Abrahamson D. (2009) Embodied design: Constructing means for constructing meaning. Educational Studies in Mathematics 70(1): 27–47. https://cepa.info/8084

Design-based research studies are conducted as iterative implementation-analysis-modification cycles, in which emerging theoretical models and pedagogically plausible activities are reciprocally tuned toward each other as a means of investigating conjectures pertaining to mechanisms underlying content teaching and learning. Yet this approach, even when resulting in empirically effective educational products, remains under-conceptualized as long as researchers cannot be explicit about their craft and specifically how data analyses inform design decisions. Consequentially, design decisions may appear arbitrary, design methodology is insufficiently documented for broad dissemination, and design practice is inadequately conversant with learning-sciences perspectives. One reason for this apparent under-theorizing, I propose, is that designers do not have appropriate constructs to formulate and reflect on their own intuitive responses to students’ observed interactions with the media under development. Recent socio-cultural explication of epistemic artifacts as semiotic means for mathematical learners to objectify presymbolic notions (e.g., Radford, Mathematical Thinking and Learning 5(1): 37–70, 2003) may offer design-based researchers intellectual perspectives and analytic tools for theorizing design improvements as responses to participants’ compromised attempts to build and communicate meaning with available media. By explaining these media as potential semiotic means for students to objectify their emerging understandings of mathematical ideas, designers, reciprocally, create semiotic means to objectify their own intuitive design decisions, as they build and improve these media. Examining three case studies of undergraduate students reasoning about a simple probability situation (binomial), I demonstrate how the semiotic approach illuminates the process and content of student reasoning and, so doing, explicates and possibly enhances design-based research methodology.

Anthony G. (1996) Active learning in a constructivist framework. Educational Studies in Mathematics 31(4): 349–369. https://cepa.info/5221

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.

Brown T. (1994) Creating and knowing mathematics through language and experience. Educational Studies in Mathematics 27: 79–100. https://cepa.info/7128

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.

Cobb P. (1994) Guest editorial [Representations: External memory and technical artefacts]. Educational Studies in Mathematics 26(2/3): 105–109.

Excerpt: The first five contributions to this Special Issue on Theories of Mathematical Learning take a cognitive perspective whereas the sixth, that by Voigt, takes an interactionist perspective. The common theme that links the six articles is the focus on students’ inferred experiences as the starting point in the theory-building process. This emphasis on the meanings that objects and events have for students within their experiential realities can be contrasted with approaches in which the goal is to specify cognitive behaviors that yield an input-output match with observed behavior. It is important to note that the term ‘experience’ as it is used in these articles is not restricted to physical or sensory-motor experience. A perusal of the first five articles indicates that it includes reflective experiences that involve reviewing prior activity and anticipating the results of potential activity. In addition, by emphasizing interaction and communication, Voigt’s contribution reminds us that personal experiences do not arise in a vacuum but instead have a social aspect.

de Freitas E. & Sinclair N. (2012) Diagram, gesture, agency: Theorizing embodiment in the mathematics classroom. Educational Studies in Mathematics 80(1–2): 133–152.

In this paper, we use the work of philosopher Gilles Châtelet to rethink the gesture/diagram relationship and to explore the ways mathematical agency is constituted through it. We argue for a fundamental philosophical shift to better conceptualize the relationship between gesture and diagram, and suggest that such an approach might open up new ways of conceptualizing the very idea of mathematical embodiment. We draw on contemporary attempts to rethink embodiment, such as Rotman’s work on a “material semiotics,” Radford’s work on “sensuous cognition”, and Roth’s work on “material phenomenology”. After discussing this work and its intersections with that of Châtelet, we discuss data collected from a research experiment as a way to demonstrate the viability of this new theoretical framework.

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.

Lochhead J. (1992) Knocking down the building blocks of learning: Constructivism and the ventures program. Educational Studies in Mathematics 23(5): 543–552. https://cepa.info/8031

Many people believe they are incapable of learning anything beyond the most trivial of mathematical ideas. Constructivism has two quite separate contributions to make in the struggle against this mistaken belief. The first is to indicate that “incompetence” is a dangerous mental construction, dangerous because it creates experiences that provide for its own confirmation. To overcome the limitations of mathematical incompetence, one must recognize it as a construction and leap to the counter-construction “all people can learn.” A second obstacle to the learning of mathematics is the naive notion that the truth of mathematical propositions is absolute, that mathematical validity is defined by an omniscient mathematical authority. Understanding the manner in which mathematical ideas are constructed by humans can help to free learners from being constrained under the terror of a mathematical thought police.