An important tenet of constructivism is that learning is an idiosyncratic, active and evolving process. Active learning, operationalized by cognitive, metacognitive, affective and resource management learning strategies, is necessary for students to effectively cope with the high level of demands placed on the learner in a constructivist learning environment. Case studies of two students detail contrasting passive and active learning behaviours. Examples of their strategic learning behaviours illustrate that having students involved in activities such as discussions, question answering, and seatwork problems does not automatically guarantee successful knowledge construction. The nature of students’ metacognitive knowledge and the quality of their learning strategies are seen to be critical factors in successful learning outcomes.

Barwell R. (2009) Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics 72(2): 255–269. https://cepa.info/3731

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.

Brown L. & Coles A. (2012) Developing “deliberate analysis” for learning mathematics and for mathematics teacher education: How the enactive approach to cognition frames reflection. Educational Studies in Mathematics 80: 217–231. https://cepa.info/6846

We illustrate and exemplify how the idea of reflection is framed by the enactive concept of “deliberate analysis.” In keeping with this frame, we do not attempt to define reflection but rather work on the question of “how do we do reflecting?” within such a frame. We set out our enactivist theoretical stance, in particular pointing to implications for how we can learn from experience and showing the role of “deliberate analysis.” We then describe, drawing on education literature, what is generally seen as the purpose of reflection and review some existing conceptualizations in mathematics education, pointing out where we draw distinctions. To illustrate how we do reflecting, we offer excerpts from two lessons of an expert teacher and the writing of a prospective teacher. We exemplify how reflecting as deliberate analysis leads to a way of working with teachers supporting them in handling multiple views and ambiguity, their actions being contingent upon their students’ actions in learning mathematics.

The radical constructivist assertion that the student constructs his or her own knowledge as opposed to receiving it “ready made” echoes the classical debate as to whether the human subject constitutes the world or is constituted by it. This paper shows how the philosophical traditions of post-structuralism and hermeneutic phenomenology offer approaches to effacing this dichotomy and how this forces a re-assertion of the teacher’s role in the student’s constructing of mathematical knowledge. It is also shown how hermeneutic phenomenology provides an opportunity to ground constructivist mathematical thinking in the material qualities of the world.

Glasersfeld E. von (1992) Guest editorial. Educational Studies in Mathematics 23: 443–444. https://cepa.info/1435

Learning theories such as behaviourism, Piagetian theories and cognitive psychology, have been dominant influences in education this century. This article discusses and supports the recent claim that Constructivism is an alternative paradigm, that has rich and significant consequences for mathematics education. In the United States there is a growing body of published research that claims to demonstrate the distinct nature of the implications of this view. There are, however, many critics who maintain that this is not the case, and that the research is within the current paradigm of cognitive psychology. The nature and tone of the dispute certainly at times appears to describe a paradigm shift in the Kuhnian model. In an attempt to analyse the meaning of Constructivism as a learning theory, and its implications for mathematics education, the use of the term by the intuitionist philosophers of mathematics is compared and contrasted. In particular, it is proposed that Constructivism in learning theory does not bring with it the same ontological commitment as the Intuitionists’ use of the term, and that it is in fact a relativist thesis. Some of the potential consequences for the teaching of mathematics of a relativist view of mathematical knowledge are discussed here.

Nemirovsky R. & Ferrara F. (2009) Mathematical imagination and embodied cognition. Educational Studies in Mathematics 70: 159–174. https://cepa.info/7133

The goal of this paper is to explore qualities of mathematical imagination in light of a classroom episode. It is based on the analysis of a classroom interaction in a high school Algebra class. We examine a sequence of nine utterances enacted by one of the students whom we call Carlene. Through these utterances Carlene illustrates, in our view, two phenomena: (1) juxtaposing displacements, and (2) articulating necessary cases. The discussion elaborates on the significance of these phenomena and draws relationships with the perspectives of embodied cognition and intersubjectivity.

Núñez R. E., Edwards L. D. & Matos J. F. (1999) Embodied cognition as grounding for situatedness and context in mathematics education. Educational Studies in Mathematics 39(1): 45–65. https://cepa.info/7134

In this paper we analyze, from the perspective of ‘Embodied Cognition’, why learning and cognition are situated and context-dependent. We argue that the nature of situated learning and cognition cannot be fully understood by focusing only on social, cultural and contextual factors. These factors are themselves further situated and made comprehensible by the shared biology and fundamental bodily experiences of human beings. Thus cognition itself is embodied, and the bodily-grounded nature of cognition provides a foundation for social situatedness, entails a reconceptualization of cognition and mathematics itself, and has important consequences for mathematics education. After framing some theoretical notions of embodied cognition in the perspective of modern cognitive science, we analyze a case study – continuity of functions. We use conceptual metaphor theory to show how embodied cognition, while providing grounding for situatedness, also gives fruitful results in analyzing the cognitive difficulties underlying the understanding of continuity.

Pixie S. & Kieren T. (1992) Creating constructivist environments and constructing creative mathematics. Educational Studies in Mathematics 23: 505–528. https://cepa.info/6093

Proulx J. (2013) Mental mathematics emergence of strategies, and the enactivist theory of cognition. Educational Studies in Mathematics 84(3): 309–328. https://cepa.info/6850

In this article, I present and build on the ideas of John Threlfall [(Educational Studies in Mathematics 50:29–47, 2002)] about strategy development in mental mathe- matics contexts. Focusing on the emergence of strategies rather than on issues of choice or flexibility of choice, I ground these ideas in the enactivist theory of cognition, particularly in issues of problem posing, for discussing the nature of the solving pro- cesses at play when solving mental mathematics problems. I complement this analysis and conceptualization by offering two examples about issues of emergence of strategies and of problem posing, in order to offer illustrations thereof, as well as to highlight the fruitfulness of this orientation for better understanding the processes at play in mental mathematics contexts.