Journal of Mathematical Behavior最新文献

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.

Research has demonstrated that problem-posing and problem-solving mutually affect one another. However, the exact nature and full extent of this relationship requires detailed elaboration. This is especially true when problem-posing arises in order to facilitate problem-solving, such as during the investigation of an unfamiliar mathematical property or phenomenon. In this study, groups of participants used scripting to record their mathematical activity as they made conjectures and justified conclusions about sums of consecutive integers. We analyze the unprompted problem-posing found within these scripting journeys using three facets of a problem-posing framework: mathematical knowledge base, problem-posing heuristics, and individual considerations of aptness. Our analysis reveals how these aspects of problem-posing emerge within a mathematical investigation, how they are related to surrounding problem-solving, and the kinds of mathematical insights and realizations that act as catalysts to promote further problem-posing activity.

There is a need for further studies on students’ learning of Differential Equations (DEs), especially in advanced undergraduate and graduate courses. Research on the mathematical education of engineers shows a conflict between students’ demands for practical, contextualized pedagogies and the need for abstract reasoning and appropriate use of mathematical results. Few papers focus on engineering students’ interpretation of theorems and their use as tools in argumentation and problem-solving. This paper takes a sociocultural stance on learning and employs dialogical inquiry – a methodology rooted in Bakhtinian theory, newly developed for collaborative inquiry and qualitative data analysis – to investigate the meanings that senior engineering students made while working on a task designed to evaluate their understanding of Existence and Uniqueness Theorems (EUTs) of solutions of DEs. We identified two important epistemological disconnections that explain the difficulties that some of our students faced in making meaning of solutions of DEs and the EUT.

This study examines the mathematical activity involved in engaging with two tasks designed for introductory linear algebra: the Vector Unknown digital game and the pen-and-paper Magic Carpet Ride task. Five undergraduate students worked on both tasks, and we qualitatively analyzed their strategies using a modified version of a framework from prior literature. In the findings, we report on the seven distinct strategies seen in our data set. We found that while our participants did use some of the same strategies on both tasks, there were also certain strategies which were more characteristic of work on one task or the other. In our discussion, we consider how the design differences in the tasks may influence the strategy differences, and how our findings can be leveraged by instructors of linear algebra in selecting tasks. Finally, we conclude by discussing broader implications for mathematics education research in comparing game-based and non-game-based tasks.

Recognizing and describing children's mathematical thinking in humanizing ways, especially when students engage in confusion, productive struggle, and mistakes, is a complex and challenging process. This paper describes an exploratory, mixed-methods study about how elementary teacher candidates (TCs) describe children's thinking as a right to exercise and to value their humanity when learning mathematics. The study analyzed transcripts from 64 TCs' summative assessments, which consisted of mock parent-teacher conferences (MPTC). Findings suggest that TCs described children's confusion, productive struggle, and mistakes (RotL 1 and 2) as: a teacher's observation, an opportunity for students to correct or clarify their thinking, an opportunity for teachers to adjust instruction or provide support, and as a normal part of the learning process. More importantly, some TCs reassured children that learners have fundamental rights when learning mathematics, especially when feeling confused and claiming a mistake. Implications for research and teacher education are discussed. Keywords: Elementary, teacher education, mathematics, mixed methods, rehumanizing, Torres’ rights of the learner