Gandell R. & Maheux J.-F. (2019) Problematizing: The Lived Journey of a Group of Students Doing Mathematics. Constructivist Foundations 15(1): 50–60. https://cepa.info/6161

Context: Mathematical problem solving is considered important in learning and teaching mathematics. In a recent study, Proulx and Maheux presented mathematical problem solving as a continuous dialectical process of small problem posing and solving instances in which the problem is continuously transformed, which they call problematizing. This problematizing conceptualization questions many current assumptions about students’ problem solving, for example, the use of heuristics and strategies. Problem: We address two aspects of this conceptualization: (a) how does problematizing evolve over time, and (b) how do the students’ problematizations interact? Method: In this study, we apply and further develop Proulx and Maheux’s enactivist perspective on problem solving. We answer our questions by applying micro-analysis to the mathematical problematizing of a group of students and, using Ingold’s pathways and meshwork as our framework, illustrate the lived practice of a group of students engaged in mathematical problem solving. Results: Our analysis illustrates how mathematical problematizing can be viewed as a complex, enmeshed and wayfaring journey, rather than a step-by-step process: in this enactive journey, smaller problems co-emerge from students’ interactions with one another and their environment. Implications: This research moves the focus on students’ mathematical problem solving to their actions, rather than strategies or direct links from problems to solutions, and provides a way to investigate, observe and value the lived practice of students’ mathematical problem solving. Constructivist content: Our work further strengthens the understanding of mathematical activities from an enactivist perspective where mathematical knowledge emerges from interaction between individual and environment.

Maheux J.-F. & Gandell R. (2019) Authors’ response: Flying Kites and the Textility of Problematizing. Constructivist Foundations 15(1): 73–77. https://cepa.info/6168

Abstract: We briefly discuss how far we take metaphors for learning or doing mathematics while challenging the descriptive-prescriptive paradigm in favor of a larger view of research and language (use) emphasizing the evocative and provocative texture of our work and Ingold’s writing. In so doing, we also bring forth an ethical dimension to research writing, which may help situate what we did not present, discuss and suggest in the target article: missing fragments, students’ backgrounds, teachers’ potential roles and direct implications or recommendations, for example. Finally, we also offer a reflection on how our study contributes to research through both its similarities to and distinctions from other conceptualisations. Jean-François Maheux & Robyn Gandell

Maheux J.-F. & Proulx J. (2015) Doing|mathematics: Analysing data with/in an enactivist-inspired approach. ZDM The International Journal on Mathematics Education 47(2): 211–221.

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.

Maheux J.-F., Roth W.-M. & Thom J. (2010) Looking at the observer challenges to the study of conceptions and conceptual change. In: Roth W.-M. (ed.) Re/structuring science education: ReUniting sociological and psychological perspectives. Springer, Dordrecht: 201–219. https://cepa.info/6145

Excerpt: In a typical study of students’ conceptions and conceptual change, researchers analyze what a student does or says in a classroom or in an interview and recognizes ideas that match or do not match their own understanding of the topic. Attributing the perspective they recognize in the student, those studies support the idea that a conception is the way by means of which an individual intrinsically conceives (of) a given phenomenon. They then hypothesize the existence of some mental structures that can be theoretically and objectively re-constructed based on what is observed in a student’s performance. Thus, researchers studying conceptions commonly assume that the observer and the observed are separate entities. However, even in the most theoretical and hardest of all sciences, physics, the independence of the measured object and the measuring subject is not taken for granted: Light, for example, will present itself as waves or as particles depending on how we examine it. The artificial sense of separation from the object(s) of study found in many accounts on students’ conceptions makes irrelevant the relationship that exists between the observer and the observed: an interdependence and co-emergence of the observer and the observed. This tight relation exists because each participant not only reacts upon what others say but also acts upon the reactions that his/her own actions give rise to. With this situation come epistemological, practical, and ethical implications for those researching in mathematics and science education. Positing or questioning the existence of an objective reality mediates how we accept or reject another human being and the worldviews s/he develops. It provides a rationale that guides our actions. This is especially important when it comes to teaching and learning at a time where the ability to deal with the plurality and diversity of human culture have emerged as significant referents for our social behavior.

Proulx J. & Maheux J.-F. (2017) Authors’ Response: On Posing|Solving Research. Constructivist Foundations 13(1): 185–190. https://cepa.info/4435

Upshot: We frame our response to the nine OPCs by illustrating how issues related to the observer are salient in research. This leads us to bring forward distinctions about what research and researchers are, for us, all about. We frame research and the researcher’s role in terms of endeavors aiming to offer ways at looking at a phenomenon, and not of providing undeniable, hard facts.

Proulx J. & Maheux J.-F. (2017) From Problem Solving to Problem Posing, and from Strategies to Laying Down a Path in Solving: Taking Varela’s Ideas to Mathematics Education Research. Constructivist Foundations 13(1): 160–167. https://cepa.info/4425

Context: There has always been a tremendous and varied amount of work on problem-solving in mathematics education research. However, despite its variety, most if not all work in problem-solving shares similar epistemological assumptions about the fact that there is a problem to be solved and that solvers make an explicit selection of a strategy and apply it to solve the problem. Problem: Varela’s ideas about problem-posing provide a means of going beyond these assumptions about problem-solving processes. We propose to explain and illustrate the way we found inspiration from these ideas in our work through a discussion grounded in data excerpts collected in our research studies on mental mathematics. Method: Concrete data and observations are referred to for discussing issues related to problem-solving processes and activity in mathematics. Results: Engaging with Varela’s work led us to revisit and reformulate many common notions in relation to mathematical problem-solving, namely concerning the meaning of a problem and of a strategy, as well as the relationship between the posing and solving of a problem. Through this, these notions are conceived as dynamic in nature and co-constitutive of one another. This leads us to engage in what we call the dialectical relationship between posing and solving. Implications: We illustrate the sort of educational insights that might be drawn from such conceptualizations, mostly in terms of affecting the way we look at students’ productions and engagements in mathematics not as pre-fixed or pre-definable entities, but as activities that emerge in the midst of doing mathematics.