Journal of Mathematical Behavior最新文献

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.

In this conceptual paper, we explore a framework of reasoning practices for decision-making in context. We extend prior work related to socioscientific issues (SSI) (Sadler et al., 2007) to consider the applicability of this framework to sociomathematical issues, specifically using the context of fairness in political (re)districting. We illustrate the usefulness of the SSI framework for sociomathematical issues drawing on student work and reflections from two undergraduate courses. We conclude by suggesting adjustments to reasoning practices of the SSI framework that might uniquely reflect the nature of sociomathematical reasoning. We discuss implications of our findings for conceptualizing reasoning practices that include mathematical perspectives within social and political contexts and for mathematics education generally.

Children are highly inclined to attend to the properties of numbers, operations and equality when given helpful tools for doing so. Our aim was to investigate early algebraic thinking with the compensation property of equality for subtraction. We provided 22 (9–11-year-old) students with physical blocks for building vertical towers and conducted individual interviews with them as they completed a sequence of 15 tasks involving subtraction as difference using concrete, numeric, and symbolic representations. Relational thinking was evidenced across a range of subtraction calculation skill levels. Those students who could use both indirect addition and take-away strategies flexibly, depending on the size of the numbers involved, were more likely to evidence attention to generality with symbolic equations. The shift to symbolic equations elicited some students’ productive attempts to connect subtraction as difference and subtraction as take way but seemed to hinder others by provoking a return to take away calculations rather than relational thinking strategies.

Explaining the connections of multiple representations can enhance students’ conceptual understanding (e.g., in the bottle-filling task for functional graphs). But it poses high discursive demands that need to be further unpacked. The design research study qualitatively investigates the potentials and demands that fourteen second-language learners face when explaining the connection between functional graphs and filling glasses. The qualitative analysis of students’ pathways towards good explanations identifies (a) demands to construct a mental contextual representation of the filling process, (b) demands to unpack the holistic perspective into more refined concept elements of covariation and correspondence approaches, (c) highly intertwined demands to identify the relevant variables in view. For each of these underlying demands, we identify scaffolds to enable students – even recent second-language learners – to engage in mathematically and discursively demanding practices and to enable teachers to support them.

Research on covariational reasoning has continued to evolve as researchers learn more about how students coordinate two (or more) quantities’ values as covarying. In this study, I examine the connection between students’ covariational reasoning and how they interpret the value of a rate of change. The findings suggest that attending to students’ quantifications of a rate of change can provide insight into their covariational reasoning and how we might better support students in reasoning at higher levels. Additionally, this manuscript provides an update to the Carlson et al. (2002) Covariation Framework that includes two additional categories of student reasoning and an additional dimension that describes students’ interpretation of a rate value at each level of the framework.

This study presents a specific case of how a teacher in China learned to teach with problem posing through a collaborative, iterative design process with a researcher. Supported by a networked improvement community, at every step of the journey that they undertook, they partnered to design, deliver, and revise a mathematics lesson that fostered students’ learning through problem posing. A detailed travelogue of their journey serves to document what research on teaching through mathematical problem posing can look like and how the teacher learned to teach using this novel approach. We explore the utility of the 3H (head, heart, and hands) model as a powerful way to think about holistic, transformative teacher learning. In addition, we consider aspects of the networked improvement community in which the teacher–researcher partnership operated that enabled capacity for sustaining this kind of effort to change practice.

Problem posing (PP) has been found to contribute to teachers’ mathematical pedagogical knowledge. However, little is known about what and how teachers learn when engaged in continuous iterative PP. We use the variation theory of learning to conceptualize what and how teachers learn during iterative PP, illustrating these processes via a case study. The main argument is that what teachers learn from engaging in iterative PP are different task variables we refer to as “dimensions of possible variation.” Awareness of these dimensions allows teachers to skillfully generate new problems or re-formulate previously posed ones to achieve desired pedagogical goals. We show how, during a collaborative design process with the PD coordinator, a teacher-designer became aware of some new-to-her dimensions and developed corresponding techniques for diversifying tasks. These awarenesses were still evident in an interview six months after the end of the PD. Recommendations for teacher educators are suggested.

This paper reports the results from a set of exploratory teaching interviews in which students constructed individualized algebraic expressions (called personal expressions) to describe their meanings for partial sums. Our analysis focused on one student, Emily, who constructed two distinct personal expressions for partial sums, one novel and one based on her image of summation notation. Emily created her novel expression to denote the process of generating the summands to compute the value of a partial sum. Emily adopted summation notation to describe the value of the partial sum. After reflecting on her inscription for a series’ general term of summation, Emily constructed a single expression to describe either the process of computing an arbitrary partial sum or the value of the sum itself. Using Emily’s story, we propose three categories for students’ coordination of their meanings for partial sums with a corresponding representation.