Working with the Anthropological Theory of the Didactic in Mathematics Education

A Comprehensive Casebook

Marianna Bosch, Yves Chevallard, Francisco García, John Monaghan, Marianna Bosch, Yves Chevallard, Francisco Javier García, John Monaghan

• Publisher: Routledge
• Year: 2019
• ISBN:
  9780429582424
• Edition: 1
• Topic: Education
• Subtopic: Education General

This book presents the main research veins developed within the framework of the Anthropological Theory of the Didactic (ATD), a paradigm that originated in French didactics of mathematics. While a great number of publications on ATD are available in French and Spanish, Working with the Anthropological Theory of the Didactic in Mathematics Education is the first directed at English-speaking international audiences.

Written and edited by leading researchers in ATD, the book covers all aspects of ATD theory and practice, including teaching applications. The chapters feature the most relevant and recent investigations presented at the 6th international conference on the ATD, offering a unique opportunity for an international audience interested in the study of mathematics teaching and learning to keep in touch with advances in educational research. The book is divided into four sections and the contributions explore key topics such as:

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• The core concept of 'praxeology', including its development and functionalities

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• The need for new teaching praxeologies in the paradigm of questioning the world

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• The impact of ATD on the teaching profession and the education of teachers

This is the second volume in the New Perspectives on Research in Mathematics Education. This comprehensive casebook is an indispensable resource for researchers, teachers and graduate students around the world.

PART 1

Unity in Diversity: ATD at Work

1

WHAT KIND OF RESULTS CAN BE RATIONALLY JUSTIFIED IN DIDACTICS?

Josep Gascón and Pedro Nicolás

1 The question of the presumed normativity in didactics

Let us start with a question about the alleged normative character of didactics:

To what extent, how and under what conditions can (or must) didactics issue value judgements and normative prescriptions about how to organise and manage study processes?

Most of the approaches and theories in didactics have an implicit position about this question. However, explicit discussions about the legitimacy of didactics to express normative prescriptions remain scarce, let alone specific deliberations about the logical form of the results one should expect. At the same time, discourses in didactics often contain value judgements concerning teaching and learning processes, which sometimes lead to proposals for rules of action—the well-known “implications for teaching”. Including value judgements and normative prescriptions in the discourse, often as the main conclusion, appears as a tacit taking of position that has a strong effect on the sort of research questions to be considered and the kind of endorsed answers. It may even lead one to believe that research results in didactics can (or even must) be stated in terms of values and rules.

How can we answer this question according to the ATD’s basic assumptions and principles? First, we will explain the ATD perspective about the object of study of didactics and the corresponding research results. This will make clear that value judgements and normative prescriptions have no place in the realm of research results in didactics. Another consequence will be the need to clarify the type of assertions didactics can formulate on solid foundations (Gascón & Nicolás, 2017).

2 Max Weber’s thesis on social sciences

The origin of one of the main points of this chapter is (Weber, 1917/2010). In this work Max Weber stands up for the following thesis:

• Social sciences can only state assertions about the rationally suitable means to achieve ends previously fixed but whose validity cannot be rationally established.
• The precept of avoiding value judgements is a consequence of the distinction between two spheres of reality, the knowledge sphere and the values sphere, each one containing questions of different nature.

In the knowledge sphere, the main questions raised relate to the behaviour of a certain portion of the world. Why is this portion of the world the way it is? What are the required conditions for this portion of the word to be modified in a certain direction, and which are the obstacles to this modification?

In the values sphere, the main questions are about what we should do in a given situation; how we value this situation; if we should do something to change this situation in a given direction, and, if so, in which one.

In the knowledge sphere, it is possible to make a critique of a given value via an analysis of the required means to attain this value, regarded as an end. Indeed, one can conclude that, given a school institution I, a certain kind of teaching (regarded as valuable, as representing a positive value) is not achievable in I because the required means to put it into practice cannot be implemented in I. As we can see, this critique cannot state whether this value is estimable or worthy. It can only say that certain means are suitable or not to achieve a certain goal, that certain conditions make it easier or more difficult to attain. Therefore, the object of study of social sciences, according to Weber, concurs with the object of study of didactics according to the ATD sense (see section 4).

The principle according to which science should not make value judgments has a positive reverse: the ability of science to provide criteria to guide us on the steps to follow. These criteria do not permit us to shun the responsibility to make decisions, but they can help us to foresee the consequences of our actions.

Ultimately, the coercive aspect of Weber’s theses points out the issues and decisions excluded from the scientific scope. Science cannot teach us what to do, only what can be done and the corresponding consequences. Thus, for instance, economy, as a science, is not qualified to decide how material goods should be distributed, in the same way, sociology cannot decide about the best social or political structure for societies or physics about what should be done with atomic energy. But, in all these cases, and many others, science can reveal some relevant consequences of each possible decision.

3 Reinterpretation of Weber’s thesis in epistemological terms

To prepare our description of the object of study of didactics, let us express Weber’s theses in terms of explanations and scientific laws. Science is important to society to the extent that it provides laws which support non-trivial and general explanations about the occurrence of objective facts. For a fact to be objective we require it to be intersubjective—perceptible by everyone in standard circumstances—and substantive—its existence does not depend upon someone’s perception or representation.

A scientific explanation can be considered as an answer to a question of the type: “Why the occurrence e is the case (instead of the occurrences d₁, d₂, …)?” Of course, deductive arguments are always welcome, but sometimes something different can be accepted as a scientific explanation, for instance, inductive or abductive arguments. The reader is invited to consult (Andersen & Hepburn, 2016; Woodward, 2017) for an overview of this topic.

Typically, a scientific explanation is a valid argument based on a so-called scientific law. What a scientific law is, remains a controversial issue in the philosophy of science. We will therefore only present a summary of the most commonly accepted features of scientific laws. The logical structure of a scientific law is as follows: “Every occurrence of type A is also of type B”. This logic structure makes it clear that the basic parts of scientific laws are occurrences-type (defined as the extension of a property, or giving an exhaustive list, or by recursion, etc.). Therefore, scientific laws are general statements and not assertions about particular occurrences localised in time-space. Another feature of scientific laws is that they seem to express a necessary relationship between types of occurrences, and are not merely accidental coincidences.

For instance, the law stating that metals are good conductors should not be regarded as pointing out a fortuitous relationship between the occurrence-type “being metal” and the occurrence-type “being conductive”. On the contrary, it seems to say that, given what we know about the world, metals are good conductors and it could not be otherwise. This is an important feature of scientific laws, which tells the difference between them and incidental true generalities, as, for example, the statement: “All the mountains on Earth have an altitude of fewer than 9,000 meters.” Indeed, the property “being a mountain on Earth” does not seem to imply the property “having an altitude of fewer than 9,000 meters”.

As we have seen, it is difficult to find the formal distinction between scientific laws and incidental generalities. Nevertheless, there are other features that distinguish them. Unlike arbitrary general statements, scientific laws are part of a scientific theory in which they are logically linked to other statements. Via its laws and its premises, a scientific theory always describes a certain part of the world. Despite being the compression of the complexity of reality, this description still allows us to master a certain part of the world: answering questions; state predictions; etc.

As an example of a law in didactics, we can consider the statement about the existence of the didactic contract (Brousseau, 1997; see also Nicolás, 2015). This law states that, under certain conditions, a study community formed by students and a teacher, along a study process (occurrence of type A), behaves according to specific (perhaps implicit) clauses which rule, among other things, the conduct that students expect from the teacher and vice versa, regarding the knowledge to be taught (occurrence of type B).

For the law of the didactic contract to be scientific, we should show, on the one hand, that both types of occurrences are objective and, on the other hand, that the law itself is objective. To claim this objectivity of the type of occurrences we should provide a sharp, precise description of them. For instance, we should answer the question: what are the precise clauses of this contract? If this is not clear enough, how could we verify the law? To ensure the objectivity of the law we could check its validity in a representative sample of the kind of study communities under consideration, and then use statistical inference. Anyway, we could also consider the law provisionally accepted if we show that, for the moment, it is the best explanation for a certain mysterious or disturbing phenomenon.

The case of the age of the captain is an example of such a phenomenon. As explained in Equipe Elémentaire IREM de Grenoble (1979), a research team noticed that students at school tend to operate with the numbers of the formulation of a problem in mathematics regardless their appropriateness for the solution. For instance, when they face the problem “A captain of a ship owns 26 sheep and 10 goats. How old is the captain?”, the majority of students give as an answer a number which results from operating with 26 and 10.

In Chevallard (1988) we find an explanation for this phenomenon: students do not pay attention to the meaning of the numbers because their behaviour is governed by a didactic contract including a clause according to which school problems always have a solution, reachable from the numbers appearing in the formulation. Therefore, the law that states the existence of the didactic contract including the clause above appears to be the best explanation for an intriguing phenomenon.

Hopefully, after this brief reflection about scientific laws we are persuaded that the questions placed by Weber in the knowledge sphere admit scientific answers (in terms of scientific laws and explanations), unlike the questions placed by Weber in the values sphere. Indeed, the occurrences involved in the questions of the values sphere are not objective.

4 The object of study of didactics

The positive part of Weber’s thesis is that we have interesting questions falling under the knowledge sphere. Let us formulate these questions to link them to didactics. According to the ATD, the theory of praxeologies is not only rich enough but also especially useful to describe all the human behaviours relevant for didactics. Therefore, in the questions included in the knowledge sphere, we will change the expression “portion of the world” by the term “praxeology” (see Glossary). Moreover, in our formulation, we will take into account one of the basic assumptions in ATD according to which the primary object of study in didactics are the processes of genesis and diffusion of institutional praxeologies.

After these considerations, the resulting questions are as follows. Which is the behaviour of a certain praxeology in a given institution? Why is this praxeology the way it is in this institution? Which are the required institutional conditions for this praxeology to be modified in a given direction, and what are the institutional obstacles to this modification?

All this is coherent with the object of study of didactics according to the ATD: ‘the ATD suggests the following definition of didactics: didactics is the science of conditions and constraints for the diffusion of praxeologies in the institutions of the society’ (Chevallard, 2011, p. 27, our translation).

To study the aforementioned conditions and constraints for the diffusion of praxeologies and, more explicitly, to account for the didactic phenomena linked to that diffusion and which appear in the different institutions of the society, didactics formulates questions belonging to the following three fundamental dimensions of a research problem: economic, ecological and epistemological (Gascón, 2011).

The economy of any system (of the body of an animal, of a plan, of a language, of a discourse, of a book, etc.) is formed by the set of rules and principles which govern the structure and the running of the system. In particular, given an institution, I, the praxeological economy of I at a certain period is given by the set of rules and principles which govern the institutional life of the praxeologies of I during this period. Thus, when we tackle the economic dimension of a didactic problem in which there are several praxeologies involved, we are concerned with those rules and principles.

The ecology of a system is given by the set of conditions and constraints that have an impact on this system. Those conditions and constraints allow an explanation, at least in part, of the evolution of the system, its current behaviour (its economy) and its possible future evolution. As a consequence, when we tackle the ecological dimension of a problem in which there are several praxeologies involved, we deal with questions concerning the conditions and constraints which:

• explain the behaviour of the corresponding praxeologies in the considered institution at a certain period (that is to say, the economy of those praxeologies at that moment in that institution),
• are required to promote or impede the life of certain kind of praxeologies in the given institution,
• facilitate or hinder the modification of certain kind of praxeologies in a given direction.

In order to study the economy and the ecology of praxeologies involved in the study of a certain didactic problem, researchers use as a reference both a model of the praxeologies relative to the knowledge at stake and a model of the correspondin...

Table of contents

• Cover
• Half Title
• Series Page
• Title Page
• Copyright Page
• Table of Contents
• List of illustrations
• List of Contributors
• Introduction An invitation to the ATD
• A Short (and somewhat subjective) Glossary of the ATD
• Acknowledgements
• PART 1: Unity in Diversity: ATD at Work
• PART 2: Praxeological and Didactic Analysis
• PART 3: Questioning the World
• PART 4: ATD and the Teaching Profession
• Index