Educational Studies in Mathematics最新文献

Whole-class mathematical discussion in a problem-solving activity is recognized as a powerful pedagogical activity but also a challenge for teachers who must consider several difficulties that learners might face, particularly in terms of an overload of Working Memory and Executive Functions. This study investigates how the use of a digital platform (Padlet) can support participatory and inclusive mathematical classroom discussion. We proposed a teaching experiment based on graphical tasks anticipating integral calculus to grade 13 students, and we examined how the use of the digital platform plays a role in the construction and interpretation of new mathematical objects emerging from the activity. The use of Instrumental Genesis and Double Instrumental Genesis frameworks allowed us to make the affordances of the tool emerge. As a result, we got evidence of how mathematical discussion may develop as a network of interactions, feedback, and connection of input and discuss examples of how active participation and inclusion are enhanced by the tool affordances. Indeed, the digital platform allowed easy interaction, with many ways to represent and express the ongoing evolution of personal and shared meanings and the possibility to manage the time of the activity. This fostered students’ participation and students which did not participate in previous discussions were actively engaged in it.

There is a burgeoning interest in the concept of identity within mathematics education research, with recent work suggesting that the interplay of identity and capital (cultural, social, and economic resources) offers a productive lens for understanding school students’ trajectories into, or away from, mathematics. This paper adds to understandings of how interactions of identity and capital play out over time, focusing on how these mediated the mathematics degree trajectories of three young men, Tom, Neb, and Gerrard, who were selected from a wider longitudinal study, as the only young people who went on to take mathematics degrees. Analysis of longitudinal data from 24 semi-structured interviews, conducted with the three young men from age 10 to 21 and two of their parents/carers, found that all three identified as being “good/exceptional at mathematics”. These identifications developed at an early age, were sustained and augmented by capital, and were closely related to pursuing the subject at university and enjoyment of mathematics. However, developing and sustaining an identity as “good at maths” relied on interactions with mathematics capital through families, school, and wider networks. Moreover, classed differences in the distribution of capital were implicated in their different degree outcomes. We argue that attending to the longitudinal interplay between capital and identity offers a rich understanding of how young people come to see themselves and be seen by others as “naturally able” at mathematics and in turn supports choosing to study a mathematics degree. Implications for mathematics education policy and practice are considered.

This study aims at elaborating a well-established theoretical framework that distinguishes three modes of thinking in linear algebra: the analytic-arithmetic, the synthetic-geometric, and the analytic-structural mode. It describes and analyzes the bundle of signs produced by an engineering student during an interview, where she was asked to recall linear algebra topics she had learned the previous year. Following the theory of conceptual blending, the presented case study is used as a model for the description of the emergence of the analytic-structural mode. It shows how the evolution of the used signs and their relationships suggests that the analytic-structural mode emerges as a blend between the other two ways of thinking.

Different types of reasoning, such as intuitive, inductive, and deductive, are used in the generalization of figural patterns, as an important part of patterns in school mathematics. It is difficult to demarcate the constructive patterns where the regularity observed in the first few sentences is generalizable to the other sentences and each reasoning can lead to the correct answer. To examine the types of reasoning used by students and related factors, this study focuses on the deconstructive figural patterns through the design of three targeted tasks. The participants were 33 female students randomly selected from three 10th grade classes in one school. The data collected showed that a consistent reasoning is not used in tasks by the same participant and various factors, such as the appearance of the figures and the concepts used in the tasks, are involved in employing a reasoning. Moreover, the intuitive reasoning under the influence of a salient irrelevant variable was more prevalent than the others in the incorrect answers.

Despite the prominence of analogies in mathematics, little attention has been given to exploring students’ processes of analogical reasoning, and even less research exists on revealing how students might be empowered to independently and productively reason by analogy to establish new (to them) mathematics. I argue that the lack of a cohesive framework for interpreting students’ approaches to analogical reasoning in mathematics contributes to this issue. To address this, I introduce the Analogical Reasoning in Mathematics (ARM) framework. Constructed from an analysis of interviews with four abstract algebra students, ARM identifies several analogical activities that serve to analyze students’ analogical reasoning with a finer grain size than was previously possible with existing frameworks. Using this framework, I present an analysis of the students’ constructions of a ring-theoretic analogy to subgroup, thus revealing that even constructing simple analogies can elicit diverse pathways of analogical reasoning across students. Implications for further research related to analogies and analogical reasoning in mathematics education are discussed.

This large-scale study characterizes the algebraic learning of a cohort of nearly 800 French students during the last year of middle school (9th grade, 14–15 years old) and examines their evolution in relation to the composition of the classes to which the students belong. Based on the Anthropological Theory of the Didactic (ATD), the study is founded both on a reference model of elementary algebra and on technological-theoretical levels characterizing the students’ reasoning and their knowledge regarding institutional expectations. These are used to code the data and define the statistical variables on which multivariate descriptive analyses are carried out. Students’ learning is analyzed through Pépite, an algebra test given at the beginning and at the end of the year. A cluster analysis resulted in three distinct clusters of similar classes and of similar students both at the beginning and end of the grade 9 school year. They are interpreted didactically and could be useful for a teacher who has to manage the heterogeneity of learning in a class. Some classes and students move from one cluster to another between the beginning and the end of the year, showing a wide variety of ways in which students’ algebra learning progresses or regresses over a school year.

School subjects differ in their histories, epistemologies, and relations to state and federal policies. Though educational policymaking is shaped by how education leaders view school mathematics—what mathematics is, how mathematics is learned, and what counts as equitable and high-quality mathematics education—policy research often takes a subject-neutral perspective, ignoring the ways in which policymaking is rooted in the subject-matter. In this article, I report the ways in which systems of meaning or discourses about school mathematics penetrate the policymaking of two school districts. Based on a discourse analysis of interviews, observations, and artifacts, I found five main discourses about school mathematics reflected in district policymaking: (1) mathematics is a core school subject; (2) mathematics is sequential, where mastery of prior learning is necessary for future learning; (3) mathematics is well-defined, with agreement over the content; (4) there are competing perspectives of high-quality mathematics pedagogy, between conceptually-oriented instruction and direct instruction focused on procedures; and (5) equity in mathematics is access and achievement. These discourses about school mathematics were written in formal policy texts, institutionalized in district-wide practices for assessment, intervention, and tracking, and reflected in leaders’ personal views and social narratives from teachers, parents, and the community. By making visible the taken-for-granted meanings about school mathematics shaping educational policymaking, it becomes possible to interrupt and challenge them with alternative discourses.

Students’ mathematical learning is bolstered when they have caring relationships with their teachers, yet there is little research on how learning environments may constrain or facilitate teacher care. In this study, I investigate how secondary mathematics teachers understood their care for students online during the COVID-19 pandemic. Through analysis of interview and focus group transcripts, I theorize the role of the learning environment in how teachers care for students and their dispositions toward that work. Findings show how an online learning environment both constrained and facilitated teacher care. An obscuration of student thinking online made it difficult for teachers to care for students’ mathematical thinking. The online learning environment contributed to a curtailment of collaborative problem-solving, which in person had supported teacher care for student–student relationships. Teachers described greater privacy with some students online but difficulty getting to know most students. Teachers also reported their care to be conditional on students’ relational initiation or reciprocity. This study highlights the need for professional development for teachers when learning environments shift and future research about learning environments and relational power.

Functions are an essential concept in mathematics. The studies that have examined functions in advanced contexts have primarily focused on students’ reasoning about specific types of functions (such as binary operations and isomorphisms) but not on the core characteristics of well-definedness and everywhere-definedness. Here, we report on a study in which we conducted task-based clinical interviews to gain insight into students’ techniques for addressing “is the given relation a function?” tasks. We found that the techniques students employed necessarily extended far beyond those reported in the literature (such as the vertical line test) and relied on the previously undocumented notions of sameness, convention, and ambiguity (for well-defined) and notions of containment, existence, and set operations (for everywhere-defined). These techniques coordinated the domain, codomain, and rule, which previous research has highlighted the importance of but stopped short of directly investigating. Two contributions of this work include identifying successful techniques (as the landscape of functions literature predominantly focuses on challenges and difficulties) and identifying techniques for everywhere-definedness (which had not previously received any direct attention in the literature).

In this contribution, we address the gap that has appeared in mathematics education research and practice with the emergence of dynamic geometry environments and build on the opportunities these environments offer to school geometry. In our qualitative empirical study, we investigate how to elaborate on the general model of conceptual knowledge to make it applicable to dynamic geometry tasks, specifically to tasks including dynamic geometric constructions. We present a design of dynamic constructions of quadrilaterals that comply with Euclidean constructions, derive an assessment instrument based on them, and study what information the instrument can provide about the quality of students’ conceptual knowledge. We present the results in the form of an assessment framework consisting of an example of the assessment instrument and an ordered system of qualitative categories serving as an assessment codebook for interpreting students’ responses in terms of the quality of conceptual knowledge. To clarify the relations between the assessment framework and the general model of conceptual knowledge, we establish a system of subdimensions of conceptual knowledge that indicates how conceptual knowledge can be understood in the context of dynamic geometric constructions and identifies the conceptual knowledge needed to achieve individual categories of the assessment framework.