Anthropological Theory Of The Didactic Research Articles

Modelling mathematics for educational research and practice: a comparison of two theoretical approaches

AbstractThe production and use of explicit models of mathematics are important elements of research in Didactics of Mathematics. In this paper, we present and compare two different European approaches to didactical modelling. The first, centred around the notion of basic idea (Grundvorstellung), arose in German-speaking countries as a development within Subject Matter Didactics (Stoffdidaktik). The second is the notion of praxeology from the Anthropological Theory of the Didactic, which originated in France. After presenting the genesis and interpretation of these notions as theoretical modelling tools, we provide a detailed case study of actual models produced with these tools, concerning the notion of limits and derivatives. Furthermore, we identify similarities and differences of both models in how they can support didactical research and practice, for instance, by identifying important phenomena or problems. We conclude that the two approaches offer important, different and, to some extent, complementary positions and perspectives.

Comparative analysis between three theoretical approaches through empirical experiences at university level

Research into mathematics education at university level includes a wide range of theoretical approaches. This poses considerable challenges to researchers in terms of understanding and harmonizing the compatibility and commensurability of those approaches. The research community has already problematised and studied these challenges using networking theories. The networking theories framework is taken as a starting point in this study to contrast different approaches and to broaden the comparison of different frameworks. In particular, three case studies framed in the Action, Process, Object, Schema Theory, in the Problem-Solving approach, and in the Anthropological Theory of the Didactic are analysed. The differences and possible similarities between the three with regard to the research questions addressed, their objects of study, their empirical bases, as well as their research ends are considered. The analysis offers an insight into the potential for collaboration and the networking of theories in the field of university mathematics education.

Mechanisms and evidence of prospective teachers’ learning through enquiry-oriented practices: the case of a lesson study intervention

Existing research has highlighted the key role of observation and reflection in teacher learning during lesson study interventions, and how challenging it can be for pre-service teachers. This study focuses on the post-lesson discussion phase and on the observation of the research lesson of a lesson study process. It is carried out with early childhood education pre-service teachers who studied the Theory of Didactical Situations. The aim of our research is two-fold: finding evidence of the activation of pre-service teacher knowledge during the post-lesson discussion, and identifying mechanisms that may explain said activation as well as the potential development of their professional knowledge. To model pre-service teacher knowledge, the notion of teachers’ praxeological equipment from the Anthropological Theory of the Didactic, specified in the Theory of Didactical Situations, is used. It is hypothesised that potential teacher learning is mainly connected to the identification of particularly relevant episodes during the observation phase, henceforth referred to as critical events. The analysis of a group discussion on three critical events shows the activation and development of mathematical and didactic components of pre-service teachers’ praxeological equipment. However, it is found that it depends on the degree of dissonance these events can provoke, and on the quality and depth of the discussion that takes place around them. It is therefore concluded that the identification of critical events followed by productive discussion is a mechanism that explains teacher learning through lesson study, which can be enhanced by the intervention of teacher educators as mediators.

The algebraic learning of middle school students’ evolution during a school year: a statistical large-scale study based on results in mathematics didactics

AbstractThis large-scale study characterizes the algebraic learning of a cohort of nearly 800 French students during the last year of middle school (9th grade, 14–15 years old) and examines their evolution in relation to the composition of the classes to which the students belong. Based on the Anthropological Theory of the Didactic (ATD), the study is founded both on a reference model of elementary algebra and on technological-theoretical levels characterizing the students’ reasoning and their knowledge regarding institutional expectations. These are used to code the data and define the statistical variables on which multivariate descriptive analyses are carried out. Students’ learning is analyzed through Pépite, an algebra test given at the beginning and at the end of the year. A cluster analysis resulted in three distinct clusters of similar classes and of similar students both at the beginning and end of the grade 9 school year. They are interpreted didactically and could be useful for a teacher who has to manage the heterogeneity of learning in a class. Some classes and students move from one cluster to another between the beginning and the end of the year, showing a wide variety of ways in which students’ algebra learning progresses or regresses over a school year.

A critical look at the anthropological theory of the didactic

ABSTRACT The aim of this paper is to discuss and challenge some basic premises of the Anthropological Theory of the Didactic, as introduced by its founder, Yves Chevallard. Using the theory of limits as a historical study as well as a number of research articles, I investigate the relation between the types of mathematical knowledge, as given in the theory of didactic transposition. I also argue that the concept of praxeology does not capture the intrinsic difficulty of the concept of limit, which many students encounter. This paper is also a call for researchers within the Anthropological Theory of the Didactic to enter into a critical discussion of its basic tenets, and to engage in dialogue and networking with other educational platforms to help improve and develop the theory.

Contributions of the anthropological theory of the didactic to the epistemological programme of research in mathematics education

In the early 1980s, the theory of didactic situations put the analysis of mathematical activity at the core of the didactic analysis, thus initiating a new research programme in mathematics education: the epistemological programme. This paper describes and interprets some of the contributions of the anthropological theory of the didactic (ATD) to the development of this programme. These contributions come from the new institutional and ecological perspective provided by the ATD to approach the didactic problem and from the significance it assigns to macrodidactic phenomena characterised by didactic transposition. They are illustrated with research works involving different fields of school mathematics in which the ATD team led by Spanish researchers played a central role: research in school algebra and its relations with functional modelling, differential calculus, and real numbers. The paper concludes by briefly describing the contributions of the ATD to the dialogue between didactic theories, another domain that has notable Spanish contributions.

Inquiry dynamics at the crossroads of descriptive and inferential statistics

The fast evolution of statistics caused by technological developments is the basis of numerous challenges in undergraduate courses. Project-based learning (PBL) and similar methodologies have been proposed by various authors to develop students’ ‘statistical sense’. This paper presents a case study regarding implementing three study and research paths (SRPs) in an undergraduate statistics course. SRPs are inquiry-based teaching devices developed in the Anthropological Theory of the Didactic. This study aims to analyse the inquiry dynamics that SRP implementations promote, the influences of teachers’ interventions and the role of external instances. The way statistical knowledge evolves and links descriptive and inferential statistics is also examined. The results of this study suggest that SRPs can contribute to the self-sustainability of the inquiry process, which is not often the case in other approaches such as PBL. This study also highlights the use of question–answer maps to analyse the dynamics of the inquiry process in terms of the connectivity of the knowledge involved. Future research points to the study of the conditions needed to generate more adidacticity in PBL proposals.

Estudio codisciplinar de la Matemática en la escuela secundaria: análisis de la gestión de una propuesta de aula para el estudio del crecimiento de plantas

This paper presents the results of the management of a didactic device formulated by a Mathematics male and Biology female teacher from a secondary school in which they sought a codisciplinary study. Based on the Anthropological Theory of the Didactic, the results of an exploratory study that sought to characterize the management of a didactic device for the study of plant growth in secondary school are reported; In particular, we analyze the relevance of mathematical knowledge in its development. The main results indicate that the study enabled the protagonists to be the students: they formulated and answered questions, achieving autonomy and responsibility for the search for information in various media; however, mathematical knowledge did not occupy a prominent place during the study.

Mathematics in a Japanese overseas school: Connecting classroom study and curriculum

The aim of this paper is to investigate how the contents related to the arithmetic of fractions, as described in the official curriculum issued by the Japanese Ministry of Education, Culture, Sports, Science, and Technology (MEXT), are implemented in actual teaching at the so-called Japanese supplementary school in Denmark. Additionally, at a methodological level, this case study demonstrates how so-called praxeological reference models can be developed at different levels of detail to describe classroom teaching at a particular grade level and the overall curriculum of primary school in a connected way. This study is based on the notions of praxeological reference models and moments of didactic processes from the Anthropological Theory of the Didactic. The findings reveal that (1) not much time is allocated to each moment and (2) the flow of gradually building up techniques, as intended by the textbook authors is to some extent disrupted in the lesson.

An anthropological point of view: exploring the Chinese and Japanese issues of translation about teaching resources

The aim of this study is to advance understading of teachers’ and researchers’ work, in particular its cultural specificities, from a resource perspective by exploring the issues and challenges faced during the translation of a theoretical framework, the Documentational Approach to Didactics (DAD), from Western (English and French) to East Asian languages (Chinese and Japanese). A basic assumption is that the challenges encountered while translating are driven by the cultural and linguistic differences that exist between the West and East. Adopting the perspective of the Anthropological Theory of the Didactic (ATD), we frame the translation work as a transposition process of research praxeology (a model of the researchers’ practices and knowledge) from a Western to an Eastern institution. We investigate this process to identify cultural elements at different levels (school, society, civilization, etc.) using the ATD framework. After translating a DAD article into Chinese and Japanese separately, we then worked collaboratively to identify similar or diverse translation issues and investigated their origins. Consequently, the results revealed a considerable difference between the West and East, and ample similarities between China and Japan, especially in terms of the researchers’ work and its relationship with the teachers’ work.

Drawing on a computer algorithm to advance future teachers’ knowledge of real numbers: A case study of task design

In our investigation of university students’ knowledge about real numbers in relation to computer algebra systems (CAS) and how it could be developed in view of their future activity as teachers, we used a computer algorithm as a case to explore the relationship between CAS and the knowledge of real numbers as decimal representations. Our work was carried out in the context of a course for university students who aim to become mathematics teachers in high schools. The main data consists of students’ written responses to an assignment of the course and interviews to clarify students’ perspectives in relation to the responses. The analysis of students’ work is based on the anthropological theory of the didactic (ATD). Our results indicate that simple CAS-routines have a potential to help university students (future teachers) to apply their university knowledge on certain problems related to the decimal representation of real number which are typically encountered but not well explained in high school.

Study and Research Paths of University Freshmen in an Online Environment: A Task Related to the Center of Population

The world witnessed a growing interest in inquiry-based learning in the past decade. Study and research path (SRP) is a model of such learning based on the anthropological theory of the didactic (ATD). This paper discusses the following research questions through [Formula: see text] maps of the SRPs of university freshmen using the notion of praxeology: (RQ1) how do students/pre-service teachers create SRPs in an online environment in terms of group work? (RQ2) How are the institutional conditions and constraints of each group characterized? In conclusions, the differences in media searches of the student groups lead to differences in final answers to the inquiry. Additionally, the media-milieu dialectic is closely related to the question–answer dialectic.

The Limit Notion at Three Educational Levels in Three Countries

This paper documents how the limit concept is treated in high school, at a university and in teacher education in England, France and Sweden. To this end we make use of vignettes, data-grounded accounts of the situation at the three levels in the three countries. These are analysed using the Anthropological Theory of the Didactic (ATD). While university praxeologies are relatively similar across the three countries, greater differences manifest themselves in high school and teacher education. For instance, at the high school level, in France a local praxeology on the limits of sequences is taught, which is not the case in England or Sweden. Results from the analysis of limits are extrapolated to comment on implications for the teaching of calculus, and for teacher education, in the three countries. The paper also raises methodological issues in our approach.

Design of 3D Metric Geometry Study and Research Activities within a BIM Framework

This paper presents research work in which an innovative didactic proposal was designed for the study of 3D metric geometry in the second year of A-level courses (secondary education) in the specialty of the sciences. The designed didactic proposal has the format of a workshop of geometry practices and is framed in the Didactic Situations Theory and in the Anthropological Theory of the Didactic. Although the mentioned theories, which the French school has developed since the seventies, are widely known and studied by secondary school mathematics teachers during their training in pedagogy, especially in Spanish- and French-speaking countries, their innovative approach has not been fully implemented in the field of algebraic geometry. Mathematics textbooks follow the traditional approach in which the teacher simply provides the contents and instructs the student, who captures these concepts and reproduces them as they have been supplied. In the presented didactic proposal, the approach proposed by Brousseau is followed, in which three fundamental elements take part: student, teacher, and the didactic environment. The teacher is the one who facilitates the environment in which the student builds his knowledge. In the proposal the didactic contract is stablished, the didactic situations are designed, and the means and didactic variables for the study of 3D metric geometry are chosen. The methodology followed in this study consisted of identifying educational problems; describing the theoretical framework; developing the didactic proposal; and analysis, reflection, and criticism of this training product. The contextualization of the geometry workshop in the field of the construction of structures and the use of deductive reasoning techniques that are applied in synthetic geometry and also in the building information modeling (BIM) methodology may be a way of showing the use of analytical geometry in the industries of engineering and architecture; they also serve as an opportunity for students to better understand this mathematical work.

Breaches of the didactic contract as a driving force behind learning and non-learning: a story of flaws and wants

AbstractIn agreement with the main tenets of the anthropological theory of the didactic (ATD), this study uncovers dependencies between what students can learn, the established curriculum and the current state of mathematicians’ mathematics (‘scholarly mathematics’). One main result is that the mathematics taught, too often taken for granted by curriculum developers and teachers, is in fact problematic not only to students but also to teachers and curriculum developers and is sometimes a challenge even to current scholarly mathematics. The mathematics taught during a given historical period within a given institution contains flaws that, when they cease to go unnoticed, generate crises, in the form of breaches of the prevailing didactic contract. The resolution of these crises allows the institution and its actors—in particular students and teachers—to learn new contents and often also leads to the more or less damaging unlearning of old contents. This key phenomenon is illustrated, at the triple level of the classroom, the curriculum and scholarly mathematics, with regard to elementary algebra and mathematical analysis, most importantly in the case of maxima and minima problems.

Understanding Chinese students’ success in the PISA financial literacy: A praxeological analysis of financial numeracy

The goal of this article was to investigate how Chinese mathematics curriculum policies and textbook tasks could help explain the results obtained by Chinese students in the 2012 and 2015 Programme for International Student Assessment (PISA) financial literacy exams. Inspired by the Anthropological Theory of the Didactic, we conducted a praxeological analysis of financial numeracy tasks from middle school textbooks and the PISA. We conceptualized the term financial numeracy as the use, production, and communication of mathematical information in financial situations. The analysis permitted us to contrast the solution to PISA tasks with that of financial tasks from middle school mathematics textbooks. Our results show that, despite the lack of attention to mathematics in the curriculum policies for financial literacy, the mathematics textbooks seem to support the performance of students in the PISA by (a) incorporating more mathematically complex content, (b) tackling equivalent financial concepts, (c) providing students with enough time to consolidate their understanding throughout middle school, and (d) designing pedagogy that revisits these concepts over the years. The implications of this study inform mathematics education research and practice. If we are to incorporate financial literacy in mathematics curricula, this process should be done with intentionality and in connection to multiple mathematical concepts and processes.

Bridging the mathematics gap between the engineering classroom and the workplace

Teaching mathematics to future engineers while keeping up with real-world scientific evolutions is a challenge with diverse constraints and difficulties. In this article, I present and analyze the design and implementation of a Study and Research Path (SRP) in a statistics class in a French chemical engineering school. An SRP is a project-based learning approach and part of the Anthropological Theory of the Didactic (ATD), the theory framing my study. In terms of data collection strategies, my research draws on observation of the SRP and on diverse monitoring tools (questionnaires, interviews, a diary, etc.). Moreover, I use the concept of mathematical praxeologies to analyze the mathematical activity that the future chemical engineers develop during the SRP and to compare the praxeologies met in the SRP with the usual praxeologies met in the workplace. The similarities observed reveal that such an SRP can contribute to bridging the gap between the engineering school and the workplace. I identify and discuss factors and conditions that played an important role in the design and implementation of the SRP.

A PRAXEOLOGICAL ANALYSIS OF PRE-SERVICE ELEMENTARY TEACHER-DESIGNED MATHEMATICS COMICS

Mathematical and didactic knowledge presented in mathematics textbooks and other resources, like mathematics comics (MCs), needs to be evaluated from a lens of appropriate theoretical framework in mathematics education before it can be used as a medium for teaching and learning mathematics.Therefore, this study investigates mathematical and didactic competencies that were reflected in MCs designed by pre-service elementary teachers. The framework for analysing mathematical knowledge embedded in these MCs is based on the Anthropological Theory of the Didactic, specifically a praxeology. This study utilized a content analysis technique within a qualitative approach. Thirteen MCs were analysed using a praxeological analysis; the type of task and techniques (praxis block) as well as the possible technology and theory (logos block). The findings demonstrate that the mathematical praxeologies embedded in MCs belong to five mathematical domains, namely numbers and operations; number theory; fractions, decimals, and percentages; ratio and proportion; as well as measurement. Additionally, the analysis revealed that seven of these MCs were related to a single domain, while the others belong to two or three mathematical domains. Concerning the mathematical praxeologies, most of MCs focus on presenting the practical blocks, the type of task and the techniques, while only a few could provide the theoretical lens to justify the practical blocks.

Study and research paths in the frontier between paradigms

Implementing inquiry-oriented instructional proposals at the university level collides with many constraints related to the current pedagogical paradigm based on the transmission of previously established works of knowledge. One such constraint comes from defining subjects’ curricula by a set of topics to study, not questions to inquire. Our research team has been working for more than 15 years in designing and implementing an inquiry-oriented instructional format called ‘study and research paths’ (SRPs) within the Anthropological Theory of the Didactic. This paper focuses on the commonalities and specificities of the implemented SRPs to identify critical strategies and didactic devices that facilitate their integration in current university settings. The experimentations also reveal key issues for the sustainability of inquiry-oriented educational proposals related to the question that generates the inquiry and the types of contracts established to approach it.

Analysis tasks based on a theorem in Nonlinear Approximation theory

The systematic design and analysis of tasks which can be implemented in first-year university courses and point to advanced inner- and extra-mathematically rich issues and their rationales is an open problem in the didactics of mathematics in higher education. Potentials of such tasks can be seen with regard to learning processes in the first year of study, which are less compartmentalized and allow for an extended acquisition of rationales of advanced mathematical practices and concepts. Against this background, subject-specific potentials of advanced mathematics are examined in this contribution. The Anthropological Theory of the Didactic (ATD) serves as the theoretical framework and in the analyses notions from its 4T-model are applied. Structural observations in praxeological terms are illustrated by examples chosen from a presentation of a classical result in Nonlinear Approximation. At the specific focus of the praxeological analyses are aspects for bridging and extending concepts within and across Analysis. On this basis, tasks designed for first-year university Analysis courses are analysed in detail. Moreover, general characteristics of the methodological approach realized in this contribution are discussed.