIONS AND IMPLEMENTATIONS AT UNIVERSITY LEVEL -EXAMPLES FROM DEMIPS' WORKSHOPS

We develop here two examples to illustrate the potentialities of discrete mathematics to engage students in learning modelling, proving, and mathematical reasoning and also to underscore the validity and the interest of keeping in mind the algorithmic point of view and the connections with computer science. These examples emphasize new perspectives for the teaching and learning of mathematics. The first example explore the links between mathematics and computer science in a problem-solving context and the second deals with a classical "divide and conquer"-type algorithm.

Example 1 -Discrete lines

The mathematical object concerned here is the discrete straight line (colouring squares, or "pixels", on a regular rectangular grid, in order to give the best possible visual impression of a straight line). The (real) straight line can act as a referent. Discrete straight lines are accessible through their representations (e.g. perceptive and analytical aspects of geometry) and their definitions and properties are noninstitutionalized (a concept is institutionalized if it has a place in a classically taught content). Computer programmers are familiar with this concept. Professional researchers in discrete geometry (both mathematicians and computer scientists) use several definitions, but the proof of the equivalence of these definitions remains worth considering. The complexity of the underlying axiomatization of discrete geometrical concepts is actually an open and interesting problem.

Ouvrier-Buffet ( 2006) has analysed the evolution potential of zero-definitions (in Lakatos' sense, zero-definitions act as working definitions) of the concept "discrete straight line" in a defining situation implemented with freshmen. She underscores several approaches dealing with this concept, namely "real straight line" (What is the "nearest" pixel to a real line? What kind of modelling should be used?), "regularity" (What are the properties of the sequence of stages (called chaincode string)?), and "axiomatization" (What about the existence of the intersection of two discrete straight lines? Is a discrete straight line unique?). Each point of view brings about several zero-definitions. To engage into an axiomatic perspective carries great difficulty. This approach deals with both the perceptive aspect of a straight line and the axiomatic perspective. We are here confronted with two markedly different defining styles: a local one and a global-theoretical one, the latter mobilizing some implicit skills and knowledge in students (e.g. building a theory and choosing among competing definitions). The main results of this experiment underscore the ability of students to engage in a defining activity with a "neutral" but complex concept.

Students do not assume an axiomatic perspective but mobilize reasoning involving approximate methods close to those used for real straight lines (and then arithmetic tools) and also the characterization of the sequence of stages of pixels (how can we modify a sequence to obtain a better regularity?) that involves recursive arguments.

From a didactical point of view, this research requires the development of a new theoretical background in order to model the defining process. From a mathematical point of view, the discrete geometrical objects, and more specifically the discrete straight lines can be approached in several ways: differential discrete analysis, the Bresenham algorithm, algorithms involving combinatorial analysis, several discretizations using algorithms which generate and study errors (Greene & Yao, Freeman & Pham, Rosenfeld), and the introduction by Reveillès of the arithmetical definition of a discrete straight line (1991). For instance, the approach to the discretization of a real straight line by checking linearity conditions is directly related to number theoretical issues in the approximation of real numbers by rational numbers. These linearity conditions can be checked incrementally, leading to a decomposition of arbitrary strings into straight substrings [START_REF] Wu | On the chaincode of a line[END_REF]. The ongoing mathematical problems in discrete geometry are intimately related to questions in other fields of mathematics and computer science. The construction and the manipulation of algorithms are important for this purpose.

Example 2 -Exponentiation by squaring

A classic algorithmic problem is that of computing for some natural n the n-th power an of real number a. A naïve solution is, starting with value 1, to multiply n times this value by a. The final value one obtains is indeed the expected result, which is not very difficult to establish. The fact that this algorithm terminates is also trivially true since it contains a single bounded repetition. Finally, the complexity of this computation is clearly in Θ(n) (i.e. asymptotically bounded above and below by n), counting for instance the number of multiplications performed, and assuming that multiplication by a is an elementary operation.

This algorithm is not very efficient, considering that its running time is actually exponential in the representation size of n (which is of the order of log(n)). A more efficient technique relies on the observation that a n =(a 2 ) n/2 if n is even, otherwise a n =a.(a 2 ) (n-1)/2 . Written as a recursive Python function, this algorithm reads as follows 3 :

def power(a, n): if n == 0: return 1 elif n % 2 == 0: return power(a * a, n // 2) else: return a * power(a * a, n // 2)

We will now study a few common questions asked about algorithms, which will allow us to illustrate examples of mathematical techniques relevant to the analysis of, and discussion about, algorithms. In the following, typewriter face (as in n) will be used for formal parameters, and italic (as in n) to denote actual values.

Termination.

A first question when it comes to analysing an algorithm is to determine whether or not it terminates, i.e. whether its execution on any instance of the problem (i.e. any pair (a, n)) yields a result after a finite number of execution steps or elementary operations. A standard technique used to prove this kind of result in non-trivial cases is the following. Assume here that there exist a 0 and n 0 such that power(a 0 , n 0 ) performs infinitely many recursive calls. Call n i the value of parameter n on the i-th recursive call. The sequence of naturals (n i ) i≥0 is strictly decreasing, because whenever n i >0, n i+1 is the quotient of n i by 2, rounded down. This contradicts the fact that infinitely many calls are made, which means that the value of n eventually has to reach 0 and the function must terminate for all values of a and n.

Correctness. It remains to prove that the result is indeed correct for any instance of the problem. This is often done using some form of induction due to the intrinsically discrete and recursive or iterative nature of algorithms. In our case, we will establish that the value returned by a call to power(a, n) is indeed a n , via a simple recurrence on the call depth, which is the maximal number, say k, of generated nested calls. The base case (k=0) is obvious: since there is no recursive call it must mean that n=0 and the returned is indeed 1 = a n . In the inductive case, assume the property holds for call depth k and consider a call of maximal depth k+1. Necessarily n must be greater than 0. If n is even, n//2 evaluates to n/2, a single nested call power(a*a, n//2) is performed and the obtained value is returned directly. This call itself has call depth exactly k, therefore by induction hypothesis its return value is (a 2 ) n/2 =a n . Similarly if n is odd, n//2 evaluates to (n-1)/2, the value returned by power(a*a, n//2) is, by induction hypothesis, (a 2 ) (n-1)/2 =a n-1 , and the value returned by the main call is a*power(a*a, n//2), which evaluates to a n . Therefore by the recurrence principle, the function returns the correct value whatever the initial value of its parameters.

Complexity. In the study of termination, we observed that in a call power(a, n), the value of n for the next call (if there is one) is divided by two (rounded down). One may observe the successive values of n more easily when it is written down in binary. Indeed, the operation of dividing a number by two and rounding down corresponds, in binary representation, to erasing its rightmost digit. The algorithm stops when n is 0, and performs one recursive call otherwise, modifying its value as we just saw. The number of nested calls for some initial value of n is therefore equal to the length, say k, of its binary representation, in other words its number of digits. Moreover, when n is even, exactly one multiplication is performed in the current call, two when it is odd. Therefore, denoting by m the number of digits equal to 1 in the binary representation of n, the total number of multiplications performed by the power(a, n) is exactly k+m, which is asymptotically bounded above by log 2 (n).

Summary. We chose this example to illustrate, on a simple problem, the type of questions which can be asked about algorithms and the methods which are likely to be used to answer them. Note that in this simple case, all three properties could have been proven simultaneously using a complete recurrence on n. For our purpose, we chose a more basic and detailed approach. It would have been interesting to show how these proofs could be rephrased in the context of an iterative function. This example also tries to advocate the necessity for students in mathematics, computer science and related topics to have at least a basic understanding of various flavours of recursion and induction (including basic properties of orderings), to be able to present rigorous proof arguments (at least informally), and to possess minimal fluency in arithmetic, in order to be able to envision algorithms as objects worth studying in their own right. It is moreover often the case that the study of algorithms provides insight on related mathematical objects (here, the relationship between the value of a number and the length of its binary representation). Finally, this example illustrates a typical preoccupation of algorithmics, which is to provide more efficient, sometimes even optimal, algorithmic solutions to problems.

DISCUSSION AND CONCLUSIONS

Discrete mathematics is now considered as an entire field of mathematics, with many links to computer science. While it has entered university curricula in many countries, its status and contour are not always clear, and there are countries (such as France) where it has difficulties finding a legitimate place. Through the two examples we have developed (the discrete line and exponentiation by squaring), we have illustrated that it is legitimate to question the place that discrete mathematics occupies in university mathematics, for different reasons:

 it allows to develop situations for mathematical reasoning, mathematical heuristics, and problem solving (by its nature, but also by contrast with traditional continuous mathematics),


Renaud Chorlay
email: renaud.chorlay@espe-paris.fr

An empirical study of the understanding of formal propositions about sequences, with a focus on infinite limits

Keywords: Teaching and learning of analysis and calculus, teaching and learning of logic, reasoning and proof, definitions, limits

In this paper, we analyze the answers of one group of high-school students and two groups of first-year University students to a questionnaire designed to test their level of recognition and understanding of the formal definition of the concept of infinite limit. Although this empirical study is ancillary to a larger project centered on didactic engineering, its analysis sheds light on the key issue of the logical prerequisites for the learning of the fundamental concepts of analysis. It also provides a new tool to investigate students' concept-image of limits, and assess the impact of teaching contexts and teaching paths.

CONTEXT AND RATIONALE

At the INDRUM 2016 conference, Cécile Ouvrier-Buffet and Renaud Chorlay presented a poster outlining a medium-scale project on definitions in analysis [START_REF] Chorlay | Comparaison de schémas de genèses didactiques de définitions, le cas de la limite d'une suite[END_REF], with a focus on the formal definition of the limit of a numerical sequence. This topic lied at the intersection of the research interests of the two researchers: Cécile Ouvrier-Buffet is a maths-education researcher with a strong epistemological background, whose work bears mainly on definitions, their use, and the conditions for their genesis in teaching-contexts [START_REF] Ouvrier-Buffet | A mathematical experience involving defining processes: in-action definitions and zero-definitions[END_REF]. Since most of her former work bore on discrete mathematics, she wanted to investigate the extent to which the theoretical tools she had developed in this context had to be adapted to deal with a teaching context with very different mathematical (continuous vs discrete) and didactical (transition from calculus to analysis) features. Renaud Chorlay is a historian of mathematics and teacher educator with a long-standing interest in the history [START_REF] Chorlay | Local -Global": The First Twenty Years[END_REF] and didactics of analysis.

We selected the topic of limits because we felt many years of didactical investigations had made it a mature topic; a topic about which knowledge has accumulated to form a sound and coherent body of knowledge. Indeed, we know a lot about limits in terms of conceptions and misconceptions [START_REF] Robert | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF]; also in terms of obstacles [START_REF] Sierpinska | Obstacles épistémologiques relatifs à la notion de limite[END_REF]. As far as the genesis or rediscovery of the (or a) definition is concerned, many attempts have been made and reported upon in details, whether in the framework of didactic engineering 1 [START_REF] Robert | L'enseignement de la notion de convergence des suites numériques en DEUG[END_REF] [START_REF] Bloch | Un modèle d'analyse des raisonnements dans les situations didactiques. Etude de niveaux de preuves dans une situation d'enseignement de la notion de limite[END_REF] or with other research tool-boxes [START_REF] Mamona-Downs | Letting the intuitive bear of the formal; A didactical approach for the understanding of the limit of a sequence[END_REF] [START_REF] Przenioslo | Introducing the concept of convergence of a sequence in secondary school[END_REF] [START_REF] Swinyard | Reinventing the formal definition of limit: The case of Amy and Mike[END_REF] [START_REF] Lecorre | Rationality and concept of limit[END_REF]) [START_REF] Roh | Designing tasks of introductory real analysis to bridge the gap between students' intuition and mathematical rigor: the case of the convergence of a sequence[END_REF]. The tricky logical aspects were studied, in particular, in [START_REF] Arsac | An epistemological and didactic study of a specific calculus reasoning rule[END_REF].

On this solid basis, our work on the genesis and use of definitions has so far been engaged along three different lines of investigation; we will distinguish between ex-ante studiesbefore students' first encounter with formal definitions of limitsand ex-post studies.

 Ex-ante 1: For year 12 (final year of secondary education), the French curriculum requires that students majoring in mathematics and the sciences study a definition of limits (finite or infinite) of numerical sequences. Students are not really expected to use this definition on their own; rather, the teacher is expected to use these definitions on a few occasions, to show that some properties of limits can actually be proved mathematically (in particular: any unbounded and increasing sequence tends to +). The underlying idea is that early encounter with a few rigorous definitions and proofs should ease the transition between highschool calculuswith its combination of algorithmic procedures and graphical intuitionand university analysis. This classroom work on the formal definition of limits is connected to another requirement of the current curriculum, namely that throughout high-school, the basic notions and the standard notations of mathematical logic be gradually made explicit. In this context, the discovery of a definition for limit, with its specific sequence of nested quantifiers, is supposed to be the culmination of this gradual process. In 2016, one of us (Chorlay) designed a teachingsession in the spirit of didactic engineering, for students to gradually formulate a formal definition of the infinite limit. We will report on this in detail in another context.  Ex-post 1: in 2015-2016 we studied howif at allprospective mathsteachers made use of the definition of limits in order to identify and analyze vague, informal or erroneous statements regarding limits. We reported on this in a poster presented at the INDRUM 2016 conference.  Ex-post 2: in 2016-2017 we designed a questionnaire in order to assess the level of recognition and understanding of the formal definition of the infinite limit. This questionnaire, and the answers collected with three groups of students are be the topic of this paper.

QUESTIONNAIRE -DATA COLLECTION

The questionnaire was of the True/False type, divided in two parts. We give below an English translation, along with indications on the correct answers.

Part I. For each one of the implications below, circle either "True" or "False". If you circle "False", justify your answer.

#1

If lim u n = + then  A  R  n A  N such that  A T -F Justification (if "False"): #2 If lim u n = + then  A  R  n  N u n  A T -F #3 If lim u n = + then  A  R  n A  N such that  A T -F #4 If lim u n = + then  A  R  n  N , u n  A T -F #5 If lim u n = + then  A  R  n A  N such that for any integer n greater than n A  A T -F Correct answers:
#1 True: Here the consequent means "not bounded above". #2 False: Here the consequent is a property which never holds; hence the implication is always invalid. #3 True: Here the consequent is always valid, hence the implication is always valid. #4 True: Here the consequent means "bounded below". #5 True: Here the consequent is the definition, worded semi-formally.

Part II. The four implications below are taken from part I. For each one of them, first state its converse, then circle "True" or "False" regarding the converse. Justify if "False". The specific form of the questionnaire derives from its original intended use. It was first designed to assess the didactic engineering, which focused on the formal definition of the infinite limit. Other forms of assessment of the ability to recognize, and of the level of understanding of the formal definition were ruled out, in particular interviews (as in [START_REF] Robert | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF]) or proof-writing (as in [START_REF] Roh | Designing tasks of introductory real analysis to bridge the gap between students' intuition and mathematical rigor: the case of the convergence of a sequence[END_REF]). We felt this questionnaire would give us feedback regarding two key features of the engineering, namely (1) the role of logic, hence the flood of formulae with nested quantifiers in this questionnaire; (2) the fact that "not bounded above" is a necessary condition for lim u n = + but not a sufficient condition, hence the importance of item #1 and its converse.

#1 If lim u n = + then  A  R  n A  N such that  A Converse : T -F Justification (if "False"): #3 If lim u n = + then  A  R  n A  N such that  A Converse : T -F #4 If lim u n = + then  A  R  n  N , u n  A Converse : T -F #5 If lim u n = + then  A  R  n A  N such that
We did not ask for justifications when the item was deemed "True" by the students, mainly to save time and keep the questionnaire feasible in about 20 minutes. In addition, the justificatory task for True statements could vary a lot across teaching-contexts and would not easily lend itself to comparison. For instance, considering item #4 (if lim u n = + then the sequence is bounded below): in some contexts citing a theorem studied in class would suffice whereas in other contexts students would have to devise and write a non trivial proof. We also chose to drop the converse of item #2, since the fact that an implication whose antecedent is False is considered valid is a purely logical matter.

In the spring of 2017, the questionnaire was administered to three groups of students: Group 1 is one of the two French Year-12 classes which had experienced the engineering; Group 2 and 3 are first-year university students in Mons University (Belgium), with high-achieving maths majors in Group 2 and medium-achieving 2 computer science majors in Group 3. In all three cases, the questionnaire was given several months after the course on limits had been taught, and students had not been asked to revise anything in particular. They were told the questionnaire was given for research purposes, and would not be graded. They were given between 20 and 30 minutes. The number of students was: 31 (group 1), 50 (group 2), and 17 (group 3).

We originally hoped a comparison between the three groups would enable us to study the effects of three teaching units: our engineering (group 1), a "standard" maths-lecturer course (group 2), and Robert's engineering (group 3, as reported upon in [START_REF] Bridoux | Introduire la notion de convergence avec une ingénierie des années 1980 : rêve ou réalité didactique pour l'enseignant[END_REF]). Unfortunately, we were not able to do that, since other factors seemed to have had a more significant impact.

FINDINGS Result #1

A first result is that this questionnaire is not unfeasible. In group 2, 14 of the 50 questionnaires were answered perfectly correctly, with relevant counterexamples for the False statements. Some of these counter-examples had been studied in class (such as for the converse of #1); in these cases, students managed to interpret " A  R  n A  N  A" as "not bounded above" and selected a relevant counter-example in a memorized repertoire. In other cases, counter-examples had not been studied in the course on limitsbecause they had nothing to do with limitsand students crafted ad-hoc counter-examples, displaying some command of logic (for instance, to prove that the negation of " A  R  n A  N  A" always holds).

Result #2

A second set of results sheds light on the role of an explicit teaching of logic.

When we collected the data we first engaged in quantitative analysis, and were pretty unhappy about the following result: in group 1 (our engineering), only 26% of the students considered #4 to be "True", compared to 86% in group 2 and 71% in group 3. A closer look at the answers showed that in group 1, a significant number of students had actually engaged in another task than the prescribed task. In Fig. 1 and 2 we translated extracts of answer-sheets from group 1: In these answer-sheets, the students did not engage in an assessment of the logical implications but in a comparison between the formal statements given as consequents (in part I) and the definition of lim u n = +. In these examples the comparison can be clumsy (as for #2 for student 29, or the "and  A" for student 3). Nevertheless, it rests on the fact that the definition is known (correct answers for #5 and its converse), and is seen as the relevant template against which other quantified formulae ought to be contrasted. Moreover, the comparison is not purely syntactical: in her assessment of #4, student 29 did not only spot that " A  R  n  N , u n  A" is not the definition, but also elicited in her own words why it could not be, namely " n  N u n  A" does not capture "beyond a certain rank", which is a key element of the definition. The reinterpretation of the prescribed task is typical of at least one third of the questionnaires from group 1.

By contrast, only one of the 67 students from Mons University reinterpreted the implication-assessment task as a comparison-with-the-definition task. A key difference between group 1, on the one hand, and groups 2 and 3, on the other hand, is that at Mons University students had studied logic in the first term, whereas the French high-school students had only occasionally been exposed to logic. The French students were familiar with the notion of converse, and had some knowledge of the meaning of quantifiers  and , but were not familiar with sequences of quantifiers; much less with the negation of such sequences. These formal aspects were not problematic for a large majority of the Mons students. This does not mean that all the logical aspects were mastered by the Mons students. In particular, when it came to proving that some formal statement was valid, many answer-sheets showed misconceptions regarding the use of  and .

This sheds some light on the standard but thorny issue of prerequisites: since the formal definition of limits involves a sequence of nested quantifiers, how much logic should be taught (either beforehand or along the way) for students to be able to do anything with it? Our results suggest that the answer depends on how "do" something with a definition is construed. Using the formal definition to design and write proofs probably requires some know-how regarding the interpretation of hitherto unknown sequences of quantifiers, and the negation of such sequences; for a significant proportion of the French student, their occasional and in-context encounters with logical notations did not allow them to acquire such know-how. However, if "do" is taken to mean "remember the definition" and even "understand the definition", then for a large majority of the French students, their command of logic was adequate. For instance, we consider the work of student 29 of group 1 (fig. 1) to display some degree of conceptual understanding of definition, namely some understanding of the specific role of each of the three quantifiers. Student 3 is clearly able to interpret " A  R  n  N u n  A". This understanding does not rest on a general ability to make sense of and formally manipulate logical formulae, but is limited to the context of the definition of limits. Since it relies on the specific connections between the concept-image and concept-definition of "limit" targeted (and, apparently, stabilized) in the didactically engineered teachingsession, this understanding is probably not only context-dependent but also pathdependent.

Result #3.

In the a priori analysis for the engineering, we studied the relations between three mathematical properties of numerical sequences:

(1) lim u n = + u n ) is not bounded above;

(3) u n ) is strictly increasing, at least from a certain rank onward.

Our hypothesis was that properties (2) and (3) were part of the concept-image of (1) for most students; of a concept-image 2 in which all three properties are considered to "go together", without any specific and explicit logical connections being part of the cognitive structure. This hypothesis was based on the didactical literature [START_REF] Robert | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF]) (Mamona-Downs 2001) [START_REF] Swinyard | Reinventing the formal definition of limit: The case of Amy and Mike[END_REF], and was perfectly confirmed during the two implementations of the engineering. For this reason, our design aimed for conceptual differentiation, to be achieved first through the study a few well-chosen sequences, and then through the formal explicitation of the logical connections between (1), (2), and (3). Consequently, we wanted our post-experiment questionnaire to help us assess to what extent students knew that (1)  (2) is valid, while its converse is not.

Due to the significant level of reinterpretation of the prescribed task in group 1, the data gathered do not easily lend themselves to quantitative comparison. However, the fact that "not bounded above" (2) is a key component of the concept image of lim u n = + (1) is again confirmed beyond doubt. Let us first compare groups 2 and 3. In group 3the medium-achieving computer science majorsall 17 students deemed the converse of #1 to be True. Leaving out 3 students whose answer-sheets show an inadequate command of the logical aspects, it seems that Aline Robert's engineering (which targeted the definition of finite limits) had no impact on the belief that if a sequence (u n ) takes on arbitrarily large values, then lim u n = +. In group 2, that of high-achieving maths majors, the results were not as striking; they were telling just as well.

Among the 50 answer-sheets, let us focus on the subpopulation of those for which all the answers to part I were correct (including relevant counterexamples for #2), and all the converses were stated correctly. Among these 33 students, 17 deemed the converse of #1 to be Falsewhich is the correct answer and all but one provided a relevant counter-example (usually whichas the lecturer confirmedhad been studied in detail). Student 25 even wrote: " A  R  n A  N such that  A means that the sequence is not bounded above, but it doesn't mean it tends to +, it may oscillate. Let's consider (…)". However, the other 16 students ticked "True" for the converse of #1. The resistance of this belief, even among students with a reasonable command of logic, who know the definition of lim u n = +(item #5 and its converse), and who had been exposed to a teaching which explicitly tackled this issue suggest that the conflation of (1) and ( 2) is an epistemological obstacle (Chorlay & de Hosson, 2016). It is probably not independent from the belief that all sequences are monotonous, at least after a certain rank [START_REF] Robert | L'acquisition de la notion de convergence des suites dans l'enseignement supérieur[END_REF]), but our questionnaire offers no new insight as to this. This confirmsin hindsightthat we were justified to take (2) into account when designing a teaching-session on the formal definition of (1). However, it does not tell us whether targeting the formulation of the definition of (1) through a process fostering the conceptual differentiation between (1) and ( 2) was didactically relevantas standard constructivist tenets suggestor just foolhardy.

The results of group 1 allow us to be cautiously optimistic. From a purely quantitative viewpoint, 58% of the students of group 1 deemed the converse of #1 to be "False"which is the correct answerbut no conclusions can be drawn from this fact beyond that this 58% stands in sharp contrast with the 0% of "False" on the subpopulation of OK-answer-sheets of group 3. In group 1, for instance, the third of the students who clearly reinterpreted the task as "compare with the definition" ticked "False", but this does not indicate that they are aware of the connections between properties (1) and ( 2), or that they were able to reformulate " A  R  n A  N such that  A" as "not bounded above". Answer-sheet 30 of group 1 shows, again, that some conceptual understanding can be achieved in a formal context in spite of a poor level of command of symbolic logic. This student systematically stated BA as converse of AB; hence one has to study her assessment of the converse of #4 instead of #1to see if she mistakes (2) for (1); which she does not, actually. Of the 31 students of group 1, only two interpreted the task correctly and provided relevant correct answers for the converse of #1, either with a formulaic counter-example or with a graphical counter-example (of the type). However, about one fourth of the students deemed the converse of #1 to be false, interpreted the task as "assess the implications" and provided arguments which we could be indicative of some conceptual understanding. In these cases, they justified their assessment not by displaying a counter-example, but by explaining why the antecedent was not strong enough to warrant the consequent: under the hypothesis " A  R  n A  N such that  A", the sequence can oscillate; or: the antecedent does not imply that the sequence is increasing. Our empirical data does not enable us to tell which of the following is the case: either, students argue on the basis of the fact that if a sequence is increasing and not-bounded above then it tends toward + (a theorem they are familiar with); or, students conflate (1) and (3).

CONCLUSIONS AND RESEARCH PERSPECTIVES

While the questionnaire studied in this paper was originally designed to compare the effectiveness of three teaching-modules on the definition of limits of sequences, it turned out that they could not serve that purpose due to the decisive impact of another factor, namely the level of familiarity with predicate calculusboth in terms of syntactic command, and in terms of ability to make sense of logical formulae involving nested quantifiers. Nevertheless, we claim that meaningful conclusions or insights can be gained from the analysis of our empirical results.

For students with some command of logica command which cannot be gained through an occasional and in-context use of logical formalismthis questionnaire does provide insight into the connections between concept-image and concept-definition for limits, thus providing a new investigative tool to study this issue; a tool which does not involve conducting interviews or studying students ability to use the definition in proofs. As far as students are concerned, the comparison between group 2 and group 3 suggest that not all teachings on limits are equivalent in this respect; the case of group 2 shows thatunder circumstances which call for further investigationfirst-year university students can display a reasonable command of the concept of limit.

As far as group 1 is concerned, the result show that the prerequisites in logic may not be as high as one might expect, if what is targeted is the ability to memorize the formal definition, and the ability to display understanding of some key features of the concept. As far as our didactic engineering is concerned, these results show that (1) it was not a complete failure, (2) some of its guiding principlessuch as the importance of the conceptual differentiation between infinite-limit and not-bounded-above, or the use of logical formalismseem relevant. However, in this context, this questionnaire is probably not the best tool for a fine-grained assessment of what the actual impact of this engineering is.

1. For introduction to didactic engineering as task-design oriented research method, see [START_REF] Bosch | Didactic Engineering as a Research Methodology: From Fundamental Situations to Study and Research Paths[END_REF].

2. This assessment of the overall level of the groups is that of the team of maths lecturers at Mons University, as communicated to us by Stéphanie Bridoux, who is both a member of that team and a mathematics education researcher (LDAR). Many thanks to her for her collaboration on this project.

3. D. Tall and S. Vinner introduced the distinction between the image and the definition of a concept to stress the difference between mathematics as a mental activity and as a formal system. "We shall use the term concept image to describe the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes. (…) it needs not be coherent (…)." (Quoted in (Tall 1991, 7)).

  3 False: the antecedent being always true while the consequent can be false, the implication is invalid. C of #4 False: being bounded below does not imply lim = +. C of #5 True: definition.

Figure

  Figure 1. Student 29 of group 1


Thérèse Dooley

scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction to the Proceedings of the Tenth Congress of the European Society for Research in Mathematics Education (CERME10)

Thérèse Dooley 1 , Ghislaine Gueudet 2 1 Dublin, Ireland, therese.dooley@dcu.ie 2 Rennes, France, ghislaine.gueudet@espe-bretagne.fr

About CERME10

The

Editorial information

These proceedings are available as a complete volume online on the ERME website and each individual text is also available on the HAL open archive, where it can be found through keywords, title or author name. This has been the practice since CERME9, to increase the visibility of the huge work done in CERME conferences.

This volume begins with texts corresponding to the three plenary activities of CERME10: the presentation by Elena Nardi on University Mathematics Education; that by Lieven Verschaffel on Early Mathematics; and the panel on Solid Findings in Mathematics Education, chaired by Marianna Bosch and involving Tommy Dreyfus, Catarina Primi and Gerry Shiel.

After the plenaries, the reader will find 23 chapters corresponding to the work done in the TWGs of CERME10 (we remind the reader that TWG7 has been cancelled; moreover, TWG14 was split in two for the conference, but all the papers are in the same section in these proceedings).

These chapters follow a similar structure: they start with an introduction; then the long contributions (8-page papers) are presentedin alphabetical order by first author's nameand finally the short contributions (2 pages). However, TWG17 has chosen a different order, corresponding to subthemes in the group.

There are two kinds of introductions to the TWGs, according to the team's choice: short introductions (4 pages) presenting the contributions; or long introductions (8 pages), which propose, in addition, an analysis of the current research on the theme of the TWG, and perspectives for the future. TWGs 6,14,15,16,17,19 and 23 have chosen this form of long introduction.

The publication of these proceedings is the result of a collaborative work, involving CERME10 IPC, the TWG leaders and co-leaders, and the LOC co-chair. We warmly thank all these people for their involvement, and hope that this volume will contribute to the development of mathematics education research in Europe and beyond.

  Tenth Congress of European Research in Mathematics Education (CERME 10) took place in Dublin (Ireland) from 1 st to 5 th February 2017. Ghislaine Gueudet (France) was the chair of the International Programme Committee (IPC) which comprised Thérèse Dooley (Ireland, chair of the

	TWG13: Early years mathematics	Ingvald Erfjord (Norway)
	TWG14a: University mathematics education	Alejandro Gonzalez Martin (Canada)
	TWG 14b: University mathematics education	Irene Biza (United Kingdom)
	TWG15: Teaching mathematics with resources and technology	Alison Clark-Wilson (UK)
	TWG16: Students' learning mathematics with resources and technology	Hans-Georg Weigand (Germany)
	TWG17: Theoretical perspectives and approaches in mathematics education research	Angelika Bikner Ahsbahs (Germany)
	TWG18: Mathematics teacher education and professional development	Stefan Zehetmeier (Austria)
	TWG19: Mathematics teacher and classroom practices	Charalampos Sakonidis (Greece)
	TWG20: Mathematics teacher knowledge, beliefs and identity	Miguel Ribeiro
		and co-chair
	respectively of the Local Organizing Committee (LOC). TWG21: Assessment in mathematics education Paola Iannone (UK)
	CERME10 hosted 23 Thematic Working Groups, listed in the table below. The TWGs 21, 22, 23 and 24 were new groups, created following a call launched just after CERME9, and a selection process TWG22: Curricular resources and task design in mathematics education Birgit Pepin (The Netherlands)
	involving the CERME10 IPC and the ERME board. They all have been very successful, and will be part of CERME11 in February 2019. TWG7 (Mathematical potential, creativity and talent) has TWG23: Implementation of research findings in mathematics education Uffe Thomas Jankvist (Denmark)
	TWG24: Representations in mathematics teaching and learning	Elisabetta Robotti (Italy)
	TWG	Leader
	TWG1: Argumentation and proof	Gabriel Stylianides (UK)
	TWG2: Arithmetic and number systems	Elisabeth Rathgeb-Schnierer (Germany)
	TWG3: Algebraic thinking	Reinhard Oldenburg (Germany)
	TWG4: Geometry	Joris Mithalal (France)
	TWG5: Probability and statistics education	Corinne Hahn (France)
	TWG6: Applications and modelling	Susana Carreira (Portugal)
	TWG8: Affect and mathematical thinking	Pietro Di Martino (Italy)
	TWG9: Mathematics and language	Núria Planas (Spain)
	TWG10: Diversity and mathematics education: Social, cultural and political challenges	Lisa Björklund Boistrup (Sweden)
	TWG11: Comparative studies in mathematics education Paul Andrews (Sweden)
	TWG12: History in mathematics education	Renaud Chorlay (France )

local Programme Committee), Andreas Eichler (Germany, co-chair), Marianna Bosch (Spain), Markku Hannula (Finland), Jeremy Hodgen (UK), Konrad Krainer (Austria), Despina Potari (Greece), Kirsti Rø (Norway), Cristina Sabena (Italy), Michiel Veldhuis (Netherland), Nad'a Vondrová (Czech Republic). Thérèse Dooley and Maurice OReilly were chair unfortunately been cancelled, due to a lack of contributions; while TWG14 has been split in two for the conference, because of the large number of papers received.

Proceedings of CERME10


E Jolimaitre
email: elsa.jolimaitre@ifpen.fr

D Delcroix

N Essayem

C Pinel

M Besson

Dihydroxyacetone conversion into lactic acid in aqueous media in the presence of metal salts: influence of the ionic thermodynamic equilibrium on the reaction performances

The catalytic conversion of dihydroxyacetone (DHA) to lactic acid (LA) via pyruvaldehyde (PA) in aqueous media was studied using different homogeneous metal salts. A kinetic model was developed and the parameters corresponding to each reaction steps were estimated. Agreement between experiments and simulated results was excellent and the performance of the different catalysts was consistent with previous studies described in the literature. Aluminium salts, which show the best performance, were tested in a whole range of concentrations and at different pH, in order to identify the catalytically active ionic species. It was confirmed that the DHA to pyruvaldehyde (PA) dehydration step is catalyzed by both Brønsted and Lewis acids whereas the consecutive reaction of PA to LA is solely catalyzed by Lewis acids. Moreover, comparing thermodynamic analysis of the reaction media and kinetic parameters demonstrated that cationic hydroxylaluminium complexes [Al(OH)h] (3-h)+ formed in situ by the hydrolysis of the aluminium aqua complexes like [Al(OH2)6] 3+ are the most active Lewis acids.

Introduction

With declining petroleum reserves as well as environmental concern for sustainable chemistry, a growing interest in the production of chemicals from renewable biomass has emerged. 1 Among these chemicals, lactic acid (LA) has been identified as a potentially interesting molecule. Not only can it be synthesized from bio-based feedstocks 2,3 but it also has numerous applications in different industries 4 (food, cosmetic, biodegradable polymers). Stereopure L-LA is currently produced by fermentation of carbohydrates. However, this process necessitates complex separation steps 4 and generates large amounts of unvaluable gypsum (CaSO 4 ). Hence, a lot of work has been focused recently on finding a more efficient chemical catalytic synthesis route, often leading to racemic LA, giving priority to water as an inexpensive and environmentally benign solvent.

2

A few studies reported in the literature 5,6 involve the use of homogeneous catalysts in hydrothermal conditions to selectively convert cellulosic feedstocks into LA via hexoses as key intermediates. It is generally reported that fructose, supplied by glucose isomerization, undergoes a retroaldol reaction to trioses dihydroxyacetone (DHA) and glyceraldehyde (GLY), which opens the way to LA as final stable product. DHA and GLY constitute reactive intermediates and their conversion to LA with high selectivities remains a challenge. In this present study, we will therefore focus on the final step of the reaction, i.e. the catalytic conversion of DHA into LA. Several homogeneous catalysts such as metal salts 7,8 or heterogeneous catalysts such as zeolites [9][10][11] and tin phosphates 9,10 were proposed.

It is well admitted that this reaction proceeds in two steps: 5,8,9,[11][12][13][14][15][16][17][18][19][20] (i) DHA is first converted into pyruvaldehyde (PA) by successive keto-enol tautomerization and dehydration; 8,9,16,18 this reaction is described to be catalyzed by both Lewis and Brønsted acids. 8,11,18 (ii) Rehydratation of PA followed by a 1,2-hydride shift yields LA; for this second step, Lewis acids are believed to lead to higher catalytic performances than Brønsted acids. 8,18,21 It has to be noted that in water, PA exists mainly in two forms: hydrated (56%) and dihydrated (44%). Since the different Obviously, the rate-limiting step depends strongly on the nature of the catalyst acid sites. Wang et al. 9,10 showed that the conversion of PA to LA was the limiting step over tin phosphate catalysts, whereas West et al. 20 found the opposite result over H-USY zeolites.

During the course of the reaction, brown soluble and insoluble products are formed, both with homogeneous 8,18 and heterogeneous 10,13,[22][23][24] , the aim of this study is to identify the most catalytically active species of the Al salts solutions -which are among the most efficient homogeneous catalysts for this reaction -for the two reaction steps involved in the DHA to LA conversion. This information is essential in order to -ultimately -optimize the reaction conditions and the catalytic systems. To do so, the following strategy was applied. First, a kinetic model was developed and its reliability was tested by simulating the concentration profiles over time for different metal salts, assumed to provide Brønsted and Lewis acid species in the aqueous phase and leading to very different catalytic performances. Influence of a pure Brønsted acid on the kinetics was also evaluated with H 2 SO 4 . The experiments were conducted at relatively low temperature (≈90°C) so as to slow down the reactions and evaluate the kinetic parameters as accurately as possible. Once the model was validated, the kinetic parameters for aluminium salts were evaluated on a large range of concentrations and with different counter anions. The effect of addition of H 2 SO 4 in the catalytic solution was also tested. To identify the most catalytically active cations for each reaction step, the kinetic parameters were finally compared with the aqueous phase aluminium cations concentrations as simulated by the OLI software. (95-97%) from Merck. All products were used as received.

Experimental

Kinetic data measurements

The reactions were conducted in a 250 mL three-neck roundbottom flask placed in a temperature controlled oil bath. The flask was equipped with a water condenser, a temperature sensor and a rubber septum. Agitation was performed with a magnetic stirrer. All reactions were conducted under ambient atmosphere and pressure. Deionized water (200 mL) and the catalyst were introduced in the flask. When the temperature of the suspension was stabilized, DHA was injected at time t=0. Samples were withdrawn with a 1 mL syringe through the septum. The amount of products were quantified by using high-performance liquid chromatography (HPLC Shimadzu Prominence system equipped with a RI differential refractometer detector and a ICSep COREGEL 107H column). Oligomers were quantified via the carbon molar balance which was compared to the initial DHA molar concentration (i.e. all the unknown species are assumed to be oligomers).

Experiments conducted without any catalyst yielded a 6% and 2% conversion (defined as the molar percentage of DHA converted) after 400 min at 90°C and 80°C, respectively. The thermal conversion of DHA was therefore thereafter neglected. No deactivation of the catalysts (e.g. by precipitation) was observed during the experiments. The pH values of the solutions were measured using a Metrohm 744 pH meter.

Kinetic model

The mechanism proposed for the conversion of DHA into LA is presented on Scheme 1. Even though both the conversion of DHA to PA and the conversion of PA to LA comprise different steps (i.e. PA has to be monohydrated before it can undergo a hydride shift), they have been considered as elementary steps and represented by a single kinetic parameter, as done in other literature kinetic studies 8,18

.

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Please do not adjust margins

For a kinetic modeling of DHA conversion to LA, the impact of DHA isomerization to glyceraldehyde (GLY) has to be taken into account. Different studies in the literature [8][9][10]20 compare batch concentration profiles when DHA or GLY are used as the starting material and all lead to the same conclusions: GLY isomerizes to DHA very quickly, the direct conversion of GLY to PA being negligible. Moreover, Liang et al. 26 reported experimental and calculated free enthalpies of reaction in the range of 2.3 to 10.9 kJ/mol for isomerization of DHA to GLY, confirming that the conversion of DHA to GLY is thermodynamically unfavourable. In the present study, nonnegligible GLY concentrations were measured in some experimental conditions probably thanks to the mild reaction conditions. The thermodynamic equilibrium between DHA and GLY was therefore accounted for in addition to the main consecutive reactions DHA→PA→LA. Furthermore, the model included the involvement of GLY in the formation of the brown insoluble oligomers. A more complex model, that included DHA and PA oligomerization was also implemented (see S.I.), but the best fit of the experimental points was obtained with the scheme proposed in Scheme 1 with only the oligomerization reaction from GLY. This scheme was therefore used for the subsequent stages of this study. Supposing that all the reaction steps are elementary steps, the following set of differential reactions is obtained for the concentrations of the species:

ௗ ௗ௧ = -݇ ଵ • ܣ -݇ ଶ • ܣ + ಲమ ಲವ • ܦ -݇ • ܦܣ (1)
ௗ ௗ௧ = ݇ ଵ • ܣ -݇ • ܤ (2)
ௗ ௗ௧ = ݇ • ܤ (3)
ௗ ௗ௧ = ݇ ଶ • ܣ - ಲమ ಲವ • ܦ -݇ • ܦܣ (4) ௗா ௗ௧ = ݇ • ܦܣ ( 5
)
where ܭ is the thermodynamic equilibrium constant :

ܭ = ಲమ ᇱ ಲమ (6)
Equations ( 1) to ( 5) were numerically solved with the SCILAB software using the ode function. Function optim was used for the parameters estimation, the objective function being the least square difference between the experimental and simulated concentrations.

Results and discussion

Comparison of different metal salts and sulphuric acid

DHA and products concentrations profiles as a function of reaction time for the different salts were measured under the same experimental conditions at 90°C (see Table 1). The concentration profiles are displayed on Fig. The proton contribution to the Al 2 (SO 4 ) 3 kinetic parameter was finally estimated from these data: k A1 = 1.18 10 -6 s -1

, i.e. approx. 1% of the overall value. The high value of k A1 obtained with aluminium sulphate is therefore solely induced by its Lewis acidity or might result from a synergy effect between Lewis and Brønsted sites. The kinetic parameters for aluminium sulphate (Table 1) are nearly an order of magnitude higher than those determined for the other Lewis acids. Only the parameter related to DHA isomerization to GLY (k A2 ) is close to that for other salts. Hence, Al 2 (SO 4 ) 3 strongly promotes the DHA conversion to PA (and then to LA), but it does not favour the parallel path leading to oligomers via GLY. As expected, the thermodynamic parameter K AD is nearly constant (given the experimental uncertainties) for all salts and gives a free enthalpy of reaction around 6 kJ/mol, in agreement with results reported in the literature for keto-enol tautomerization between DHA and GLY (2.3 to 10.9 kJ/mol). 26 The activation energies were also evaluated for aluminium sulphate and sulfuric acid (see S.I.). Comparable values (respectively 98 kJ/mol and 92 kJ/mol) were found for reaction A1 (DHA to PA conversion). The aluminium sulphate activation energy for reaction B (PA to LA conversion) is 82 kJ/mol, i.e. nearly 16 kJ/mol smaller than that of reaction A1, highlighting the ability of aqueous phase aluminium species to specifically activate the pyruvaldehyde molecule.

Aluminium salts

In order to identify the catalytically active species, it is necessary to consider which aqueous species of aluminium is most likely to be present in the studied solutions. Over the years, numerous studies have been dedicated to the complex behaviour of aluminium ions in aqueous solution, using either Please do not adjust margins Please do not adjust margins analytical methods (X-Ray, 27 Raman, 28,29 NMR 30 spectroscopy, mass spectrometry 31,32 ), macroscopic thermodynamic properties 30,33 and DFT simulations 27,29,32,34,35 . Although a great variety of chemical species have been identified, including polymeric Al complexes, 31,32 it is generally admitted that under acidic conditions, the predominant aluminium species are the aqua complex formed by interaction of the Al 3+ cation with water molecules (Eq 7), the hydroxyl complexes resulting from water hydrolysis (Eq 8), and other complexes formed with anionic species potentially present in the aqueous solution (Eq 9).

It is well known that dissolved aluminium is essentially in the form of a cationic aqua complex with 6 water molecules, when pH is below neutrality 27,29,34 : ݈ܣ ଷା + ܪ6 ଶ ܱ → ሾ݈ܣሺܱܪ ଶ ሻ ሿ ଷା (7) Hydroxides or other anions (such as sulphate) can displace water molecules to form additional complexes, following different equilibrium reactions (for clarity, water molecules are not included in the formula of the complexes thereafter). Hydrolysis of aqua complexes takes place according to the following reaction, with a loss of proton 33,34,36,37 :

ሾ݈ܣ ଷା ሿ + 2݄ • ܪ ଶ ܱ ⇔ ሾ݈ܣሺܱܪሻ ሿ ሺଷିሻା + ݄ • ܪ ଷ ܱ ା (8)
Under acidic conditions (pH < 5), only ሾ݈ܣሺܱܪሻሿ ଶା and ሾ݈ܣሺܱܪሻ ଶ ሿ ା are present in significant concentrations in solution. 34,36,37 In the presence of the sulphate anion, the formation of sulphato complexes has been demonstrated 28,30 :

ሾ݈ܣ ଷା ሿ +  • ܱܵ ସ ଶି ⇔ ൣ݈ܣሺܱܵ ସ ሻ ൧ ሺଷିଶሻା (9)
On the contrary, the nitrate and chloride anions are much less inclined to form thermodynamically stable complexes with aluminium in aqueous solutions.

29,31

Also, in acidic conditions, the equilibrium between the sulphate and hydrogen sulphate anions has to be taken into account:

ܪ ଷ ܱ ା + ܱܵ ସ ଶି ⇔ ܱܵܪ ସ ି + ܪ ଶ ܱ (10)
Equations ( 8) to ( 10) are governed by their equilibrium constant, i.e. depend on the activity of each ionic species.

From equation ( 8), it is clear that when aluminium salts are dissolved into water (with no addition of Brønsted acids), the concentration of ሾ݈ܣሺܱܪሻ ሿ ሺଷିሻା cations will be inversely proportional to the pH of the aqueous solution. Reversely, addition of protons to the solution will reduce the concentration of the hydroxyl complexes. Kinetic experiments were performed with solutions containing 0-0.1 M aluminium salt. The evolution of the kinetic parameters as a function of Al 2 (SO 4 ) 3 concentration are displayed on Fig. 3a.

In theory, the kinetic parameters should vary linearly with the catalyst concentration. Two different trends could be observed: the evolution of k A1 is linear, whereas k B and k A2 vary non-linearly as a function of Al 2 (SO 4 ) 3 concentration. The confidence in these results is supported by the excellent agreement with the values obtained by Lux et al. 18 at 99°C (Fig. 3b). Presumably, the different reactions are not catalyzed by the same ionic species: the DHA to PA dehydration reaction is catalyzed by a cation whose concentration is proportional to the amount of Al 2 (SO 4 ) 3 introduced in solution whereas both the conversion of PA to LA and the DHA isomerization to GLY are catalyzed by another complex cation, whose relative concentration drops as Al 2 (SO 4 ) 3 concentration increases.

In order to identify the nature of these species, the influence of pH was studied. Experiments were carried out with a constant concentration of Al 2 (SO 4 ) 3 (0.09 +/-0.005 M) and addition of various concentrations of H 2 SO 4 . The effect of H 2 SO 4 addition on the kinetic rate constants is shown in Fig. 4. The addition of the inorganic acid into the reaction medium gives a rapid decrease of k B , k A2 and k AD i.e. the reactions that are presumably catalyzed by Lewis acidity. The formation of the ionic species active for these two reactions is therefore hindered at higher pH.

As shown by eq. ( 8), this should be the case when aqua complexes hydrolyze into hydroxyl complexes, which seems to indicate that the ሾ݈ܣሺܱܪሻ ሿ ሺଷିሻା cations are the most active Lewis catalytic species in solution.

The effect of sulphuric acid on kinetic constant k A1 is less clear: there is a small decrease of k A1 upon addition of 0.02 M H 2 SO 4 , but an inverse tendency is observed for higher concentrations. Since this reaction is catalyzed both by Lewis and Brønsted acidity, precise knowledge of the nature and repartition of each Al cation species present in the reaction medium is needed to interpret these results.

To do so, the concentrations of the different Al cations were calculated using the OLI software. The mechanism proposed by the authors -complex formation between two adjacent oxygen groups and the metallic center, followed by hydride transfer from the C2 to C1 carbon and back proton transfer to O1can be applied to the (hydrated) PA to LA and DHA to GLY isomerization steps. The fact that the kinetic parameters corresponding to these two reactions -k B and k A2 -follow the same evolutions as a function of Al 2 (SO 4 ) 3 concentration (Fig. 3) and when H 2 SO 4 is added to the aqueous solution (Fig. 4) confirm this hypothesis. Please do not adjust margins Please do not adjust margins

From this analysis, it is also possible to conclude that the Lewis acid ionic species that catalyze reaction A1 in the absence of strong acid (cf. Fig. 3) are very probably the aqua and hydroxyl complexes of Al 3+ .

Since the conversion of DHA to LA is the result of two successive reactions that are catalyzed by different cationic species, the rate limiting step is strongly dependent on the nature and concentration of the catalyst. The effect of the counter anion on the rate limiting step is illustrated on Fig. 7. The global tendencies are not affected by the nature of the counter anion: both the addition of a strong acid and increasing concentration of aluminium salts result in a decrease of the k B /k A1 ratio. However, the introduction in the reaction medium of different anions changes the absolute values of this ratio. For less complexing anions 29,31,32 such as Cl and NO 3 -, the kinetic of the second reaction step step k B is promoted. This may be ascribed to the less favorable formation of aluminium nitrate and aluminium chloride complexes, resulting in higher concentrations in hydroxylaluminium complexes.

Conclusions

Different homogeneous catalysts have been tested for the conversion of DHA to LA and a kinetic model has been developed, which enables a very satisfactory representation of the experimental concentration profiles for all catalysts.

In agreement with previous studies, the aluminium salts are identified as very efficient catalysts for this reaction, with a selectivity to LA of over 80% at 90% conversion of DHA. An attempt was made to simulate the ionic equilibrium concentrations in the reaction medium using the commercial software OLI. Discrepancies between calculated and measured pH values of different Al 2 (SO 4 ) 3 solutions lead us to the conclusion that the model, although useful to predict general tendencies, is not sufficiently accurate to quantitatively interpret the present results. In order to optimize the formulation of the catalytic system, disposing of reliable thermodynamic models would be needed. Moreover, given the concentrations needed in industrial processes, the models have to be valid for very high ionic strengths, i.e. in strongly non ideal thermodynamic conditions. Nonetheless, a qualitative analysis of the evolution of the different kinetic parameters at different Al 2 (SO 4 ) 3 concentrations and for 2 (SO 4 ) 3 /H 2 SO 4 mixtures, showed that the two successive reactions (DHA to PA and PA to LA) are catalyzed by different ionic species:

1. the dehydration of DHA to LA is catalyzed by Brønsted acidity, and also very probably by ሾ݈ܣሺܱܪ ଶ ሻ ሿ ଷା , 2. the further conversion of LA to PA is catalyzed by the hydroxyl-aluminium ሾ݈ܣሺܱܪሻ ሿ ሺଷିሻା complexes formed by hydrolysis of aqua complexes. These results were confirmed by complementary experiments with different counter-anions. For less complexing anions, which generate more hydroxyl-aluminium complexes in solution, the first reaction step is even more limiting.

25

 25

Figure 2 :

 2 Figure 2: : Evolution of kA1 kinetic parameter as a function of proton concentration (90°C, addition of H2SO4).

Figure 3 :

 3 Figure 3: Evolution of the kinetic parameters with Al2(SO4)3 concentration (a: this work at 90°C, lines are guide for the eyes ; b: comparison with Lux et al. (T=99°C)

Figure 6 :

 6 Figure 6:

  Regarding their capacity to produce LA with a very high yield in water, previous studies have shown that aluminium and chromium salts were very promising homogeneous Lewis acid catalysts. Using different Cr and Al salts at 140°C, Rasrendra et al.8 obtained selectivity to LA in the range 80-90% at total conversion of DHA. A small effect of the nature of the anion was observed, with LA yields varying in the order Cl

			-≈NO 3	->
	SO 4	2-. Lux et al. 18 obtained a 77% yield of LA at 97% DHA
	conversion using aluminium sulphate at 99°C. The unique
	performance of Al	3+ and Cr 3+ (compared to all the other metal
	tested) are not explained at the moment. Interestingly, Lux et
	al. 18 also observed that the rate constant of the second step of
	the reaction (conversion of PA to LA) varied non-linearly with
	the catalyst concentration, without providing any explanation.
	In that respect, the study by Choudhary et al. 25 on the CrCl 3 -
	catalyzed transformation of glucose to fructose to yield 5-HMF
	in aqueous media might provide useful indications. Indeed,
	some parallels can be drawn between glucose-fructose and
			2+
	cation, although less concentrated than [Cr]	3+ in the aqueous
	phase, was the most active Cr species for glucose
	isomerization, thanks to strong interaction of glucose in the Cr
	first coordination sphere.
	Following the methodology proposed by Choudhary et al.

catalysts. These unwanted products that are attributed to either DHA or PA oligomers can lead to a drastic reduction of the LA yield and deactivation of catalysts. This journal is © The Royal Society of Chemistry 20xx

Please do not adjust margins Please do not adjust margins GLY-DHA / PA-LA isomerization as they may involve similar active species for the aldose-ketose isomerization. By using the commercial OLI Systems software (OLI Systems Stream Analyzer Software, OLI Systems, 2012), the concentrations of the different cations generated from the dissolution of CrCl 3 in water were evaluated by the authors. Comparison of the glucose initial conversion rates with the concentrations of the different cations indicated that the pentahydrated

[CrOH]

Table 1 .

 1 1. It is noteworthy that due to oligomerization reactions, the total concentration of products in solution (expressed in mol/L) drops during the course of the reaction. The simulated and experimental concentrations are in very good agreement, which means that the reaction mechanism proposed in is valid for all the tested salts. Moreover, the concentration profiles are consistent with literature results involving the same catalysts for DHA conversion to LA. For Al 2 (SO 4 ) 3 and H 2 SO 4 , the k A1 parameter is in the same order of magnitude. Aluminium salts can generate both Lewis acidity from cationic aqua complexes like [Al(OH 2 ) 6 ]

	3+ and Brønsted

8,15,17,18,20,22

Indeed, H 2 SO 4 is very selective towards the formation of the intermediate PA, meaning that Brønsted acids are active neither for the conversion of PA into LA nor for the DHA to GLY isomerization. The absence of oligomers is consistent with the reaction scheme proposed on Scheme 1: the condensation with DHA cannot take place in the absence of GLY. Furthermore, aluminium sulphate is by far the most selective catalyst to LA. In that case, PA and GLY appear as intermediates and the amount of oligomers formed is very low. By contrast, ErCl 3 and ZnSO 4 display similar poor performances: at the end of the experiment, PA and oligomers are the main products. The outstanding performances of ErCl 3 for the hydrothermal conversion of cellulose to lactic acid described by Lei et al.

6

are not found here. This may be explained by the huge gap of temperature (240°C for Lei et al. 6 vs. 90°C in this work). The resulting kinetic parameters are gathered in Scheme 1: Proposed reaction mechanism for the conversion of DHA into LA via PA This journal is © The Royal Society of Chemistry 20xx Please do not adjust margins Please do not adjust margins

Table 1 :

 1 experimental conditions and corresponding parameters for different salts

J. Name., 2013, 00, 1-3 | 3

This journal is © The Royal Society of Chemistry 20xx J. Name., 2013, 00, 1-3 | 5

J. Name., 2013, 00, 1-3 | 7

Acknowledgements

The authors would like to thank deeply Dr. Hélène Olivier-Bourbigou for her careful reading of this work and for her constructive suggestions.