What evolutionary account explains our capacity to reason mathematically? Identifying the biological provenance of mathematical thinking would bear on education, because we could then design learning environments that simulate ecologically authentic conditions for leveraging this universal phylogenetic inclination. The ancient mechanism coopted for mathematical activity, I propose, is our fundamental organismic capacity to improve our sensorimotor engagement with the environment by detecting, generating, and maintaining goal-oriented perceptual structures regulating action, whether actual or imaginary. As such, the phenomenology of grasping a mathematical notion is literally that – gripping the environment in a new way that promotes interaction. To argue for the plausibility of my thesis, I first survey embodiment literature to implicate cognition as constituted in perceptuomotor engagement. Then, I summarize findings from a design-based research project investigating relations between learning to move in new ways and learning to reason mathematically about these conceptual choreographies. As such, the project proposes educational implications of enactivist evolutionary biology.

Key words: constructivism, design, education, enactivism, mathematics

A distinction is made between weak and strong research programs in cognitive science, the latter being characterized by an emphasis on the development of runnable computer programs. The paper focuses on the strong research program and initially considers situations in which it claims to have advanced our understanding of mathematical activity. It is concluded that the program’s characterization of students as environmentally driven systems leads to: (a) a treatment of mathematical activity in isolated, narrow, formal domains; (b) a failure to deal with relevance, common sense, and context, and (c) a separation of conceptual thought from sensory-motor action. Taken together, these conclusions imply a failure to deal adequately with the issue of mathematical meaning. In general, the program’s primary focus appears to be on programmable mechanisms rather than fundamental problems of mathematical cognition. The purview of the discussion is then widened to consider the strong program’s difficulties in dealing with social interaction, intellectual communities, and the hidden curriculum. It is noted that instructional implications derived from this program typically involve the organization of mathematical stimuli that make explicit or salient the relevant properties of a propositional mathematical environment. Finally, it is argued that some members of the strong program have recently acknowledged that it has limitations. The possibility of a rapprochement in which the strong program is supplanted by a form of social constructivism is discussed.

In this chapter, I focus on one of the aspects of constructivist theory that Glasersfeld (Ch. 1) identifies as in need of further development. This aspect of the theory involves locating students’ mathematical development in social and cultural context while simultaneously treating learning as a process of adaptive reorganization. In addressing this issue, I illustrate the approach that I and my colleagues currently take when accounting for the process of students’ mathematical learning as it occurs in the social context of the classroom. In the opening section of the chapter, I clarify why this is a significant issue for us as mathematics educators. I then outline my general theoretical orientation by discussing Glasersfeld’s constructivism and Bauersfeld’s interactionism. Against this background, I develop criteria for classroom analyses that are relevant to our interests as researchers who develop learning environments for students in collaboration with teachers. Next, I illustrate the interpretive framework that I and my colleagues currently use by presenting a sample classroom analysis. Finally, in the concluding sections of the chapter, I reflect on the sample analysis to address four more general issues. These concern the contributions of analyses of the type outlined in the illustrative example, the relationship between instructional design and classroom-based research, the role of symbols and other tools in mathematical learning, and the relation between individual students’ mathematical activity and communal classroom processes.

The primary purpose of this chapter is to clarify the basic tenets of activity theory and constructivism, and to compare and contras instructional approaches developed within these global theoretical perspectives. This issue is worthy of discussion in that research and development programs derived from these two perspectives are both vigorous. For example, the work of sociocultural theorists conducted within the activity theory tradition has become increasingly influential in the United States in recent years. One paradigmatic group of studies conducted by Lave (1988), Newman, Griffin, and Cole (1089). and Scribner (1984) has related arithmetical computation to more encompassing social activities such as shopping in a supermarket, packing crates in a dairy, and completing worksheets in school. Taken together, these analyses demonstrate powerfully the need to consider broader social and cultural processes when accounting for children’s development of mathematic cal competeuce.

Key words: mathematics education, activity theory, mathematical activity, mathematical practice, instructional development.

Excerpt: For the past six years we, together with Erna Yackel and Terry Wood, have conducted a classroom-based research and development project in elementary school mathematics.’ In this paper, we draw on our experiences of collaborating with teachers and of analyzing what might be happening in their classrooms to consider three interrelated issues. First, we argue that the teacher and students together create a classroom mathematics tradition or microculture and that this profoundly influences students’ mathematical activity and learning. Sample episodes are used to clarify the distinction between the school mathematics tradition in which the teacher acts as the sole mathe-matical authority and the inquiry mathematics tradition in which the teacher and students together constitute a community of validators. Second, we consider the theoretical and pragmatic tensions inherent in the view that mathematical learning is both a process of individual cognitive construction and a process of acculturation into the mathematical practices of wider society. In the course of the discussion, we contrast constructivist attempts to cope with this tension with approaches proposed by sociocultural theorists. Finally, we use the preceding issues as a backdrop against which to consider the development of instructional activities that might be appropriate for inquiry mathematics classrooms.

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.

Open peer commentary on the article “From Problem Solving to Problem Posing, and from Strategies to Laying Down a Path in Solving: Taking Varela’s Ideas to Mathematics Education Research” by Jérôme Proulx & Jean-François Maheux. Upshot: The target article prioritises the emergence of pupils’ mathematical ideas. Other constructivist approaches have focussed on how teachers might act to facilitate pupils’ mathematical activity. How might teachers be helped to use Varela’s insights into the uncontrollable emergence of ideas while teaching in the context of dominant intentional problem-solving approaches?

In an 8-month teaching experiment, the author aimed to establish mathematical caring relations (MCRs) with 4 6th-grade students. From a teacher’s perspective, establishing MCRs involves holding the work of orchestrating mathematical learning for students together with an orientation to respond to energetic fluctuations that may accompany student″teacher interactions. From a student’s perspective, participating in an MCR involves some openness to the teacher’s interventions in the student’s mathematical activity and some willingness to pursue questions of interest. Analysis revealed that student″teacher interactions can be viewed as a linked chain of perturbations; in MCRs, the linked chain tends toward perturbations that are bearable for both students and teachers. This publication is relevant for constructivist approaches because it examines how attention to affective responses (specifically, emotion and vital energy) can be included in a radical constructivist approach to knowing and learning.

Open peer commentary on the article “From Problem Solving to Problem Posing, and from Strategies to Laying Down a Path in Solving: Taking Varela’s Ideas to Mathematics Education Research” by Jérôme Proulx & Jean-François Maheux. Upshot: Proulx and Maheux’s view of problem-posing|solving compels insights about roles and lived experiences of teachers. Living and reporting co-emergence of teaching activity and mathematical activity are discussed.

In this paper, we emphasize the methodological challenges of analyzing data with/in an enactivist framework in which (1) all knowing is doing is being, while focusing on, (2) how an entry by the observer transforms usual ways of analyzing data. In this sense, the association of knowing with doing suggests a shift in attention away from what students might “know” toward attending to the active, dynamic, enacted mathematical activity as knowing, bringing to bear the local and emergent character of the mathematical activity. In addition to illustrating data analysis along those lines, we discuss the methodological significances and challenges of, and paradigm shifts required for, replacing questions of knowledge and acquisition with ones that concern mathematical doing alone.