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Noticing the strengths in students' mathematical thinking is a critical skill that teachers need to develop, but it can be challenging due to the prevalence of deficit-based thinking in mathematics education. To address this challenge, a teacher education course was designed to encourage prospective teachers to engage in critical reflection on their own and others' framings of students' thinking and shift their focus towards noticing students' strengths. The study analyzed written responses from the prospective teachers, collected at the beginning and end of the course, to investigate their framing and noticing of students' mathematical thinking. The analysis focused on the aspects of students' thinking that the prospective teachers paid attention to, the stances they took when interpreting students' thinking, and the instructional moves they proposed in response to their thinking. Furthermore, the study established a spectrum of deficit-based and strength-based framings on students' mathematical thinking. This spectrum allowed for the identification of each participant's written noticing responses within a range of possibilities, contributing to a more nuanced understanding of the changes in teachers' framing and noticing of students' thinking over time.

This paper examines how different approaches in mathematics education conceptualise the relationship between school mathematics and university mathematics. The approaches considered here include: (a) Klein's elementary mathematics from a higher standpoint; (b) Shulman's transformation of disciplinary subject matter into subject matter for teaching; and (c) Chevallard's didactic transposition of scholarly knowledge into knowledge to be taught. Similarities and contrasts between these three approaches are discussed in terms of how they frame the relationship between the academic discipline and the school subject, and to what extent they problematise the reliance and bias towards the academic discipline. The institutional position implicit in the three approaches is then examined in order to open up new ways of thinking about the relationship between school mathematics and university mathematics.

This paper problematizes the metaphors of unpacking, deconstructing, and decompressing mathematics that are prevalent in discourses about mathematics teacher knowledge. It is argued that the logic of these metaphors is based on the reversibility of packing, constructing, and compressing mathematics and seems to suggest the view that knowledge is in some ways similar to physical objects in the material world that can be taken apart and put back together. Although such a view may be intuitively compelling, it is problematic and misleading in light of existing research on mathematical knowing and learning. Moreover, these metaphors can be understood as playing a role in the ongoing process of colonization, and as such must be conscientiously and critically understood, applied, and analyzed.

This paper explores how different schools of thought in mathematics education think and speak about preparing mathematics for teaching by introducing and proposing certain metaphors. Among the metaphors under consideration here are the unpacking metaphor, which finds its origin in the Anglo-American school of thought of pedagogical reduction of mathematics; the elementarization metaphor, which has its origin in the German school of thought of didactic reconstruction of mathematics; and the recontextualization metaphor, which originates in the French school of thought of didactic transposition. The metaphorical language used in these schools of thought is based on different theoretical positions, orientations, and images of preparing mathematics for teaching. Although these metaphors are powerful and allow for different ways of thinking and speaking about preparing mathematics for teaching, they suggest that preparing mathematics for teaching is largely a one-sided process in the sense of an adaptation of the knowledge in question. To promote a more holistic understanding, an alternative metaphor is offered: preparing mathematics for teaching as ecological engineering. By using the ecological engineering metaphor, the preparation of mathematics for teaching is presented as a two-sided process that involves both the adaptation of knowledge and the modification of its environment.

Earlier approaches to sense-making in mathematics have looked at the ways students comprehend a mathematical concept. Recent research suggests that some students make sense not only of mathematical objects that have a being, but also of objects that have yet to become. In such cases, learning mathematics is not just an act of comprehending a given mathematical concept, but rather an act of creating a mathematical concept that has yet to come into being. In this paper, we argue that this act of creation is better referred to as meaning-making than sense-making, bringing to the fore the distinction between acts of comprehension and acts of creation. Recognising learning mathematics as both sense-making and meaning-making allows better acknowledging the reciprocal relationship between mathematical concept and mathematical conception, a relationship constituted by the interplay of comprehension and creation.

This paper examines fundamental assumptions about the notion of transforming subject matter, which is widely regarded as a core practice of teacher work, a crucial feature of teacher knowledge and a measure of teacher expertise. First, the notion of transforming subject matter and the ways it has been taken up in Anglo-American discourses of teacher knowledge are discussed in relative detail. Second, the paper examines and questions fundamental, yet mostly unexplored, assumptions, including the individual teacher as the locus of transformation, the possessor of the content knowledge at stake and the gatekeeper who enables students to access subject matter content. Finally, these widespread assumptions are problematised against the background of French and German traditions of didactics. These traditions do not regard the capacity to transform subject matter as a characteristic of an individual teacher, but rather of social and cultural systems that are institutionally contextualised and oriented towards normative conceptions of education.

Teacher noticing has become increasingly acknowledged as a fundamental aspect of teacher professional competence. Teacher education scholars have examined how the development of noticing might be supported both in initial teacher education and in professional development. In mathematics teacher education, several studies have explored the use of video as a supporting tool for teacher noticing. It remains unclear how this body of work builds on the various theoretical perspectives of noticing prevalent in the literature, thus broadening our understanding of noticing. Furthermore, the field has not examined systematically the extent to which research has leveraged the affordances of digital video technologies, and whether scholars have employed different research methods to answer questions that are critical to teacher educators. This survey paper reviews studies published in the last two decades on programs centered on mathematics teacher noticing that used video as a supporting tool for teacher learning. Thirty-five peer-reviewed papers written in English were identified and coded along three dimensions: (1) theoretical perspectives; (2) use of video technologies; and (3) research questions and methods. This review summarizes important findings and highlights several directions for future research. Most studies involved pre-service teachers, and only a few centered on in-service teachers. Developers of the large majority of programs took a cognitive psychological perspective and focused on the attending/perceiving and interpreting/reasoning facets of noticing. Few studies used video-based software and few studies used grouping, and even fewer used randomized grouping. Evidence of program effects on responding and decision making, and on instructional practice, is limited and should be extended in the future.

Teacher noticing has been widely understood as a kind of seeing or way of making sense of classroom events and instructional details. Such notions of teacher noticing often construe noticing as a disembodied, purely mental form of seeing and position the teacher as separated or separable from the observing environment. They rely on intuitive models that adopt the usual divide between mind, body, and matter, and that fuel the dualism between the individual and the environment. In this paper, I attempt to work towards a more comprehensive model of teacher noticing that instead proposes an entanglement of the cultural-historical, embodied-ecological, and social-material. Teacher noticing, in such a proposal, includes culturally and historically constituted forms of framing classroom events, embodied ways of accessing and exploring the classroom world, and active shaping and interaction with the classroom setting's social and material structure.

A great deal of progress has been made in dealing with the multiplicity and diversity of theories in mathematics education. However, relatively little attention has been paid to the opportunities offered by conflicts, tensions, and paradoxes among accepted yet opposing theoretical perspectives for theory building and theory advancement. In this paper, four modes of dealing with opposing perspectives are outlined: (1) taking contrasting theoretical perspectives as incommensurable; (2) holding opposites not as conflicting but as complementary; (3) dissolving or surpassing oppositions by blending perspectives; and (4) preserving paradoxes by recognizing the interdependence of constitutive oppositions. These four modes are illustrated by application to the long-standing debate of knowledge-in-structures versus knowledge-in-pieces and further exemplified by turning to the research literature on students' understanding of limit.

This article describes emerging perspectives on contextualizing, complementizing, and complexifying--three processes involved when individuals ascribe meaning to mathematical objects of their thinking. The article is oriented toward a dialectic between theory and empirical research and is structured in two parts. The first part focuses on an evolving theoretical framing that acknowledges the significance of these three processes in mathematical cognition. In the second part, the evolving theoretical framing is used to analyze one student's knowing of the limit concept of a sequence. This analysis directs one's attention to the emergence and function of this student's knowledge resource, which was generic in usage and complex in structure, allowing the activation of productive ideas and contextual meaning-making as needed. Through this analysis, theoretical and interpretative possibilities were generated that inform research on mathematical cognition and elucidate the emerging theoretical perspectives of contextualizing, complementizing, and complexifying.

The purpose of this article is to contribute to the discussion of mathematics teacher knowledge, and the question of what makes it specialized. In the first part of the article, central orientations in conceptualizing mathematics teacher knowledge are identified and the more serious limitations of the grounds on which they stand are explicated. In the second part of the article, alternative views are offered to each of these orientations that direct attention to underexplored issues about what makes mathematics teacher knowledge specialized. Collectively, these alternative views suggest that specialization in mathematics teacher knowledge cannot be comprehensively accounted for by 'what' teachers know, but rather by 'how' teachers' knowing comes into being. We conclude that it is not a kind of knowledge but a style of knowing that signifies specialization in mathematics teacher knowledge.

This paper comments on the theoretical formulations and usage of the construct of teacher noticing in a selection of the papers in this special issue of "ZDM Mathematics Education." The analysis of how the notion of teacher noticing is used in the papers suggests that it draws attention to several interdependencies involved that have not been attended to in the past. However, the contributions in this special issue have only partially accounted for the dynamic interactions in teacher noticing, suggesting that there is potential for enriching our understanding of the complexities involved in the realm of teacher noticing. The purpose of this commentary is to stimulate the current discussion on teacher noticing by providing insights from cognitive science and the applied science of human factors, which have the potential to challenge the current understanding of noticing. In doing so, the paper sets the stage for several related constructs from these research disciplines to raise awareness of aspects that recent conceptualizations of teacher noticing may have blinded rather than enlightened.

The initial assumption of this article is that there is an overemphasis on abstraction-from-actions theoretical approaches in research on knowing and learning mathematics. This article uses a critical reflection on research on students' ways of constructing mathematical concepts to distinguish between abstraction-from-actions theoretical approaches and abstraction-from-objects theoretical approaches. Acknowledging and building on research on knowing and learning processes in mathematics, this article presents a theoretical framework that provides a new perspective on the underlying abstraction processes and a new approach for interpreting individuals' ways of constructing concepts on the background of their strategies to make sense of a mathematical concept. The view taken here is that the abstraction-from-actions and abstraction-from-objects approaches (although different) are complementary (rather than opposing) frameworks. The article is concerned with the theoretical description of the framework rather than with its use in empirical investigations. This article addresses the need for more advanced theoretical work in research on mathematical learning and knowledge construction.