Scheming and Re-scheming: Secondary Mathematics Teachers’ Use and Re-use of Resources

In this article, we examine secondary mathematics teachers’ work with resources using the Documentational Approach to Didactics lens. Specifically, we look at the resources and a teacher’s scheme of use (aims, rules of actions, operational invariants, and inferences) of these resources across a set of lessons (macro-level analysis) that aim towards students’ preparation for the examinations and how this use emerges in a set of three lessons on the same topic (micro-level analysis) as a response to contingent moments. We propose the terms scheming—a teacher’s emerging scheme of use related to the same set of resources used for the same aim—and re-scheming, namely, shifts in such scheming. Our analysis of lesson observations and the teacher’s reflections on his actions from a post-observation interview demonstrate the interplay between the stable characteristics of the scheme of use and the scheming and re-scheming in individual lessons. We conclude this article with a discussion on the methodological potential of using both macro- and micro-level analyses in the investigation of teachers’ use of resources.

Implementation Studies in Mathematics Education: What Theoretical Resources?

This article is devoted to the theoretical resources likely to support implementation studies. After the introductory section, we focus on the potential offered by resources internal to the field of mathematics education, especially those offered by design-based research, by the systemic and ecological perspective underlying the anthropological theory of the didactic, and by the documentational approach to didactics regarding the teachers’ relationship with educational resources. We then examine particular examples of implementation studies, paying specific attention to the way they combine internal and external resources. For that purpose we consider on the one hand, European projects aiming at the large-scale implementation of inquiry-based learning and teaching in STEM education, especially the PRIMAS project, and on the other hand, the contributions to the thematic working group devoted to implementation and replication studies at CERME 10 and CERME 11. We conclude with more general reflections.

Theoretical approaches to teachers’ lesson designs involving the adaptation of mathematics textbooks: two cases from kyouzai kenkyuu in Japan

In this paper, we explore theoretical approaches to Japanese teachers’ lesson designs involving the adaptation of mathematics textbooks for instructional change. In Japan, although most teachers use textbooks as the main resource for lesson planning, called kyouzai kenkyuu, the ways in which they use textbooks can differ. In a kyouzai kenkyuu practice, Japanese teachers sometimes attempt to design innovative tasks and/or learning trajectories beyond or parallel to the textbooks and curriculum guidelines. Such practice is often invisible to international researchers, since it is culturally situated in the local context. To make it more visible, in this study we address and advance theoretical approaches using two frameworks which are well-established for research in mathematics education, namely, Anthropological Theory of the Didactic (ATD) and Documentational Approach to Didactics (DAD). Two cases of lesson design regarding fractions using the same textbook chapter for 3rd graders in primary schools in Japan, are used as examples to illustrate crucial elements of each framework and ways of combining and coordinating both frameworks. As a result, we conceptualise teachers’ kyouzai kenkyuu practice using two approaches: from a praxeological analysis based on ATD, the difference in teachers’ mathematical and didactic knowledge was emphasised, while within DAD, the difference in the instrumentalisation process was emphasised. This analysis implies that the two theoretical approaches can be compatible, and used to gain deeper insight into the relationship between lesson design using the textbook and teacher knowledge.

Revisiting the French Didactic Tradition through Technological Lenses

There is, in France, a long-term tradition of research in the didactics of mathematics. In this paper, we revisit this tradition using, as a specific lens, the research carried out on the educational use of digital tools and resources for teaching and learning mathematics. We first briefly introduce this tradition and the three main theories at the base of it: the theory of didactical situations, the theory of conceptual fields, and the anthropological theory of the didactic. Then, considering three different technological lenses, i.e., dynamic geometry environments, computer algebra systems, and digital resources, we show how these theories and the long-term connection established in this tradition with the fields of cognitive ergonomics and computer sciences have influenced technological research and its outcomes. We also show that, conversely, didactic technological research has led to original and influential theoretical constructions, such as the instrumental approach and the documentational approach to didactics, and that it has contributed in a substantial way to the opening of this didactic tradition to other didactic cultures and other communities, beyond the didactic one.

A abordagem dos recursos no processo de ensino do conceito de derivada

ResumoNeste artigo apresentamos ideias preliminares de nossa tese de doutorado e elementos de um novo campo de pesquisa em educação matemática: The ‘Resources’ Approach to Mathematics Education (RAME) – A Abordagem dos “Recursos” para a Educação Matemática (TROUCHE, GUEUDET E PEPIN, 2019). A Abordagem Documental do Didático (GUEUDET e TROUCHE, 2010) desempenha um papel particular neste novo campo. Na revisão da literatura verificamos que o conceito de derivada, inserido em um campo conceitual (VERGNAUD, 1990), apresenta dificuldade de aprendizagem para os estudantes, sendo que esses se saem melhores no uso da técnica do que no entendimento e atribuição de significado a esse conceito matemático. A proposta da pesquisa é de efetivar um estudo de caso com estudantes de Licenciatura em Matemática de uma universidade brasileira, desenvolvendo uma ação coletiva entre pesquisador e professor na concepção de recursos para o processo de ensino e aprendizagem do conceito de derivada no Ensino Superior. Palavras-chave: Abordagem dos Recursos; Abordagem Documental do Didático; Conceito de Derivada; Campo Conceitual; Estudo de Caso.AbstractIn this article we present preliminary ideas for our doctoral thesis and elements of a new field of research in mathematics education: The ‘Resources’ Approach to Mathematics Education - RAME - (TROUCHE, GUEUDET & PEPIN, 2019). The Documentational Approach to Didacts (GUEUDET & TROUCHE, 2010) has a particular role in this new field. In the literature review, we found that the concept of derivative, inserted in a conceptual field (VERGNAUD, 1990), presents learning difficulties for students, because they do better in the use of the technique than in the understanding and attribution of meaning to this mathematical concept. The research proposal is to carry out a case study with Mathematics undergraduate students from a brazilian university, developing a collective action between researcher and professor in the conception of resources for the process of teaching and learning the concept of derivative in Higher Education.Keywords: Resource Approach; Documentational Approach to Didacts; Concept of Derivative; Conceptual Field; Case Study.

Investigating Teachers’ Work with Digital Resources. A Comparison Between the Teaching of Mathematics and English

The abundant digital resources on the Internet are leading to profound changes in teachers’ activity. We refer to the theoretical framework and methodology of the documentational approach to didactics to investigate these changes. We compare two case studies: an English teacher, and a mathematics teacher, both working in the same upper secondary school in France. We have observed significant use of digital resources in both cases, but of a different nature. We present and discuss these differences that shed light upon teachers’ documentational work. We use these cases to draw more general conclusions on designing, using and sharing digital teaching resources.