# Key word "mathematics education"

Research in mathematics education is a discursive process: It entails the analysis and production of texts, whether in the analysis of what learners say, the use of transcripts, or the publication of research reports. Much research in mathematics education is concerned with various aspects of mathematical thinking, including mathematical knowing, understanding and learning. In this paper, using ideas from discursive psychology, I examine the discursive construction of mathematical thinking in the research process. I focus, in particular, on the role of researchers’ descriptions. Specifically, I examine discursive features of two well-known research papers on mathematical thinking. These features include the use of contrast structures, categorisation and the construction of facts. Based on this analysis, I argue that researchers’ descriptions of learners’ or researchers’ behaviour and interaction make possible subsequent accounts of mathematical thinking.

Context: During the 1980s, Ernst von Glasersfeld’s reflections nourished various studies conducted by a community of mathematics education researchers at CIRADE, Quebec, Canada. Problem: What are his influence on and contributions to the center’s rich climate of development? We discuss the fecundity of von Glasersfeld’s ideas for the CIRADE researchers’ community, specifically in didactique des mathématiques. Furthermore, we take a prospective view and address some challenges that new, post-CIRADE mathematics education researchers are confronted with that are related to interpretations of and reactions to constructivism by the surrounding community. Results: Von Glasersfeld’s contribution still continues today, with a new generation of researchers in mathematics education that have inherited views and ideas related to constructivism. For the post-CIRADE research community, the concepts and epistemology that von Glasersfeld put forward still need to be developed further, in particular concepts such as subjectivity, viability, the circular interpretative effect, representations, the nature of knowing, errors, and reality. Implications: Radical constructivism’s offspring resides within the concepts and epistemology put forth, and that continue to be put forth, through a large number of new and different generations of theories, thereby perpetuating von Glasersfeld’s legacy.

Constructivism is a theory of learning, which claims that students construct knowledge rather than merely receive and store knowledge transmitted by the teacher. Constructivism has been extremely influential in science and mathematics education, but much less so in computer science education (CSE). This paper surveys constructivism in the context of CSE, and shows how the theory can supply a theoretical basis for debating issues and evaluating proposals. An analysis of constructivism in computer science education leads to two claims: (a) students do not have an effective model of a computer, and (b) computers form an accessible ontological reality. The conclusions from these claims are that: (a) models must be explicitly taught, (b) models must be taught before abstractions, and (c) the seductive reality of the computer must not be allowed to supplant construction of models.

This article discusses the central question of how to deal with the diversity and the richness of existing theories in mathematics education research. To do this, we propose ways to structure building and discussing theories and we contrast the demand for integrating theories with the idea of networking theories.

Context: The paper utilizes a conceptual analysis to examine the development of abstract conceptual structures in mathematical problem solving. In so doing, we address two questions: 1. How have the ideas of RC influenced our own educational theory? and 2. How has our application of the ideas of RC helped to improve our understanding of the connection between teaching practice and students’ learning processes? Problem: The paper documents how Ernst von Glasersfeld’s view of mental representation can be illustrated in the context of mathematical problem solving and used to explain the development of conceptual structure in mathematical problem solving. We focus on how acts of mental re‑presentation play a vital role in the gradual internalization and interiorization of solution activity. Method: A conceptual analysis of the actions of a college student solving a set of algebra problems was conducted. We focus on the student’s problem solving actions, particularly her emerging and developing reflections about her solution activity. The interview was videotaped and written transcripts of the solver’s verbal responses were prepared. Results: The analysis of the solver’s solution activity focused on identifying and describing her cognitive actions in resolving genuinely problematic situations that she faced while solving the tasks. The results of the analysis included a description of the increasingly abstract levels of conceptual knowledge demonstrated by the solver. Implications: The results suggest a framework for an explanation of problem solving that is activity-based, and consistent with von Glasersfeld’s radical constructivist view of knowledge. The impact of von Glasersfeld’s ideas in mathematics education is discussed.

Discusses the implications of information processing psychology for mathematics education, with a focus on the works of schema theorists such as D. E. Rumelhart and D. A. Norman and R. Glaser and production system theorists such as J. H. Larkin, J. G. Greeno, and J. R. Anderson. Learning is considered in terms of the actor’s and the observer’s perspective and the distinction between declarative and procedural knowledge. Comprehension and meaning in mathematics also are considered. The role of abstraction and generalization in the acquisition of mathematical knowledge is discussed, and the difference between helping children to “see, ” as opposed to construct abstract relationships is elucidated. The goal of teaching is to help students modify or restructure their existing schema in predetermined ways by finding instructional representations that enable students to construct their own expert representations.

Problem: Ernst von Glasersfeld’s radical constructivism has been highly influential in the fields of mathematics and science education. However, its relevance is typically limited to analyses of classroom interactions and students’ reasoning. Methods: A project that aims to support improvements in the quality of mathematics instruction across four large urban districts is framed as a case with which to illustrate the far-reaching consequences of von Glasersfeld’s constructivism for mathematics and science educators. Results: Von Glasersfeld’s constructivism orients us to question the standard view of policy implementation as a process of travel down through a system and to conceptualize it instead as the situated reorganization of practice at multiple levels of a system. In addition, von Glasersfeld’s constructivism orients us to understand rather than merely evaluate policies by viewing the actions of the targets of policies as reasonable from their point of view. Implications: The potential contributions of von Glasersfeld’s constructivism to mathematics and science education have been significantly underestimated by restricting the focus to classroom actions and interactions. The illustrative case of research on the application of these ideas also indicates the relevance of constructivism to researchers in educational policy and educational leadership.

The representational view of mind in mathematics education is evidenced by theories that characterize learning as a process in which students modify their internal mental representations to construct mathematical relationships or structures that mirror those embodied in external instructional representations. It is argued that, psychologically, this view falls prey to the learning paradox, that, anthropologically, it fails to consider the social and cultural nature of mathematical activity and that, pedagogically, it leads to recommendations that are at odds with the espoused goal of encouraging learning with understanding. These difficulties are seen to arise from the dualism created between mathematics in students’ heads and mathematics in their environment. An alternative view is then outlined and illustrated that attempts to transcend this dualism by treating mathematics as both an individual, constructive activity and as a communal, social practice. It is suggested that such an approach might make it possible to explain how students construct mathematical meanings and practices that, historically, took several thousand years to evolve without attributing to students the ability to peek around their internal representations and glimpse a mathematically prestructured environment. In addition, it is argued that this approach might offer a way to go beyond the traditional tripartite scheme of the teacher, the student, and mathematics that has traditionally guided reform efforts in mathematics education.

Part 1 of a three-part article analyzing radical constructivism (as one interpretation of Piaget) and the socio-cultural perspective (as one interpretation of Vygotsky), including major principles, primary contributions to mathematics education, and potential limitations. Introduces an integration of the two theories through a feminist perspective.

Problem: What is it that Ernst von Glasersfeld brought to mathematics education with radical constructivism? Method: Key ideas in the author’s early thinking are related to ideas that are central in constructivism, with the aim of showing their importance in math education. Results: The author’s initial thinking about constructivism began with Toulmin’s view of thinking as evolving. Ernst showed how Piaget’s genetic epistemology implied an epistemology that was not about ontology. Continuing with an analysis of the way radical and trivial constructivism were received by the mathematics education community, implications of Ernst’s ideas are considered. Implications: These include the need to consider major changes in ways content is introduced to children, to consider carefully the language used to describe children’s emerging mathematical ideas, and to consider new conjectures and also how we think about the foundations of mathematics. Ultimately the value of RC is the way it reinspires belief in the possibility and importance of human growth.

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