Journal of Mathematical Behavior最新文献

This study uses Action-Process-Object-Schema theory (APOS) to examine students’ understanding of two-variable function optimization. A genetic decomposition (GD) based on the notion of Schema is proposed. This is a conjecture of mental structures and relations that students may construct to understand the optimization of these functions. The GD was tested with semi-structured interviews with eleven students who had just finished an introductory multivariable calculus course. Results show that giving explicit attention during instruction to the topological structure of the domain of the function to be optimized and the use of GD-based activities was effective in promoting students’ understanding of two-variable function optimization. On the theoretical side, the study contributes to a better understanding of the APOS notions of Schema, Schema-triad, and types of relations between Schema components that have not been used extensively in the literature and that proved to be a powerful tool to model students’ learning.

Many empirical studies documented students’ challenges with operation-choice problems, in particular for multiplication and division with rational numbers. The design principle of problem variation was suggested to overcome these challenges by engaging students in making connections between inverse operation-choice problems of multiplication and division, and between problems with natural numbers and fractions/decimals, but so far, this approach was hardly investigated empirically. In this study, we investigate 17 sixth graders’ modelling pathways through sets of operation-choice problems that are systematically designed according to the variation principle. In the qualitative analysis, we identify five pathways by which students solve the problems and sometimes connect them. While one pathway uses deep relational connections, others only draw superficial and operational connections and others stay with informal strategies without connecting them to formal operations.

The concept of inverse is threaded throughout K-16 mathematics. Scholars frequently advocate for students to understand the underlying structure: combining an element and its inverse through the binary operations yields the relevant identity element. This ‘coordinated’ way of reasoning is challenging for students to employ; however, little is known about how students might reason en route to developing it. In this study, we analyze a teaching experiment with two beginning abstract algebra students through the lens of three ways of reasoning about inverse: inverse as an undoing, inverse as a manipulated element, and inverse as a coordination of the binary operation, identity, and set. In particular, we examine the implications of these ways of reasoning as students work to develop inverse as a coordination. We identify pedagogical tools and facets of instructional design that appeared to support students’ development of inverse as a coordination. We further suggest that all three ways of reasoning can support productive activity with inverses.

Inverse is a critical topic throughout the K–16 mathematics curriculum where students encounter the notion of mathematical inverse across many contexts. The literature base on inverses is substantial, yet context-specific and compartmentalized. That is, extant research examines students’ reasoning with inverses within specific algebraic contexts. It is currently unclear what might be involved in productively reasoning with inverses across algebraic contexts, and whether the specific ways of reasoning from the literature can be abstracted to more general ways of reasoning about inverse. To address this issue, we conducted a standalone literature review to explicate and exemplify three cross-context ways of reasoning that, we hypothesize, can support students’ productive engagement with inverses in a variety of algebraic contexts: inverse as an undoing, inverse as a manipulated element, and inverse as a coordination of the binary operation, identity, and set. Findings also include explicating affordances and constraints for each of these ways of reasoning. Finally, we reflect on when and how standalone literature reviews can serve the purpose of unifying fragmented and obscured insights about key mathematical ideas.

This study compares pre-service mathematics teachers’ (PSMTs) and in-service mathematics teachers’ (ISMTs) noticing of argumentation at the secondary-school level. Thirty-five PSMTs and 32 ISMTs engaged in analyzing argumentation classroom situations (ACSs) using an ACS-report format emphasizing two sub-skills of noticing: attending and interpretation. Analysis of the participants’ ACS reports revealed differences between the two research groups. The ISMTs paid a high level of attention to all four aspects: ‘co-constructing of arguments’, ‘critiquing arguments’, ‘mutual respect’, and ‘working toward consensus-building’, whereas the PSMTs paid a high level of attention to ‘mutual respect’ and ‘co-constructing of arguments’ aspects only. In terms of interpretation, the ISMTs outperformed the PSMTs in interpreting the argumentation through the lenses of ‘task characteristics’, ‘teaching strategies’, and ‘student cognitive characteristics’. The findings are interpreted in light of both theory and practice.

A reader of mathematical text must often switch between reading mathematical symbols and reading words. In this study, five different categories of structural connections between symbols and language, which invite such switches, are presented in a framework. The framework was applied in a study of Swedish mathematics textbooks, where 180 randomly selected pages from different educational stages were analyzed. The results showed a significant change in communication patterns as students progress through school. From a predomination of connections based on proximity found in year two, there is a gradual change to a predomination of symbols interwoven in sentences in year eight. Furthermore, more qualitative investigations of the different connections complemented the quantification, both through further explanations of the quantitative results, and through more examples of differences in communication patterns. The implications for readers of mathematics texts are discussed.

This study used three pairs of problem-posing tasks to examine the impact of different prompts on students’ problem posing. Two kinds of prompts were involved. The first asked students to pose 2–3 different mathematical problems without specifying other requirements for the problems, whereas the second kind of prompt did specify additional requirements. A total of 2124 students’ responses were analyzed to examine the impact of the prompts along multiple dimensions. In response to problem-posing prompts with more specific requirements, students tended to engage in more in-depth mathematical thinking and posed much more linguistically and semantically complex problems with more relationships or steps required to solve them. The findings from this study not only contribute to our understanding of problem-posing processes but also have direct implications for teaching mathematics through problem posing.

We addressed the call for explorations of how BIPOC students’ “experiences in secondary mathematics classrooms might advance transformative, equity-focused, pedagogical models” (Joseph et al., 2019, p. 149) by exploring how a nested, equity-directed approach created different kinds of opportunities for students to take up, shift, or resist what it means to teach, learn, and do mathematics. Specifically, we looked at efforts to engage equity-directed dominant and critical approaches through a series of three mathematics projects aimed at investigating food insecurity as a social (in)justice issue using geometry. Our analysis focused on a subset of data generated during three projects from different times of the year. Findings revealed that the teacher more readily enacted critical equity-directed practices than dominant ones; that students more readily embraced those critical practices; and that students expected their use of mathematics and exploration of social issues to align with authentic, real-world situations.

Although interest in using videos in educational settings has surged in recent years, researchers know little about what mathematical meanings students develop from watching these videos or how they do so. To contribute to this gap in the research, we examined how two students appropriated mathematical meanings from instructional videos. In contrast to typical instructional videos, which rely heavily on an expert’s exposition, the videos in our study featured the unscripted conversation of two high school students as they engaged with novel mathematical problems. This allowed us to examine how other students watching the videos coordinated meanings expressed by both the video participants and each other, including meanings that were initially incorrect or incomplete. To analyze these data, we adopted a Bakhtinian-inspired lens, which allowed us to conceptualize meaning as emerging from the relationships among multiple voices. Additionally, the appropriation of meanings from the videos was not straightforward. Instead, we found evidence of the repetition (mimicry) of words and actions from the video participants, revision, resistance, and the invocation of previously-appropriated voices, before the students were able to make the meanings expressed in the videos their own.