Journal of Mathematical Behavior最新文献

Early algebra can prepare elementary students for the transition they will need to make from arithmetic and algebra. Although teacher preparation programs emphasize the teaching of early algebra, research on how to prepare elementary pre-service teachers (PSTs) to teach early algebra is still scarce. The replication study reported in this article was a conceptual replication study designed to examine Iranian PSTs’ reasoning about pre-symbolic early algebra by looking at what was more, somewhat, and less challenging. The aims of the replication study aligned with the original study (Hohensee, 2017). Results from the replication study show that participating PSTs (N = 15) found the early algebra approach to variables and functions more challenging, indeterminable unknowns somewhat challenging, and equivalence and equations less challenging. We make comparisons with the original study, as well as offer implications and suggestions for preparing PSTs to teach early algebra.

Dynamic geometry software (DGS) has long been studied in mathematics education as a way for students to explore and interact with geometric objects and figures. Recent advances in Augmented Reality (AR) technologies that allow dynamic three-dimensional mathematical objects to appear in students’ environment as holograms have changed the nature of what is possible for a DGS, particularly with respect to embodiment. New forms of embodied interactions may arise in AR-based DGS, as students gesture and move their bodies through their environment, taking different perspectives to interact with these immersive shapes projected in three dimensions. In the present study, we examine videos of 28 high school students interacting with an AR-based version of the DGS GeoGebra, while wearing the Microsoft HoloLens 2 headsets. We document the novel kinds of embodied interactions that the AR environment affords, relating to (1) perspective and orientation, (2) scale, (3) three dimensions. Based on our analysis, we give important directions for future research on DGS and implications for the design of the next generation of holographic DGS.

Group isomorphism and homomorphism are core concepts in abstract algebra, and student understanding of isomorphism has received extensive attention in line with the centrality of this topic. However, limited work has directly examined student conceptions of homomorphism or what metaphors students use to express their thought processes while problem solving. Based on interviews with four students, we contrast two students who used predominantly formal definition and mapping-centered metaphors for homomorphism with two who additionally used sameness-centered metaphors and note that the usage or non-usage of sameness-centered metaphors was not indicative of successful problem solving. Implications include the alignment between students’ metaphors and those used in instruction, indicating the importance of attending to metaphors when teaching, and the importance of discussing what is intended by some sameness-based metaphors, such as operation-preservation.

This non-interventional study investigates the contribution of university mathematics to teaching high-school mathematics. Data sources included interviews with five teachers who taught high-school mathematics before, during, and after their academic mathematics studies. All teachers provided tangible examples of fundamental changes in instructional practices that they explicitly linked to new knowledge acquired in the academic mathematics courses. The domain of analysis in general, and the topics of integrals and derivatives in particular, were central in the teachers’ illustrations of changes they made in their teaching, although other mathematical topics and domains were also mentioned. The reported changes were mostly associated with emphasis on mathematical explanations, exposition of two key elements of the deductive structure of mathematics: definition and proof, an increased focus on formal mathematics, and portrayal of mathematics as a wide and varied discipline. The study results are discussed in light of the relevant literature.

Background

This comparative case study examined the use of math walks with middle grade youths and adult facilitators in an informal STEM learning space. Math walks are place-based walking tours where youths and facilitators critically examine and ask math-related questions about their environment.

Method

Drawing on situated theories of learning and frameworks for understanding group participation, we examined how facilitators constrained or supported youths’ mathematical thinking as they participated in math walks at the local zoo.

Results

Using interaction and stance analysis, we identified, analyzed, and compared three contrasting cases: In the first case, the facilitator may have overly constrained youths’ mathematical thinking by asking leading questions and not providing time for youths to discuss their personal interests. In the second case, the facilitator may have underly constrained youths’ mathematical thinking by allowing youths to ask too many new questions without refining or developing any one specific question. In the third case, the facilitator supported mathematical thinking by praising youths’ work, layering on mathematical terminology, and providing clear and actionable instructions for how youths could refine their mathematical questions.

Conclusions

Findings support efforts to understand how adult facilitators can support youths in seeing mathematics within and asking mathematical questions about the world around them.

The literature has highlighted the significant role of definitions and defining in mathematics learning and teaching. Furthermore, non-prototypical figures are particularly important when teaching geometry, but teachers and pre-service teachers still have problems defining them. For these reasons, we investigated whether there were differences in the way that pre-service mathematics teachers constructed and selected definitions for prototypical and non-prototypical solids. In particular, the commognitive framework was employed to investigate the differences in the discourse of 33 pre-service secondary-school teachers when constructing and selecting definitions in task situations that involved prototypical and non-prototypical solids. Moreover, we studied if some commognitive conflicts appeared in task situations involving non-prototypical solids but not in similar task situations involving prototypical solids. The findings show some differences between the pre-service teachers’ discourses in both types of task situations. Additionally, some commognitive conflicts appeared only in task situations with non-prototypical solids. Lastly, we classified those commognitive conflicts.

Based on the theory of the Cognitive Structure of the Emotions (OCC), this article has a dual objective: 1) to document negative attitudes toward mathematics generated during the academic formation of Manuel, an adult who ended his school education 15 years before this investigation and today is a professional voice-over actor; and 2) to analyze the reflection of those attitudes in the emotions that emerged during the process of solving mathematical problems related to his work. Manuel was chosen from among ten voice-over professionals who, before the study began, manifested a strong aversion toward mathematics. Manuel was interviewed twice: once before, and then after solving the mathematical problems posed. Results indicate that his negative attitude toward mathematics was reflected in the emotions — frustration, fear, anger, and nervousness — he experienced while solving the problems.

This study is a qualitative case study that focuses on the instructional practice of a white woman teacher with three Black girls. We draw upon literature that describes racial storylines for Black girls to focus on how a white Woman teacher navigates racial storylines to enact instructional practices that are humanzing for Black girls. Specifically, we show how interactions the teacher has with three Black girls disrupt racial storylines about invisibility/hypervisibility, behavior, and ability. Implications of this work pushes back against the idea that Black girls are monolithic and illustrate practices in mathematics that support the success of Black girls.

Linguistic features as a task-related feature influence the difficulty of mathematical tasks. To reduce this influence (e.g., in testing situations), studies on linguistic simplification focus on modifying linguistic features. These studies show little or no effect on increasing test performance. An open question is whether a quantitative–exploratory approach with texts from a specific domain can be an additional model for reducing the linguistic influence on mathematical tasks. To answer this question, generalized linear mixed models were used to determine the effects of linguistic factors, the requirements of the items, and the effects of linguistic factors when differentiating the requirements of the items, while controlling for further person- and item-related effects. The results show that linguistic factors can have either a negative or positive influence on test performance. The findings indicate that for mathematics assessments and teaching, it might be essential to consider the influence of language factors and task requirements.

The Math-LIGHT program is directed at promoting literacy-rich mathematical instruction in middle school. A team of designers with different types of expertise pose Math-Light problems. We perform comparative analysis of problem-posing activities by experts with different types of expertise. We demonstrate that Activity Theory (Leontiev, 1978) is a powerful theoretical framework for the analysis of the structure of problem posing activity. Framed by activity theory we ask “Why?” questions to understand the main goals of posing problems; “What?” questions are directed at the characteristics of the PP process and PP products; and “How?” questions are aimed at identifying the tools used by the designers to fit the conditions in which the problems are implemented. We find that the three designers’ problem-posing activities are complimentary and suggest that the cooperative problem posing process is essential for posing problems that integrate different perspectives and thus allow more goals to be attained.