Task-based interviews are an established method in mathematics education research, to investigate students’ mathematical thinking. While interviews are a social practice, principles for design and enactment of task-based interviews have offered limited insight into navigating power dynamics inherent in interviews. Drawing on the theorizing of positioning, we examine the role of communication acts before and during task-based interviews to position students as experts in their mathematical thinking. We offer three “positioning principles” for conducting task-based interviews: set the stage to surface interviewees’ brilliance; be alert to the influence of power on interviewees’ responses; and, redistribute power to interviewees. To support our claims, we provide excerpts from a videoconference interview with a college algebra student interacting with a graphing task. This work illustrates how researchers can draw on different theories to advance research methods developed in conjunction with another theoretical perspective.

This paper explores the role of both theoretical and practical mathematics in university curricula. While math instruction often focuses on theoretical reasoning, it is equally important to successfully integrate both kinds of knowledge. As the landscape of mathematics education changes and the demand for practical mathematical skills increases across various fields, this discussion is especially timely. This paper begins by examining Sierpińska’s pressing challenge regarding the fragility of theoretical thinking and situates this issue within a broader intellectual context. We incorporate views from Aristotle, Poincaré, Sierpińska, Tall, and Freudenthal, each representing a unique tradition regarding the relationship between practical and theoretical knowledge. Additionally, the paper references findings from collaborative research with mathematicians to connect theoretical insights to current teaching practices. The goal is to show how these perspectives, combined with research evidence, can inform teaching methods, curriculum development, and classroom approaches, opening an exciting area of exploration. Its contribution lies in fostering a dialogue that needs further research, especially on how students can transition smoothly between practical and theoretical mathematics.

By building on Vergnaud’s theory of conceptual fields and his description of concepts as conglomerates of situations, invariants, and representations, we stipulate that mathematical concepts can have precisely three roots: relations in situations, relations in iconic imagery, or invariants in symbol systems. Through conceptual polysemy, several classes of situations and iconic imagery will regularly map to the same mathematical symbol system, which by mathematical virtue must be free of contradictions. Mathematical concepts in school are almost always introduced through situations and iconic imagery and labeled with symbol systems. We stipulate that successful conceptual progression is characterized by an epistemological shift, where the dominant meaning transfers from residing in situations or iconic imagery to residing in relations in symbol systems. We explain how this theory can be put to work when planning primary school teaching, focusing on long-term conceptual progress, by exemplifying with the multiplicative conceptual field.

Sociocultural research has been successful in helping us to better understand how social, cultural, and political contexts shape the teaching and learning of mathematics. Against rationalist and empiricist accounts of knowledge, sociocultural research has forcefully argued that knowledge is culturally situated. The results contributed by ethnomathematics in recent years support this idea. However, there still remains the problem of theoretically explaining how knowledge is anchored in culture. Such an explanation requires a clear theoretical account of the ontological constitution of knowledge. In the first part of this paper, I explore this problem by offering an overview of knowledge as conceived in one of the contemporary sociocultural theories in mathematics education: the theory of objectification. In the second part I turn to the importance of clarifying the conceptions of knowledge we use in mathematics education research. My argument is that, since learning is always about learning something, it is virtually impossible to understand learning from an educational viewpoint if the question of knowledge has not been elucidated first.

This study aims to deepen the understanding of researchers’ theorising work concerning task design principles in mathematics education. To achieve this, we employ a praxeological analysis to explain and characterise the designing and theorising work of researchers and teachers within a Japanese project on task design. By utilising two distinct praxeologies—designing praxeology and research praxeology—along with the notion of theory elements, we illustrate the results of the case study conducted within the research project, revealing how design principles were produced and elaborated as outcomes of successive theorising steps. Furthermore, cultural issues related to the nature of theorising work are also discussed, providing insights into the subtle distinction between ‘development’ and ‘research’ within Japanese mathematics education research.

The concept of didactic transposition expands the analytical scope of educational research by including the transformations of knowledge orchestrated by diverse actors, from academic researchers to curriculum developers and teachers. Researchers adopt distinct positions regarding the institutions involved in this transposition process, fundamentally shaping which aspects of knowledge become questionable and which remain naturalised. Through examining three theoretical approaches—Stoffdidaktik, pedagogical content knowledge, and the anthropological theory of the didactic—this study illuminates how researcher positioning influences investigations of mathematical knowledge transformation. We use the mathematisation of quantities as an analytical case to demonstrate these distinctions, revealing how each position enables particular forms of epistemological inquiry whilst constraining others. This exploration establishes didactic transposition as a powerful meta-theoretical lens for delimiting analytical units across different perspectives, stimulating further research into their theoretical foundations. The analysis contributes to mathematics education research by providing a refined framework for understanding how epistemological assumptions embedded in researcher positioning influence both problem formulation and solution development in educational contexts.

Considering some implicit assumptions concerning people, such as an unspecified majority, are mathematics education theories consistently effective only in some specific fields, and is their growth also limited? Using case studies in which the authors were involved—criteria for students’ knowing geometrical objects and students’ strength for understanding mathematics—, this article suggests that the theory-building process in mathematics education is a heteronomous and interdependent activity. From an enactivist perspective, the authors highlight that theories evolve through progressive individualization, encounters with theories in various domains, and subsequent collaboration. In particular, collaborating with theories from other fields can generate new questions and offer valuable conceptual perspectives. This collaboration is vital in developing interdisciplinary research in mathematics education.

Knowledge in Pieces (KiP) and Commognition are two influential heuristic epistemological frameworks. While the two approaches have different historical origins and distinct theoretical foci (encoding of intuitive knowledge and a complex systems approach to understanding conceptual development in the case of KiP and mathematical discourse and its development in individual and communal forms in the case of Commognition) they both share an accountability to the nuances of the moment-by-moment details of records of cognition. Both perspectives draw insights from fine-grained analysis of reasoning from video data of learning interactions and reasoning processes. In terms of the kinds of theoretical machinery produced, both KiP and Commognition share a skepticism towards common sense terms about knowledge and they both emphasize the need for precision in new theoretical constructs and models. This theoretical paper explores both the differences and similarities between Commognition and Knowledge in Pieces and begins to map out the possibilities and potential benefits for networking these perspectives.

This paper examines the challenges of connecting two kinds of inquiry-based didactic approaches: the study and research paths (SRPs), developed in the framework of the anthropological theory of the didactic, and the structured problem-solving lessons in the Japanese educational context. At first glance, both didactic proposals appear similar, as they emphasize the centrality of questions in instructional situations. However, a deeper analysis considering the management of the didactic time, the didactic contract, and the didactic milieu, reveals significant differences between their organizations, that is, the particular teaching patterns they promote. Furthermore, analyzing how SRPs have been adapted to the Japanese mathematics classroom also highlights ecological differences, while shedding light on some institutional constraints in the Japanese educational system that hinder their implementation by schoolteachers. Finally, our study uncovers a particular form of institutional transposition of SRPs, shaped by the specific demands of the teaching profession.

Fundamental questions about what constitutes valuable theoretical work remain underexplored in mathematics education research. In this paper, I propose an axiological approach to understanding how theoretical contributions gain recognition and influence within our scholarly community. Drawing from critical philosophical investigations of value and systematic analysis of theoretical developments in the field, I argue that theoretical recognition emerges through dynamic engagement with four interconnected value dimensions: epistemic (advancing knowledge), practical (enhancing educational practice), cultural (respecting diversity and disciplinary norms), and ethical (promoting justice and well-being). These dimensions operate as historically situated commitments that both shape and are reshaped by emerging theoretical work within broader socioeconomic and political contexts. Through critical examination of pivotal theoretical developments—from constructivism’s ascendance to contemporary sociopolitical frameworks—I demonstrate how axiological dynamics illuminate both historical patterns and contemporary debates about theoretical priorities whilst revealing the material conditions that influence theoretical production. This axiological perspective shows that theoretical recognition emerges through community negotiations about value and worth that are enacted within political and historical contexts, encompassing questions about whose knowledge counts and what kinds of transformation theoretical work should promote.