Matteo Frigo

Isa Costantini

Rachid Deriche

Samuel Deslauriers-Gauthier

Resolving the crossing/kissing fiber ambiguity using Functionally Informed COMMIT

Keywords: Tractography, False positives, Diffusion MRI, Resting state functional MRI

The architecture of the white matter is endowed with kissing and crossing bundles configurations. When these white matter tracts are reconstructed using diffusion MRI tractography, this systematically induces the reconstruction of many fiber tracts that are not coherent with the structure of the brain. The question on how to discriminate between true positive connections and false positive connections is the one addressed in this work. State-of-the-art techniques provide a partial solution to this problem by considering anatomical priors in the false positives detection process. We propose a novel model that tackles the same issue but takes into account both structural and functional information by combining them in a convex optimization problem. We validate it on two toy phantoms that reproduce the kissing and the crossing bundles configurations, showing that through this approach we are able to correctly distinguish true positives and false positives.

Introduction

The increasing interest in connectomics shed light on the intrinsic limitations of tractography based connectomes [START_REF] Zalesky | Connectome sensitivity or specificity: which is more important?[END_REF][START_REF] Maier-Hein | The challenge of mapping the human connectome based on diffusion tractography[END_REF][START_REF] Daducci | Microstructure informed tractography: pitfalls and open challenges[END_REF]. Tractography itself is an ill-posed problem that aims at reconstructing the white matter fiber tracts from diffusion MRI (dMRI) data and is shown to generate a significant number of false positive connections between brain regions [START_REF] Maier-Hein | The challenge of mapping the human connectome based on diffusion tractography[END_REF]. This is due to the fact that whenever two bundles of axons are organized in a crossing or kissing configuration, in-vivo tractography is not able to distinguish their actual trajectories from diffusion data. This implies the presence of many spurious entries in the structural connectome obtained from a tractogram. Moreover, specificity of brain networks has been demonstrated to be twice as important as sensitivity when estimating properties of the brain networks [START_REF] Zalesky | Connectome sensitivity or specificity: which is more important?[END_REF]. The re-establishment of an equilibrium between highly sensitive and highly specific networks is a challenge in brain connectomics and is the goal of this work. The task of extracting only true positive connections from a tractogram is also called tractography optimization or tractogram filtering [START_REF] Frigo | Hierarchical tractography optimisation[END_REF][START_REF] Frigo | Reducing false positive connection in tractograms using joint structure-function filtering[END_REF]. Several tools to perform this task have been proposed in the last years, including SIFT2 [START_REF] Smith | Sift2: Enabling dense quantitative assessment of brain white matter connectivity using streamlines tractography[END_REF], LiFE [START_REF] Pestilli | Evaluation and statistical inference for human connectomes[END_REF] and COMMIT [START_REF] Daducci | Commit: convex optimization modeling for microstructure informed tractography[END_REF]. While the first has the goal of matching a tractogram with the fiber densities estimated using the spherical deconvolution diffusion model, the two last are designed in such a way that they identify a subset of streamlines that explains the acquired diffusion MRI through a linear forward model. This model takes into account only the intra-cellular compartment in the case of LiFE and more complex tissue microstructure models in the case of COMMIT. Recent studies showed that anatomical priors can be taken into account while detecting false positives [START_REF] Frigo | Hierarchical tractography optimisation[END_REF]. Daducci et al. [START_REF] Daducci | Reducing false positives in tractography with microstructural and anatomical priors[END_REF] analyzed how the intrinsic bundle structure of both the brain and tractograms can be used to tackle the false positives issue, while Girard et al. [START_REF] Girard | Axtract: microstructure-driven tractography based on the ensemble average propagator[END_REF] exploited microstructural information to solve the kissing bundles problem. Still, both these approaches showed some limitations. What we propose in this work is to overtake the purely structural-based approach and consider both structural and functional information in the formulation of the tractography optimization problem. We do this by showing that the injection of functional priors coming from resting-state functional MRI (rs-fMRI) gives an effective answer to the false positives issue. The functional information is exploited in the form of static functional connectivity and it plays the role of promoting the involvement of bundles that connect highly correlated cortical regions in the fitting of the dMRI signal. We encapsulate all of this within the COMMIT framework, which provides a flexible tool for defining a convex optimization problem that, for a given tractogram, simultaneously promotes sparsity among the bundles, takes into account the considered functional information and selects the streamlines that are sufficient to explain the diffusion MRI signal. We call this novel framework Functionally Informed COMMIT (FIC). This formulation is tested on TVB [START_REF] Sanz Leon | The virtual brain: a simulator of primate brain network dynamics[END_REF] and Phantomas [START_REF] Caruyer | Phantomas: a flexible software library to simulate diffusion mr phantoms[END_REF] based synthetic phantom datasets that reproduce the two problematic configurations of crossing and kissing bundles, which are the source of most of the biases in connectivity estimation.

Theory

As the name suggests, tractography optimization implies the solution of a minimization problem. The functional to be minimized takes into account both the structural information coming from dMRI and the functional information given by rs-fMRI. The space in which we want to optimize it is the space of streamlines, hence we want to assign to each of them a coefficient x i ≥ 0. This coefficient quantifies the contribution of streamline i to the definition of the dMRI signal, which means that if x i = 0 then the i-th streamline does not explain the diffusion signal and will be marked as a false positive. The fitting of the dMRI signal is tackled by means of the COMMIT framework. In particular, we use the provided linear forward model for diffusion MRI that predicts the signal generated by a tractogram through a given microstructure model. The one we are going to consider during this work is the Stick-Zeppelin-Ball model [START_REF] Panagiotaki | Compartment models of the diffusion mr signal in brain white matter: a taxonomy and comparison[END_REF]. This leads to the definition of the optimization problem as

x * = argmin x∈R n + 1 2 Ax -y 2 2 + Ω(x) (1)
where n is the number of streamlines, y ∈ R m is the vectorized dMRI signal, A ∈ R m×n is the linear operator that projects streamline weights on the diffusion data according to the microstructure model and Ω : R n → R is a regularization term that for Ω(x) = 0 returns the classical formulation of COMMIT, which will be used as a first reference for our results.

Regularization term. Our objective is to promote sparsity in a structured way, namely we want to promote the preservation of the streamline bundles, instead of individual streamlines, that are sufficiently significant to explain the diffusion MRI signal. This leads to the definition of the following regularization term

Ω(x) = λ g∈G w g x g 2 (2)
where G is a collection of groups of indexes (i.e. streamline bundles) forming a partition of [1, n], w g is a positive weight associated to each group g, x g is the restriction of x ∈ R n to the entries indexed by g and λ ≥ 0 is the regularization parameter. This Ω is a classical choice [START_REF] Bach | Structured sparsity through convex optimization[END_REF] in structured sparsity regularization and is also known as 1,2 regularization. The groups can be both decoupled and overlapping, leading to different strategies in the solution of the optimization problem. See [START_REF] Jenatton | Proximal methods for sparse hierarchical dictionary learning[END_REF] for an exhaustive discussion on the mathematical details of this formulation and [START_REF] Frigo | Hierarchical tractography optimisation[END_REF] for an application of this concept in tractography optimization. A common choice for the specific weight associated to each group is

N -1/2 g
, where N g is the cardinality of group g. This normalization avoids preserving a group just because it involves a smaller number of variables. In the context of our problem this means that we try to avoid penalizing bigger bundles more than smaller bundles. This choice of w g is called group sparsity and will be used as a second reference to compare our results. Notice that group sparsity is purely based on a structural prior, hence it does not take into account the fact that a bundle could connect two regions that are more or less functionally correlated. We can quantify the functional correlation between cortical regions i and j by means of the functional correlation coefficient c i,j . If bundle g connects regions i and j, we associate to it the coefficient c g = c i,j . How to compute this functional correlation coefficient is the question addressed in Section 3.1. Our objective is to favor more bundles that are associated to high functional correlation than the ones associated to low functional correlation. We designed a strategy for determining the weight w g that reads as follows:

w g = 1 N g 1 c 2 g exp 1 -c 2 g 2s 2 -1 . (3)
The function of c g defined in equation ( 3) enjoys the property of being equal to zero when c g = 1 and going asymptotically to infinity when c g → 0 + . The undefined parameter s controls the speed with which w g goes to infinity. In particular it affects more the behavior in the case of low functional correlation than in the case of high functional correlation by acting as a smooth threshold parameter, as shown in Figure 1.

Combining the COMMIT optimization problem defined in equation ( 1) together with the designed joint structural-functional prior offered by equations ( 3) and (2), we are now able to give the mathematical formulation of FIC.

Definition 1 (Functionally Informed COMMIT). Let S be a set of n fiber tracts and let G be the bundle structure over these fiber tracts. Moreover, let A and y be as in equation 1, λ > 0, N g equal to the cardinality of bundle g, s > 0 and let c g > 0 be the functional correlation coefficient between the regions connected by bundle g. The solution of the following minimization problem is called FIC filtering of S.

x * = argmin

x∈R n + 1 2 Ax -y 2 2 + λ g∈G 1 N g 1 c 2 g exp 1 -c 2 g 2s 2 -1 x g 2 . ( 4
)
Theorem 1. The FIC filtering of a tractogram exists, is unique and can be obtained in polynomial time.

Proof. The target functional of equation ( 4) is strictly convex and lower semicontinuous, hence its solution exists and is unique. Moreover it can be split in a smooth part with L-Lipschitz continuous gradient and a non-smooth part for which we are able to compute the proximal operator [START_REF] Jenatton | Proximal methods for sparse hierarchical dictionary learning[END_REF], hence the solution can be approximated in polynomial time [START_REF] Beck | A fast iterative shrinkage-thresholding algorithm for linear inverse problems[END_REF].

Methods

The two studied datasets were both generated with Phantomas [START_REF] Caruyer | Phantomas: a flexible software library to simulate diffusion mr phantoms[END_REF] for the structural part (dMRI) and The Virtual Brain (TVB) [START_REF] Sanz Leon | The virtual brain: a simulator of primate brain network dynamics[END_REF] for the functional part (rs-fMRI). The diffusion MRI signal was simulated for a geometry of 94x94x94 voxels of (2.0mm) 3 size in 64 sampling points uniformly distributed [START_REF] Caruyer | Design of multishell sampling schemes with uniform coverage in diffusion mri[END_REF] on a single shell at b = 1500 [s/mm 2 ]. The phantom consists of two streamline bundles either crossing or kissing in the center of the image. This is an extremely challenging configuration for tractography because voxels in the crossing/kissing region are identical for both configuration. See Figures 2 and3 for a visual description of the simulated phantoms. On each of these datasets we ran a probabilistic anatomically constrained tractography algorithm seeding from the white matter.

For each of the cortical regions we used TVB [START_REF] Sanz Leon | The virtual brain: a simulator of primate brain network dynamics[END_REF] to simulate the BOLD time series with repetition time (TR) of 1 second, obtaining a 15 minute time series for each region. The chosen noise model is Gaussian and additive with standard deviation equal to 5 • 10 -4 , which corresponds to the one suggested in [START_REF]The Virtual Brain: The resting state network scripting tutorial[END_REF].

Dynamic Functional Connectivity

The wanted functional correlation coefficient corresponds to the dynamic functional connectivity (dFC) between two regions and is estimated using slidingwindow correlation. The window is defined by its length, its shape and the offset time. As far as the window length is concerned, we set it to 100 seconds following the strategy described in [START_REF] Leonardi | On spurious and real fluctuations of dynamic functional connectivity during rest[END_REF]. For the window shape we consider a rectangular box which gives unitary weight to each involved time point. The window offset is set to 1 TR and the functional correlation between each pair of windows is estimated using the Pearson correlation coefficient. Finally, the maximal correlation among all the windows is elected to represent the wanted dFC coefficient.

Parameters choice

The experiments were run for the following values of parameters λ and s:

λ = [1.0, 1.5, 2.0, 2.5] , (5)
s = [0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 1.0, 1.

1, 1.2, 1.3, 1.4, 1.5] . (6)

Results

For each phantom we show the ground truth geometry and the solutions obtained with classical COMMIT, group sparsity (GS) and FIC in Figures 2 and3. The ideal result preserves only the bundles that belong to the ground truth. It can be noticed how COMMIT and GS are not able to distinguish between true positive (TP) and false positive (FP) bundles, while FIC correctly detects the FPs and returns a properly filtered tractogram. This holds true for both the kissing bundles configuration and the crossing bundles configurations. For the result given by each combination λ and s we computed two types of connectivity matrix. The first one is a classical streamline count and is indicated as T . The second one is what we call the FIC connectome, which is defined as H i,j = x * g 2 , where bundle g connects regions i and j and x * is the solution obtained with FIC. Matrix T is normalized by the total number of streamlines while H is normalized by its maximal entry. Both of them share the same sparsity pattern, which reads as follows:

G c =     0 0 * 0 0 0 0 * * 0 0 0 0 * 0 0     G k =     0 0 0 * 0 0 * 0 0 * 0 0 * 0 0 0     (7)
where G c is the sparsity pattern of the crossing configuration, while G k is the sparsity pattern of the kissing configuration. Due to different kind of normaliza-

Conclusions

We proposed an evolution of the the recently proposed [START_REF] Daducci | Reducing false positives in tractography with microstructural and anatomical priors[END_REF] bundle sparsity formulation of COMMIT that takes into account the functional information coming from rs-fMRI. The results obtained on two simple but very problematic phantoms prove that the FIC approach is effective for the detection of false positive connections in tractograms. Moreover these results are stable with respect to the two parameters to be tuned. We expect that the more complex configurations given by real data will need a deeper study of these parameters. Finally, we introduced a new concept of connectome that can't be considered neither purely structural nor purely functional but inherits features both from structural and functional information, hence it represents a first step towards the construction of a joint structural and functional connectome.

5 Fig. 1 .

 51 Fig.1. Each line represents the weight wg as a function of cg for Ng = 1 and four different values of the shape parameter s. This plot suggests that our weighting strategy gives high control over the low functional correlation case.

Fig. 2 .

 2 Fig. 2. Kissing. The shown results were computed for λ = 2.5 and s = 1. Subplot a shows the ground truth geometry of the phantom, where each bundle has a distinctive color. Subplots b, c, and d show every bundle belonging to the filtered tractogram.Those marked as FP should not be present after the tractography optimization while those marked as TP should be. Bundles connecting regions 0 ↔ 1 and 2 ↔ 3 were not generated by the tractography algorithm.

Fig. 3 .

 3 Fig. 3. Crossing. The shown results were computed for λ = 2.5 and s = 1. Subplot a shows the ground truth geometry of the phantom, where each bundle has a distinctive color. Subplots b, c, and d show every bundle belonging to the filtered tractogram.Those marked as FP should not be present after the tractography optimization while those marked as TP should be. Bundles connecting regions 0 ↔ 1 and 2 ↔ 3 were not generated by the tractography algorithm.

Acknowledgement

The authors would like to thank Rebecca Bonham-Carter for the help in simulating resting state networks. This work received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (ERC Advanced Grant agreement No 694665 : CoBCoM -Computational Brain Connectivity Mapping).


Nadia Azrou
email: nadiazrou@gmail.com

Meta-mathematical knowledge about proof

Keywords: Proof, meta-knowledge, theorem, undergraduate students

We deal in this paper with a particular difficulty with proof and proving at the undergraduate level, which concerns knowledge about proof at a meta-level. Some undergraduate students' difficulties or mistakes observed in their proof texts have been related to lack of

Introduction

Most mathematics teaching at all school levels is concentrated on teaching content; at the university level, students learn about functions, differential equations, matrices and integrals, by manipulating definitions and theorems. In order to assimilate the content, students are asked to solve problems and prove statements. The difficulties of students with proof have been largely investigated in research [START_REF] Moore | Making the transition to formal proof[END_REF][START_REF] Epp | The role of logic in teaching proof[END_REF][START_REF] Selden | Overcoming students' difficulties in learning to understand and construct proofs[END_REF][START_REF] Harel | Students' proof schemes: Results from exploratory studies[END_REF]; some of these difficulties are related with the fact that students do not know mathematics at the meta-level, particularly as it concerns proof [START_REF] Morselli | Sui fattori culturali nei processi di congettura e dimostrazione[END_REF][START_REF] Hemmi | Students' encounter with proof: the condition of transparency[END_REF]. Several students do not see clearly the difference between a definition and a theorem, the difference between an example and a counter example. Knowledge about proof at the meta-level is neither presented in textbooks nor in courses of specific mathematical disciplines, but it makes one of the most important differences between mathematicians and students. In this paper, we will consider in particular the meta-knowledge about proof (MKP), such as the knowledge of the notion of proof and the rules related to how a proof must be organized. Many researchers acknowledge the fact that high school and university students do not understand what is meant by "proof" and "proving" (e.g. [START_REF] Schoenfeld | Exploration of students' mathematical beliefs and behaviors[END_REF][START_REF] Harel | Students' proof schemes: Results from exploratory studies[END_REF]. "To most undergraduates, convincing their teacher (and thereby earning satisfactory grades) is typically the most important reason for constructing a proof" (Weber, 2004, p. 429) and "unfortunately, many students believe that they either know how to solve a problem (prove a theorem) or they don't, and thus, if they don't make progress within a few minutes, they give up" (Selden & Selden 2007, p. 96); students often believe that non-deductive arguments constitute a proof, or "an argument is a proof if it is presented by or approved by an established authority, such as a teacher or a famous mathematician" (Weber, 2003, p. 3); other different interpretations and conceptions of students regarding proof are described in [START_REF] Harel | Students' proof schemes: Results from exploratory studies[END_REF] and in [START_REF] Recio | Institutional and personal meanings of proof[END_REF]. Meta-knowledge about proof is used implicitly by mathematicians when they construct proofs, "what may be assumed contextually and what needs to be explicitly proved, using logical deduction and previously established results, is highly non-trivial and, I would suggest, is implicit rather than explicit in the minds of most mathematicians" (Tall, 2002, p. 3). Our focus in this paper will be on the lack of MKP and what it might cause as difficulties to students when constructing proofs. The present study is developed using a past empirical study with undergraduates that consisted of investigating students' difficulties by analyzing their proof texts responding to different tests [START_REF] Azrou | Proof writing at the undergraduate level[END_REF]. We would like to examine if the following hypothesis is supported by an interview analysis: Is lack of MKP one of the reasons behind the messy proof texts? Moreover, we would like to answer the following question: Why students do not develop MKP?

Theoretical framework

We choose the definition of proof stated by [START_REF] Durand-Guerrier | Examining the role of logic in teaching proof[END_REF], inspired from the Vergnaud's conceptual fields [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF]. According to Vergnaud, a concept (in our case proof) is learnt by acquiring three components: the set of different representations (oral, written, formal, etc), the situations of reference (proof in geometry, in algebra, in calculus, etc.) and the operatory invariants (related to the logical structure of proof: legitimated inference rules, status of hypotheses, thesis, axioms, etc). Mastering the MKP is mastering the concept of proof according to Vergnaud, as it was stated by [START_REF] Durand-Guerrier | Examining the role of logic in teaching proof[END_REF].

We are interested in comparing MKP that students acquire with how they have been presented proofs and how they have been taught MKP by their teachers. As we consider the relationship with the teaching regarding proof and proving, we will be referring to the didactic contract [START_REF] Brousseau | Le contrat didactique: Le milieu[END_REF] that is defined as a set of rules framing the mathematical practices of teachers and students under the constraints of the teacher-students institutional relationships. Most of these rules regarding how, why and what teachers do mathematically (and students should learn to do) are implicit and thereby not declared by teachers, who often suppose that students would assimilate them over time and practice. Often, teachers use some particular intentions and rules with the proofs exposed to students, without being aware of and without feeling the importance of explaining them; consequently, sometimes students are misguided to make correct proofs. Let us take the example of proof writing: 'the processes used by mathematicians are often rough and informal, but students typically see proofs in their final forms, and rarely witness the process of creating a "rough draft", as a result, students often do not know where to begin when writing their own proofs' [START_REF] Moore | Making the transition to formal proof[END_REF]. We will examine what kind of MKP students learn from their teachers and how they manifest it.

Methods

We are more interested in examining in students' proof texts their MKP considering the three components of the definition of [START_REF] Durand-Guerrier | Examining the role of logic in teaching proof[END_REF]; students' behaviors will be checked regarding definitions and other used mathematical statements (mathematical argument), regarding their modes of reasoning and argumentation and how they expressed them and presented them. Especially, how lack of MKP is manifested through students' proof texts and their interviews. A test composed of open questions (to which the proof cannot be procedural but rather syntactic or semantic (Weber, 2004)) have been submitted to 98 undergraduates during their third academic year, for a complex analysis course in a high level school of engineers in Algeria. The written language is French, but often, Arabic dialect is used, along with French, in the oral form. The analysis of students' proof texts indicates that one of the difficulties behind writing messy and disorganized proof texts to open questions was lack of MKP [START_REF] Azrou | Proof writing at the undergraduate level[END_REF]. To receive further evidence for our findings, interviews were performed with a sample of students. We have chosen fourteen students to interview whose proof texts contained well-organized, less organized and very disorganized proof texts. Our aim was to investigate, by analyzing students' words, whether they master the concept of proof at a conscious level, in other words if they have a mastery of its operatory invariants [START_REF] Vergnaud | La théorie des champs conceptuels[END_REF].

A-priori analysis of the test

The test contained three questions, but in scope of this paper we can only include the first question.

1-Is it possible to find a holomorphic function that admits 0 as a simple pole such that Residue of f at 0 is 0 (Res (f, 0) =0)?

By designing such questions, we aimed at ascertaining if students were able to construct the proof, based on known definitions and theorems, in a clear argumentation form by providing their own way of expressing the answering to the question. The question is about the possibility of having (P Q) and its negation (P and ) at the same time, which results in a contradiction and thus it is impossible. Logically speaking: the fact that the residue at a point is not 0 is a direct consequence of that point being a simple pole for the function. We have chosen to refer to the point 0 to simplify the formula. There was no doubt that students knew all these concepts because they had used them many times before, but always when performing direct calculations and procedures. However no request of identifying and exploring the links between concepts had been made, especially in a written form.

A preliminary analysis of students' proof texts

We have observed in students' proof texts, among others, the following behaviors related to the concept of proof:

-Lack of justifications: students do not know when the justification is necessary and when it is not; they might give a justification for an obvious fact and miss to justify a non-obvious statement.

-Students turn around confusing the hypothesis and the thesis (forwards and backwards between the premise and the result).

-An example is given instead of a justification to prove that the statement is true.

-Incomplete mathematical statements and/or formulas.

-Missing details that make holes in the proof.

-Lack of organization of proof steps.

-Disconnection between statements.

-Writing the proof text like a draft or a sketch.

Examples of students' proof texts

The following excerpts show some of the difficulties cited before; the language used is French.

 Q Proof text 1
In this proof, the student responds by saying that such function does not exist and gives an example of a function that does not verify the conditions given by the statement. Clearly, the existence of a function that does not verify the conditions does not tell why these conditions cannot go together. This student considers that giving such example is the proof of the inexistence of functions verifying the two conditions.

Proof text 2

The student does not provide an answer to the question, the proof is a series of statements; each one derived from the previous one by an implication, but without any justification; moreover the last three ones are similar but incorrect, they present the simple pole definition, but the limit should be not 0.

Interviews

Based on the analysis of students' written productions, in the wider investigation this study belongs to, we have conducted interviews to address our previous questions, but also to receive more information about students' points of view. We will present only the questions of the interviews that deal with MKP about proof. The main interview contained three questions, each with three or four sub-questions.

Q1 Results of interviews Q1: Four students said 'yes', while the rest (ten) said 'no'. They intended their responses to be given especially to their own teacher, so they made their responses intentionally focusing on what is important to the teacher which is, according to them, their reasoning and showing that they got the idea of the process and understood enough the concept at stake in order to get the credit or a part of it. 'I know that my teacher will understand it even if it's not complete'.

Q1.a: All students responded that the teacher would check in a proof whether a student got the whole idea of the solution or not: 'the teacher would see always the method'; 'the reasoning'; 'the process of the proof'.

Q1.b:

Half of the students said 'yes' and the others said 'no': 'no, because we are used to respond to get the credit, so we address the response to our teacher'.

Q1.c: All of them responded mentioning the reasoning of the student (method, the logic in his response, whether it is convincing, if there is no contradiction): 'the teacher would follow the reasoning of the student to find out if his understanding is clear or not about the concept'.

Q1.d:

Twelve students answered 'no' and only two students said 'yes'. They would keep the idea or the method the same but make better the organization or the presentation: 'I would have changed the way I wrote, … the organization'; 'I would have given more details'.

Q3: Two students (good ones) said 'the same'; one didn't answer clearly, six said with more details and/or better organization; two said with better reasoning and three said with more symbols: 'a mathematician would have another goal, mine is to give the response and get the credit'; 'he would use only symbols till getting the final result, you see, I wrote a lot of comments'.

Q3.a:

Twelve students said that they would contain more symbols and less comments; with an academic rigorous style: 'it's different'; 'my answer is addressed to the teacher while mathematics books are addressed to all'; 'with more symbols and less comments'.

Q3.b:

Four students among fourteen answered by keeping their text as they are. Five said they would improve the organization, three said they would add more details and two said they would make the explanation better: 'I might keep the idea, but I will give more details'; 'I would write it better'.

Data analysis

Analysis of the written texts

Different students' weaknesses emerge from the analysis of students' written texts (difficulty of communication, lack of justification, using incomplete mathematical statements (or formulas) and lack of organization of the proof); the last three are of particular interest for MKP. Failing to give justification may be caused by the didactic contract supposing that the teacher would not mind it, by a lack of concept mastery or by a lack of meta-knowledge about proof. Mathematical statements are given incomplete because students might suppose that they are clear for the teacher, or because they are not well mastered by them or not important to be given complete in a proof text, which is related to MKP. The lack of organization of the statements displayed by students might be originated in didactic contract, in lack of concepts mastery, but also in lack of MKP.

Results of interviews analysis

The answers to Q1, Q1.b and Q1.c. confirm that students, when writing their proof texts, intend to address it particularly to their own teacher. The answers to Q1.d show that students are aware of their unclear text and possible missing details. According to them, they have to focus on two important points that have the same objective: how to get the most part of the credit and show to their teacher that they understood the concept at stake by presenting the main idea or the method of the proof; because they believe that the teacher will focus on that. This shapes their metaknowledge about proof writing. Most answers to the third question and to Q3.a support more details would be given by a rigorous mathematician and mathematics textbooks, students mention that the organization would be better in both cases than theirsbut they reveal how their conception about proofs in mathematics concerns superficial aspects when they say that proofs in textbooks contain more symbols and less comments and words in comparison with their proofs and do not mention the structure of the proof. Answers to question Q3.b confirm that students are aware that their proof texts need improvementbut it must be related to previous consideration about superficial aspects.

Conclusion

Students' texts and interviews offered strong evidence for students' lack of MKP and its influence on proof writing. Students have many situations of reference for proofs at their disposal but do not master the operatory invariants of the proof concept and the form of the proof texts as conscious objects. Findings suggest particularly that the influence of the didactic contract is strong. Teachers generally write proofs in a direct linear way, making unfolding the steps till the conclusion. Students learn to do the same: when they first set some ideas about how to solve a problem, they write their first exploratory draft as a final text because they were never shown how to go further to the written proof text. Here, the didactic contract works against to the development of MKP because the contractual knowledge substitutes the knowledge about the concept of proof. An important element emerged in the interviews, which is the intention of the students to write the proof text only for their teachers, which supports our hypothesis of lack of MKP. Students acknowledge that their texts miss details, but do not see that these missing details would make the organization of the different parts of the proof clearer. This shows that the MKP and the didactical contract are strongly related. When students compare their texts with mathematicians' or textbooks' proofs, they only point out to symbols and comments, they do not see that in these perfect proofs, the statements are linked through a deductive process from the hypothesis to the proof end, the proof text is organized, not only in its form, but also in the structure; avoiding holes, disconnections and missing justifications. This is evidence of students' superficial perception of proof texts, which indicates lack of mastery of proof structure and representation as a concept, which is related to lack at the operatory invariants level. As MKP is also built up through language, we hypothesize that students' weak mastery of French language, especially in the oral form, which should be translated to the written form, might have contributed to their unclear written texts.

Let us examine now why students are not able to develop their MKP; it seems that they are stuck in a constant perception of proof that does not help them to overcome their difficulties, if not causing some of them, and as long as there are not alternative ways of presenting proofs, they will hold on it. "Students need to understand that proofs are not generally conceived of in the order they are written" (Selden & Selden, 2007, p. 114) and that "successful reasoning can be carried out both by relying on the logic and formal structures of syntactic reasoning, and by relying on the informal representations of mathematical objects of semantic reasoning" (CadwalladderOlsker, 2011, p. 48).

Changing or adjusting the didactic contract may favor students' autonomy to understand and make proofs; university teachers often mistakenly think that undergraduates understand what a proof is and how to make proofs by following the standard presented proofs. In fact, "while a traditional definition-theorem-proof style of lecture presentation may convey the content in the most efficient way, there are other ways of presenting proofs that may enable students to gain more insight" (Selden & Selden, 2010, p. 411). Teachers should provide samples of proof construction instead of final products, to be clear about what do they expect from students when they are asked to prove and to provide an opportunity to learn how to make proofs. "In general, professors should avoid "dumbing down" their assessments by asking routine questions that can be answered by mimicking. One needs to modify the "didactic contract" in order to achieve this; otherwise, questions requiring genuine problem solving and proving will be considered "unfair" " (Selden & Selden, 2010, p.414). We support that "university teachers should consider including a good deal of student-student and teacher-student interaction regarding students' proof attempts, as opposed to just presenting their own or textbook's proofs" (cf. Selden & Selden, 2007, p. 114). Finally, in order to gain control, students need to master meta-knowledge about proof; "the difficulty of students to reach a structural axiomatic proof scheme suggests that a capstone course including some attention to metamathematics as a topic might be of value to mathematics majors" (Harel and Sowder, 1998, p. 280).

  After this time, looking again to your answer, is there something you would add or change in yours answer or would you keep it as it is?

	-If your answer would have been addressed to another teacher, would you have written it the
	same way?
	a-What is important, to a teacher, to see in a student response to questions like this one?
	b-Do you think another teacher, not familiar with the course, would have understood the answer?
	c-How can the teacher know if the answer is right or not?
	d-If the question has been proposed in homework, would you have presented it in a different way?

Q3-If a rigorous mathematician would have answered to this question, how would he presented his answer? a-What is the difference from your answer text and those we find in mathematics books? b-

Proceedings of CERME10