Journal of Mathematical Behavior最新文献

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.

This research focused on understanding the variables inherent in the design and implementation of a mathematical problem-posing task. We developed a single case study of a problem-posing lesson by an Early Childhood Education teacher in a classroom with 4- to 5-year-old children who were unfamiliar with such activities. The results of this study show the potential of considering five variables serving as critical points that pose dilemmas linked to the design and implementation of problem-posing tasks. We found that the task changed from its original design during implementation, implying that the choices the teacher made about the variables were not static and were strongly linked to the purpose of the problem-posing task as well as to the contextual characteristics of the early childhood classroom. This study provides a potentially useful framework for analyzing the design and implementation of problem-posing tasks as a dynamic process.

In this study we address two issues related to problem-posing tasks in teacher education: (i) the characterization of the specialized knowledge mobilized by prospective teachers when carrying out these tasks and (ii) the identification of the prospective teachers’ pedagogical intentions in making adaptations to textbook problems. We asked prospective teachers to outline their suggestions for transforming a multiplicative problem so as to “promote the understanding” of their potential pupils. We then carried out a content analysis of their responses using the Mathematics Teachers’ Specialized Knowledge model of teachers’ specialized knowledge and identified their pedagogical intentions by means of the constant comparison method. The results show that prospective primary teachers mobilized both mathematical and pedagogical content knowledge in their responses to the problem reformulation task. Further, four distinct pedagogical intentions emerged that drew on different interpretations of the task prompt, and this influenced the type of transformation the prospective primary teachers suggested and the knowledge they mobilized in their answers.

Problem posing—generating one’s own problems—is considered a powerful teaching approach for fostering students’ motivation such as their interest. However, research investigating the effects of task variables of self-generated problems on students’ interest is largely missing. In this contribution, we present a study with 105 ninth- and tenth-graders to address the question of whether the task variables modelling potential, assessed by openness and authenticity, or complexity of self-generated problems have an impact on students’ interest in solving them. Further, we investigated whether the effect of task variables of self-generated problems on students’ interest differed among students with different levels of mathematical competence. High modelling potential had a positive effect on interest in solving the problem for students with low mathematical competence, whereas it had a negative effect for those with high mathematical competence. However, complexity of self-generated problems did not affect students’ interest in solving problems.