Artigue & Winsløw

8/20/2019 Artigue & Winsløw

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International comparative studies on mathematics education:

A viewpoint from the anthropological theory of didactics

Michèle Artique, University of Paris Diderot, FranceCarl Winsløw, University of Copenhagen, Denmark

1. Introduction

The last decades have seen the rise of interest – political as well as academic – in large internationalsurveys of students’ mathematical performances on meant-to-be “neutral” test items, which may ormay not take into account the variability of emphases in the participating educational systems. Atthe same time, the internationalisation of research on mathematics education has led tocollaborations and studies which involve perspectives and practices from otherwise very distant

contexts of teaching. This, as much as the large scale surveys, has led scholars to notice, and reflecton, differences. Investigating differences and their causes is one simple definition of whatcomparative studies are all about.

Comparative studies may be based on different assumptions – ranging from presumably universalmodels of “mathematical competency” to in-depth comparisons of local practices in two specificcontexts, in which one pays the highest attention to, for instance, the linguistic, cultural, societal andintellectual specificities of the two environments. One therefore ends up with comparisons which,metaphorically, are based on everything between “areal photography” to “microscopy” – and theneed arises for a common framework which permits one to relate and integrate different purposesand methods of comparison, involving different – but not necessarily unrelated – objects of study.

Didactics is the science which takes, primarily, the teaching of disciplined knowledge as its object.Such a study may equally be more or less comprehensive, and more or less focused on localconditions and constraints. In didactics of mathematics, several theoretical frameworks are availableto organise and, indeed, make possible, different kinds and levels of research. Such research maywell be (indeed, is very often) taking place within fairly homogenous cultural and institutionalsettings, and then these are often subject to less attention. It is clear that a didactical theory thatcould help informing and organising comparative studies would have to take a less naïve viewpointon cultures and institutions. This is why, in this paper, we propose – and exemplify – the use ofelements from the anthropological theory of didactics (ATD), founded by Y. Chevallard and

developed over the last decades with a number of other researchers, primarily from France andSpain. In ATD research, institutional and more widely cultural conditions are always crucial.

We first introduce some of the main elements of ATD, and the way we make use of them toconsider different kinds of comparative research (Section 2-3). Then, in Section 4-7, we consider avariety of recent comparative studies on mathematics education, in order to demonstrate how theviewpoint of ATD helps articulating their results and, indeed, understand the differences andrelations between them. Finally, in Sec. 8, we discuss briefly some more general clusters of

problems which emerge from the cases considered – in particular, certain gaps and shortcomingswhich, according to us, require further work, but to which the viewpoint if ATD could bringimportant means.


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2. The anthropological theory of didactics : some preliminaries

The anthropological theory of didactics (ATD) offers a bold vision of human activity, based on thefollowing simple assumptions (cf. Chevallard, 1999):

• Human activity can be analysed in terms of its purpose from the point of view of thoseinvolved (or from other points of view, such as the analyst). In ATD, all human activity isconsidered as arising from tasks (e.g. ‘how much is 56/37’; ‘clean this carpet’).

• Together with the identification of the task to be carried out, one notices also ways in whichthis is done. In ATD this is called a technique (in the two examples, long division and use ofa vacuum cleaner could be relevant). Techniques can usually be used for a variety ofdifferent, yet “similar” tasks; so, in fact, they pertain to a type of task (such as ‘evaluate thesize of a fraction’; ‘floor cleaning’). Notice how the type of task is abstracted from the

particular circumstances (context, situation…) in which actual tasks are always carried out.• Techniques and types of tasks may be described, as we just did it for the examples, using a

certain “language” or a specific theorem to formulate it; this is called a technology (so inATD, “technology” refers to a shared form of discourse – the logos pertaining to techniques

– and not to computers or other machines!).• Finally, techniques may be justified within a theory, which may be described as a system of

shared principles which serve to justify techniques (using a technology, of course; the theorymay include specific forms of technology, e.g. to state its principles). In the case of solvingequations, a theory that could justify solving techniques would include general principles

pertaining to algebraic equations (and thus constitute or be part of an algebraic theory).

The minimal unit of human activity is thus a practice block , formed by a type of task and atechnique. A complete praxeology contains, besides this, a corresponding theory block. The latter

consists of a technology explaining the practice block, and a theory which allows systematicreasoning about (e.g. justifying) the practice block, in relation to other ones. A graphic illustration isgiven in Fig. 1, which is meant to show how the elements of a praxeology are built up from the typeof task which identifies it.

It is apparent that praxeologies often occur in larger systems, sharing some of the explicit elements(i.e., of theory blocks). A technology may serve to explain more than one practice block; acollection of praxeologies sharing the same technology (and theory) is called a local organisation(of praxeologies). Such an organisation is characterised by the common technology, such as analgebraic “language” for the example. Finally, a theory may serve to reason about several localorganisations, which then are said to form a regional organisation (of praxeologies), characterised

by a unifying theory, such as a theory of solving equations. Fig. 2 and 3 show how suchorganisations are constituted by several praxeologies sharing either a technology (and hence, potentially, a theory) or just a theory (but involving several technologies to describe different kindsof practice blocks).

It is normally easy to associate school mathematical practices (as the example given) with theory blocks, and to group these according to the shared technologies or theories (leading to referencesmodels of the corresponding mathematical organisations). It may be less straightforward for other

practices – such as teaching a given mathematical organisation – even if these practices containdefinite types of tasks that are carried out using techniques. This form of praxeological organisation

– which may of course be considered for the teaching of any praxeology – is called a didactic

organisation. It comes with didactical technologies and theories to describe and justify the


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corresponding practices, in which the types of task are of the form ‘teach a given mathematicalorganisation to a given group of students’. For such tasks, the techniques will depend on both ofthese elements, as well as on the context in which the task is carried out; in fact, mathematical anddidactical praxeologies in a given school context can be said to determine each other, since the

realisation of mathematical organisations are also shaped by the available didactical techniques andso on. In fact, for very broad ranges of didactic practice blocks (not necessarily confined tomathematics teaching) one may be able to identify regularities related to more general types ofteacher tasks, for instance in the way lessons are structured in time, how students are addressed bythe teacher, and so on.

One of the main challenges we face is that didactic organisations typically have less developedtechnologies, especially when teachers mainly work individually and, as a consequence, do not needa shared, precise discourse for communicating about the techniques they use to teach. Thedescription of didactical techniques may be done with different purposes and technologies, from theinformal discussion among colleague teachers, to the coding of video data in large scale

comparative studies (cf. sec. 4). It is an important function of didactic theories like ATD to developmore precise forms of discourse (technology) with which one may articulate didactic techniques,and to develop, within such a discourse, systematic tools to reason about the practices.

.Fig. 1. A praxeology. Fig. 2. A local organisation. Fig. 3. A regional organisation.

A fundamental observation of ATD is that praxeological organisations arise and develop notthrough the activity of isolated individuals or groups, but always in the setting of institutions whichorganise human activity according to certain goals and rules. These may not always be entirelyexplicit in the activity. Institutions can neither be identified with groups of individuals (who passwhile the institution remains) or with the praxeologies they organise (as these are potentiallydeveloping). An institution is better described as the complex of conditions and constraints whichenable a (large) organisation of praxeologies to live and develop, as a kind of ecology for them (ametaphor referring to human activity systems as living, ever changing “organisms”). Just like

praxeological organisations, institutions often exhibit ‘nested structure’, i.e. one may contain or‘dominate’ the other. This is quite obvious when we consider mathematical and didacticorganisations (cf. Sierpinska, 2006 for a number of interesting references and examples regardingthe notion of institution and its use in ATD).

Didactic and mathematical praxeologies and their development cannot be understood fully withoutconsidering broader contexts than the ones in which they are directly occurring. This is becausethey are determined by a whole hierarchy of institutional levels which successively condition andconstrain each other. It is an important contribution of ATD to provide a detailed model of theselevels of didactic determination, as shown in Fig. 4 (together with simple examples; for furtherdetails, we refer to Chevallard, 2002); it is important to remember that we are talking about

discipline, domain and so on, as they are realised in the educational context. Many of the

Theory

Technology

Tech- ni- ques

Types of tasks

Theory

Technology

Technique

Type of task

Theory

Tech- nolo- gies


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conditions, particularly those arising from the higher levels, cannot be changed by individualteacher; some of them may still be modified by others, such as school principals, curriculumdevelopers or politicians, and more generally what Chevallard (1985) calls the “noosphere” of theeducational system.

The higher levels cover many types of content, the most well known “grand types” being what iscommonly referred to as disciplines. But ATD also allows us to identify certain intermediate levels,specific to the contents of teaching within a discipline, and associated with the unifying elements ofthe praxeologies they determine. A domain is a collection of regional organisations, involvingseveral theories forming a larger part of the discipline (such as algebra within mathematics, both to

be understood within the institutional context of a school). A sector is characterised by a unifyingtheory of one or more regional organisations. So, for instance, when we study the “sector” of

polynomials within secondary school, we will encounter praxeological organisations unified by atheory which allows us to examine precise assertions about polynomials, e.g. about their number ofreal valued roots. Different technologies may be in use, from informal discourses in which

polynomials appear just as “equations” determining a curve, to formal ones in which they arealgebraic elements with their own composition rules. This gives rise to different themes, unifiedroughly by a technology. Finally, a subject is concentrated on a type of task and technique,articulated within a larger theme.

Fig. 4. Three factors potentially to be compared: levels of didactic determination, mathematical praxeologies

realised in school (through didactic organisations enacted by teachers) and the knowledge of students related to

the taught mathematical organisations.

These different levels, which determine or condition the actual activity in teaching (didactic praxeologies), are all exterior to the teaching situation itself. Students meet didactic organisations inteaching situations, through mathematical and other praxeologies which the organisation seeks to

engage the students in. The personal knowledge (including both practical and theoretical forms)

9. Civilisation

8. Society

7. School

6. Pedagogy

5. Discipline

e.g. “Western culture”

State, ministry, region

Teaching institution

Local teaching principles

e.g. Mathematics

4. Domain e.g. Algebra

3. Sector e.g. “Polynomials”

2. Theme e.g. “Pol.equations”

1. Subject e.g. Quadratics

0. Student knowledge (related to

raxeolo ical or anisations

Theory

Technologies

Techniques

Types of task


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which they construct through participation in these situations may, of course, vary between studentsand also be different from what was intended by the teacher, the school, society and so on. It seemsreasonable and useful to our purposes in this paper to include, in our model as illustrated in Fig. 4,this ultimate “product” of the teaching enterprise. From top to bottom, we thus have a system of

processes through which didactic practices and their outcome are successively determined andrealised by agents within the different institutional levels. The model provides a fine grained,structured view of what is called didactic transposition, the processes through which knowledge istransposed in and between institutions (cf. Chevallard, 1985), and ultimately reconstructed by thenext generation.

3. Levels of comparison suggested by ATD

Comparative studies address some of the levels in Fig. 4, but rarely all of them. For instance, astudy may consider a number of different schools within the same society, and be focused on someof the underlying levels, such as pedagogies or overall aspects of students’ practical knowledge

within a given subject – a kind of comparison which may be of interest to parents about to choose aschool for their child. Of course, comparison between different societies (understood as nations) isthe hallmark of international comparison. In general, comparative studies of mathematics educationmay involve the following ten levels of comparison between two or more contexts (cf. Fig. 5):

0. Knowledge of students within one or more specific subjects, situated and articulated withincertain themes or sectors (and inevitably observed through a technology, which may to someextent be observed along with elements of students’ theoretical knowledge);

1. Praxeologies of specific subjects, as prescribed by programs or official evaluations, oractually found in the didactic practices of the contexts;

2. Local organisations of specific themes in the didactic practices of the contexts, as prescribed by programmes, as described by teachers or as inferred from observation of several subjectswithin the theme;

3. Regional organisations of specific sectors in the didactic practices of the contexts, as prescribed by programmes, as described by teachers or as inferred from observation ofseveral themes within the sector (studies at level 2);

4. Global organisations of specific domains in the didactic practices of the contexts, within agiven discipline, as prescribed by programmes, as described by teachers or inferred fromobservation of several sectors (studies at level 3):

5. Organisations of a discipline in domains or more globally, based on programmes and otherevidence (including observation and assertions by teachers);

6. Pedagogies in the contexts, as prescribed by schools or programmes, as observed or as

described by teachers (this includes principles for teaching that may transcend or be specificto various disciplines taught, prescribed or observed interactions among disciplines and soon);

7. Conditions and characteristics of specific teaching institutions, e.g. the obligations andautonomy of teachers;

8. Conditions and characteristics of whole societies, including in particular the way in whichschools are governed, funded and systemically organised;

9. Larger cultural contexts or “civilisations”, their principles for human society in particular asregards the role and meaning of education.


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These ten “levels” of comparison are illustrated in Fig. 5, emphasising also that the comparison mayconcern both mathematical and didactic organisations, as observed (in real teaching) or prescribed(e.g. through national programmes and exams, text books and so on).

It should be emphasised that concrete comparative studies may include several such levels and acrucial feature of such studies – and the ways in which they used – is to what extent, and how, theselevels of comparison are related , in particular what causal relations are inferred or assertedbetween these levels of comparison. For instance, differences identified at level 0 (studentknowledge) could be claimed to be explained by differences at higher levels.

Moreover, it is crucial to consider if the question of comparability is really addressed, in particularto determine if and how technologies of the two contexts – at the various levels of institutional

practice – are critically examined. One may, of course, find studies in which a universal technology(or just one particular to one context) is simply assumed, or even implicitly used; then it is aseparate issue to see if this is justified. In the line of this, notice that what we have introduced in this

section is a technology to describe comparative studies, to be “tested” in use on concrete studies.

Fig. 5. Possible levels of comparison of the determination of mathematical organisations (MO) and didactic

organisations (DO) in two contexts (the contexts might coincide from some level upwards, e.g. same school).

In short, to identify and characterise a comparative study on mathematics education, we propose toconsider the following four questions:

a) At which of the levels 0-9, described above, does comparison take place? b) Assuming that the comparison includes didactic or mathematical organisations, what kinds

of evidence is used to study them? (E.g. direct observations, interview with teachers,textbook analyses,…)

c) What methodological tools (in particular, technology in the sense of ATD) are used tointerpret these observations or descriptions, to ensure that the comparison makes sense at agiven level? (e.g., at level 2, how are comparable “themes” identified and how iscomparability argued for? In particular, how are language differences dealt with?)

Civilisation

Society

School

Pedagogy

Discipline

Domain

Sector

Theme

Subject

8

7

6

5

4

3

2

1

9

Studentknowledge

Studentknowledge0

Civilisation

Society

School

Pedagogy

Discipline

Domain

Sector

Theme

Subject P r e s c r i b e d / r e a l i s e d M O a n d D O

P r e s c r i b e d / r e a l i s e d M O a n d D O


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d) How does the study relate horizontal comparison (between two contexts at the same level)to the vertical relations between levels, found within each context? In particular, arecausalities (between horizontal levels of the comparison) claimed?

Not all of the questions may be equally important for a given study. But in the following sections

we demonstrate their pertinence for a wide selection of major comparative studies – and more broadly, the interest of considering comparative studies from the point of view of ATD.

4. TIMSS and PISA

We begin by considering, in this section, two of the most well known cases of internationalcomparative studies in mathematics education, namely TIMSS (trends in mathematics and sciencestudy) and PISA ( programme for international student assessment ). They belong to a lineage oflarge scale studies, beginning with the first international mathematics study in 1964. Their main aimis to measure students’ mathematical capacities across a large number of countries. In this paper,we consider neither the history nor the many critiques that have been made of such studies (see e.g.

Keitel and Kilpatrick, 1999; Hoppmann et al., 2007). The complexity of the studies is such that wecan only provide rough answers to our questions in this case. We base these primarily on the recentframeworks (Mullis et al., 2005; OECD, 2006) and on recent general (world wide) reports of theresults (Mullis et al., 2008; OECD, 2004).

TIMSS builds on a curriculum model with three components: intended , implemented and attained curriculum, all studied at two grade levels exclusively (4th and 8th year of schooling). The intendedcurriculum is studied through official programs. Based on this, the model used to describe theimplemented and attained curricula is constructed, in order to “ensure that goals of mathematics andscience education regarded as important in a significant number of countries were included”. Thismodel involves, in fact, four levels which correspond neatly to the levels 2-4 of didacticdetermination of ATD (in parenthesis we give indicative examples from the TIMSS framework

pertaining to the fourth year of schooling):4. Content domains (Number)3. Topic areas (Whole numbers)2. Objectives (Compare and order whole numbers)1. Items (examples of tasks that correspond to the objectives)

TIMSS then compare national curricula at levels 2-4, both for the intended and the implementedcurriculum. As far as level 5 is concerned, TIMSS operates with mathematics as a type of disciplinewhich is simply identified with actual school disciplines in participating countries; this gives rise toless ambiguity than for science disciplines. Information on the implemented curriculum is collected

through questionnaires to school principals, teachers and students; for the two former groups,information of a more general nature, concerning levels 6 and 7, are also included; finally, publicdata concerning national features of education (level 8) can also be drawn upon.

The attained curriculum – located at level 0 but obviously influenced, if not determined, by thehigher levels – is measured by submitting large samples of students in each country to a testcontaining items that pertain to objectives identified in the intended curriculum of that country; asignificant feature of TIMSS is thus the possibility of omitting entire chunks of items based on theirweakness or absence in the intended curricula. These items are classified according to three levelsof “cognitive domains” (knowing, applying and reasoning, with further sub-categories which showthat the three levels cover successively more “advanced” forms of mathematical activity). We

notice in passing that the category “student knowledge” of our model is meant to encompass every


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exercise of mathematical practice and theory blocks and so is much broader than knowing in theTIMSS sense. For all of these categories, universal validity is assumed, based on the participation ofnational experts in developing the framework. English is used as a common language, and so thisimplies considerable implicit assumptions regarding national experts’ consideration of translation

problems and of the (English language based) categories’ relations with notions proper to thenational languages and contexts.

The development of the PISA framework is not based on examining curricula in the participatingcountries. Instead, a new model of “mathematical literacy” is elaborated within the project, more

precisely by the “mathematics expert group”. Given this approach to formulate a framework fordescribing the mathematics related part of “how well students, at age 15, are prepared to meet thechallenges they may encounter in future life” (Mullis, 2005, 3), it is interesting to note twoinfluences which appear quite clearly in the framework itself: the Dutch research tradition of“realistic mathematics education” (cf. de Lange, 1996) and related German and Scandinavianresearch ideas on mathematical competencies, in particular related to modelling (egg. Niss, 2003). It

is no coincidence that de Lange and Niss were both members of the mathematics expert group(Mullis, 2005, 187). The “mathematical literacy” model consists of three components: situations orcontexts (in which mathematics is handled, from personal life to work or science); the mathematicalcontents, organised in overarching ideas (namely: space and shape; change and relationships;quantity; uncertainty); and finally competency clusters (namely: reproduction, connections andreflection). Besides test items constructed to measure students’ mathematical literacy along thesedimensions, PISA also use questionnaires for students and school principals (but not teachers) inorder to determine certain features of the students’ situation and the school context, and in somecountries a questionnaire for parents. That is, certain variables at level 6 and 7 (concerning

pedagogy and school) can be related to the outcome of students’ responses to the test items and thequestionnaire.

PISA’s main aim is to confront the described model of “mathematical literacy”, which is supposedto be desirable in any society (level 8), with students’ performance on corresponding items. But thedata give no means to evaluate whether high or low performance is caused by regulations at levels3-5 (such as a mathematics curriculum with different components than those assessed), levels ofefficiency in teachers’ realisation of the curricular goals (mainly levels 1-3), or something else.Other studies (cf. egg. Hopmann et al., 2007) have considered this question for particular countries

On the other hand, it is possible to analyse the PISA framework itself as suggesting certain types ofmathematical praxeologies to be particularly relevant for students’ future life and activity in society.

The competency clusters, which at first sight resemble the cognitive domains in TIMSS, are in factfurther described in terms of Niss’ competency categories (Niss, 2003) which include both technicaland technological components (for a further discussion of the links between competence categoriesand praxeologies, see Winsløw, 2005). It is also possible to interpret the overarching ideas asmathematical domains, and compare them with what is found or prescribed in schools. It may bethat the test items can be ordered according to a kind of sectors and subjects. But since the details ofsuch an analysis would require an access to the test items which we do not have, we can onlyconclude that the framework itself outlines an independent structure of mathematical domains andthemes which may or may not correspond to parts of the levels 3 and 4 of didactical determination,for a given country.


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The outcome of both TIMSS and PISA is a huge collection of data pertaining to levels 0 and 2-8(for PISA, mainly level 0 and 6-8) which can be to some extent be considered at and betweenlevels, in order to identify correlations across levels. In both surveys, students’ performance on testitems (pertaining to level 0) is clearly the main data object, while the rest is considered “background

data”, a term clearly suggesting that they are determinants for level 0. The main correlations soughtfor are thus between students’ performance on a certain group of items – such as all items related tomathematics – and other parts of the data that can be linked to the students (egg. country,characteristics of their school’s pedagogy, intended curricula, social or ethnic specificity, and soon). The studies clearly assume that one can interpret vertical correlations between level 0 (student

performance) on the one hand, and background data (from higher levels) on the other, in directlycausal ways signalled by terms like “impact”, “influence”, “results in”, and so on (cf. Fig. 6). Thisis apparent already from the statement of their overall purposes:

By participating in TIMSS, countries can (…) understand the contexts in which student learn best.TIMSS enables international comparisons among the key policy variables in curriculum, instruction,and resources that result in the higher levels of student achievement (Mullis, 2005, 10)

PISA is designed to collect information through three-yearly cycles and presents data on the reading,mathematical and scientific literacy of students, schools and countries. It provides insights into thefactors that influence the development of skills and attitudes at home and at school, and examineshow these factors interact and what the implications are for policy development. (OECD, 2006, 9)

In the presentation of results, one finds many implicit and explicit claims about how differencesat level 0 (e.g. concerning average mathematics performance) may be related to differences athigher levels (e.g. at school level, the number of out-of-school mathematics activities the schooloffers):

Schools’ offering of activities to promote student engagement with mathematics, such as mathematicscompetitions, mathematics clubs or computer clubs related to mathematics, show a positive impact aswell, over and above all other factors. Each additional such activity that is offered by schools isassociated with an average performance advantage of 7 score points. (OECD, 200, 259)

The key word here is, again, “impact” and synonyms of it. The problems with asserting such“impacts” are twofold: first, are the horizontal comparisons both meaningful and valid (this relatedto the framework, questionnaires, items, data sampling and treatment)? And secondly, assuming thehorizontal differences are “real”, does one produce the other? It is indeed possible that bothdifferences are instead produced by a third factor, belonging to the same or a higher level, beingsimilarly different for the two contexts. In the example given by the above quotation, one couldthink of factors like “school budget”, “teachers’ affective relation to mathematics”, and so on. If infact one such factor correlates strongly with both student performance and offering of activities to

promote student engagement with mathematics, then the last sentence in the quotation becomesdubious, at least if it is read to say that adding an activity of the said type, a school could improveits students’ performance on PISA items by an average of 7 points. The collection of data in PISA isso huge that it is very difficult to control for this type of covariation, even if discarding the issue ofdata validity.

An important technique to produce “findings” from the complex data material consists in groupingstudents according to one variable (typically country) and then presenting the average of anothervariable for the groups (like test performance or expressed interest in mathematics). In the tableswhich result, the assumption that the second variable depends somehow on the first may be implicit

but still strongly suggested. We say “somehow” because it is not always the first variable which is

seen implicitly as the “cause”. In the case of “country” (level 8) the tendency is to view differences


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in performance as caused by the sum of the underlying levels, as it would be evidently meaninglessto claim that, for instance, performance or interest in mathematics depends directly on nationality.

National policies (regarding curricula, pedagogy principles, teacher education and so on) appearthen as candidates for causes, and may be further investigated through new groupings across nations

(related to variables at these lower levels). However, all correlations identified remain weakhypotheses, given the amount of variables available and the large variations one find withindifferent groups, like the students in several of the participating countries.

PISA Level TIMSS

9876

543210

Fig. 6. PISA and TIMSS compare several contexts horizontally, at a given level – the primary level being student

knowledge as measured by test items. The figure just shows the comparison of two contexts, for simplicity.Causal effects are sought in the background data.

Indeed, many aspects of both surveys remain highly intricate and controversial: the details ofquestionnaires and test items (the latter mainly secret), data processing methods and so on. Thesecrecy surrounding test items prevents us from confronting the measurement at level 0 directlywith the mathematical and didactical practices in schools (level 1) which, by the way, are notobserved directly in TIMSS and PISA (but it some follow-up studies, like the ones we consider inSec. 5). It is therefore impossible to evaluate – within concrete mathematical themes, sectors ordomains, or for mathematics in general – the validity of claimed links between measurements“student performance” and what happens in mathematics classes in schools. However, as we justsaw, such links tend to be assumed in the reports, which mainly correlate student performance tohigher level factors determining the mathematical practices in school classes. And whatever theitem details are, one may contend that “multiple choice” answers give very limited information on

students’ knowledge in the sense of praxeologies they master individually, even at the level of practice blocks (since a given answer could usually be found using more than one technique). Giventhe results from these tests, we can infer next to nothing about the (social, institutionally situated)

praxeologies students encounter in school mathematics. Conversely, to the extent information isavailable about the school mathematical practices (particular at level 1) in which the tested studentshave been involved, effects at level 0 remain highly speculative.

Nevertheless, both surveys provide a rich source of material on which one may base, at leastinitially, more detailed discussions of the effects of didactical determination, and which canmotivate more fine grained studies of particularly intriguing hypotheses as far as effects on level 0is concerned. This is not least due to the fact that the frameworks (which serve as a technology for

the comparison) are very explicit, especially at higher levels. The involvement of national experts in

Suggestedcausaleffects offounddifferences

Comparison of background data

Comparison of primary data


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their elaboration warrants a certain degree of compatibility with more local discourses at thesehigher levels. One problem that these frameworks share, and which may be hard to avoid given thenumber of contexts involved, is the lack of discussion of linguistic details and references to researchspecific to each context. Despite the involvement of national experts, the translation of frameworks,

test items and so on are bound to be highly approximate – even as regards the technologicalelements of mathematical organisations as they appear in schools. We return to these questions asthey appear in binary comparisons (Sec. 6 and 7).

5. Large-scale international studies of classroom activity

The 1995 TIMSS video study (cf. Kawanaka et al., 1999) was among the first attempts to get moredirect information on classroom practices, in order to supplement and explain the results of surveysof the type considered in Sec. 4 (of course, mainly TIMSS). The basic unit here is the lesson; in

principle, 100 “random” lessons in 8th grade classes are videotaped in each country (in the 1995 run,USA, Germany and Japan). In practice some compromises were made, like just 50 lessons from

Japan. The video recordings are subtitled in English so that they can be analysed by Anglophoneresearchers. For each lesson, a table is produced which shows the phases of the lesson, as regards

- time interval (with in the lesson, of each phase)- organisation of interaction (class work, individual seatwork, … )- activity (e.g. setting up, working on tasks)- short description of contents (Task 1: “Find the angle…”; Student solution 1: …)

The main focus of the study – as evidenced by the coding schemes – is on structures in the lessonwhich are supposed to be generic for the discipline or country, that is at level 6 (pedagogy) or level5 (mathematics): how are lessons organised time wise (phases), who present solutions to tasks, howmany techniques are presented, to what extent are the tasks of a routine nature, do “proofs” appearin the lesson or not, and so on. Striking results include, for instance, the average percentages ofseatwork time that is spent on general task types: “practicing routine procedures”, “invent newsolutions/think”, “applying concepts in new situations” (the last code being defined as the negationof the other two). Praxeologies enacted in classrooms are considered under these general angles andmore recently also with respect to domains (level 4), such as ‘algebra’ and ‘geometry’, combinedwith the general task types such as the above (Neubrand, in Leung et al., 2006, 300).

In short, the aim is to find differences between general (country specific) patterns of howmathematics lessons proceed. The hypothesis of the study it that certain aspects of pedagogy (level6) and of the global aspects of school mathematics (level 5) are stable within a society (country)where they remain valid for all schools (level 7) and in particular for mathematics teaching in

general (level 5 and below). This hypothesis seems to be validated by the study data, for instance asregards general “lesson scripts” or the distribution and quality of students’ and teachers’interventions. The methods and results of the study, particularly concerning national “scripts” formathematics lessons, remain controversial (cf. egg. Keitel and Kilpatrick, 1999; Clarke et al., 2007).But assuming mathematics teaching in a country has qualities which may be compared to what isfound in other countries, the TIMSS video study has certainly been pioneering in advancing aserious and operational alternative to measuring these in terms of students’ responses tostandardized tests – or, if you want, to complement such measurements with hypotheses for morewell-founded causes of the differences found.

The learners’ perspective study (Clarke, 2006) aims at investigating more deeply the practices in

classrooms, through videos of a small set of 10 consecutive lessons in each country, also in grade 8,


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and complemented with interviews with students and teachers, textbook, and other data which permits to take into account a wide variety of perspectives. The video data itself is richer in thesense that three angles (teacher’s, students’ and whole-class) are filmed in a given lesson. Thelevels 1-4 of students’ praxeologies may thus be considered for each sequence of lessons, but as

they are merely samples – even if they are aimed to be representative of the curriculum – theycannot be directly compared. Also, there is no claim to “national representativeness” for the lessonsfilmed; instead, rather good teachers are deliberately selected, out of the idea that one could learnmore from studying their lessons. Indeed this study is not preoccupied like the TIMSS video studieswith direct horizontal comparison between national contexts, but instead with in-depth analysis ofthe vertical relations (from level 7 down to level 0) within each classroom considered (cf. Fig. 6).The deliberate focus on learners’ practice (level 0, related to levels 1 and 2) is a significant featureof this study, making it in some ways an organisation of parallel studies of single classrooms,carried out in different contexts without aiming, a priori, at common measures. The goal, however,is clearly that one may learn from studying the material and analyses from other contexts (be itcountries, societies or civilisations). It is interesting to notice the involvement of local experts of the

relevant school curriculum in analysing these relations, and also the requirement that any publishedinterpretations of data from a given country “must be validated by member researchers from thecountry providing the data” (Clarke, 2003, 176).

The learners’ perspectives study thus differs from the studies previously considered in the followingsense : the horizontal comparison across countries is not a primary goal, and to the extent it is done,it is based on the primary study of vertical relations within one country or school, i.e. the levels ofdetermination as considered in the primary part. Indeed, many of the papers published from the

project present only results from one country. Indeed, in the research questions and the researchdesign (Clarke, 2006), the main emphasis is on studying relations that are internal to the contexts:

between observed practices of students and teachers (level 0 and 1, but also level 2 and 3 ascoherent sequences of lessons are considered) and the relation to “local curricular goals” (level 2and higher). International differences or cultural specificities are discussed based on these parallel,in-depth studies of vertical relations within each context; but, given the few and relatively randomcontent areas which are encountered in each country, only at the higher levels (5 and above).

TIMSS Video Study Level Learners’

perspective study

98

76543210

Fig. 6. TIMSS video studies compares pedagogical principles from contexts directly. LPS seeks to first establish

vertical relations (from data at levels 0-8) and then possibly compare the result among contexts.

Primarycomparison

Possiblesecondarycomparison


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This difference between the two video studies may also help to explain the divergence as regardsthe results obtained, especially on the issue of lesson structure (cf. Clarke et al., 2007). In theTIMSS video study, looking at large numbers of random lessons, we saw that certain “scripts”regarding the pedagogical structure of lessons (level 6) were identified. These structures seem to

disappear in the learners’ perspective study, and this could, somewhat paradoxically, be seen as aconsequence of the more fine grained data available:

An inevitable consequence of any nationally representative sample of individual lessons is to averageover the distinctive lesson elements, whose location in the lesson is a direct and informative reflectionof the lesson’s location in the topic sequence. The detailed LPS data set supports a fine-grainedanalysis of both the form and the function of lesson events (Clarke et al., 2007, 292).

Metaphorically, regularities which appear at large distances may disappear in a close-up look. More precisely, studies with a strong focus on relating levels vertically, and with a strong focus thedetermination of the sub-disciplinary levels, may not always support conclusions from surveysaiming at horizontal comparisons at the higher levels (obtained by averaging over large data sets).

6. The civilisation level: the East-West case (ICMI Study 13)

Ever since the second international study of mathematics was completed in 1981, it has been clearthat such surveys appear to show systematic differences that go beyond countries and educationalsystems. The most striking of these is the superiority of students from East Asia (Japan, HongKong, Korea…), according to virtually all available measures of their “overall performance” inmathematics. At the same time, it has been observed that culturally mixed countries like the USAdisplay relatively large variations among students, which are moreover linked to ethnicity more thanto other sociologic variables; so that for instance, Asian American students’ performance is quitesimilar to that of students in East Asian countries, despite the lower average of American students at

large (cf. Clarke in Leung et al. 2006, 354f). This leads to the hypothesis that level 9 of our model –“civilisation” in the sense of larger cultural context – may be held accountable for some of thedifferences observed at lower levels of determination and, through them, in students’ performance:

Our contention is that cultural divisions are much more meaningful than political or geographicdivisions in explaining differences of educational practices in mathematics. East Asia and the West inthis study are therefore cultural demarcations rather than cultural divisions, roughly identified as theChinese/Confucian tradition on one side, and the Greek/Latin/Christian tradition on the other. (Leunget al., 2006, 4)

The assumption that one civilisation or culture “dominates” a society (built into the model we usehere) is not invalidated by the existence of multicultural societies. However, the results alluded to

above suggest that the determining force of cultural factors may well be independent from those of“official society”, and conceivably bypass most of the levels in Fig. 5, to exercise its influence moreor less directly on students’ activities and learning outcomes. To investigate such hypotheses clearlyrequire deeper studies of well delimited forms of civilisation, and this was indeed what the ICMI“comparative study of East Asia and the West” set out to do – following the call from which theabove quotation comes.

The study resulted in a rich body of contributions, with a reasonable balance of authors comingfrom East Asian and Western countries. The overall impression is that viewpoints of a Westernorigin (which are certainly permeating the present paper!) tend to dominate – perhaps in part as aneffect of a Western language being the norm for all chapters. Part of the comparative challenge,

indeed, is related to the different sets of analytic frameworks which scholars from different cultural


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traditions bring with them – in fact, frameworks from the two “sides” are combined in somechapters, particularly in those with authors from different traditions. It is evident that comparison ofcivilisations cannot be done from a “higher”, neutral viewpoint; measuring one in terms of the otherinevitably leads to a colonial position which does not further our understanding. As a consequence,

we (as Western readers) can perhaps learn most from those part of the study which are substantially based on East Asian models for describing and analysing differences; these are therefore privilegedin the following sample.

The first two chapters are written from manifest East Asian perspectives; they provide portraits ofcultural and historical tendencies linked to education in Japan (clearly at level 9). In particular,Hirabayashi (in Leung et al., 2006, 51-65) points out the importance of Japanese notions like dô(道, roughly translated as “way”) and justsu (techniques) which, to the Japanese tradition, areimportant factors in educational goals. He contends that the prevailing role of jutsu over dô may beresponsible for central shortcomings in modern Japanese mathematics education (implicitly at level5 and lower). A related institutional structure – that of iemoto (家元, roughly “spiritual familyhouse”, evolving around common goals) – is described by Emori and Winsløw (in Leung et al.,2006, 553-566). They use it to interpret observations from secondary mathematics classrooms andcompare with Western counterparts to show the pertinence of this idea. While the main conclusionsconcern levels 5 (the meaning of mathematical sign systems and discourse, learning goals ofmathematics), the cases considered are clearly situated within subjects and themes (levels 1 and 2).Similar lines of thought were developed by Wong (in Leung et al., 2006, 111-121), for the Chinesecontext (but probably within the same civilisation): again, a notion of “way” (translated here as“right way to do something”) is central to describe what education is supposed to “initiate” thelearner into; some examples at level 1 and 2 are considered to show the meaning of the deeper

principles (at level 6) for such initiation: repeating and varying techniques related to a subject.

Several contributions to the study are concerned with investigating teachers’ values and beliefsrelated to mathematics (determination level 5), and to pinpoint systematic differences between twocultural traditions (such as China and Australia). An interesting variation from the more obviousways to organise such studies is presented by Tiong and Bishop (in Leung et al., 2006, 523-535),who report on a qualitative study with two East Asian mathematics teachers who moved toAustralia and are now teaching there. They identify a number of value difference related to the waymathematics is taught (level 5) and what they see as underlying factors in schools and in societies(mainly level 6 and 7); the authors identify these with certain “universal dimensions” of culturalvariability such as “individualism/collectivism”. The opinions of the two teachers are interesting,not because they are “representative” of anything, but because of the teachers’ privileged viewpoint

on the boundaries between the two cultures of teaching, and the cultures more broadly. A similaridea can be found in the chapter by Isoda, McGrae and Stacey (in Leung et al., 2006, 397-408); theyreport on the benefits and challenges experienced by Japanese and Australian school kids’, as theydiscussed their ideas and methods for solving a set of mathematics problems. Such an experiment isobviously difficult to set up in a reasonable way, especially given the language barriers; but it could

provide an interesting supplement to the methods found in studies of videotaped lessons.

Another set of studies concern the analysis of global differences between textbooks and curricula in East Asia and the West. As a strong example, Park and Leung (in Leung et al., 2006, 227-238)compared major 8th grade text books in China, Japan, Korea, UK and the USA, and foundsignificant differences between the Asian and Western books (and commonalities within each

group). For instance, they found that the Western books reflected a more individualised view of the


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learner, with their stronger emphasis on attractive design, “real world” examples to motivate newtopics, and the provision of different levels of approach. Asian text books at this level are moreterse or “minimal” in their presentation, as “East Asian culture believes in orthodoxy, and studentsare believed to adhere to the orthodoxy despite their individual differences” (p. 236). Here the word

“orthodoxy” comes, for the Westerner, with a number of connotations which are not likely to beintended by the authors; the text as a whole suggests that it refers to the larger “cultural idea” ofteaching a common “way”, mentioned above, corresponding to the value of “collectivism” and theideal of social harmony in society. Thus the manifest differences of textbook styles seem to fit intoa larger picture of determining forces coming from level 9.

Finally, several of the studies remind us that East Asian and Western cultures, societies and systemsof education have interacted in various ways and cannot, therefore, be considered as independentsystems. Even in East Asian countries like Japan, which was never colonized, Western methods inmathematics teaching have been at times systematically imported, and of course subsequentlyadapted, as explained e.g. by Ueno (in Leung et al., 2007, 65-79). Moreover, within each of the two

“civilisation groups”, one can find enormous variations which cast further doubt on any easygeneralisations. Kaiser, Hino and Knipping (in Leung et al., 2006, 318-350) proposed eight“empirically grounded” variables for comparing mathematics teaching in different countries (all atlevel 5 or 6) which they describe in some details for four countries (England, France, Germany andJapan). Their discussion is mainly based on previous studies of the first three countries and morelimited evidence from Japan (due “mainly [to] time and capacity restrictions”). For most of thevariables, they found that England and France had “polar mathematics educations, with Germanyand Japan situated somewhere in between”. This is, for instance, the case as regards “the role ofreal-world examples”, which is found to be strongest in England and weakest in France, withintermediate positions occupied by Germany and Japan.

In fact, “Western mathematics teaching” may not be a meaningful category. Bessot and Comiti (inLeung et al., 2006, 159-179) provide a multilevel comparison of mathematics teaching in Vietnamand France, going from brief historical sketches of how mathematics curricula developed in the twocountries (level 8) to a study of how graphs may be used by students in investigating properties offunctions (level 3 or 2). The study of the genesis of educational practices in these countries showshow competing ideologies and philosophies (at level 6 or 8) as well as institutional systems (level7) have succeeded each other in ways which cannot be accounted for in terms of a generalised “EastAsian” or “Western” civilisation.

7. Binary comparative studies in recent doctoral theses

Up to now, we have considered the diversity of widely published comparative studies onmathematics education and its outcomes, and put to the test the use we propose of ATD for makingsense of this diversity. In this section, we use the same frame for introducing and situating a body ofcomparative research which may be less widely known, developed within the French didactictradition. More precisely we studied 11 doctoral theses produced within the last decade at (or incollaboration with) French universities and present some of their main characteristics. We note in

passing that these 11 theses are just a sample and that doctoral theses on comparative issues areincreasingly common in many countries, beginning with the widely known thesis of Ma (1999).

Doctoral projects are of course more modest in scope that the large scale projects evoked in the previous sections, and they face different constraints. For our sample of 11 theses we will mainly

consider the following questions: to what extent is the complexity of comparative studies, as


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illustrated in previous sections, approached in doctoral theses? How do the characteristics of thedoctoral academic exercise influence the type of comparative research carried out, and is ourframework able to capture this influence? What do these theses add to the kind of comparativeresearch evoked so far?

We think our sample is appropriate for approaching these questions. A first reason is that the corpusat our disposal, even if reduced, presents a considerable diversity in terms of the countries involved,of the way comparative research themes are constructed and methodologically addressed.Moreover, it contains both isolated pieces of research from a variety of countries, and a “group ofrelated theses”, developed in the context of long term collaboration between France and Vietnam(more precisely between University Grenoble 1 and Ho-Chi-Minh Pedagogical University).Finally, the fact that these doctoral theses were elaborated within the common research tradition ofFrench didactics enabled a coherent analysis and synthesis of the results appropriate for the tightspace of an article. Fig. 7 provides a general presentation of the sample.

Author Year Countries Research theme (rough outline)

Nadia Amra 2004 France-Palestine The teaching of functions in uppersecondary schools

Valentina Celi 2002 France-Italie Teaching of geometry for 11-16 years oldstudents, and its effects

Richard Cabassut 2005 Allemagne-France

Reasoning and proof in secondarymathematics education

Pascal Stoelting * 2008 Allemagne-France

Functional thinking of students 10 to 16years old and its teaching in France andGermany

Lê Thi Hoai Châu* 1997 France-Vietnam The teaching of vectors at grade 10 levelDoan Huu Hai* 2001 France-Vietnam The teaching of 3D geometry in upper

secondary schoolLê Van Tien * 2001 France-Vietnam The relationships between functions and

equations in upper secondary school Nguyen ChiThanh*

2005 France-Vietnam Introduction in secondary schools ofalgorithms and programming withcalculators

Nguyen Hai Quoc* 2006 France-Vietnam The resolution of quadratics in secondaryeducation

Tran Luong CongKhanh*

2006 France-Vietnam The notion of integral at upper secondaryschool

Lê Thai Bao ThienTrung*

2007 France-Vietnam Relationships between the teaching of thelimit notion and the “decimalization” ofreal numbers in a calculator environment

Fig. 7. The 11 theses of our sample. Thesis with an * had two co-supervisors, one in each country.

Most of the theses may be accessed online at the database http://ccsd.cnrs.fr

One first characteristic emerging from the table is that the theses focus the comparison at the levelof some particular mathematical domain or sector (level 3 or 4), or around a type of paradigmatic

practice of the mathematical discipline, such as proof (level 5). The titles of the thesis confirm thisclaim, being for instance (in English translation): « Proof, reasoning and validation in thesecondary level teaching of mathematics in France and Germany », « A didactical andepistemological study of the teaching of vectors in two institutions: 10th grade in Vietnam andFrance », « The contributions of comparative didactic analysis of the resolution of quadratics in


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secondary education in Vietnam and France ». These titles also suggest that horizontal comparison is the main aim. But in order to understand the reasons motivating the comparisons undertaken, howthey are achieved, how the relationships between the different levels of determination are taken intoaccount, and up to what point “comparability” is critically examined, one needs to enter in the

details of the comparative work. For this purpose, we have constructed a specific grid of analysis asshown in Fig. 8. For the synthetic presentation of the results, we split the corpus into two sub-corpuses, the French-Vietnamese theses (forming a coherent series) and the rest.

Fig. 8. The framework used to analyse the theses listed in Fig. 7.

The French-Vietnamese theses.

A. Bessot and C. Comiti were at the origin of the French side of the French-Vietnamesecollaboration underlying the theses we consider here. In a recent article (Bessot and Comiti, toappear) they complement the information and results provided by them in the ICMI study (and

presented at the end of Sec. 6). The article provides a synthetic vision of the way comparison has been approached in these theses, and points out the coherence of the whole enterprise. It is madeclear, for instance, that within this collaboration, the choice of favoring comparative issues fordoctoral thesis was made quite early. Its main aims were: i) to reinvest and put to the test didacticknowledge built in the French context, ii) to answer the needs of the Vietnamese educationalsystem, and iii) to favor the integration of new doctors into the scientific community of theircountry. Moreover, comparative studies were seen as a privileged means for questioning thetendency we all have to consider our own particular context as “normal” or “natural”, that is the

phenomenon which Chevallard describes as “naturalization of mathematics and didactic praxeologies”. These comparative studies have thus specific characteristics expressed by the twoauthors in the following terms:

Far from a ‘cognitivist’ approach centred on the particularities of students and teachers, this type ofresearch situates the problem of students’ and teachers’ difficulties in the totality of conditions andconstraints which bear upon the teaching of mathematics (Bessot and Comiti, to appear ; our translation)

This explains why institutional analysis, relying on curricular documents and textbooks, plays afundamental role in the methodology developed for these comparisons. In general, institutionalanalysis and comparison has both a synchronic and diachronic dimension. The importance given tothe diachronic dimension is coherent with the reference made to the notion of didactic transposition,understood as a long term process. In the diachronic work, specific importance is given to themoments of curricular reform. At these points of change, one may more clearly observe thetechnological and theoretical discourse produced by the noosphere in order to explain and justify

the choices made in terms of both mathematics and didactic organizations.

Basic data of thesis (author, title, year of defense, university, supervisor(s))

Motivation of the comparison (outline)

General characteristics: countries compared, level(s) of teaching considered, scale of contexts

of data, the object(s) compared, main level at which comparison is done (cf. Fig. 5), secondary

levels (if relevant), methods used to collect and compare data.

Theoretical and methodological dimensions: theoretical frameworks and constructions on

which the comparison is founded, the most important notions and their function in the project;

the methodological tools specifically elaborated within the project

The results obtained: situated, as far as possible, at the levels of comparison (cf. Fig. 5) and the

extent to which comparisons at different levels are related, possibly causally; difficulties

encountered in the course of comparison


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Bessot and Comiti (to appear) provide an example which illustrates the type of result obtainedthrough such diachronic comparative analysis. It deals with the different status given toapproximation in Vietnamese and French teaching of Analysis. This difference was identified firstin the thesis by Lê Van Tien studying the relationships between functions and equations (our level

3) in upper secondary school. Differences observed at this level were interpreted by Lê Van Tien byconsidering more globally the historical changes in the epistemological vision of Analysis (level 4),and in the status given to experimental activities (levels 5 and 6), in the two countries. Notice thattheses of Nguyen Chi Thanh, Tran Luong Cong Khanh and Lê Thai Bao Thien Trung buildsubstantially on the thesis by Lê Van Tien, which adds a cumulative effect to the work within thisgroup of theses (with its virtually novel research context).

Institutional analysis of mathematical and didactical praxeologies plays a predominant in themethodologies used in the theses, but connections with implemented and achieved curricula are also

present. On the one hand, students’ questionnaires are used for clarifying how identified differencesin mathematics praxeologies impact on students’ practices. On the other hand, institutional

comparison can lead to the conception, implementation and analysis of didactic engineering products (for use in the Vietnamese system).

This is the case for instance in the two theses by Nguyen Chi Tanh and Lê Thai Bao Thien Trung,which are less focused on comparison than the other ones. In their projects, the institutionalcomparison is put at the service of didactical design, and as expressed by Bessot and Comiti:

The realisation and analysis of a didactical engineering [based on such an institutional comparison, ournote] in one of the systems, allows not only to raise questions about this system but also on the othersystem, and it favours thus the emergence of generic questions whose pertinence are suggested by theother system. (Bessot and Comiti, to appear; our translation)

It is worth noticing that a direct study of the implemented curriculum through the observation and

analysis of classroom practices comparable to what has been described in section 5 is not part of themethodologies used in this sub-corpus.

This set of thesis offers thus a quite coherent picture. The main focus is generally situated at levels3 and 4. The comparison explicitly takes into consideration various levels of determination, mainlyup to 7, even if societal and cultural factors are regularly invoked especially through reference tohistory and to the different foreign countries which have successively imposed their views onVietnamese education, beyond the long-lasting influence of the Confucius heritage (cf. Sec. 6).

Theoretical frames are mainly provided by ATD, including the theory of didactic transposition, andthe theory of didactical situations when a dimension of design is present. The use of ATD naturally

leads to consider upper levels of determination even when this notion is not explicitly used. What isat stake is the understanding of an ecological system and its possible dynamics. In accordance withthe theoretical frames, students and teachers are not considered in their individuality (as pointed outin the first quotation above). Questionnaires and experimentations involving them are generallysmall scale qualitative studies. There is no direct study of the implemented curriculum. We suspectthat some characteristics of the Vietnamese context contribute to this methodological choice: theexistence of one official textbook and a detailed teacher guide, and the dominating culture oftransmissive pedagogy (cf. Bessot and Comiti, in Leung et al.., 2006, p. 169). They can leadresearchers to consider that the implemented curriculum is rather uniform, and close to the intendedcurriculum. Moreover, even when comparison is the apparent focus of the study, the goal of theenterprise is a better understanding of the rationality of the two systems at stake and of their

possible evolution.


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Finally, it should be noted that in this group of theses studies, the two countries are far from playinga symmetric role. The doctoral projects are all done by Vietnamese students, and their final goal isto serve the educational system of their country. On the other hand, theoretical frames and

methodology mainly come from the French tradition of research in didactics.

The DIDIREM comparative thesis.

The second part of the corpus consists of isolated theses, all of which were prepared with theDIDIREM research group at the University Paris-Diderot; however among the authors, just one (R.Cabassut) is French. The overall image we can give of these theses is not as homogeneous as withthe first sub-corpus, but some common traits are nevertheless observed.

Like in the first sub-corpus, curricular analysis carried out through the study of syllabi andtextbooks plays a fundamental role. It generally follows an epistemologico-mathematical analysis ofthe domain at stake and a research review, which support the framing of the didactic analysis.

Except for one thesis (Stölting), the main theoretical frame is ATD and it is complemented by morelocal constructs considered especially relevant for the mathematical domain at stake, orcomplementing the constructs provided by ATD. From this point of view it is worth noticing thateven if ATD has a semiotic dimension structured around the dialectics between ostensive and non-ostensive, semiotic phenomena are mainly addressed through the constructs and categories provided

by Duval’s more elaborate theory concerning semiotic systems of representations.

As with the first sub-corpus, the results of the institutional analysis carried out at levels from 3 to 5are often interpreted by referring to higher levels of determination. For instance, to explain the bigdifferences existing between French and Palestinian mathematical and didactical praxeologiesrelated to the notion of function, Amra points out the diverse colonial influences which have shapedthe Palestinian mathematics syllabi, and also the transmissive vision of pedagogy which prevailsthere. Similarly, in order to interpret the differences she observes, Celi refers to generalcharacteristics of the curricular organisations in Italy, such as the greater autonomy given toteachers, the different role given to textbooks, and the historical culture of Euclidian geometry inmathematics education.

Finally – again much like in the first sub-corpus – one can observe a jump from the study ofintended curriculum to the study of achieved curriculum, implemented curriculum being themissing link. Institutional analysis appears as a first and necessary step in the comparative work,necessary for making sense of students’ behavior. The constraints of a doctoral study (developing

one’s first, personal research work, to be carried out in a few years) do not permit one to seriouslyaddress both the implemented curriculum and the achieved curriculum. Within the current state ofdevelopment of comparative studies, it is then understandable that the prevalent choice is toexamine the effects on students. Within this general trend, differences are nevertheless observed inthe importance given to this dimension of the research (from one chapter to an importancecomparable to that given to institutional analysis) and in the methodologies developed (self madequestionnaires, interviews, use of data coming from other sources such as PISA, national orregional studies, competitions involving two different states).

This is also a place where issues of comparability often crystallize. For instance, in Celi’s thesis,selecting a theme appropriate both for French and Italian grade 10 students, and formulating the

precise questions posed to the students, raise difficult problems due to the difference in the


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geometrical approaches and tools between France and Italy, and also the fact that in contrast toItalian students, French students are used to very detailed and guiding formulations of tasks. Thesituation is even worse in Amra’s thesis, the French approach to functions being co-variationalwhile the Jordanian approach is set theoretical. This leads the author to built tasks which are

supposedly unfamiliar for students in both countries, and look at the way they make sense ofunfamiliar tasks.

Contrary to what could have been expected, linguistic issues do not necessarily emerge as a majorconcern. Cabassut’s thesis is an exception in this sub-corpus. In order to cope with the diversity ofterms currently used for addressing issues related to validation and proof in the educationalcommunities in France and Germany, he develops a systematic comparative analysis of thevocabulary used in the two countries. Moreover, motivated by the differences between French andGermany as regards didactic contracts related to the formulation of proofs, he develops a specifictool for comparing what he calls the discursive expansion of proofs.

One of the theses (Stölting) in this group thesis has supervisors from two countries. It is interestingto notice that this situation has an evident impact on the theoretical frames used. The domain atstake is that of functions and it is approached through the notion of “functional thinking”,introduced in Germany at the beginning of the 20th century. The study of functional thinking in thethesis combines general constructs coming from the two didactic cultures: the notion of conceptualfield due to Vergnaud; semiotic registers in the sense of Duval; Grundvorstellungen (fundamentalideas) due to vom Hofe; and Grundkenntnisse, a notion created in the thesis. A similar combinationor integration of theories can be observed in Celi’s thesis where notions that are part of theeducational discourse in Italy (e.g. the distinction between an attitude of rational intuition and arational attitude) are connected and combined with the notion of geometrical paradigms due toHoudement and Kuzniak. These examples show that comparative studies also have to address theissue of compatibility and connection between the theoretical frames and notions supportingdidactical research in the involved countries, and more globally the ideas involved in theireducational discourses. In our opinion, it is not a coincidence that this is especially visible in studieswhich compare the teaching in countries with well established research cultures in didactics.

8. Final remarks

International comparative studies of regulations and practices of mathematics education come inmany forms and they are carried out with a large variety of purposes and means. In this paper wehave presented what seems to us a promising way to relate and contrast the domains that such

studies investigate and compare, as well as the relations they establish or claim to find betweenthem. While the large scale studies considered in Sec. 4 (and to some extent also sec. 5) havecertainly drawn more public attention than the close-up studies of mathematics teaching in a few(often just two) countries considered in Sec. 6 and 7, it seems clear to us that the deeper questionsinvolved in comparison require the affordances of research with a sharp focus on particularcontexts, including the institutional analysis of how mathematical domains, sectors and themes areunderstood and articulated with each educational culture. This is in particularly salient for the

purpose of comparing contexts in which radical differences exist at every level (from civilisation tothe way mathematical subjects are approached), and where the risk of drawing simplistic conclusionas to their mutual relations or impacts is almost inevitable without such a closer consideration ofconcrete conditions at every level. The model we have proposed in this paper offers a means of


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identifying missing elements in a given analysis but also to situate it with respect to studies thatcould might supply those elements.

More specifically, we wish to point to four problematic observations emerging from our work with

the research products analysed in this paper:

1. While the integration of studies with data from several levels should certainly be possible,there is a tendency of most studies to rely entirely on their own data, perhaps as aconsequence of particular aims but possibly also out of a lack of a framework that couldhelp identifying what kind of other research it could be important to draw on.

2. There is a clear tendency in large-scale studies to “average” over the lower levels (belowlevel 5 in particular) in ways which are rarely justified by the data; moreover, these studiesgenerally presents “snap-snots”, with little or no view of how conditions develop over time.On the other hand, of binary, small-scale studies to draw hasty conclusions for higher levelsof determinations of a smaller number of contexts, based on close-up studies of fairly

peculiar (often contingently “chosen”) contexts, with particular sectors or themes and the possibility to consider also a diachronic dimension. Researchers are clearly inclined to drawmaximal conclusions from their own research, but combinations of data of the kind alludedto in the previous point, may help to close this gap between evidence and inferences.

3. Language issues – which affects the formulation of conditions at all levels – are rarely takeninto serious consideration. Given the cultural and institutional situatedness of mathematicaland didactical organisations in the context of teaching, this may lead to very partial or evenmisconstrued conclusions which are not free of colonialist tendencies in the (frequent) casewhere the models of one culture are taken for granted as a pseudo-neutral reference.

4. As most studies are concerned with more than one level of determination, and indeed withhypotheses of how they interact and determine each other, it is very important that theselevels be explicit and that the aims of comparison – vertical and horizontal – are made

clear. The way in which results are used and interpreted cannot be entirely controlled byresearchers, but we should at least not be naïve when it comes to the massive interests whichare attached to the objects of research – in the first place, at the level of society, but when itcomes to the conclusions that may be drawn, at each and every of the other levels as well.

When studying didactical and mathematical practices in a cultural-institutional light – which is allthe more inevitable when considering such phenomena in different institutional and culturalcontexts – we are necessarily constrained by our own experiences and contexts. While ATD as awhole is certainly not free of marks from the (European, more precisely francophone) context in

which is arose, we do think that it provides useful means of identifying important assumptions andtraditions well beyond its origins; but also that its adaptation and development for such anenterprise will necessarily require collaborative efforts of the kind some of the studies considered inthis paper give examples.

References

Bessot, A. and Comiti, C. (to appear). Apport des études comparatives en recherches en didactiquedes mathématiques. Actes du Séminaire National de l’ARDM .

Bishop, A. (1988). Mathematical enculturation. Dordrecht : Kluwer.

Chevallard, Y. (1985). La transposition didactique. Du savoir savant au savoir enseigné.

Grenoble : La Pensée Sauvage.


22/22

Chevallard, Y. (1999). L’analyse des pratiques enseignantes en théorie anthropologique dudidactique. Recherches en Didactique des Mathématiques 19 (2), 221-265.

Chevallard, Y. (2002). Organiser l’étude 3. Écologie & régulation. In: Dorier, J. L. et al. (Eds.), Actes de la 11

e école de didactique des mathématiques. Grenoble : La Pensée Sauvage.

Clarke, D. (2003). International comparative research in mathematics education. In: Bishop, A. etal., Second international handbook of mathematics education, pp. 143-184. Dordrecht: Kluwer.

Clarke, D., Mesiti, C., O’Keefe, C., Xu, L., Jablonka, E., Chee Mok, I., and Shimizu, Y. (2007).Addressing the challenge of legitimate international comparisons of classroom practice.

International Journal of Educational Research 46, 280–293.

Clarke, D. (2006). The LPS research design. In D. Clarke, C. Keitel & Y. Shimizu (Eds.)Mathematics in twelve countries: the insider’s perspective (15-36). Rotterdam: Sense Publishers.

Hopmann, S., Brinek, G. and Retzl, M. (Eds) (2007). PISA according to PISA – does PISA keepwhat it promises? Wien: LIT Verlag.

Kawanaka, T., Stigler, J. and Hiebert, J. (1999). Studying mathematics classrooms in Germany,Japan and the United States: lessons from the TIMSS videotape study. In G. Kaiser, E. Luna and I.Huntley (Eds), International comparison in mathematics education, pp. 86-103. London: FalmerPress.

Keitel, C. and Kilpatrick, J. (1999). The rationality and irrationality of international comparativestudies. In G. Kaiser, E. Luna and I. Huntley (Eds), International comparison in mathematicseducation, pp. 241-256. London: Falmer Press.

Lange, J. de (1996). Using and Applying Mathematics in Education. in: A.J. Bishop, et al. (eds).1996. International handbook of mathematics education, Part one, pp. 49-97. Dordrecht: Kluwer.

F. Leung, K.-D. Graf and F. Lopez-Real (2006) (Eds). Mathematics education in different culturaltraditions – a comparative study of East Asia and the West . New York: Springer Science.

Ma, L. (1999). Knowing and teaching elementary mathematics. Mahwah: Lawrence Erlbaum.

Mullis, I. et al. (2005). TIMSS 2007 assessment frameworks. Boston: TIMSS & PIRLS internationalstudy center.

Mullis I. et al. (2008). TIMSS 2007 International Mathematics Report: Findings from IEA’s Trendsin International Mathematics and Science Study at the Fourth and Eighth Grades. Boston: TIMSS& PIRLS international study center.

Niss, N. (2003). Quantitative literacy and mathematical competencies. In B. L. Madison and L. A.

Steen (Eds), Quantitative literacy : why numeracy matters for schools and colleges, pp. 215-220.Princeton: National Council on Education and the Disciplines.

OECD (2006). Assessing scientific, reading and mathematical literacy. A framework for PISA2006. Paris: OECD.

OECD (2004). Learning for tomorrow’s world: first results from PISA 2003. Paris: OECD.

Sierpinska, A. (2006). Looking at mathematics education from an institutional perspective. Lecturenotes, retrieved 26.02.2009 from http://www.asjdomain.ca/institutional_perspective.html

Winsløw, C. (2005). Définir les objectifs de l’enseignement mathématique : La dialectiquematières-compétences. Annales de didactique et des sciences cognitives 10 (2005), 131-156.

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