Tommy Dreyfus

HAL is

Introduction

Mathematics education as a research community has grown over the past approximately 50 years: ERME, The European Society for Research in Mathematics Education is approaching its 20 th anniversary in 2018 -CERME1, the first conference took place in Osnabrück, Germany, in August, 1998. PME, the International Group for the Psychology of Mathematics Education has held its 40 th annual conference in 2016 -the first one took place in 1977 in Utrecht, The Netherlands. JRME, the Journal for Research in Mathematics Education, is now producing its 48 th annual volume, and ESM, Educational Studies in Mathematics is currently in its 50 th year of publication since Volume 1 appeared in 1968. One of the characteristics of research results in (mathematics) education is that they depend on the context in which the research has been designed and carried out. Nevertheless, after 50 years, one would expect the community to be able to make statements that go beyond "it depends on the context and the learning environment", which is often implicit in the results of even high quality research articles. Review articles could be expected to remedy this situation to some extent but few review articles are published in the domain.

What are the results of research in mathematics education -ICMI Study 8

The question whether we, as a research community, have obtained results with a certain scope, range or breadth of validity and what these results are, has been approached at least twice, once in the framework of the study conference of ICMI Study 8 in 1996 (ICMI stands for the International Commission on Mathematical Instruction), and a second time in the framework of the Education Committee of the European Mathematical Society (EMS) in 2011.

The task assigned by ICMI to the Study 8 program committee was to discuss what is research in mathematics education and what are its results. The title of the book published two years later as outcome of the study is Mathematics Education as a Research Domain: A Search for Identity (Sierpinska & Kilpatrick, 1998). Maybe significantly, the word 'results' has disappeared in the process. Nevertheless, one of the working groups at the study conference dealt with results (Dreyfus & Becker, 1998). One of the questions the working group dealt with was what counts as result; the term 'solid' did not appear. Rather, 'result' was interpreted as 'significant result'.

Working group members agreed that without a question, there can only be facts but no results. Results are more than data: They are based on data collected with questions in mind that have been asked within a theoretical framework, and consist of findings interpreted in that theoretical framework. Effects alone (e.g., statistical differences in achievements between different groups) are not results. In mathematics education, we need to explain the differences, not only show them. We need to identify the variables of the didactic situation in order to combine the different facts into a coherent network of reasons, which informs the circular process of understanding the learning of mathematics and thus improving its teaching. Hence, results are often theoretical as well as experimental. Many of our theoretical frameworks are mathematics-specific (e.g., process, object, procept), and therefore our research questions and results are often domain-specific.

Context was seen as relevant with respect to theory as well as beyond: mathematical context (contents, concepts, symbols, representations and epistemological status), the community, the educational system, among others. Results might be tied not only to the theoretical framework but

Introduction

Mathematics education as a research community has grown over the past approximately 50 years: ERME, The European Society for Research in Mathematics Education is approaching its 20 th anniversary in 2018 -CERME1, the first conference took place in Osnabrück, Germany, in August, 1998. PME, the International Group for the Psychology of Mathematics Education has held its 40 th annual conference in 2016 -the first one took place in 1977 in Utrecht, The Netherlands. JRME, the Journal for Research in Mathematics Education, is now producing its 48 th annual volume, and ESM, Educational Studies in Mathematics is currently in its 50 th year of publication since Volume 1 appeared in 1968. One of the characteristics of research results in (mathematics) education is that they depend on the context in which the research has been designed and carried out. Nevertheless, after 50 years, one would expect the community to be able to make statements that go beyond "it depends on the context and the learning environment", which is often implicit in the results of even high quality research articles. Review articles could be expected to remedy this situation to some extent but few review articles are published in the domain.

What are the results of research in mathematics education -ICMI Study 8

The question whether we, as a research community, have obtained results with a certain scope, range or breadth of validity and what these results are, has been approached at least twice, once in the framework of the study conference of ICMI Study 8 in 1996 (ICMI stands for the International Commission on Mathematical Instruction), and a second time in the framework of the Education Committee of the European Mathematical Society (EMS) in 2011.

The task assigned by ICMI to the Study 8 program committee was to discuss what is research in mathematics education and what are its results. The title of the book published two years later as outcome of the study is Mathematics Education as a Research Domain: A Search for Identity (Sierpinska & Kilpatrick, 1998). Maybe significantly, the word 'results' has disappeared in the process. Nevertheless, one of the working groups at the study conference dealt with results (Dreyfus & Becker, 1998). One of the questions the working group dealt with was what counts as result; the term 'solid' did not appear. Rather, 'result' was interpreted as 'significant result'.

Working group members agreed that without a question, there can only be facts but no results. Results are more than data: They are based on data collected with questions in mind that have been asked within a theoretical framework, and consist of findings interpreted in that theoretical framework. Effects alone (e.g., statistical differences in achievements between different groups) are not results. In mathematics education, we need to explain the differences, not only show them. We also to the institution that asked the question. It is not the result itself, but the conditions under which it was obtained, that make it significant.

The contextual nature of results implies that results are neither universal nor eternal, that their validity is situated in space and time, and that we have to be careful when trying to generalize. The validity of a result depends on the interpretation within a theory, and the theory might change with time and place, with mathematical content, learning environment, and so on. Hence results are permanent but their relevance might be ephemeral.

Characterising solid findings

It is on this background that the members of the Education Committee (EC) of the European Mathematical Society (EMS) asked themselves what solid findings mathematics education has produced. While the question was motivated by the intention of the committee to present mathematics education to mathematicians, in particular to EMS members, with an interest in mathematics education, committee members were well aware that the exercise of identifying solid findings contributes to establishing and organizing mathematics education as scientific discipline and to determining where we come from, where we are and where me might go as a research community.

Of course, the first, and possibly most difficult task of the EC was to discuss, agree and explain what they meant by 'solid findings'. A major difficulty in defining what it means that a result it solid is the context dependence, mentioned above. A second and related difficulty is complexity. As we know well, things are more complex than one might think; we know, for example that the mathematics taught and learned in parallel classes with a similar population according to the same curriculum may be quite different (e.g., Even & Kvatinsky, 2010;Pinto, 2013). A third, and of course also related difficulty is that much of the research in mathematics education is qualitative. Since qualitative empirical research cannot be repeated in a strict sense, reproducibility is replaced by the question how close the results are that one obtains in similar contexts; and the answer to this question of course depends on the metric used to measure closeness. This lack of reproducibility may appear as a serious drawback of mathematics education's claim to be a scientific discipline; however, reproducibility has recently been shown to be very low even in many hard sciences such as physics, chemistry and engineering (Baker, 2016).

Aware of these difficulties and with the ICMI 8 study characterization of (significant) results as background, the EC has observed that results with the potential of being considered solid usually do not stand alone but have emerged from a line of research consisting of a larger set of related studies. Solid findings are typically yielded by such a line of studies. Next, the EC has built on three properties of research quality proposed by Schoenfeld (2007see there for a much more detailed discussion): trustworthiness, generality and importance. Each of these contributes to the solidity of research results. A characterisation adapted to the purposes of the EC was agreed upon and published in the Newsletter of the EMS (Education Committee of the European Mathematical Society, 2011a). I summarize this characterization here, adapting and supplementing it for the purposes of the present CERME panel. need to identify the variables of the didactic situation in order to combine the different facts into a coherent network of reasons, which informs the circular process of understanding the learning of mathematics and thus improving its teaching. Hence, results are often theoretical as well as experimental. Many of our theoretical frameworks are mathematics-specific (e.g., process, object, procept), and therefore our research questions and results are often domain-specific.

Context was seen as relevant with respect to theory as well as beyond: mathematical context (contents, concepts, symbols, representations and epistemological status), the community, the educational system, among others. Results might be tied not only to the theoretical framework but also to the institution that asked the question. It is not the result itself, but the conditions under which it was obtained, that make it significant.

The contextual nature of results implies that results are neither universal nor eternal, that their validity is situated in space and time, and that we have to be careful when trying to generalize. The validity of a result depends on the interpretation within a theory, and the theory might change with time and place, with mathematical content, learning environment, and so on. Hence results are permanent but their relevance might be ephemeral.

Characterising solid findings

It is on this background that the members of the Education Committee (EC) of the European Mathematical Society (EMS) asked themselves what solid findings mathematics education has produced. While the question was motivated by the intention of the committee to present mathematics education to mathematicians, in particular to EMS members, with an interest in mathematics education, committee members were well aware that the exercise of identifying solid findings contributes to establishing and organizing mathematics education as scientific discipline and to determining where we come from, where we are and where me might go as a research community.

Of course, the first, and possibly most difficult task of the EC was to discuss, agree and explain what they meant by 'solid findings'. A major difficulty in defining what it means that a result it solid is the context dependence, mentioned above. A second and related difficulty is complexity. As we know well, things are more complex than one might think; we know, for example that the mathematics taught and learned in parallel classes with a similar population according to the same curriculum may be quite different (e.g., Even & Kvatinsky, 2010;Pinto, 2013). A third, and of course also related difficulty is that much of the research in mathematics education is qualitative. Since qualitative empirical research cannot be repeated in a strict sense, reproducibility is replaced by the question how close the results are that one obtains in similar contexts; and the answer to this question of course depends on the metric used to measure closeness. This lack of reproducibility may appear as a serious drawback of mathematics education's claim to be a scientific discipline; however, reproducibility has recently been shown to be very low even in many hard sciences such as physics, chemistry and engineering (Baker, 2016).

Aware of these difficulties and with the ICMI 8 study characterization of (significant) results as background, the EC has observed that results with the potential of being considered solid usually do not stand alone but have emerged from a line of research consisting of a larger set of related studies.

Trustworthiness includes the explanatory power of research, its rigor and specificity, and whether it makes use of multiple sources of evidence. However, a study may be trustworthy but trivial, in terms of generality or importance. Generality (or scope) refers to the question: What is the scope or generality of a research result? How widely does this finding, this idea, or this theory apply across content domains, learning contexts, cultures, etc.? For example, did researchers, in different countries and school systems obtain comparable or related empirical results? Do theoretical constructs turn out to be useful beyond the bounds of the individual studies in which they were developed?

Trustworthiness and generality together are expected to impart some predictive power to a result. A result that has no predictive power cannot be considered solid. On the other hand, the difficulties mentioned above, such as context dependence, will usually limit this predictive power. If a result is used to predict an outcome in a new context, and the prediction failed, a trustworthy explanation of the failure may in fact increase the solidity of the result.

Importance addresses the question: Does it matter? What is the (actual or potential) contribution of the research to theory and practice. Of course, importance is to a large extent a value judgment. As in any other field of study, beliefs about what is essential and what is peripheral are not static but change over decades, reflecting trends both within and beyond the discipline. Hence recognition of the significance of the result by experts contributes essentially to the solidity of a result.

The term 'solid' may remind the reader of the term 'robust' often used in related situations. 'Robust' often has a technical meaning that refers to a finding having been repeatedly observed or confirmed in many studies reporting the same or similar results leading to the same (general) conclusions (see Primi, below). The term solid has been chosen intentionally, to refer to results rather than findings, and imply that robustness in the technical sense is not possible, nor maybe desirable, in mathematics education.

While robustness can be defined and hence (dis-)proved, solidity cannot. The above is a characterization or descriptionnot a definition. Hence, solidity cannot be proved but it can definitely be argued by on the basis of the above criteria of significance, trustworthiness, generality and adaptability to context.

Examples

The second major task of the EC with respect to solid findings was to provide a variety of examples of findings that are solid according to the EC's characterization. While the selection of the examples to be presented was somewhat eclectic and partly determined by EC members who were willing to write about a topic, the EC as a whole discussed and approved the proposed topics; the EC also revised every draft several times. As result a sequence of brief articles has been published presenting a rather representative selection including solid findings about cognition and about affect, about teaching and about learning, about elementary school and about university, about specific mathematical contents and about cross domain issues such as the use of technology, and maybe most importantly about theoretical and about empirical results. Most of the issues of the EMS newsletter from Issue 82 (December 2011) to Issue 95 (March 2015) present such a solid finding.

Plenary Panel

Proceedings of CERME10 59

Solid findings are typically yielded by such a line of studies. Next, the EC has built on three properties of research quality proposed by Schoenfeld (2007see there for a much more detailed discussion): trustworthiness, generality and importance. Each of these contributes to the solidity of research results. A characterisation adapted to the purposes of the EC was agreed upon and published in the Newsletter of the EMS (Education Committee of the European Mathematical Society, 2011a). I summarize this characterization here, adapting and supplementing it for the purposes of the present CERME panel.

Trustworthiness includes the explanatory power of research, its rigor and specificity, and whether it makes use of multiple sources of evidence. However, a study may be trustworthy but trivial, in terms of generality or importance.

Generality (or scope) refers to the question: What is the scope or generality of a research result? How widely does this finding, this idea, or this theory apply across content domains, learning contexts, cultures, etc.? For example, did researchers, in different countries and school systems obtain comparable or related empirical results? Do theoretical constructs turn out to be useful beyond the bounds of the individual studies in which they were developed?

Trustworthiness and generality together are expected to impart some predictive power to a result. A result that has no predictive power cannot be considered solid. On the other hand, the difficulties mentioned above, such as context dependence, will usually limit this predictive power. If a result is used to predict an outcome in a new context, and the prediction failed, a trustworthy explanation of the failure may in fact increase the solidity of the result. The term 'solid' may remind the reader of the term 'robust' often used in related situations. 'Robust' often has a technical meaning that refers to a finding having been repeatedly observed or confirmed in many studies reporting the same or similar results leading to the same (general) conclusions (see Primi, below). The term solid has been chosen intentionally, to refer to results rather than findings, and imply that robustness in the technical sense is not possible, nor maybe desirable, in mathematics education.

While robustness can be defined and hence (dis-)proved, solidity cannot. The above is a characterization or descriptionnot a definition. Hence, solidity cannot be proved but it can definitely be argued by on the basis of the above criteria of significance, trustworthiness, generality and adaptability to context.

Examples

The second major task of the EC with respect to solid findings was to provide a variety of examples of findings that are solid according to the EC's characterization. While the selection of the examples to be presented was somewhat eclectic and partly determined by EC members who were willing to write about a topic, the EC as a whole discussed and approved the proposed topics; the EC also Here, I briefly present two of these, one reason for my choice again being personal preference and the other representativeness, at least in the empiricaltheoretical dimension.

Do theorems admit exceptions?

Empirical studies on students' conceptions of proof have found that many students provide examples when asked to prove a universal statement. Universality refers to the fact that a mathematical claim is considered true only if it is true in all admissible cases without exception. A student who seeks to prove a universal claim by showing that it holds in some cases is said to have an empirical proof scheme. The same student is also likely to expect that a statement, even if it has been 'proved', may still admit exceptions. There is considerable evidence that many mathematics students, and some mathematics teachers, rely on validation by means of one or several examples to support general statements. The majority of students who begin studying mathematics in high school have empirical proof schemes, and many students continue to act according to empirical proof schemes for many years, sometimes into their college years.

While the issue of empirical proof schemes has already been mentioned by Polya (1945), Bell (1976) may have been the first to report an empirical study about students' proof schemes. Following Fischbein's (1982) seminal investigation on universality, the issue has been re-examined many times, usually with similar results. For example, findings by Sowder and Harel (2003) indicate the appearance of empirical proof schemes among university mathematics graduates.

The phenomenon of empirical proof schemes is general in the sense that it has been found in many cultures, countries, school systems, and age groups. It is persistent in the sense that many students continue to do so even after explicit instruction about the nature of mathematical proof. However, it is also complex. For example, the London proof study (Healy and Hoyles, 2000) showed that even for relatively simple and familiar questions, 14-15 years old high-attaining students' most popular approach was empirical verification but that many students correctly incorporated some deductive reasoning into their proofs and most valued general and explanatory arguments.

How can this pervasive phenomenon be explained? The notion of a "universally valid statement" is not as obvious as it might seem to mathematicians. Mathematical thought concerning proof is different from thought in all other domains of knowledge, including the sciences, as well as everyday experience. In everyday life, the "exception that confirms the rule" is pertinent. Students, in particular young children, have little experience with mathematics as a wonderful world with its own objects and rules. According to Fischbein (1982), the concept of formal proof is completely outside mainstream thinking, and we require students to acquire a new, non-natural basis of belief when we ask them to prove. These explanations contribute to the trustworthiness of the findings on empirical proof schemes.

In summary, the studies on empirical proof schemes, only a few of which have been referred to here, firmly establish the solidity of the phenomenon of empirical proof schemes. (For a more detailed exposition, see Education Committee of the European Mathematical Society, 2011b.)

Concept images in students' mathematical reasoning

Vinner and Hershkowitz (1980) were the first ones to point out that students' geometrical thinking is frequently based on prototypes rather than on definitions. They have shown, for example, that Here, I briefly present two of these, one reason for my choice again being personal preference and the other representativeness, at least in the empiricaltheoretical dimension.

Do theorems admit exceptions?

Empirical studies on students' conceptions of proof have found that many students provide examples when asked to prove a universal statement. Universality refers to the fact that a mathematical claim is considered true only if it is true in all admissible cases without exception. A student who seeks to prove a universal claim by showing that it holds in some cases is said to have an empirical proof scheme. The same student is also likely to expect that a statement, even if it has been 'proved', may still admit exceptions. There is considerable evidence that many mathematics students, and some mathematics teachers, rely on validation by means of one or several examples to support general statements. The majority of students who begin studying mathematics in high school have empirical proof schemes, and many students continue to act according to empirical proof schemes for many years, sometimes into their college years.

While the issue of empirical proof schemes has already been mentioned by Polya (1945), Bell (1976) may have been the first to report an empirical study about students' proof schemes. Following Fischbein's (1982) seminal investigation on universality, the issue has been re-examined many times, usually with similar results. For example, findings by Sowder and Harel (2003) indicate the appearance of empirical proof schemes among university mathematics graduates.

The phenomenon of empirical proof schemes is general in the sense that it has been found in many cultures, countries, school systems, and age groups. It is persistent in the sense that many students continue to do so even after explicit instruction about the nature of mathematical proof. However, it is also complex. For example, the London proof study (Healy and Hoyles, 2000) showed that even for relatively simple and familiar questions, 14-15 years old high-attaining students' most popular approach was empirical verification but that many students correctly incorporated some deductive reasoning into their proofs and most valued general and explanatory arguments.

How can this pervasive phenomenon be explained? The notion of a "universally valid statement" is not as obvious as it might seem to mathematicians. Mathematical thought concerning proof is different from thought in all other domains of knowledge, including the sciences, as well as everyday experience. In everyday life, the "exception that confirms the rule" is pertinent. Students, in particular young children, have little experience with mathematics as a wonderful world with its own objects and rules. According to Fischbein (1982), the concept of formal proof is completely outside mainstream thinking, and we require students to acquire a new, non-natural basis of belief junior high school students tend to think that the altitude has to reach the base (rather than its extension). Hence, they draw the altitude inside the triangle, even in a triangle with an obtuse base angle. Students' prototype altitude is one that is inside the triangle. This is so, even if the students know and can recite the (general) definition of altitude in a triangle.

Authors from many countries have reported, over the past 35 years, analogous patterns in students' reasoning in other areas of mathematics, even among talented students in elementary school, high school and college. For example, and in spite of 'knowing' the appropriate definitions, students tend to act according to rules such as multiplication makes bigger, inflection points have horizontal tangents, definite integrals must be positive, and sequences are monotonous.

A commonality in these and parallel studies is that students do not base their reasoning on the definition of the concepts under consideration (even though they are often aware of these definitions and can recite and explain them) but rather on what Tall and Vinner (1981) have called their concept image: "the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes" (p. 152). A student's concept image need not be globally coherent and may have aspects which are at variance with the formal concept definition.

The notion of a student's concept image is complex since it is influenced by all of this student's experiences associated with the concept. These include examples, problems the student has solved, prototypes the student may have met substantially more often than non-prototypical examples, and different representations of the concept including visual, algebraic and numerical ones. Images may deeply influence concept formation. As a consequence, the concept image is personal and continuously changing through the student's mathematical experiences.

How can this pervasive phenomenon be explained? While it is not possible to introduce a concept without giving examples, particular instances of the concept never suffice to fully determine the concept. As a consequence, specific elements of the examples, even if not pertinent to the mathematical definition of the concept, become for the student key elements characterizing the concept. And even if at the stage a concept is introduced a teacher might make an effort to present a rather varied set of examples, as the concept is being used over the coming months or years, some of these properties tend to be reinforced because they appear much more frequently than others that may recede. Examples abound, and the height of a triangle being vertical in the sense explained above is a typical one. Students may see many triangles in which the altitude is inside the triangle, and few in which it is not. They might consider these few cases as exceptions (Lakatos might say monsters). This explanation contributes to the trustworthiness of the findings on empirical proof schemes.

In summary, a solid finding of mathematics education research, supported by dozens of studies in many difference contexts, is that students' mathematical reasoning is frequently based on their concept images rather than on a mathematical concept definition. A more detailed exposition of this solid finding has been published elsewhere (Dreyfus, on behalf of the EC of the EMS, 2014).

Plenary Panel

Proceedings of CERME10 61

when we ask them to prove. These explanations contribute to the trustworthiness of the findings on empirical proof schemes.

In summary, the studies on empirical proof schemes, only a few of which have been referred to here, firmly establish the solidity of the phenomenon of empirical proof schemes. (For a more detailed exposition, see Education Committee of the European Mathematical Society, 2011b.)

Concept images in students' mathematical reasoning Vinner and Hershkowitz (1980) were the first ones to point out that students' geometrical thinking is frequently based on prototypes rather than on definitions. They have shown, for example, that junior high school students tend to think that the altitude has to reach the base (rather than its extension). Hence, they draw the altitude inside the triangle, even in a triangle with an obtuse base angle. Students' prototype altitude is one that is inside the triangle. This is so, even if the students know and can recite the (general) definition of altitude in a triangle.

Authors from many countries have reported, over the past 35 years, analogous patterns in students' reasoning in other areas of mathematics, even among talented students in elementary school, high school and college. For example, and in spite of 'knowing' the appropriate definitions, students tend to act according to rules such as multiplication makes bigger, inflection points have horizontal tangents, definite integrals must be positive, and sequences are monotonous.

A commonality in these and parallel studies is that students do not base their reasoning on the definition of the concepts under consideration (even though they are often aware of these definitions and can recite and explain them) but rather on what Tall and Vinner (1981) have called their concept image: "the total cognitive structure that is associated with the concept, which includes all the mental pictures and associated properties and processes" (p. 152). A student's concept image need not be globally coherent and may have aspects which are at variance with the formal concept definition.

The notion of a student's concept image is complex since it is influenced by all of this student's experiences associated with the concept. These include examples, problems the student has solved, prototypes the student may have met substantially more often than non-prototypical examples, and different representations of the concept including visual, algebraic and numerical ones. Images may deeply influence concept formation. As a consequence, the concept image is personal and continuously changing through the student's mathematical experiences.

How can this pervasive phenomenon be explained? While it is not possible to introduce a concept without giving examples, particular instances of the concept never suffice to fully determine the concept. As a consequence, specific elements of the examples, even if not pertinent to the mathematical definition of the concept, become for the student key elements characterizing the concept. And even if at the stage a concept is introduced a teacher might make an effort to present a rather varied set of examples, as the concept is being used over the coming months or years, some of these properties tend to be reinforced because they appear much more frequently than others that may recede. Examples abound, and the height of a triangle being vertical in the sense explained above is a typical one. Students may see many triangles in which the altitude is inside the triangle, and few in which it is not. They might consider these few cases as exceptions (Lakatos might say

Conclusion

The list is of solid findings presented by the EC of the EMS is, of course, not exhaustive but limited by the time of service of the committee and the people who served on it. I would like to encourage CERME members (and other researchers) to write and publish articles about solid findings they are aware of and consider important. This might have the desirable effect of producing a type of article lacking almost completely from our literaturereview articles. Let me make just one suggestion: Work to raise the awareness of issues and of research on teaching and learning among university lecturers and tutors is necessary; it usually improves students' attitudes but effects on learning are limited. Research in at least four countries (USA, Germany, England, Finland) has shown that work with students has more potential for large scale effects. It seems to me that a suitable review article might not only inform many mathematics educators of an important line of research but might have a considerable effect on university teaching centres, an effect that a single study report could (and should) never have.

In conclusion, the researchers and teams referred to above have shown that mathematics education has, over the past 50 years, produced theoretical and empirical results that are solid in the sense that they have explanatory and predictive power, that they can be applied in contexts beyond those involved previous studies, and that they are recognised as important contributions that have significantly influenced the research field, for example by providing a theoretical lens that allows to see an observed phenomenon differently from how it was seen before.

Acknowledgment

I am grateful to Günter Törner, chair of the EC of the EMS for initiating and sustaining the discussion about solid findings in mathematics education at and in between the six committee meetings from 2010-2014; I would also like to acknowledge the contribution of the committee members to my thinking about what makes findings solid and about what solid findings mathematics education has produced.

Plenary Panel

Proceedings of CERME10 62 monsters). This explanation contributes to the trustworthiness of the findings on empirical proof schemes.

In summary, a solid finding of mathematics education research, supported by dozens of studies in many difference contexts, is that students' mathematical reasoning is frequently based on their concept images rather than on a mathematical concept definition. A more detailed exposition of this solid finding has been published elsewhere (Dreyfus, on behalf of the EC of the EMS, 2014).

Conclusion

The list is of solid findings presented by the EC of the EMS is, of course, not exhaustive but limited by the time of service of the committee and the people who served on it. I would like to encourage CERME members (and other researchers) to write and publish articles about solid findings they are aware of and consider important. This might have the desirable effect of producing a type of article lacking almost completely from our literaturereview articles. Let me make just one suggestion: Work to raise the awareness of issues and of research on teaching and learning among university lecturers and tutors is necessary; it usually improves students' attitudes but effects on learning are limited. Research in at least four countries (USA, Germany, England, Finland) has shown that work with students has more potential for large scale effects. It seems to me that a suitable review article might not only inform many mathematics educators of an important line of research but might have a considerable effect on university teaching centres, an effect that a single study report could (and should) never have.

In conclusion, the researchers and teams referred to above have shown that mathematics education has, over the past 50 years, produced theoretical and empirical results that are solid in the sense that they have explanatory and predictive power, that they can be applied in contexts beyond those involved previous studies, and that they are recognised as important contributions that have significantly influenced the research field, for example by providing a theoretical lens that allows to see an observed phenomenon differently from how it was seen before.

Acknowledgment

I am grateful to Günter Törner, chair of the EC of the EMS for initiating and sustaining the discussion about solid findings in mathematics education at and in between the six committee meetings from 2010-2014; I would also like to acknowledge the contribution of the committee members to my thinking about what makes findings solid and about what solid findings mathematics education has produced.

  Importance addresses the question: Does it matter? What is the (actual or potential) contribution of the research to theory and practice. Of course, importance is to a large extent a value judgment. As in any other field of study, beliefs about what is essential and what is peripheral are not static but change over decades, reflecting trends both within and beyond the discipline. Hence recognition of the significance of the result by experts contributes essentially to the solidity of a result.

  revised every draft several times. As result a sequence of brief articles has been published presenting a rather representative selection including solid findings about cognition and about affect, about teaching and about learning, about elementary school and about university, about specific mathematical contents and about cross domain issues such as the use of technology, and maybe most importantly about theoretical and about empirical results. Most of the issues of the EMS newsletter from Issue 82 (December 2011) to Issue 95 (March 2015) present such a solid finding.

	Plenary Panel
	Proceedings of CERME10	60

Proceedings of CERME10


l M LD (d), q M LD (d) = arg max l,q∈E s d k d:d k I co (d)q(d)p ch (y | d)l(d) with optimal solution l M LD (d) = q M LD (d) = δ d MLD (d)
but still untractable...

n convenient APPROXIMATIONS: C1 . . . Ck . . . Cn ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✭ ✘ ✘ ✘ ✘ ✘ ✘ ✘ ✆ ✆ ❳ ❳ ❳ ❳ ❳ ❳ ❳ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ ❤ C M LD specific for each bit-marginal d k d:d k I co (d)q(d)p ch (y | d)l(d) bit-marginals of a product (C MLD , optimal) ≈ d k d:d k I co (d)q(d) d:d k p ch (y | d)l(d)
product of the bit-marginals ( Ck , suboptimal)

• Suggests a distributed optimization strategy • Average cost function: C = n i=1 Ck ⇒ Score function for the distributed optimization strategy • C is a weighted sum:

C(l, q) = nC M LD + n-1 r=1 (n -r)N r (l, q) where N r (l, q) = (d,d ′ )∈S r I co (d)q(d)p ch (y | d ′ )l(d ′ ) and S r denote the set of pairs (d, d ′ ) such that d H (d, d ′ ) = r (Hamming distance)

Distributed Optimization

Maximize separately the n cost functions Ck

l k , q k = arg max l k ,q k ∈F d k q k (d k )l k (d k ) d:d k I co (d)Π i =k q i (d i ) d:d k p ch (y | d)Π i =k l i (d i ) Linear in q k (d k )l k (d k ) ⇒ l k q k ∈ {0, 1}: HARD decision SOFT decision preferred: ⇒ q k l k ∝ d:d k I co (d)Π i =k q i (d i ) extrinsic (decoder) d:d k p ch (y | d)Π i =k l i (d i ) extrinsic (demapper) A Posteriori Probability β
Equivalently with LLR :

λ l,k + λ q,k = β log d:d k I co (d)Π i =k q j (d j ) d:d k I co (d)Π i =k q i (d i ) + β log d:d k p ch (y | d)Π i =k l i (d i ) d:d k p ch (y | d)Π i =k l i (d i ) (1)
Can be recast as a distributed optimization process with strictly concave utility functions:

U k (λ l,k , λ q,k , λ l,-k , λ q,-k ) = -λ l,k + λ q,k -β log f d k (q, I co )f d k (l, p ch (y | d)) f d k (q, I co )f d k (l, p ch (y | d) 2 (2)

Iterative Decoding as a Game

Definition 1 (Game). A game G is defined as a triplet G = (K, {S i } i∈K , {u i } i∈K ) where K = {1, 2, ..., n} is a finite set of players, ∀i ∈ K, {S i } is the set of strategy of player i and u i its utility function.

Proposition 1. Iterative decoding is a game G sof t with n players (encoded bits), set of strategy {S i } i∈K = R 2 , and utility function {U i } i∈K defined in (??).

Definition 2 (Nash Equibrium

). A profile s * is a (pure) NE for G if ∀i ∈ K, ∀s ′ i ∈ S i , u i (s * i , s * -i ) ≥ u i (s ′ i , s * -i ).
Proposition 2. Let (λ l * , λ q * ) be a solution of (??). Then, (λ l * , λ q * ) is a NE of the game G sof t .

Proposition 3. Let (λ l * , λ q * ) be a solution of (??). Then, (λ l * , λ q * ) is an induced equilibrium of the game G hard with K = {0, 1, ..., n}, {S i } i∈K = [0, 1] × [0, 1] and with utility function Ci meaning that ∀i ∈ K,

∀s ′ i = (l i (d i ), q i (d i )) ∈ S i , Ci (H(s * i ), s * -i ) ≥ Ci (s ′ i , s * -i ) where H(s * i ) is a hard decision operator.
Definition 3 (Social Welfare). The social welfare W of a game is defined as the sum of the utilities of all players, W = n i=1 u i .

• Game G sof t ⇒ W sof t = 0 for any NE (USELESS).

• Game G hard ⇒ W hard = C ⇒ a posteriori evaluation of the optimization process and of the vicinity to the MLD solution. • There always exists at least one NE (proved in (?) when β = 1).

Convergence

Iterative decoding = (??) + Jacobi/Gauss-Seidel scheduling (?). Convergence proofs based on the Jacobian ∇F (?) → Key property for ∇F : STRICTLY DIAGONAL DOMINANCE. • β useful to improve convergence rate

• Convergence is not an issue

••

  Processing Conference (EUSIPCO 2011) Laboratoire des Signaux et Systèmes Supelec, Gif-sur-Yvette, France Goal: Iterative Decoding as a distributed optimization strategy -Interpretation of turbo-like Iterative Decoding (Game) -Results on Convergence. y k = h k s k + n k s k = M apping(d kM +1 , d kM +2 , ..., d (k+1)M ) n b information bits, n encoded bits Maximum Likelihood Decoding (MLD) Searching for optimal sequence of information bits b M LD b M LD = arg max b∈{0,1} n b p(y | b) • Searching for optimal sequence of encoded/interleaved bits d M LD d M LD = arg max Searching for optimal weighting p M LD p M LD (d) = arg max p∈E s d I co (d)p ch (y | d)p(d) E s : fully-factorized PMFs Solution is p M LD (d) = δ d MLD (d) Implementation untractable ⇒ Suboptimal optimization Suboptimal Decoding Numerical value of n ⇒ Solution 1: Compute bit-marginals Interleaver ⇒ Solution 2: Consider separately decoding and demapping Leads to another expression for the MLD

Definition 4 .

 4 Let A denote a n × n matrix with elements a ij in R. A is a strictly diagonal dominant matrix if|a ii | > j =i |a ij | ∀i ∈ {1, ..., n} Proposition 4. It always exist β 0 > 0 such that ∀β ≤ β 0 , ∇F (λ l , λ q )) is a strictly diagonally dominant matrix for all (λ l , λ q ) ∈ R n × R n .IllustrationBICM -(5, 7) convolutional code of rate 1/2 -n b = 400 (a frame). Set partitioning -16-QAM. EbN0=5dB -β 0 distribution among non-converging sequences in classical BICM , (right) EbN0=7dB among converging sequences in classical BICM Conclusion • C: a posteriori score function (optimization process / MLD solution)


Lessons for mathematics higher education from 25 years of mathematics support

Duncan Lawson, Tony Croft

To cite this version:

Lessons for mathematics higher education from 25 years of mathematics support

Duncan Lawson 1 and Tony Croft² 1 Newman University, United Kingdom; ²Loughborough University, United Kingdom

INDRUM KEYNOTE PRESENTATION

The scale and scope of mathematics support within UK universities have grown significantly since the 1990s. Mathematics support has evolved from a 'cottage industry' initiated by enthusiasts into a main-line student support provision overseen by institutional senior managers. Over this 25+ year period, the importance of the mathematical sciences in other disciplines has similarly boomed. No longer is it just engineering and physics undergraduates who need to acquire highly developed mathematical skills. Today geographers, bioscientists, sociologists and political scientists (to name but a few) have to be more skilled than ever before with understanding mathematical and statistical models and methods, particularly if they are to be able to access the international research literature and compete in the international employment market. Just as in the 1980s and 1990s, the Engineering Council produced reports warning of 'the mathematics problem', so in the 2000s and 2010s, the British Council and Royal Society of Arts have done the same. This presentation will outline how mathematics support has developed throughout the UK to meet this increasing demand.

Whilst the initial impetus for mathematics support came from a desire to improve the mathematical learning of students from other disciplines, it is an indisputable fact that a significant proportion of the users of mathematics support has been, and remains, mathematics undergraduates. This gives us cause to reflect: why is mathematics support so attractive to mathematics undergraduates? To answer this question, we explore the views of mathematics undergraduate students as expressed through the National Student Survey and in focus groups and individual interviews. The views the students express shed light on the reasons why many of them find mathematics support to be an attractive resource to support their learning.


Florence Alberge

Ziad Naja

Pierre Duhamel

New criteria for iterative decoding

Keywords:

Context:

System model (Bit Interleaved Coded Modulation) [START_REF] Zehavi | 8-PSK trellis codes for a Rayleigh fading channel[END_REF][START_REF] Li | Bit interleaved coded modulation with iterative decoding and 8-PSK signaling[END_REF] AWGN Channel

y k = s k + n k n k N (0, σ 2 n ) s k = M

Information geometry and iterative decoding

Basic tools and Notations [START_REF] Walsh | Turbo decoding as Iterative Constrained Maximum Likelihood sequence Decoding[END_REF])

(B i ) ∈ {0, 1} N binary representation of integer i gathered into matrix B = (B 0 , B 1 , ..., B 2 N -1 ) T . PMF η = (Pr[χ = B 0 ], Pr[χ = B 1 ], ..., Pr[χ = B 2 N -1 ]) T Log-coordinates of PMF η (θ i ) 0≤i≤2 N -1 = ln(P r[χ = B i ]) -ln(P r[χ = B 0 ]) Bitwise log-probability ratio (λ j ) 0≤j≤N -1 = log( P r[χ j =1] P r[χ j =0] )
For factorisable probability measures, the log-coordinates take the form θ = Bλ.

Link with iterative decoding

• Demapping subblock

p AP P (d km+i = b) = K ′ m s k ∈Ψ i b p(y k | s k ) j p(d km+j ; I)     Log-coordinates θ m Bλ 1 p AP P (d km+i = b) = K ′′ m p(d km+i ; I) p(d km+i ; O)     Log-coordinates Bλ 1 Bλ 2
The demapping sub-block solves, with respect to λ 2 , p B(λ 1 +λ 2 ) = p Bλ 1 +θ m

• Decoding sub-block

p AP P (c l = b) = K ′ c c∈R l b I C (c) j p(c j ; I)     Log-coordinates θ c
Bλ 2

p AP P (c l ) = K ′′ c p(c l ; I) p(c l ; O)     Log-coordinates Bλ 2 Bλ 1
The decoding sub-block solves, with respect to λ 1 , p B(λ 1 +λ 2 ) = p Bλ 2 +θ c

• Global criterion

Let D F D (p, q) = j p j ln p j q j + j (1p j ) ln 1-p j 1-q j denote the Fermi-Dirac entropy (Kullback-Leibler distance for bit probabilities).

The demapping sub-block solves the minimization problem min λ 2 D F D (p Bλ 1 +θ m , p B(λ 1 +λ 2 ) ) .

The decoding sub-block solves the minimization problem min λ 1 D F D (p Bλ 2 +θ c , p B(λ 1 +λ 2 ) ) .

New criteria An hybrid proximal point algorithm

Goal: Link the two (independent) criteria using proximal point technique [START_REF] Luque | Asymptotic convergence analysis of the proximal point algorithm[END_REF].

Demapping min λ 2 J θ m (λ 1 , λ 2 ) = min λ 2 D F D (p Bλ 1 +θ m , p B(λ 1 +λ 2 ) ) + µ m D F D (p B(λ (k) 1 +λ (k) 2 ) , p B(λ 1 +λ 2 ) ) Decoding min λ 1 J θ c (λ 1 , λ 2 ) = min λ 1 D F D (p Bλ 2 +θ c , p B(λ 1 +λ 2 ) ) + µ c D F D (p B(λ (k) 1 +λ (k+1) 2 ) , p B(λ 1 +λ 2 ) )
Optimal choice of the step-size :

Choose µ m such that J θ m (λ (k) 1 , λ (k+1) 2 ) ≤ J θ c (λ (k) 1 , λ (k) 2 ) Choose µ c such that J θ c (λ (k+1) 1 , λ (k+1) 2 ) ≤ J θ m (λ (k) 1 , λ (k+1) 2
) Both criteria decrease with the iterations. Convergence towards the same stationary point than the classical iterative decoding.

An hybrid minimum entropy algorithm

Goal: Improve the performance of the iterative decoding.

Rationale:

The entropy of the APP gives a measure of the reliability of the decision : E B(λ 1 +λ 2 ) → 0 means that the iterative decoding is confident about its decisions [START_REF] Kocarev | Nonlinear dynamics of iterative decoding systems: Analysis and Applications[END_REF].

Demapping min λ 2 D F D (p Bλ 1 +θ m , p B(λ 1 +λ 2 ) ) + η m E B(λ 1 +λ 2 ) Decoding min λ 1 D F D (p Bλ 2 +θ c , p B(λ 1 +λ 2 ) ) + η c E B(λ 1 +λ 2 )

Simulation

The generator polynomial of the encoder is g = [111; 001; 100]. The bits are mapped using subset partitioning to a 8-PSK modulation. The length of the coded bit sequence is L c = 6000. The step-sizes η m and η c in the HMEA are both chosen equal to 0.05.

  apping(d km+1 , d km+2 , ..., d (k+1)m ) BICM-ID receiver with soft-decision feedback ) p AP P (d km+i = b) = K ′ m s k ∈Ψ i b p(y k | s k ) j p(d km+j ; I) Prior × Channel probability → Marginalization Extrinsic (Demapping) p(d km+i = b; O) = K m s k ∈Ψ i b p(y k | s k ) j =i p(d km+j ; I) APP (Decoder) p AP P (c l = b) = K ′ j ; I) (I C indicator f unction of the code) Prior × Indicator function → Marginalization Extrinsic (Decoder) p(c l = b; O) = K c

  versus EbN0 -Right: Iteration number versus EbN0


David Houdoux

Binh Thai

Axelle Nguyen
email: axelle.amon@univ-rennes1.fr

Amon

Jérôme Crassous

Plastic flow and localization in an amorphous material: experimental interpretation of the fluidity

Keywords: numbers: 83.50.-v, 83.80.Fg, 62.20.F-

We present a thorough study of the plastic response of a granular material progressively loaded. We study experimentally the evolution of the plastic field from a homogeneous one to an heterogeneous one and its fluctuations in term of incremental strain. We show that the plastic field can be decomposed in two components evolving on two decoupled strain increment scales. We argue that the slowly varying part of the field can be identified to the so-called fluidity field introduced recently to interpret the rheological behavior of amorphous materials. This fluidity field progressively concentrates along a macroscopic direction corresponding to the Mohr-Coulomb angle.

I. INTRODUCTION

A physical description of the elementary mechanisms underlying the plasticity of amorphous materials has emerged in the past few years based on experimental evidences and numerical investigations [START_REF] Barrat | Heterogeneities in amorphous systems under shear in Dynamical Heterogeneities in Glasses[END_REF]. One of the basic ingredient is the fact that at an elementary level, plasticity occurs through local plastic events implying only a few number of constituents [START_REF] Argon | [END_REF]3], typically a few tens [4][5][6]. Such events are schematically represented in Figure 1(a). When such events occur, they redistribute stress [7][8][9]. This redistribution can trigger other rearrangements generating avalanches of events. An important point of this description is its universality as the stress redistribution is independent of the local interaction between the constituents. The mechanical properties that intervene in this description are the elastic properties which characterize the amorphous material on a large scale as an effective medium. The rearrangement can be treated theoretically as a small inclusion in an elastic matrix and the stress redistributed can be computed as shown by J. D. Eshelby [10]. The coupling between the plastic events is then quadrupolar.

While there is now an agreement about the validity of this picture in the community, the question of how this microscopic picture builds up in a macroscopic flow is still open.

For athermal materials, an important point is to model how the flow can be self-sustained. A possible approach is at a coarse-grained level. In order to describe rheology of yield stress material, Hebraud and Lequeux [11] introduced a local internal variable which represents the spatial and temporal density of plastic events occurring at a given time and at a given position. At a given point, this quantity evolves due to some stress relaxation, macroscopic loading and a background of mechanical noise due to plastic events occurring everywhere in this sample. The use of this internal variable, usually called fluid- ity, have been very fruitful to explain the rheology of yield-stress fluid. However, numerous studies evidence that local constitutive equations for the rheology are not compatible with the observations [12]. A plastic flow in a point of a material has an effect on the response of the material at some distance. A new class of nonlocal constitutive models have then been introduced [12,13].

The length characterizing the range of this interaction between events is called the cooperativity length and is supposed to depend on the distance of the local stress to the yield stress. Such a model has been adapted to granular materials and describes the stationary response of granular materials in numerous configurations [14,[START_REF] Henann | Proc. Natl. Acad. Sci[END_REF]. An important feature usually observed in amorphous materials is the fact that strain localization are observed at a large scale [START_REF] Schall | [END_REF]. When a yield stress material is sheared homogeneously the deformation is not homogeneously allocated in the sample but is concentrated in thin parts, called shear bands, in which the strain rate is large while the other parts of the material experience small strain rate (Fig. 1(b)). A description of how those stationary shear bands emerge from microscopic elementary plastic events or from an homogeneous fluidity field is still missing. At a macroscopic point of view, to describe shear bands is to consider a stress-based failure criterion. When the local stress is larger than a threshold, called the yield stress, the material flows. In the particular case of granular materials, the failure criterion is given by the Coulomb threshold: failure occurs when the ratio of the local shear stress to the pressure is larger than tan φ where φ is the angle of internal friction. Experimentally the internal friction angle is defined from the value of the yield stress. It may also been determined from the angle between the failure plane and the principal stress direction using a Mohr-Coulomb construction. Such a description of plasticity of shear bands has several limitations and in particular the fact that no lengthscale is introduced in the failure criterion so that the width of the shear band cannot be predicted.

In summary, the picture emerging from the literature is the following. At a microscopic scale the particles move. At a mesoscopic scale, individual plastic events may be defined, and such events are coupled by elasticity. At a macroscopic scale, the rate of such events may be theoretically represented by a variable called fluidity, which varies at the scale of the flow. Finally, macroscopic experimental observations show that strain localization occurs. Experiments or numerical simulations showing simultaneously those different behaviors are missing. We propose in this study an experiment that evidences many features of those plastic flow behaviors. For this we performed experiments on a shear flows of athermal spheres. Using an interferometric technique, we are able to follow the fluctuations of plasticity. Those fluctuations evidence some features of individual plastic events, such as the coupling of events by elasticity. If those fluctuations are averaged, a slowly varying field of deformation emerges, that may be identified with the fluidity field defined theoretically. Depending of the temporal scales at which the plasticity field is observed, different behaviors at mesoscopic and macroscopic scales may be evidenced simultaneously.

The manuscript is organized in the following way. In section II we present the experimental setup. In section III, the behavior of the correlation functions of the plasticity field is investigated. Those correlations functions can be separated into a slow and a fast component as shown in section IV. Finally, in section V, we discuss the identification of the slow component of the plastic flow to the fluidity field.

II. EXPERIMENTAL SET-UP

The experimental setup consists of a biaxial compressive test in plane strain conditions already described extensively in [17]. This kind of plane-strain compression is also very common in geomechanics [18]. The granular material (dry glass beads of diameter d = 70 -110 µm, initial volume fraction ≈ 0.60) is placed between a preformed latex membrane (85 × 55 × 25 mm 3 ) and a glass plate. A pump produces a partial vacuum inside the membrane, creating a confining stress -σ xx 30 kPa (see Fig. 2). The sample is placed in the biaxial apparatus where displacement normal to the xy plane is prevented by the front glass plate and a back metallic one, ensuring plane-strain conditions. At the bottom the sample is blocked by another metallic plate, while an upper plate is displaced vertically by a stepper motor. The stress applied at the top of the sample is then -σ yy = -σ xx + F/S, where F is the force measured by a sensor fixed to the upper plate, and S the section of the sample. The velocity of the motor is 1 µm • s -1 , leading to a deformation rate ε = 1.2 × 10 -5 s -1 (where the deformation is defined as ε = -ε yy = δ/L -see inset Fig. 3). The corresponding loading curve is presented in Fig. 3(b). We work in the quasistatic regime. Note that the stress gradient due to gravity is negligible, and the value of the confining stress is too low to expect a crushing of particule. To break the symmetry when failure occurs, for instance, to study the behavior of the shear band (see [19]), the metallic plate at the bottom of the sample can freely translate in the x-direction thanks to a roller bearing. In the following, experiments that are presented are made in this latter configuration.

Deformations are observed through the front glass plate using a Diffusing Wave Spectroscopy (DWS) method already described before in [20,21]. A laser beam at 532 nm is expanded to illuminate the entire sample. The light undergoes multiple scattering inside the granular material and we collect the backscattered rays. The latter interfere and form a speckle pattern. The image of the front side of the sample is recorded by a 7360 × 4912 pixels camera. Speckle images are subdivised in square zones of size 16 × 16 pixels and compared using a correlation method explained elsewhere [20,22]. For each zone the correlation between two successive images 1 and 2 is computed as follows:

g (1,2) I = I 1 I 2 -I 1 I 2 I 2 1 -I 1 2 I 2 2 -I 2 2
(1)

where I 1 and I 2 are intensities of the pixels of two successive images and the averages . are done over the 16 × 16 pixels of a zone. We obtain correlation maps where each pixel is calculated from eq. ( 1) and corresponds to a volume of area in the front plane 4d × 4d and depth of a few d. Examples of such maps of correlation are given in Fig. 3(a). The normalisation used in (1) leads to values for g I in the interval [0, 1] (see the colorscale). The decorrelation of the backscattered light (i.e. low value of g I ) comes from relative beads motions as, for example, combinations of affine and nonaffine bead deplacements or rotation of nonspherical beads. In the following, maps of correlation are calculated between two successive images with a fixed axial deformation increment δε = 3.5×10 -5 .

III. SPATIO-TEMPORAL CORRELATION FUNCTIONS

Observations made on the correlation maps during the loading are the same as those already presented in [17,19,23]. In the sample, deformations are distributed inhomogeneously (see Fig. 3(a)). At the beginning of the loading, plastics events are randomly distributed throughout the sample. After a deformation of a few percent (ε 3 %) these plastic activities are organized along intermittent micro-bands until one or two permanent shear bands are established. We relate the former to avalanches of local rearrangements predicted in the microscopic descriptions of the plasticity of amorphous materials. The latter are in agreement with the Mohr-Coulomb model. In the following, we present a method (see [START_REF] Barrat | Heterogeneities in amorphous systems under shear in Dynamical Heterogeneities in Glasses[END_REF]) is calculated on a square area of (4d × 4d) at position r. I 1 and I 2 are intensities of pixel images taken respectively at ε 1 = ε and ε 2 = ε + δε. Thus a(r, ε) represents the activity at position r around the deformation ε.

We define also a spatio-temporal correlation function as follow :

C(ε, dε, dr) = a(r , ε ) • a(r + dr, ε + dε) -a(r , ε ) a(r + dr, ε + dε) (2)
where r and r + dr have to be in the region of interest (ROI, see Fig. and consequently a(r, ε) are calculated over an increment δε = 3.5 × 10 -5 . So there is no integration of mean displacement over [ε ; ε + ∆ε]. Averages are made over all pairs of pixels for which r and r + dr are in the ROI and over a deformation span ∆ε. In the following, the value of ∆ε is fixed at ∆ε = 0.35 %. This value is large enough to have good averages but remains small compare to the total deformation applied. Note that in all the following the deformation plays the role of time. Our experiments are in quasi-static conditions. A process which can be observed only for small deformation increments will be called a fast process in the following while a feature that lasts for large strain increments will be called a slow process.

B. Temporal scale of fluctuations

Throughout the loading, we can observe on the correlation maps that micro-bands are fluctuating rapidly while shear bands evolves slowly. To quantify the temporal scale of the fluctuating part, we compute temporal correlations as follow:

C T (ε, dε) = C(ε, dε, dr = 0) (3)
where C T can be interpreted as a spatial autocorrelation function giving only the temporal information.

C. Separation method

The aim is to separate the two plastic phenomena throughout the loading. Our method is based on their different temporal properties. As discussed in section III B, the fluctuating part is no longer correlated for a gap of deformation larger than dε * . In the case when dε dε * , the spatio-temporal correlation function C(ε, dε, dr) corresponds only to the slow part. In the following, we define the correlation function for the slow part by taking arbitrary dε = ∆ε/2 = 0.175 % dε * for the spatio-temporal correlation function:

C slow (ε, dr) = C ε, ∆ε 2 , dr (4)
In the same way, for dε dε * the spatio-temporal correlation function C(ε, dε, dr) contains all the fluctuations. Thus, we define the total correlation function by taking dε = 0 as follows:

C tot (ε, dr) = C(ε, 0, dr) (5)
Finally, the correlation function describing the fluctuating part is obtained as follows: Fig. 5 shows the correlations functions C tot , C slow and C fast defined previously. Those correlations functions are computed at different macroscopic deformation ε = 2.80 % (well before failure), ε = 5.95 % (close to failure), and ε = 8.75 % (post-failure).

C fast (ε, dr) = C tot (ε, dr) -C slow (ε, dr) (6)

IV. EXPERIMENTAL RESULTS

A. Behavior of the slow and fast parts

In this subsection, we discuss the results obtained by the calculation of the slow and fast parts of the correlation functions, and we evidence the differences of behavior between the long and the short time correlation functions.

We first discuss the correlation function near failure (middle column in Fig. 5) where the characteristic behavior is clearly visible. The total correlation function C tot (see Fig. 5(a2)) displays two different spatial features. Near dr = 0, we observe a high correlation along two directions, forming a small cross at the center of the figure. At large distance, the correlation prevails along a large inclined band spanning the full image. The fast part and the slow part of the total correlation function displays each only one of those two spatial behaviors. The fast part displays only the small cross near dr = 0 (Fig. 5(c2)), whereas the large inclined band belongs to the slow part (Fig. 5(b2)).

In the pre-failure stage, Fig. 5(c1) shows a fluctuating contribution at short distances, but the slow part of the correlation function (Fig. 5 For this, we need to determine the angles at which the correlation enhancement occurs. We use a projection method inspired by what has been done previously [19]. As the correlation maps are centered (see Fig. 5), we compute the mean value of the pixels intersected by a line passing through the center and with an angle θ with the x-axis. Next we plot these mean values as a function of the angle θ between -π/2 and +π/2. Fig. 6 As it has been shown previously, we clearly observe two different angles for the correlation function. The fast part of the correlation function is quadrupolar, with an average inclination θ E 48 • . This orientation may be directly linked to the direction of maximum stress redistribution given by the Eshelby tensor. We can interpret the fast correlation of the plastic events as small cascades of events as represented schematically in Figure 8. Such fluctuations has also been reported in few numerical studies [7,24].

The slowly varying part of the correlation function is oriented at an angle θ M C

63 • . This orientation may be related to the formation of a macroscopic shear band into the material. The orientation of the shear band depends on the value of internal friction ratio accordingly to a Mohr-Coulomb construction. This slow variation is related to large scale cooperative effects. It is important to note that two scales, and the two orientations coexist during the approach to failure (see Fig. Figure 9 summarizes schematically our results. At the microscopic level, some grains reorganize when the applied stress is increased: this is one "elementary event", that is represented as a dot in Fig. 9(a1). This behavior can be observed at the very beginning of the experiment. On a larger time-scale, we observe correlated cascades of events which fluctuate rapidly Fig. 9(b1). Those microavalanches are inclined along a direction θ E given by the Eshelby stress tensor (see Fig. 8). Increasing further the deformation field on a larger time scale, we observe a continuous field of plasticity (see Fig. 9(a3)).

At the beginning of the loading, the events repartition is roughly homogeneous at the sample scale (Fig. 9(a1)) and the corresponding mean plastic field is also homogeneous (Fig. 9(c1)). When the loading progresses, the events mainly accumulate on an active zone (Fig. 9(a2) and (a3)), and the plasticity field then becomes heterogeneous (Fig. 9(c2) and ( c3)). This zone of activity is the precursor zone of the final shear band, and is inclined at an angle θ M C . Finally, at failure (Fig. 9(c4)), all the activity is concentrated on a narrow and stationary shear band.

C. Fluidity as slow part of the plastic field.

The physical interpretation of what is called fluidity is still a debated issue [25]. The literature agrees on the fact that it corresponds to a coarse-grained field directly linked to the local plastic activity. Among the hypotheses underlying the Kinetic Elasto-Plastic model of Bocquet et al. [13] an important point is a Boltzmann Stosszahlansatz-like hypothesis of the decoupling of the plastic-event dynamics. This means that the fluidity field is defined on a time scale large enough for the memory of the coupling between the underlying events to be lost.

We interpret the slow part of the plastic field (Fig. 9(c1) to (c4)) as the field of the so-called fluidity. Indeed, this field evolves smoothly spatially and has lost the quadrupolar symmetry of the underlying fluctuations of the plasticity. It corresponds to a coarse-grained field reflecting the local plastic activity and thus matches the theoretical definition of fluidity.

In the specific case of sheared granular materials, it has been proposed recently that a granular fluidity g may be defined which is proportional to a local shear rate γ and which is also proportional to the velocity fluctuations δv [26]. Experimentally (see [22] for details), the speckle decorrelation is a function of the affine and the non-affine displacements of particles ξ 2 0 f (ε)+ < δξ 2 >, where ξ 0 is a length of the order of the grain diameter, f (ε) is a quadratic function of the affine deformation field ε, and < δξ 2 > is the mean square non-affine displacement. The spatial scale on which the affine displacement field is defined is few bead diameters [22]. For a sheared granular material, during a time τ , we expect

θ E θ E MC θ θ E MC θ MC θ MC θ MC θ (a1) (a2) (a3) (a4) (b3) (b1) (b2) (b4) (c1) (c2) (c3) (c4)
absolute axial strain incremental strain ε dε FIG. 9. Schematic representation of the plastic activity at different values of the axial loading ε (horizontal axis) and for different strain increments dε. At an elementary level, plasticity is awaited to occur through local plastic events represented as dots on the first line. When integrating those spots on a larger strain increment, a correlated micro-strucutre emerges governed by the elastic properties of the bulk material (second line). The angle characterizing this elastic coupling, θE, is given by the Eshelby kernel. On a larger strain increment this fluctuating pattern is lost and only a slowly evolving coarse-grained field remains. Shear band formation can be observed on this slow field as the emergence of an orientation θMC in the mean plastic field. The grey area displays the features actually observed experimentally. The reason why (b4) is not observed experimentally is linked to the nonlinearity in deformation of the measurement methods: the observed field saturates in the band. (a1)-(a4) are not observed because the acquisition rate is too small to reach small enough dε.

that f (ε) ∼ ( γτ ) 2 [22] and < δξ 2 >∼ (δv τ ) 2 . It follows that ξ 2 0 f (ε)+ < δξ 2 > ∝ g. So our experimental probe of plastic activity seems closely related to the above-mentionned definition of the granular fluidity.

An important point in the definition of fluidity is the nature of the coarse-graining at play to obtain a smooth field. Theoretically fluidity is a variable obtained by a coarse-graining in space and time. It is unclear in the literature if this last coarse-graining is in "true" time or in strain increment. In our experiment, our technique of measurement gives intrinsically a coarse-grained measurement in space because it is based on multiple scattering of light (see [20,22] for a thorough discussion of the spatial resolution of the method). The measurement is also time-averaged as the acquisition rate of the camera is always much smaller than the inverse of the time of propagation of sound waves in the system. In addition to those resolution-based averages, we have studied in the present article the effect of a coarse-graining in strain increment on the observed plastic field. An important conclusion drawn from our observations is the fact that for a large enough strain increment dε the plastic field lose the memory of the "fast" correlation between the plastic events. The mean field thus obtained has its own dynamics on a slow scale. Consequently, there exists a strain increment above which a fluidity field can be defined as the slowly evolving part of the plastic field. Our experiment allows to follow a fluidity field which is homogeneous at the beginning of the loading, and which condensates on a inclined band. The fact that strain localization occurs in bi-and tri-axial tests performed on granular materials is well known since many decades. However, a clear understanding of this feature is still missing [5]. Some authors tried to relate strain localization with properties of the fluidity-field. In a first study, Goyon et al. [12] proposed a mechanism of formation of shear band in flows of emulsion. The authors assumed that walls that are at flow boundaries act as sources of fluidity. This fluidity then diffuses inside the sample, forming a fluid layer close to the wall. In Couette flow of granular material, fluidity increases preferentially close to the rotor because of the inhomogeneity of the stress field. In those cases, the fluidity source is localized, and the fluidity tends to spread into the sample.

In our experiment the situation is totally opposite. Fluidity accumulates on a band although the applied stress is increased homogeneously, i.e. the fluidity is initially homogeneous and finally heterogeneous. The single property that the fluidity diffuses inside the sample is not sufficient to explain this feature. Recently, Benzi and coworkers [27] have shown that shear band formation starting from an initially homogeneous fluidity in a fluidity model should be possible.

Another possible approach to understand this localiza-tion process can be to consider that there is some memory of where plasticity has already occurred. In the case of cohesive materials, the introduction of a modification of the local elastic modulus in mesoscopic models is known to lead to some localization [28]. But the memory of previous local plastic activity is not specific to cohesive materials and can take different forms. It may described any local structural modification due to plasticity as for example a modification of the local structure or of the local packing fraction as well as the wear of some frictional contacts. In the case of shear band formation in granular materials, it is known that the packing fraction inside the flowing band takes a particular value (the socalled critical state) [START_REF]Granular media: between fluid and solid[END_REF]. The damage internal variable to take into account might then be the local packing fraction which should evolve when plastic events take place. The introduction of some memory or history-dependent effect is known to generate shear-banding in mesoscopic models [START_REF] Martens | [END_REF]31].

E. Fluidity and Mohr-Coulomb theory

As mentionned in the Introduction, the Mohr-Coulomb theory gives a first order description of failure threshold and shear bands orientation for frictional materials [START_REF]Granular media: between fluid and solid[END_REF][START_REF]Statics and Kinematics of Granular Materials[END_REF]. This theory introduces a frictional internal angle which is closely linked to the angle of repose for a granular material. Localisation is predicted to occur along a line which inclination is given by the internal friction. This picture of failure is very simplified but captures the prominent features observed experimentally.

In the previous parts we have shown that a shear band can be described by an inhomogeneous concentration of fluidity. Thus, it is very appealing to describe the emergence of a shear-band with a spontaneous orientation in the framework of fluidity theory. To our knowledge, in the current versions of this theory, shear bands orientation is always given by the loading geometry. It is very likely that a theory relying on a single internal variable is insufficient to predict the emergence of an angle in an homogeneous stress field. As discussed in the previous part, the introduction of another internal variable describing a form of damage might be necessary to display localization along an inclined line. This damage variable is likely to be linked to the local packing fraction. Nevertheless, the existence of a macroscopic friction in the absence of dilantancy for frictionless particles [START_REF] Peyneau | [END_REF] suggests that packing fraction might not be the right variable.

Very recently, a mesoscopic model using a local Mohr-Coulomb criterion and taking into account the tensorial nature of the stress redistribution have been able to reproduce macroscopic localization in the absence of a damage variable [34] as well as an orientation for avalanches different from the local coupling. Taking into account the redistributed pressure in addition to the deviatoric stress suggests also to introduce another internal variable to describe the effect of the pressure redistribu-tion.

VI. CONCLUSION

In this work we have discussed the nature of the spatiotemporal correlations observed in the plastic response of a granular material submitted to a biaxial test. We have detailed a procedure allowing us to separate a fluctuating component of the plastic field at small strain increments from a slowly evolving component. Those two components correspond to two different types of behaviors: the two fields have independent and coexisting characteristic orientations. We discuss the interpretation of those two behaviors in the framework of the present debated theories of plasticity and rheology of amorphous materials. We argue that the slowly varying component of the plastic field is a good candidate to be interpreted as the so-called fluidity field introduced in the last ten years to describe nonlocal effects in the flow of amorphous materials. We underline the importance of coarse-graining in strain increment in the definition of this field.

The question of the independence or correlation between the fast and slow components of the field is still open. Even if the two orientations observed are apparently unrelated it could be possible that the slow field inherits its characteristics from the fluctuations in a non trivial manner.

FIG. 1 .

 1 FIG. 1. Schematic pictures of a plastic flow in athermal materials. (a) At a microscopic scale, a small goup of elements reorganizes and redistributes mechanical stress in the material. This initial event can trigger a cascade of other rearrangements represented schematically with other colors to indicate their order of occurence. (b) At a macrocsopic scale, the flow concentrates into a narrow zone of the material such as a shear band.

FIG. 2 .

 2 FIG.2. Schematic representation of a biaxial setup. The granular material is enclosed between a latex membrane and a glass plate (not represented here). A partial vacuum inside the sample creates a confining stress -σxx = 30 kPa. At the top, a moving plate exerts a compression of the sample at fixed velocity along the y-axis. The back plate as well as the front glass one forbid displacement along the z-direction ensuring plane-strain conditions.

FIG. 3 .

 3 FIG. 3. (a) Correlation maps gI for three different values of ε (2.80 %; 5.95 % and 8.75 %). The region of interest used for image analysis is shown in dashed line. (b) Loading curve. Dots symbolize the positions of the three correlation maps. Inset : notations.

  3(a)) and the deformation ε in the interval [ε; ε+ ∆ε]. Keep in mind that g (1,2) I

FIG. 4 .

 4 FIG. 4. (a) Temporal correlation function as a function of dε at the beginning (ε = 2.80%) and at the end (ε = 8.75%) of the loading. (b) Rescaled correlation functions showing the initial drop over a characteristic gap dε * ≈ 0.01 %.

Fig. 4

 4 Fig. 4 shows C T (ε, dε) as a function of dε at two points in the loading ε = 2.80 % and ε = 8.75 %. A rapid decorrelation for a gap dε * ≈ 0.01 % is initially observed, followed by a second smoother decay which tends to a nonzero finite value because of the heterogeneity of the mean field. The scale in deformation increment over which the initial drop takes place, dε * ≈ 0.01 %, gives us an estimation of the temporal scale of the fluctuating part.

FIG. 5 .

 5 FIG. 5. Spatial representation of the different parts of the correlation function for different values of the loading. All maps are centered at dr = 0. The spatial scale, in unit of beads diameter, is represented in (b1). Top row ((a1), (a2) and (a3)): total correlation function Ctot; Middle row ((b1), (b2) and (b3)): slow part C slow of the correlation function; Bottom row ((c1), (c2) and (c3)): fast part C fast of the correlation function. Left column, the system is well before failure, ε = 2.80 %; Middle column, it is close to the failure ε = 5.95 %; Right column, after failure ε = 8.75 %. Scale represented by intervals corresponds to white and black pixels respectively.

  (b1)) remains spread on the full image and displays no structure. After failure, the slow part of the correlation function appears clearly correlated along two bands (Fig. 5(b3)), whereas the quadrupolar structure of the fluctuating part near the origin is no longer visible. B. Geometrical characterisation of the correlations.

Fig. 5

 5 Fig.5evinces that the fast and slow components of the correlation functions are both anisotropic, but with slightly different directions. We now use this observation to quantify the duality of the correlation functions with the time-scale.For this, we need to determine the angles at which the correlation enhancement occurs. We use a projection method inspired by what has been done previously[19]. As the correlation maps are centered (see Fig.5), we compute the mean value of the pixels intersected by a line passing through the center and with an angle θ with the x-axis. Next we plot these mean values as a function of the angle θ between -π/2 and +π/2. Fig.6(a) shows the result of such projection of the fast part of the correlation C fast for an axial deformation ε = 5.95 %. Fig.6(b)shows the projection of the slow part of the correlation C slow for ε = 8.75 %. From a projection profile as the ones obtained in Fig.6, we determine the angle characterizing a response as the mean of the absolute values of the angles at which occur the maxima of the curve. Error bars are determined as the difference between the angles at which the maxima occur and the ones obtained from gaussian interpolations of the projection profile on each halves of

  Fig.5evinces that the fast and slow components of the correlation functions are both anisotropic, but with slightly different directions. We now use this observation to quantify the duality of the correlation functions with the time-scale.For this, we need to determine the angles at which the correlation enhancement occurs. We use a projection method inspired by what has been done previously[19]. As the correlation maps are centered (see Fig.5), we compute the mean value of the pixels intersected by a line passing through the center and with an angle θ with the x-axis. Next we plot these mean values as a function of the angle θ between -π/2 and +π/2. Fig.6(a) shows the result of such projection of the fast part of the correlation C fast for an axial deformation ε = 5.95 %. Fig.6(b)shows the projection of the slow part of the correlation C slow for ε = 8.75 %. From a projection profile as the ones obtained in Fig.6, we determine the angle characterizing a response as the mean of the absolute values of the angles at which occur the maxima of the curve. Error bars are determined as the difference between the angles at which the maxima occur and the ones obtained from gaussian interpolations of the projection profile on each halves of

FIG. 6 .

 6 FIG. 6. Angular projection of the spatial correlation functions, i.e. mean value of the pixels intersected by a line passing through the center of a spatial correlation function and making an angle θ with the x-axis as a function of θ for (a) C fast at ε = 5.95 % and (b) C slow at ε = 8.75 %.

Fig. 7

 7 Fig.7shows the values of the angles at which the fast and the slow part of the correlation function are respectively maximals as a function of the macroscopic deformation in the ε ∈ [2.45 %; 9.80 %] range. The failure occurs for ε 6.65 %. We observe that angles at which correlation occurs are clearly different for the fluctuating and the slow parts. For a deformation in the range ε ∈ [5.60 %; 6.65 %] the coexistence of a fast and slow correlation oriented at different angles is clearly visible. Outside this range of deformation, we were not able to define two different angles simultaneously in the correlation functions.

FIG. 7 .

 7 FIG. 7. Angles of the fast and the slow parts of the spatial correlation function. A coexistence of two distinct directions is observed just before the failure represented by the dashed line. εc stands for the critical value of the strain when the failure occurs.

FIG. 8 .

 8 FIG. 8. Schematic representation of our interpretation of the microbands as cascades of elementary events. (a) Local rearrangements coupled by elasticity. (b) Resulting deformation integrated on a larger time-scale.

  7): the fluctuations of plasticity are oriented along θ E whereas the mean plasticity is oriented along θ M C . The two angles seem relatively constant during the loading process, and no intermediate orientations are observed, even close to failure.

B

  . Schematic interpretation of the experiment.

D

  . Inhomogeneous fluidity field and localization.

VII. ACKNOWLEDGEMENTS

The authors thank Sean McNamara and Jérôme Weiss for many scientific discussion and acknowledge funding from Agence Nationale de la Recherche (ANR "Relfi", ANR-16-CE30-0022).