Barwell R. (2009) Researchers’ descriptions and the construction of mathematical thinking. Educational Studies in Mathematics 72(2): 255–269. Fulltext at 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.

Beer R. (2003) The dynamics of active categorical perception in an evolved model agent. Adaptive Behavior 11(4): 209–243. Fulltext at https://cepa.info/5188

Notions of embodiment, situatedness, and dynamics are increasingly being debated in cognitive sci ence. However, these debates are often carried out in the absence of concrete examples. In order to build intuition, this paper explores a model agent to illustrate how the perspective and tools of dynam ical systems theory can be applied to the analysis of situated, embodied agents capable of minimally cognitive behavior. Specifically, we study a model agent whose “nervous system” was evolved using a genetic algorithm to catch circular objects and to avoid diamond-shaped ones. After characterizing the performance, behavioral strategy and psychophysics of the best-evolved agent, its dynamics are analyzed in some detail at three different levels: (1) the entire coupled brain/body/environment sys tem; (2) the interaction between agent and environment that generates the observed coupled dynam ics; (3) the underlying neuronal properties responsible for the agent dynamics. This analysis offers both explanatory insight and testable predictions. The paper concludes with discussions of the overall picture that emerges from this analysis, the challenges this picture poses to traditional notions of rep resentation, and the utility of a research methodology involving the analysis of simpler idealized mod els of complete brain/body/environment systems.

Cristea S. (2015) The fundaments of constructivist pedagogy. Procedia – Social and Behavioral Sciences 180: 759–764. Fulltext at https://cepa.info/5864

Our study underlines the specific study object, the specific normativity and the specific research methodology for constructivist pedagogy. he specific study object of constructivist pedagogy constitutes the difference between the inner system of the one who learns and the external environment objective, natural, community, cultural, civic, political, religious etc., at which the pupil refers to subjectively. The specific normativity of constructivist pedagogy engages four central principles: a) the principle of self-preserving the trainee’s resources; b) the principle of the subjective reporting of the educated to the objective reality; c) the principle of the viable, efficient pedagogic activity / action; d) the principle of subjective valorizing of the objective reality.

Larochelle M. & Désautels J. (2007) Concerning Ernst von Glasersfeld’s Contribution to Intellectual Freedom: One Interpretation, One Example. Constructivist Foundations 2(2-3): 90–97. Fulltext at https://cepa.info/35

Purpose: According to the constructivist perspective tirelessly promoted by Ernst von Glasersfeld for more than 40 years now, the world we see is of a piece with our way of understanding and locating ourselves within it; ultimately, whenever we claim to describe the world-in-itself, we in fact are describing the product of the mapping process that has enabled us to make our way in this world and to actualize our projects within it. Obviously, this kind of perspective has consequences for the way both educational action and research on this theme are conceived of and accomplished. That, at least, is what we shall attempt to show in this article. Implications: In keeping with the claim that knowledges are constituted not in reference to reality “itself” but to practices and activities, constructivism advocates examining cognition in action – that is, in terms of how the latter is enacted in the field. Accordingly, constructivism also seeks to prompt teachers to: (1) scrutinize the processes and distinctions by which students chart out the world; (2) and to personally devise, on the basis of this experience, a model – or models, rather – of their students’ future relationship to the universes of knowledge intended for learning. Likewise, constructivism also aims to prompt researchers to perform some very careful detective work into the ways in which this charting process is played out and thus to opt for a comprehensive rather than an experimentalist approach. Conclusion: To adopt the constructivist perspective also means to “de-siloize” knowledge production and to recognize that this production occurs in all spheres of society. From this point of view, constructivism can thus be viewed as a way of challenging the claims of a certain scientific establishment to alone possess the requisite standing for interpreting the world.

Reid D. A. (1996) Enactivism as a methodology. In: Puig L. & Gutiérrez A. (eds.) Proceedings of the Twentieth Annual Conference of the International Group for the Psychology of Mathematics Education (PME-20), Volume 4. PME, Valencia: 203–210. Fulltext at https://cepa.info/2519

As research is learning, theories for learning and research methodologies in mathematics education overlap. For the Enactivist Research Group, enactivism is both the theoretical framework and the methodology for our research. Key ideas such as autopoesis, structure determinism, structural coupling, and coemergence are used to make sense of the learning of all participants in research, researchers included. This paper describes these key ideas and enactivist research methodology in mathematics education.

Roth W.-M. (1994) Experimenting in a constructivist high school physics laboratory. Journal of Research in Science Teaching 31: 197–223.

Although laboratory activities have long been recognized for their potential to facilitate the learning of science concepts and skills, this potential has yet to be realized. To remediate this problem, researchers have called for constructivist learning environments in which students can pursue open inquiry and frame their own research problems. The present study was designed to describe and understand students’ experimenting and problem solving in such an environment. An interpretive research methodology was adopted for the construction of meaning from the data. The data sources included videotapes, their transcripts, student laboratory reports and reflections, interviews with the students, and the teacher’s course outline and reflective notes. Forty‐six students from three sections of an introductory physics course taught at a private school for boys participated in the study. This article shows the students’ remarkable ability and willingness to generate research questions and to design and develop apparatus for data collection. In their effort to frame research questions, students often used narrative explanations to explore and think about the phenomena to be studied. In some cases, blind alleys, students framed research questions and planned experiments that did not lead to the expected results. We observed a remarkable flexibility to deal with problems that arose during the implementation of their plans in the context of the inquiry. These problems, as well as their solutions and the necessary decision‐making processes, were characterized by their situated nature. Finally, students pursued meaningful learning during the interpretation of data and graphs to arrive at reasonable answers of their research questions. We concluded that students should be provided with problem‐rich learning environments in which they learn to investigate phenomena of their own interest and in which they can develop complex problem‐solving skills.

Simmt E. & Kieren T. (2015) Three “moves” in enactivist research: A reflection. ZDM Mathematics Education 47: 307–317. Fulltext at https://cepa.info/4481

In this paper the authors reflect on the contents of this current issue of ZDM and ask why focus an entire issue on enactivism as a research methodology in mathematics education. In their synthesis of the papers they distinguish and explicate what they observe as three moves in the enactivist research discussed. The first move (and the one that receives much of the attention in the papers) is that of the observer. Enactivism proposes the observer is one who arises in the act of observing and whose knowing is explained through the mechanism she describes. The second move is an understanding that all knowing is perceptually guided action that brings forth a world of significance. The third is a consequence of the first two: All knowing has implications. Hence that third move is towards ethics. The observer is not neutral; her observations bring forth worlds of significance that intersect with the worlds of others. They conclude, that the strength of enactivism as a methodological frame for mathematics education research is that it is a form of research that is incomplete. Incomplete in that from this framework there is necessarily always more to be said and different grounds for the saying about the phenomena under investigation in mathematics education.

Steffe L. P. (1991) The constructivist teaching experiment: Illustrations and implications. In: Glasersfeld E. von (ed.) Radical constructivism in mathematics education. Kluwer, Dordrecht: 177–194. Fulltext at https://cepa.info/2098

In an epistemology where mathematics teaching is viewed as goal-directed interactive communication in a consensual domain of experience, mathematics learning is viewed as reflective abstraction in the context of scheme theory. In this view, mathematical knowledge is understood as coordinated schemes of action and operation. Consequently, research methodology has to be designed as a flexible, investigative tool. The constructivist teaching experiment is a technique that was designed to investigate children’s mathematical knowledge and how it might be learned in the context of mathematics teaching (Cobb & Steffe, 1983; Hunting, 1983; Steffe, 1984). In a teaching experiment, the role of the researcher changes from an observer who intends to establish objective scientific facts to an actor who intends to construct models that are relative to his or her own actions.

Steffe L. P. & Ulrich C. (2013) Constructivist teaching experiment. In: Lerman S. (ed.) Encyclopedia of mathematics education. Springer, Berlin: 102–109. Fulltext at https://cepa.info/2959

In an epistemology where mathematics teaching is viewed as goal-directed interactive communication in a consensual domain of experience, mathematics learning is viewed as reflective abstraction in the context of scheme theory. In this view, mathematical knowledge is understood as coordinated schemes of action and operation. Consequently, research methodology has to be designed as a flexible, investigative tool. The constructivist teaching experiment is a technique that was designed to investigate children’s mathematical knowledge it might be learned in the context of mathematics teaching (Cobb & Steffe, 1983; Hunting, 1983; Steffe, 1984). In a teaching experiment, the role of the researcher changes from an observer who intends to establish objective scientific facts to an actor who intends to construct models that are relative to his or her own actions.

Steier F., Gyllenpalm B., Brown J. & Bredemeyer S. (2008) World Cafe: Förderung der Teilhabekultur (The world cafe: Fostering cultures of participation). In: Kersting N. (ed.) Politische Beteiligung: Einführung in dialogorientierte Instrumente politischer und gesellschaftlicher Partizipation. VS Verlag Für Sozialwissenschaften, Wiesbaden: 167-180.

While presenting principles of The World Cafe as a collaborative way of meeting, strong possibilities for recognizing The World Cafe as a generative and highly participatory research methodology for knowledge generation are discussed. At its core, The World Cafe as research methodology embodies and extends principles of radical constructivism, including a shift from methodologies as “discovering” to a methodology as “harvesting” knowledge.