Journal of Mathematical Behavior最新文献

Our study explores boundaries and boundary crossing between communities of mathematics teaching (MT) and visual art teaching (AT). We focus on teacher collaboration, draw on communities of practice and examine the existing boundaries and the way collaborating members handle them. We present data from 17 group meetings of secondary school art and mathematics teachers who try to develop ways of linking ΜΤ and ΑΤ. Results indicate the emerging boundaries by means of discontinuities regarding mathematical practices and tools used in MT and AT, and the disciplines’ teaching and curriculums. Via analysis of the central boundary – the analytical or visual ways of thinking – we present members’ boundary handling that indicates the development and developmental process of integrated practice. The study incorporates boundary crossing processes and learning mechanisms in which the two separate ways of thinking are gradually transformed into an integrated one and reveals possible ways of integration in teaching and learning.

Analogical reasoning is an important mathematical process for undergraduate students. However, it is unclear how students understand analogies that are presented to them, and more importantly, how students understand and create their own analogies. In this paper, we present a case study of four students as they reason analogically about several structures in abstract algebra. In particular, we expand on the notion of structure sense to include a wider range of structures in advanced mathematics and attend to each students’ analogical structure sense associated with each structure. Findings suggest that although students may possess a strong structure sense for group-theoretic structures, it is not necessarily the case that they possess a comparatively strong analogical sense of structure for ring-theoretic structures. In addition, those students with weaker senses of structure for group-theoretic structures are still able express productive reasoning about ring-theoretic analogies. Implications for future research and instructional practice are discussed.

This study contributes to understanding the influence of different problem posing tasks on the performance of in-service teachers in posing important and worthwhile mathematical problems. The problem posing tasks pertain to the part-whole concept of fraction which presents ongoing challenges for teachers and students. The study sample was comprised of 40 in-service primary school teachers who completed an electronic problem posing test. The problem posing tasks included different problem situations and prompts that addressed: (a) four types of problem posing processes (editing, selecting, comprehending, and translating), and (b) four levels of complexity (uni-structural, multi-structural, relational, extended abstract). The results suggested that in-service teachers’ performance is mainly influenced by the process involved in a problem posing task, being higher in problem situations that are more closed structured compared to more open structured. The level of complexity was not found to influence in-service teachers’ performance.

We report a case study of scaling a research-based curriculum and professional development innovation. We describe the Pathways Precalculus Curriculum and Professional Development (PPCPD) project and provide an overview of its development and components. In doing so, we detail how research informed its development and refinement, illustrate why we claim the PPCPD innovation is research-based, and document ways in which it is educative for both instructors and students. We describe results from a case study in which the PPCPD scaled to 13 sites that piloted the innovation, 12 of which locally scaled the innovation and attempted to sustain its use. We report findings from survey and interview data that reveal key variables that led to the sites’ sustaining or not sustaining the PPCPD innovation. We further highlight the importance of conceptualizing curricular scaling as an opportunity for continuous learning among the project leaders, local leaders, and precalculus instructors during all phases (considering, piloting, locally scaling, and sustaining) of the PPCPD.

Despite its almost four-decade history, research remains in the early stages of understanding the phenomenon of mathematical problem posing. In particular, the quality of problems created by beginning posers has been recognized as a persistent challenge. In this conceptual paper, I endorse an approach that identifies posing and solving as co-emergent components of steps learners take in a mathematically problematic situation. I further argue that some of the initial responses to the situation may constitute productive ingredients for creating problems that are personally meaningful and interesting to the learners. Drawing on the literature, I offer two principles through which the process of transitioning from initial responses to fully-fledged problems can be supported: making the didactical contract of the problem-posing activity transparent and immersing learners in socio-mathematical settings that are conducive to “good” problems. The approach is illustrated with fragments from a workshop for in-service teachers. The concluding discussion focuses on how the presented approach addresses some common issues in problem posing research.

Researchers have identified three stages of units coordination that influence a range of domains of student reasoning. The primary foci of this research have been students’ reasoning in discrete, non-combinatorial whole number contexts, and with fractions, ratios, proportions, and rates represented using length quantities. This study extends this prior work by examining connections between eight high school students’ combinatorial reasoning and their representation of this reasoning using 3-D arrays. All students in the study were at stage 3 of units coordination. Findings include differentiation between two student groups: one group had interiorized three-levels-of-units, but had not interiorized four-levels-of-units; and the other group had interiorized four-levels-of-units. This differentiation was coordinated with differences in how they reasoned to produce 3-D arrays. The findings from the study indicate how combinatorics problems can support quantitative reasoning, where combinatorial and quantitative reasoning are framed as a foundation for algebraic reasoning.

Signified by a familiar phrase, teaching to the test, the effect of high-stakes exams on mathematics instruction is widely accepted. It is, however, largely unknown how the discourse of a high-stakes exam is brought into everyday classroom interaction and shapes ways of doing mathematics in situ. This study examines details of social interaction in one Advanced Placement (AP) Calculus classroom, in which preparing for a high-stakes exam—the AP Calculus Exam—is highly relevant. By applying positioning theory and conversation analysis, the analysis shows that the discourse of the AP Exam shapes a hierarchical authority relation on the basis of what is believed to be on the AP Exam at the cost of the diminished authority of students. This study suggests further examination of the role of high-stakes exams in classroom settings and alternative perspectives on instructional supports for AP Calculus teachers and students.

Current research and curriculum documents from several countries have highlighted the importance of problem posing. However, some elements of this activity in mathematics classes still need to be further explored and understood. The objective of this article is to investigate how situations and problem-posing prompts affect teaching through problem posing. Three teaching cases are presented and analyzed, including instructional techniques guiding teaching through problem posing. The results indicate that teachers must align the problem-posing task with the intended goals for the lesson, analyzing how the prompt can enhance or limit particular aspects of the problems posed by the students.

In light of the growing interest in research examining problem posing, this study explores an under-researched aspect – group problem posing. It examines the role of problem situations and prompts, introduced within a problem posing intervention, on the nature of the problems posed by groups of pre-service elementary teachers for children in two 4th grade classes. The impact of the group on the problem posing process was also explored. The findings reveal improvements in the quality of the problems posed due to a number of factors including the use of textbook problems as initial problem situations that supported problem reformulation, the constructive role of prompts to focus attention on desirable problem features and the valuable role of feedback in determining problem aptness. A challenge when problem posing was difficulty coordinating attention to multiple problem features, however, this was mediated by the affordance of working in groups which served to share the responsibility for, and enhance, problem quality alongside the use of the F-PosE problem posing framework which focused attention on desirable problem features.

In problem-posing research, the influence of task variables on problem-posing outcomes is a relatively new endeavor. To systematically vary task variables, we designed eight problem-posing tasks by crossing two problem-posing situations (unstructured vs. structured) with two problem-posing prompts (open vs. closed) and two mathematical contexts (patterns vs. geometry). Using this design, we investigated the influence of these task variables on (1) creative problem-posing performance, (2) problem-posing self-efficacy, and (3) the relationship between self-efficacy and creative problem-posing performance in 187 pre-service teachers. The analyses show that (1) the influence of the situation and prompt is small and topic-specific, (2) that self-efficacy is significantly lower in unstructured situations with an open prompt than in the other tasks, and (3) that creative problem-posing performance and self-efficacy are correlated negatively. The findings imply a need for more detailed investigations regarding the influence on creative problem-posing performance and for which subjects it is relevant.