LOOKING AT MATHEMATICS EDUCATION

FROM AN INSTITUTIONAL PERSPECTIVE

FROM AN INSTITUTIONAL PERSPECTIVE

NOTE: This document was developed from my lecture notes for the course MATH 645 for students in the Master in the Teaching of Mathematics (M.T.M.) graduate program at Concordia University, September-December 2006.

It has long been accepted that learning has not only a cognitive dimension, but also emotional and social dimensions (Illeris, 2004). The social dimension could be split into two sub-dimensions: one focused on the spontaneous social interactions, and the other - on interactions that are structured by an institution.

There is a difference between spontaneous learning in the so-called "everyday life" and learning in school (see, e.g., Lave, 1988; Walkerdine, 1988; Sierpinska, 1995; Brenner & Moschkovich, 2002). Different things are learned and different ways of thinking are developed. Learning mathematics in a mathematics course may also be different from learning mathematics "on the job” in financial or engineering institutions, or in technical positions in factories or offices, where it is mostly tacit knowledge.

Moreover, the mathematics learned in one course can be very different from the mathematics learned in another course even if the same names are used in both to describe their content. For example, an undergraduate statistics course for mathematics majors can teach different theory and procedural skills than a statistics course for social sciences students. The differences are also stricking if one looks at the contents of courses over time. For example, logarithms in upper secondary schools in 1950s were mainly computational tools; their most important property was that of transforming multiplication into addition (log_{b} (xy) = log_{b}x + log_{b}y). Today, the main concept of school mathematics is that of function; existence of inverse function is one of the structural concepts studied; logarithmic functions are studied as inverses of exponential functions and the main role of both is to model certain growth phenomena.

A student's learning experience - in its cognitive and emotional aspects - in a mathematics course could also differ depending on whether he or she takes the course by choice or because the administration of the institution forces the student to take the course as a prerequisite for a non-mathematical program of study.

In fact, it may be difficult to describe what it means to know this or that domain of mathematics without mentioning the institution to which the knowing subject is being held accountable, and the historical period of the subject's participation in the institution. Academic knowledge depends on the space and time in which it is being considered as academic knowledge. This is not unrelated with rules and norms governing the requirements of academic degrees. We could take the example of the notion of vector space, taught, today, in many linear algebra university courses. When, in the 1840s, Grassmann developed a way of thinking about vectors that was close to the axiomatic approach in present-day vector space theory, it was not recognized as an obvious addition to the existing mathematical knowledge but rather as an idiosyncratic product of an eccentric mind. The “institutionalization” of vector spaces came almost a century later, after the axiomatic approach to defining concepts became an accepted practice in mathematics, and it was accompanied by the development of a clearer terminology and a simpler notation (Dorier, 2000). Vector space theory started to be taught at universities in the 1930s, at the graduate level. Today, it is often taught at the undergraduate level and it would be certainly very difficult to qualify as a mathematician (i.e. obtain a master’s or doctoral degree in mathematics) without thinking of vectors as elements of a vector space and using the general, structural properties of operations on vectors and transformations of vector spaces in routine techniques of solving theoretical or applied mathematical problems.

Moreover, the mathematics learned in one course can be very different from the mathematics learned in another course even if the same names are used in both to describe their content. For example, an undergraduate statistics course for mathematics majors can teach different theory and procedural skills than a statistics course for social sciences students. The differences are also stricking if one looks at the contents of courses over time. For example, logarithms in upper secondary schools in 1950s were mainly computational tools; their most important property was that of transforming multiplication into addition (log

A student's learning experience - in its cognitive and emotional aspects - in a mathematics course could also differ depending on whether he or she takes the course by choice or because the administration of the institution forces the student to take the course as a prerequisite for a non-mathematical program of study.

In fact, it may be difficult to describe what it means to know this or that domain of mathematics without mentioning the institution to which the knowing subject is being held accountable, and the historical period of the subject's participation in the institution. Academic knowledge depends on the space and time in which it is being considered as academic knowledge. This is not unrelated with rules and norms governing the requirements of academic degrees. We could take the example of the notion of vector space, taught, today, in many linear algebra university courses. When, in the 1840s, Grassmann developed a way of thinking about vectors that was close to the axiomatic approach in present-day vector space theory, it was not recognized as an obvious addition to the existing mathematical knowledge but rather as an idiosyncratic product of an eccentric mind. The “institutionalization” of vector spaces came almost a century later, after the axiomatic approach to defining concepts became an accepted practice in mathematics, and it was accompanied by the development of a clearer terminology and a simpler notation (Dorier, 2000). Vector space theory started to be taught at universities in the 1930s, at the graduate level. Today, it is often taught at the undergraduate level and it would be certainly very difficult to qualify as a mathematician (i.e. obtain a master’s or doctoral degree in mathematics) without thinking of vectors as elements of a vector space and using the general, structural properties of operations on vectors and transformations of vector spaces in routine techniques of solving theoretical or applied mathematical problems.

Walkerdine, V.: 1988, *The Mastery of Reason*, Routledge: London.

Lave, J.: 1988, *Cognition in Practice: Mind, Mathematics and Culture in Everyday Life*, Cambridge, NY: Cambridge University Press.

Sierpinska, A.: 1995, 'Mathematics: 'in content', 'pure' or 'with applications'? A contribution to the question of transfer in the learning of mathematics, *For the Learning of Mathematics* 15.1, 2-15.

Brenner, M.E. & Moschkovich, J.N. (Eds): 2002, Everyday and Academic Mathematics in the Classroom, *Monograph No. 11 of the Journal for Research in Mathematics Education*, Reston, VA: NCTM.

Dorier, J.-L.: 2000, ‘Epistemological analysis of the genesis of the theory of vector spaces’, in J.-L. Dorier (ed.), *On the Teaching of Linear Algebra*, Dortrecht: Kluwer Academic Publishers, pp. 1-82.

Illeris, L.: 2004, *The Three Dimensions of Learning*, Malabar, Florida: Krieger Publishing Co.

In mathematics education research papers it is not uncommon to formulate proposals of changing the contents and methods of teaching as if it was up to the teacher reading these papers to make the proposed changes. The proposals often ignore the institutional constraints of teaching mathematics. The instructor has to respect these constraints or he or she risks losing the job.

The constraints are many and varied. Here are some examples:

- The curriculum, including the goals of mathematics education at the given level (e.g. primary, secondary, etc); the expected outcomes (which can be formulated in terms of general and specific "competencies" and/or in terms of mathematical concepts and techniques to be mastered).

- Approved sources of knowledge (e.g., a textbook considered conform to the curriculum).

- The number of class hours per week.

- Teacher's degree of freedom in the choice of additional sources of knowledge (internet, software, collections of exercises, etc.).

- Teacher's administrative duties relative to the course (responsible or not for keeping records of students' attendance, for preparing the exams, etc.).

- Assessment: external vs internal; oral vs written; formative vs evaluative; kinds of exam questions; criteria of evaluation; grading schemes; schedule, etc.

- Resources (human resources, e.g. the number of students per teacher; financial resources, e.g. funds for technological equipment, funds for field trips, etc.).

- Available technology (e.g., photocopiers; over-head and data projectors; computers; hand-held calculators; tables of functions, etc.)

We could easily give more examples of institutional constraints of mathematics teaching.

But how do we*systematically *study the processes of teaching and learning mathematics without losing sight of the fact that they occur within a particular institution? We need a framework that would direct our attention to crucial aspects of this institution and structure our data and our analyses.

If we are interested in describing the institutional aspects of mathematics education at the scale of a country or compare the systems of mathematics education of two or more countries, than perhaps a definition of an institution in general, together with a framework for analyzing any institution could be useful. For example, we could use Ostrom's framework for the analysis and development of institutions (ADI, see Ostrom, 2005).

If the object of our research is a particular mathematical domain (e.g. algebra, geometry or calculus), and we are interested in knowing how the various elements of this domain have been institutionalized in the history of mathematical research, or in the history of mathematics curricula at a certain level (elementary, secondary, tertiary), or in the customary didactic practices in a given culture or country, then Chevallard's Anthropological Theory of Didactics (ATD) could be useful (Chevallard, 1992; 1997; 1999; 2002; for applications of this theory see, e.g., Barbe et al., 2005; Castela, 2004).

If we are looking at didactic relations between a teacher, his or her students and the mathematical content planned to be taught and learned in a particular lesson or sequence of lessons, then we could be inspired by the work of sociologists of organizations such as Crozier & Friedberg (1977) and the framework of the Theory of Didactic Situations in mathematics, developed especially for mathematics education by Brousseau (1997).

As it turned out in my own research, a combination of such macro and micro institutional perspectives can be necessary for some research problems. I and my colleagues were looking at the phenomenon of frustration among adult students forced to take elementary mathematics courses as a condition for their admission into university programs such as Commerce or Psychology. Our subjects' frustrations were closely related with the fact that they were learning mathematics within a specific institutional context, namely the context of "prerequisite mathematics courses". This institution defined them at the outset as inadequate in some respect: "lacking" knowledge that other candidates already possessed (Sierpinska, Bobos & Knipping, 2007). The framework we used in planning the research, designing a questionnaire and analyzing the data, was based on variables describing the rapport of the students with both the macro elements of the institution of the prerequisite mathematics courses (e.g. the fact that courses were administered at the university, the rule forcing candidates to commerce to take calculus), and micro elements such as teachers' pedagogical and didactic practices and students' own techniques and strategies of learning mathematics.

The constraints are many and varied. Here are some examples:

- The curriculum, including the goals of mathematics education at the given level (e.g. primary, secondary, etc); the expected outcomes (which can be formulated in terms of general and specific "competencies" and/or in terms of mathematical concepts and techniques to be mastered).

- Approved sources of knowledge (e.g., a textbook considered conform to the curriculum).

- The number of class hours per week.

- Teacher's degree of freedom in the choice of additional sources of knowledge (internet, software, collections of exercises, etc.).

- Teacher's administrative duties relative to the course (responsible or not for keeping records of students' attendance, for preparing the exams, etc.).

- Assessment: external vs internal; oral vs written; formative vs evaluative; kinds of exam questions; criteria of evaluation; grading schemes; schedule, etc.

- Resources (human resources, e.g. the number of students per teacher; financial resources, e.g. funds for technological equipment, funds for field trips, etc.).

- Available technology (e.g., photocopiers; over-head and data projectors; computers; hand-held calculators; tables of functions, etc.)

We could easily give more examples of institutional constraints of mathematics teaching.

But how do we

If we are interested in describing the institutional aspects of mathematics education at the scale of a country or compare the systems of mathematics education of two or more countries, than perhaps a definition of an institution in general, together with a framework for analyzing any institution could be useful. For example, we could use Ostrom's framework for the analysis and development of institutions (ADI, see Ostrom, 2005).

If the object of our research is a particular mathematical domain (e.g. algebra, geometry or calculus), and we are interested in knowing how the various elements of this domain have been institutionalized in the history of mathematical research, or in the history of mathematics curricula at a certain level (elementary, secondary, tertiary), or in the customary didactic practices in a given culture or country, then Chevallard's Anthropological Theory of Didactics (ATD) could be useful (Chevallard, 1992; 1997; 1999; 2002; for applications of this theory see, e.g., Barbe et al., 2005; Castela, 2004).

If we are looking at didactic relations between a teacher, his or her students and the mathematical content planned to be taught and learned in a particular lesson or sequence of lessons, then we could be inspired by the work of sociologists of organizations such as Crozier & Friedberg (1977) and the framework of the Theory of Didactic Situations in mathematics, developed especially for mathematics education by Brousseau (1997).

As it turned out in my own research, a combination of such macro and micro institutional perspectives can be necessary for some research problems. I and my colleagues were looking at the phenomenon of frustration among adult students forced to take elementary mathematics courses as a condition for their admission into university programs such as Commerce or Psychology. Our subjects' frustrations were closely related with the fact that they were learning mathematics within a specific institutional context, namely the context of "prerequisite mathematics courses". This institution defined them at the outset as inadequate in some respect: "lacking" knowledge that other candidates already possessed (Sierpinska, Bobos & Knipping, 2007). The framework we used in planning the research, designing a questionnaire and analyzing the data, was based on variables describing the rapport of the students with both the macro elements of the institution of the prerequisite mathematics courses (e.g. the fact that courses were administered at the university, the rule forcing candidates to commerce to take calculus), and micro elements such as teachers' pedagogical and didactic practices and students' own techniques and strategies of learning mathematics.

Ostrom, E.: 2005, *Understanding Institutional Diversity*, Princeton, New Jersey: Princeton University Press.

Chevallard, Y.: 1992, ‘Fundamental concepts in didactics. Perspectives provided by an anthropological approach’, in R. Douady & A. Mercier (eds), *Research in Didactique of Mathematics, Selected Papers, extra issue of Recherches en Didactique des Mathématiques*, Grenoble: La Pensée Sauvage editions, pp. 131-167.

Chevallard, Y.: 1997, ‘Familière et problématique, la figure du professeur’, *Recherches en Didactique des Mathématiques* 17/3, 17-54.

Chevallard, Y.: 1999, ‘L'analyse des pratiques enseignantes en théorie anthropologique du didactique’, *Recherches en Didactique des Mathématiques* *19*/2, 221-266.

Chevallard, Y.: 2002, ‘Organiser l’étude 1. Structures et fonctions’, in J.-L. Dorier et al. (eds), *Actes de la 11e Ecole d’Eté de Didactique des Mathématiques - Corps 21-30 Août 2001*, Grenoble: La Pensée Sauvage editions, pp. 3-22.

Castela, C.: 2004, ‘Institutions influencing mathematics students’ private work: A factor of academic achievement’, *Educational Studies in Mathematics* 57.1, 33-63.

Barbé, J., Bosch, M., Espinoza, L., Gascón, J.: 2005, ‘Didactic restrictions on the teacher’s practice: the case of limits of functions in Spanish High Schools’, *Educational Studies in Mathematics 59*, 235-268.

Brousseau, G.: 1997, *Theory of Didactical Situations in Mathematics*, Dortrecht: Kluwer Academic Publishers.

Crozier, M. & Friedberg, E.: 1977, *L’Acteur et le système. Les contraintes de l’action collective*, Paris: Editions du Seuil.

Sierpinska, A, Bobos, G. & C. Knipping: 2007, ‘Presentation and partial results of a study of university students’ frustration in pre-university level, prerequisite mathematics courses: Emotions, positions and achievement’. Manuscript posted on the web, at __<http://www.asjdomain.ca/>__

Some theorizations discuss the difficulties and contradictions related to institutionalization of practices. For Wenger, often quoted by mathematics educators, "institutionalization must be in the service of practice. Practice is where policies, procedures, authority relations, and other institutional structures become effective. Institutionalization in itself cannot make anything happen." (Wenger, 1998: 243).

But some mathematics educators stress the fact that institutions are sometimes established to curtail certain practices, and impose others. They do make things happen, although these might not be the things officially declared as desirable for all concerned. For example, the rule of the prerequisite mathematics courses, mentioned in the Introduction, selects some people

It is perhaps these constraining aspects of educational mathematical institutions, which affect the lives of so many people, that have drawn the attention of researchers to the works of Bourdieu (e.g., 1972/2002; 1994) and Bernstein, (1973, a, b; 1977); see, for example, (Wedege, 1999; Sfard, 2001; Zevenbergen & Valero, 2004; Lerman & Zevenbergen, 2004; Gorgorio et al., 2004; Evans et al., 2006).

The institutional aspects are central in Chevallard’s

Although the institutional perspective on teaching and learning is explicit in ATD, this theory does not define the notion of institution (it is taken as a primitive term of the theory, Chevallard, 1992: 142) but constructs a framework for the study of institutional practices. An important element of this framework is the concept of “praxeology”. Praxeology is a system of types of tasks (which are accomplished within a given institution), techniques for solving them, technologies for the justification of techniques and theories for the justification of technologies. Institutions that have the same tasks but differ by the techniques they routinely use for accomplishing them are considered different institutions. This framework is sufficient in “didactics of mathematics”, that is, the area of research in mathematics education focused on “didactic systems”, whose task is to teach a specific subject matter (by constructing “mathematical” and “didactic” organizations). These systems exist within educational institutions as sub-institutions.

One could say that ATD takes a “macro-level” perspective on the mathematics classroom. In the

Processes of institutionalization of knowledge in the mathematics classroom have interested also other researchers who tried to explain them using concepts such as

1. Introduction

2. Overview of institutional perspectives in mathematics education research

3. Reading theories of institutions, elaborated within sociology or political science - some excerpts

4. Proposal of a synthesis of theories of institutions for the purposes of mathematics education research: SIF

5. Theory of Didactic Situations framed within the proposed SIF framework

6. Examples of research from an institutional perspective: Titles of MATH 645 students' term papers

Boaler, J.: 1997, *Experiencing School Mathematics, Teaching Styles, Sex and Setting*, Buckingham: Open University Press.

Abreu, G., Bishop, A.J., Presmeg, N. (Eds): 2002, *Transitions between Contexts of Mathematical Practices*, New York: Springer.

Venturini, P., Amade-Escot, C., Terrisse, A. (Eds): 2002, *Etude des Pratiques Effectives: l'Approche des Didactiques*, Grenoble: la Pensée Sauvage Editions.

Wenger, E.: 1998, *Communities of Practice: Learning, Meaning and Identity*, Cambridge, U.K., New York, NY: Cambridge University Press.

Bourdieu, P.: 1972/2002, *Esquisse d’une théorie de la pratique*, Paris: Editions du Seuil.

Bourdieu, P.: 1994,*Raisons pratiques. Sur la théorie de l’action*, Paris: Editions du Seuil.

Bourdieu, P.: 1994,

Bernstein, B.: 1973a, *Class, Codes and Control, vol. 1*, London: Routledge & Kegan Paul.

Bernstein, B.: 1973b,*Class, Codes and Control, vol. 2*, London: Routledge & Kegan Paul.

Bernstein, B.: 1977,*Class, Codes and Control, vol. 3*, London: Routledge & Kegan Paul.

Bernstein, B.: 1973b,

Bernstein, B.: 1977,

Wedege, T.: 1999, To know or not to know - mathematics, that is a question of context, *Educational Studies in Mathematics* 39. 1-3, 205-227.

Sfard, A.: 2001, ‘There is more to discourse than meets the ears: Looking at thinking as communicating to learn more about mathematical learning’, *Educational Studies in Mathematics 46*,13-57.

Zevenbergen, R., Valero, P. (Eds): 2004, *Researching the socio-political dimensions of mathematics education: Issues of power in theory and methodology*, New York: Springer Verlag, pp. 1-16.

Lerman, S. & Zevenbergen, R.: 2004, ‘The socio-political context of the mathematics classroom. Using Bernstein’s theoretical framework to understand classroom communications’, in R. Zevenbergen & P. Valero (Eds), *Researching the socio-political dimensions of mathematics education: Issues of power in theory and methodology*, New York: Springer Verlag, pp. 1-16.

Gorgorio, N. Planas, N. & Bishop, A.J.: 2004, ‘Dichotomies, complementarities and tensions. Researching mathematics teaching in its social and political contexts’, in R. Zevenbergen & P. Valero (Eds), *Researching the socio-political dimensions of mathematics education: Issues of power in theory and methodology*, New York: Springer, pp. 107-123.

Evans, J., Morgan, C. & Tsatsaroni, A.: 2006, ‘Discursive positioning and emotion in school mathematics practices’, *Educational Studies in Mathematics* 63.2, 209-226.

Douglas, M.: 1986, *How Institutions Think*, Syracuse, NY: Syracuse University Press.

Salin, M.-H., Clanché, P. & Sarrazy, B.: 2005, *Sur la Théorie des Situations Didactiques*, Grenoble: La Pensée Sauvage.

Brousseau, G.: 1986, Fondéments et méthodes de la didactique des mathématiques,* Recherches en Didactique des Mathématiques* *7*.2, 33-115.

Krummheuer, G.: 1991, ‘Argumentations-formate in Mathematikunterricht’, in H. Maier and J. Voigt (eds), *Interpretative Unterrichtsforschung*, Köln: Aulis.

Krummheuer, G.: 1995, ‘The ethnography of argumentation’, in P. Cobb & H. Bauersfeld (eds), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures, Hillsdale, NJ: Lawrence Erlbaum Associates, Publishers, 229-270.

Krummheuer, G.: 1995, ‘The ethnography of argumentation’, in P. Cobb & H. Bauersfeld (eds), The Emergence of Mathematical Meaning: Interaction in Classroom Cultures, Hillsdale, NJ: Lawrence Erlbaum Associates, Publishers, 229-270.

Sierpinska, A.: 1997, ‘Formats of interaction and model readers’, *For the Learning of Mathematics* 17.2, 3-12.

Voigt, J.: 1998, ‘The culture of the mathematics classroom: Negotiating the mathematical meaning of empirical phenomena’, in F. Seeger, J. Voigt, U. Waschescio (Eds), *The Culture of the Mathematics Classroom*, New York: Cambridge University Press, pp. 191-220.

Yackel, E. & Cobb, P.: 1996, ‘Sociomathematical norms, argumentation and autonomy in mathematics’, *Journal for Research in Mathematics Education* 27/4, 458-477.

Luhmann, N.: 1990, *Essays on Self-Reference*, New York: Columbia University Press.

Steinbring, H.: 2005, *The Construction of New Mathematical Knowledge in Classroom Interaction: An Epistemological Perspective*, New York: Springer.

Twentieth century has seen a proliferation and diversification of theoretical approaches to institutions (Peters, 1999). Looking for commonalities among this variety, Peters identified four features that are widely accepted as characterizing an institution, and distinguish it from other kinds of social activity:

1) an institution is a structural feature of a society, where the structure may be formal (as in a legal framework) or informal (as in networks of organizations or in sets of shared norms);

2) an institution has some stability over time;

3) an institution constraints the individual behavior of its members (participants), through more or less explicit and formal rules and norms;

4) members of an institution share certain values and goals and give common meaning to the basic actions of the institution (summarized from Peters, 1999: 18).

Crozier, Michel & Friedberg, Erhard: 1980,

Institutions are regarded as forms of organizing and regulating interactions amongst individuals engaged in a collective action, aimed at achieving certain goals and producing certain outcomes. Collective action is always a problem for the society, because it is not a natural phenomenon; it is a social construct. As such, the emergence and sustainability of any organization is problematic; it has to be explained.

“... the phenomenon of organization appears... as a political and cultural construct forged by the actors to ‘regulate’ their interactions so as to obtain a minimum of cooperation necessary to carry on with the collective objectives, while at the same time, preserving their autonomy as relatively free agents” (Crozier & Friedberg, 1977: 197 , my translation).

“When we say that the

“Analyses show us.... that while any real change is always accompanied by

Crozier & Friedberg (1977) propose a conjecture: organizations where crises are more likely to produce innovations are systems which are capable of “tolerating a diversity, an inconsistency, a greater openness, and - of managing the inevitable tensions” (p. 383, my translation).

Can we think of an example of a rupture and crisis in an institution of mathematics education that has triggered regress?

“Le jeu est l’instrument que les hommes ont élaboré pour régler leur coopération. C’est l’instrument essentiel de l’action organisée. Le jeu concilie la liberté et la contrainte. Le joueur reste libre, mais doit, s’il veut gagner, adopter une stratégie rationnelle en fonction de la nature du jeu et respecter les règles de celui-ci.... S’il s’agit d’un jeu de coopération, comme c’est toujours le cas dans une organisation, le produit du jeu sera le résultat commun recherché par l’organisation. Ce résultat n’aura pas été obtenu par la commande directe des participants, mais par l’orientation qui leur a été donnée par la nature et les règles de jeux que chacun d’eux joue et dans lesquelles ils cherchent leur propre intérêt. Ainsi défini, le jeu est un construit humain.” (Crozier & Friedberg, 1977: 113)

Douglas considers institutions as a consequence of properties of social behavior such as

- (cognitive) the individual demand for order and coherence and control of uncertainty...

- (transactional) ... the individual utility maximizing activity (Douglas, 1986: 19).

According to Ostrom (2005), an institution is a system of regularized social interactions. (Ostrom, 2005: 3)

“Broadly defined, institutions are the prescriptions that humans use to organize all forms of repetitive and structured interactions including those within families, neighborhoods, markets, firms, sports leagues, churches, private associations, and governments at all scales. Individuals interacting within rule-structured situations face choices regarding the actions and strategies they take, leading to consequences for themselves and others. The opportunities and constraints individuals face in any particular situation, the information they obtain, the benefits they obtain or are excluded from, and how they reason about the situation are all affected by the rules or absence of rules that structure the situation. Further, the rules affecting one situation are themselves crafted by individuals interacting in deeper-level situations." (Ostrom, 2005: 3).

Like Crozier & Friedberg (1977) Ostrom also conceptualizes the participation of an individual in an institution in terms of playing a

Ostrom’s framework for analyzing and developing institutions (ADI) can be described in terms of questions to ask about an institution, for example:

Who are the participants?

What are the positions they are assigned to?

What actions do they perform?

How are these actions dependent on the biophysical conditions and the characteristics (cultural and other) of the larger community to which the participants belong?

What outcomes do the participants expect or desire?

What outcomes do the actions actually produce?

What information participants have about the (potential) outcomes?

What control participants have over the outcomes?

What are the obstacles that make it hard to obtain the desired outcomes?

What criteria and tools do the participants use to evaluate the outcomes?

What rules regulate the actions and their relations with the outcomes? (A rule is an institutional statement of the form: In such and such circumstances, such and such group of participants, shall do so and so, or else such and such sanctions apply).

What norms guide the actions and the outcomes? (Norms describe when, who will normally do what; no sanctions are mentioned)

What strategies are available to or used by the participants?

How do the participants estimate the costs and benefits (the “utility”) of an action?

How can the utility of an action be estimated from a theoretical point of view?

For Ostrom, institutional analysis aims at facilitating prediction of the consequences of undertaken actions. It tries to identify the relations between rules, actions and their outcomes.

“If the individuals who are crafting and modifying rules do not understand how particular combinations of rules affect actions and outcomes in a particular ecological and cultural environment, rule changes may produce unexpected and, at times, disastrous outcomes”. (Ostrom, 2005: 3)

Can we name a situation in the history of reforms in mathematics education, where not understanding the rules and norms governing this institution has lead to "disastrous outcomes"?

Institutions can be described from a macro-societal point of view: identifying

the participants, the positions they occupy, the official outcomes, the explicit rules, the implicit norms and strategies, etc. But one can also be interested in describing the subjective experience of the institution, as it is experienced by its participants. An institution as a whole may be interested in finding strategies to produce outcomes with the highest utility value or at least attempt to maintain a balance between the costs and the benefits of using various human, financial, temporal, material, cognitive, emotional and other resources. On the other hand, an individual's participation in an institution can be seen as a "fight for power", whether it is the power to dictate to others what they should do, or the power to decide about one's own contribution to the institution's collective actions.

In Ostrom’s definition, “the ‘

Here is a framework for research in mathematics education, where teaching and learning of mathematics are taken in their institutional context. It has been inspired by theories of Crozier & Friedberg (1977), Ostrom (2005) and Chevallard (2002).

We label it with the acronym "SIF": synthetic institutional framework.

We assume that institutions are the result of conscious social and legislative effort; they are not just spontaneous behavior or practice that has acquired the status of “normal” or “customary” in a type of situation.

For the practice to be sanctioned and regularized, the society must be a lot more explicit about it. The society must develop a discourse to speak about the practice, to justify it and to teach it to others.

The justification is necessary, because institutions are not natural phenomena; they are social artifacts and, as such, their reason of being and ways of functioning are forever put into question.

While the main tasks (e.g. “to educate an informed and active citizen of a democratic state”) can be formulated in everyday language, the techniques of accomplishing them, together with their underlying rules, norms and strategies, require specialized discourses and perhaps even more formal analytical tools. This leads to the construction of “technologies” and “theories”.

Techniques reduce the diversity of tasks by dividing them into a manageable set of types. An institution provides its members with techniques appropriate for each type of tasks, so that lack of means and knowledge cannot be used as an excuse for not fulfilling the tasks.

Institutions do not require their members to be particularly creative or ingenious in fulfilling its tasks. Their aim is to routinize their solution in an efficient way.

However, the more knowledgeable the members are about the utility of a type of task, the reasons why and when the proposed techniques work, and the better able they are to recognize and seize an opportunity to use these techniques, the more power they have to bend and change the routines and create new ones.

Thus, we will assume that an institution is an organized collective action, defined by:

- its position within a network of larger and smaller institutions;

- its praxeologies, which can be described by

-- their objectives (tasks, outcomes);

-- tools for reaching them (techniques governed by rules, norms and strategies);

-- discourses for explanation, justification, validation and evaluation of the tasks, techniques, and expected and actual outcomes;

- its participants and

-- the tasks they are assigned;

-- the positions they occupy or think they occupy with respect to these tasks and the institution's objectives, tools and discourses;

-- the power (=control * opportunity) these positions afford actually or afford in the eyes of the participants;

-- the interpretations they hold of the praxeologies of the institution, including their interpretation of the rules, norms and strategies used by the institution to regulate their actions and make them both possible and necessary;

-- the games they play to negotiate their positions and power relations with other participants in the institution, where the games can be described in terms of their possible outcomes as well as their rules, norms and strategies.

Below is a Figure from Ostrom, 2005:15 with annotations showing how elements of Brousseau's theory of didactic situations ("milieu", "action situations") can be seen as corresponding to a general framework for studying institutions.

In the Theory of Didactic Situations, the action arena is the mathematics class. The participants/positions are: teacher, students, mathematics as object of learning.

TDS distinguishes several kind of situations:

- out-of-class situations (e.g. when the teacher prepares the lesson or reflects upon his or her less)

- in-class situations with a didactic intention of teaching pupils some mathematics

-- action situations

-- formulation situations

-- communication situations

-- validation situations

-- institutionalization situations

TDS distinguishes several kind of situations:

- out-of-class situations (e.g. when the teacher prepares the lesson or reflects upon his or her less)

- in-class situations with a didactic intention of teaching pupils some mathematics

-- action situations

-- formulation situations

-- communication situations

-- validation situations

-- institutionalization situations

LEFT: figure from (Brousseau, 1997: 248), Chapter 5, representing the structure of the didactic milieu, particularly the different roles of the teacher and the student

Interpretation of the roles of teacher and student in the figure:

P1: Teacher prepares for class

S1: Student prepares for class

P2: Teacher in class

S2: Student in class

P2<->S2 Teacher and student negotiate the terms of a task (the “rules of the game”, the terms of the “didactic contract”)

S3: Student prepares to engage with a task

S4: Student engages with the task

S5+ [the innermost square] : the context and content of the task

P1: Teacher prepares for class

S1: Student prepares for class

P2: Teacher in class

S2: Student in class

P2<->S2 Teacher and student negotiate the terms of a task (the “rules of the game”, the terms of the “didactic contract”)

S3: Student prepares to engage with a task

S4: Student engages with the task

S5+ [the innermost square] : the context and content of the task

In TDS, participants play their games with the “milieu” of the situation.

We can ask: what are the main components of the milieu for P1, i.e. of the teacher preparing a class?

- P1 is an employee of The School (Schoolboard) and engages in and with practices (i.e. tasks, techniques and discourses) specific to this school.

- P1 is a legal subject of The Curriculum, which is a law passed by a state, a provincial government or a schoolboard.

- P1 is a client of textbook publishers.

- P1 is an epistemic subject of The School Mathematics for a given level.

- P1 is an epistemic subject of the pedagogical, psychological and didactic theories he or she learned in pre- and in-service courses for teachers.

- P1 is constrained by the biophysical, socio-cultural and material (financial and technical) conditions of the school system in which he or she works.

- P1 is the manager of the process of participation of his or her students in the educational project and engages in the planning phase of the management.

We can also ask: what are the main components of the milieu for S1, i.e. of the student preparing for a class?

- S1 is a legal subject of the school administration.

- If the school has tuition fees, S1 is a client of the school owners.

- S1 is a client of textbook publishers.

- S1 is an epistemic subjet of The School Mathematics.

- S1 is a member of the class as a social group.

- S1 is constrained by the biophysical, socio-cultural and material (financial and technical) conditions of the school system in which he or she is enrolled.

- S1 is a disciple of P1: S1 engages with the tasks assigned by P1.

*What happens in a didactic situation, in terms of SIF?*

In the position of P1, the teacher has thought of setting up a milieu specific for the mathematics he wants the students to learn and of engaging students in such games with this milieu that, he believes, will result in students’ learning of the target mathematical knowledge.

In the position of P2, the teacher describes the milieu and negotiates the rules of the games to be played with the students. Students (in the position of S2) respond by questions of clarification and then accepting or not accepting the rules of the game.

TDS specifies the conditions of success of the teacher's plan?*When do students actually learning the aimed-at knowledge? *

Aiming at a particular mathematical knowledge, the teacher will try to set up the content of the milieu and the rules of the student-milieu games so that this knowledge appears to be the best means for understanding the rules of the game and elaborating a winning strategy.

If the milieu is set up so that students have no choice but to use the target knowledge (e.g. the knowledge to be used is explicitly indicated or strongly suggested in the rules of the game), then students do not learn the knowledge, in the sense of incorporating it in the repertoire of their spontaneous strategies in playing games with mathematics. They only learn to “recite a mathematical text” or execute a procedure by rote. (Brousseau, 1997: 40-42)

Examples of didactic contracts and their descriptions in terms of SIF:

**The contract of the ACTION SITUATION: opportunistic teaching and learning**

Participants act as persons in a natural situation, not as students executing the instructions of a teacher.

The focus is on getting something done, not on reflecting on it, or on talking about.

If a problem arises for a participant, he or she tries to solve it as they see fit, given the circumstances; there is no teacher to tell them if they are right or wrong; they know from the results of their actions (the feedback of the milieu) if they have solved the problem or not or if their strategy was effective or not.

Participants figure out the rules, norms and strategies that govern the milieu by acting on it and receiving feedback.

If the action situation occurs in school, the participants who are students in the school are not given grades for their solution of the problems that occur in the situation. They are neither punished nor rewarded in this institutional conventional manner.

*Example of an action situation in school*: a math class takes place in a computer lab. At some point, the mathematical software used in the class does not respond as expected. The teacher has trouble seeing what’s wrong and gets upset because time is passing by and she will not be able to finish the material planned for the class. A student notices a syntax error in a command and thus solves the problem. The student doesn’t get a mark for this, but only a "thank you" from the teacher.

**The contract of the FORMULATION SITUATION: there is a person in the position of an organizer of discussions**

Under this contract, participants may still act as persons in a natural situation, not as students executing the instructions of a teacher, but in a school situation, in, e.g. a whole class discussion, the teacher may hold the position of a chairman of a session, not allowing all to speak at the same time, and making sure that, when a student speaks, the others keep quiet.

The focus is on description: some action took place and now participants reflect on it and describe it, in the aim of understanding what happened.

Participants describe the situation in any words they know, the best they can and there is no teacher to tell them that they are using an incorrect terminology. The speakers can only be asked to explain more clearly what they mean.

Participants figure out the rules, norms and strategies that govern the milieu by acting on it and receiving feedback: they speak and find they are understood or not. If they are not, they try to improve their description.

If the action situation occurs in school, the participants who are students in the school are not given grades for their descriptions. They are neither punished nor rewarded in this institutional conventional manner.

Examples? It is difficult to find an example of a spontaneous formulation situation in school that wouldn’t be linked with a communication situation. Students describe what they did, because they want to communicate to others their problem or how they solved a problem, and they cannot do it otherwise than by speaking or writing. Formulation situations without the intention of communication occur more often in situations of intellectual work or research: the thinker writes because this helps organizing his or her thoughts.

**The contract of the VALIDATION SITUATION: there is a leader who engages the group in developing criteria of validity of the outcomes of their work**

Under this contract, participants may still act as persons in a natural situation, not as students executing the instructions of a teacher, but in a school situation, in, e.g. a whole class discussion, the teacher may hold the position of a chairman of a session.

The focus is on evaluation (of actions, statements): why something is correct, true and something else isn’t? The discussion requires developing criteria of evaluation.

Participants use any arguments they want to prove or disprove statements, or to give credit or discredit actions; but the leader will press to establish clear criteria of validity and to use them in the evaluation.

Participants figure out the rules, norms and strategies that govern the milieu by acting on it and receiving feedback: they argue and find their arguments are considered convincing or not, sound or not and they learn the criteria of validity in so doing.

If the validation situation occurs in school, the participants who are students in the school are not given grades for their arguments. They are neither punished nor rewarded in this institutional conventional manner.

**The contract of the INSTITUTIONALIZATION SITUATION: there is a participant with the authority to communicate or establish the rules, norms and strategies conform with the institutional praxeology **

Under this contract, participants are in positions conferred to them by the institution; in a school situation, some are students, subjects of the institution and disciples of their teacher; there is a teacher, representing the institutional praxeologies.

The focus is on establishing which outcomes of students’ actions, which formulations and which arguments will count as correct by the institution.

From now on, students who produce outcomes not satisfying these established standards will receive poor grades.

Participants should not have to figure out the rules, norms and strategies that govern the milieu by acting on it spontaneously and receiving feedback: these rules, etc. must be made clear and explicit by the teacher. If rules are clear and explicit but a student still does not abide by them, this is considered as the student’s own fault. Not having learned the rules is not an acceptable excuse.

We can ask: what are the main components of the milieu for P1, i.e. of the teacher preparing a class?

- P1 is an employee of The School (Schoolboard) and engages in and with practices (i.e. tasks, techniques and discourses) specific to this school.

- P1 is a legal subject of The Curriculum, which is a law passed by a state, a provincial government or a schoolboard.

- P1 is a client of textbook publishers.

- P1 is an epistemic subject of The School Mathematics for a given level.

- P1 is an epistemic subject of the pedagogical, psychological and didactic theories he or she learned in pre- and in-service courses for teachers.

- P1 is constrained by the biophysical, socio-cultural and material (financial and technical) conditions of the school system in which he or she works.

- P1 is the manager of the process of participation of his or her students in the educational project and engages in the planning phase of the management.

We can also ask: what are the main components of the milieu for S1, i.e. of the student preparing for a class?

- S1 is a legal subject of the school administration.

- If the school has tuition fees, S1 is a client of the school owners.

- S1 is a client of textbook publishers.

- S1 is an epistemic subjet of The School Mathematics.

- S1 is a member of the class as a social group.

- S1 is constrained by the biophysical, socio-cultural and material (financial and technical) conditions of the school system in which he or she is enrolled.

- S1 is a disciple of P1: S1 engages with the tasks assigned by P1.

In the position of P1, the teacher has thought of setting up a milieu specific for the mathematics he wants the students to learn and of engaging students in such games with this milieu that, he believes, will result in students’ learning of the target mathematical knowledge.

In the position of P2, the teacher describes the milieu and negotiates the rules of the games to be played with the students. Students (in the position of S2) respond by questions of clarification and then accepting or not accepting the rules of the game.

TDS specifies the conditions of success of the teacher's plan?

Aiming at a particular mathematical knowledge, the teacher will try to set up the content of the milieu and the rules of the student-milieu games so that this knowledge appears to be the best means for understanding the rules of the game and elaborating a winning strategy.

If the milieu is set up so that students have no choice but to use the target knowledge (e.g. the knowledge to be used is explicitly indicated or strongly suggested in the rules of the game), then students do not learn the knowledge, in the sense of incorporating it in the repertoire of their spontaneous strategies in playing games with mathematics. They only learn to “recite a mathematical text” or execute a procedure by rote. (Brousseau, 1997: 40-42)

Examples of didactic contracts and their descriptions in terms of SIF:

Participants act as persons in a natural situation, not as students executing the instructions of a teacher.

The focus is on getting something done, not on reflecting on it, or on talking about.

If a problem arises for a participant, he or she tries to solve it as they see fit, given the circumstances; there is no teacher to tell them if they are right or wrong; they know from the results of their actions (the feedback of the milieu) if they have solved the problem or not or if their strategy was effective or not.

Participants figure out the rules, norms and strategies that govern the milieu by acting on it and receiving feedback.

If the action situation occurs in school, the participants who are students in the school are not given grades for their solution of the problems that occur in the situation. They are neither punished nor rewarded in this institutional conventional manner.

Under this contract, participants may still act as persons in a natural situation, not as students executing the instructions of a teacher, but in a school situation, in, e.g. a whole class discussion, the teacher may hold the position of a chairman of a session, not allowing all to speak at the same time, and making sure that, when a student speaks, the others keep quiet.

The focus is on description: some action took place and now participants reflect on it and describe it, in the aim of understanding what happened.

Participants describe the situation in any words they know, the best they can and there is no teacher to tell them that they are using an incorrect terminology. The speakers can only be asked to explain more clearly what they mean.

Participants figure out the rules, norms and strategies that govern the milieu by acting on it and receiving feedback: they speak and find they are understood or not. If they are not, they try to improve their description.

If the action situation occurs in school, the participants who are students in the school are not given grades for their descriptions. They are neither punished nor rewarded in this institutional conventional manner.

Examples? It is difficult to find an example of a spontaneous formulation situation in school that wouldn’t be linked with a communication situation. Students describe what they did, because they want to communicate to others their problem or how they solved a problem, and they cannot do it otherwise than by speaking or writing. Formulation situations without the intention of communication occur more often in situations of intellectual work or research: the thinker writes because this helps organizing his or her thoughts.

Under this contract, participants may still act as persons in a natural situation, not as students executing the instructions of a teacher, but in a school situation, in, e.g. a whole class discussion, the teacher may hold the position of a chairman of a session.

The focus is on evaluation (of actions, statements): why something is correct, true and something else isn’t? The discussion requires developing criteria of evaluation.

Participants use any arguments they want to prove or disprove statements, or to give credit or discredit actions; but the leader will press to establish clear criteria of validity and to use them in the evaluation.

Participants figure out the rules, norms and strategies that govern the milieu by acting on it and receiving feedback: they argue and find their arguments are considered convincing or not, sound or not and they learn the criteria of validity in so doing.

If the validation situation occurs in school, the participants who are students in the school are not given grades for their arguments. They are neither punished nor rewarded in this institutional conventional manner.

Under this contract, participants are in positions conferred to them by the institution; in a school situation, some are students, subjects of the institution and disciples of their teacher; there is a teacher, representing the institutional praxeologies.

The focus is on establishing which outcomes of students’ actions, which formulations and which arguments will count as correct by the institution.

From now on, students who produce outcomes not satisfying these established standards will receive poor grades.

Participants should not have to figure out the rules, norms and strategies that govern the milieu by acting on it spontaneously and receiving feedback: these rules, etc. must be made clear and explicit by the teacher. If rules are clear and explicit but a student still does not abide by them, this is considered as the student’s own fault. Not having learned the rules is not an acceptable excuse.

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