Educational Studies in Mathematics最新文献

Using Balacheff’s (2013) model of conceptions, we inferred potential conceptions in three examples presented in the spanning sets section of an interactive linear algebra textbook. An analysis of student responses to two similar reading questions revealed additional strategies that students used to decide whether a vector was in the spanning set of a given set of vectors. An analysis of the correctness of the application of these strategies provides a more nuanced understanding of student responses that might be more useful for instructors than simply classifying the responses as right or wrong. These findings add to our knowledge of the textbook’s presentation of span and student understanding of span. We discuss implications for research and practice.

Efficient recognition of geometric shapes is an important aspect of proficiency in geometry. Building theoretically on the cultural-historical approach enriched by the physiology of activity, we investigate theoretical perception in geometry—the ability to recognize conceptual geometric aspects of visual figures. Aiming to understand the development of theoretical perception, we investigate how sensory-motor processes of eye movements differ between adults and children when perceiving geometric figures. In an empirical study, we explored the variety of perceptual strategies used by first-grade students and compared them with the adults’ perception. The results reveal the contraction of eye movements: with growing expertise, foveal analysis—namely, an inspection of the figures by directing the gaze to their parts—is substituted by extrafoveal analysis—namely, perceiving without looking directly. The variety of the observed children’s perceptual strategies demonstrates that theoretical perception of different figures is heterogeneous. From the suggested theoretical perspective, the direct foveal inspection of particular figures is critical for the development of general anticipatory images of geometric shapes. Our theoretical analysis and empirical findings lead to distinguishing several functions of sensory-motor processes in theoretical perception in geometry. Those functions include positioning the retina in the best way for the comparison of sensory feedback with the geometric shape’s anticipatory image, advancing an anticipatory image based on visual experience, and regulating covert attention. All of these functions need to be taken into account when interpreting the results of eye-tracking studies in mathematics education research. Notably, our research highlights the limitations of the eye-mind hypothesis: direct fixations on a figure are not always needed for its theoretical perception and, the other way around, a fixation position may indicate the comparison of a broad extrafoveal region with an anticipatory image.

Social institutions function not only by reproducing specific practices but also by reproducing discourses endowing such practices with meaning. The latter in turn is related to the development of the identities or subjectivities of those who live and thrive within such institutions. Meaning and subjectivity are therefore significant sociological categories involved in the functioning of complex social phenomena such as that of mathematical instruction. The present paper provides a discursive analysis centered on these categories of the influential OECD’s PISA mathematics frameworks. As we shall see, meaning as articulated by the OECD primarily stresses the utilitarian value of mathematics to individuals and to society at large. Furthermore, molding students’ subjectivities towards endorsing such articulation of meaning is emphasized as an educational objective, either explicitly or implicitly, as connected to the OECD’s definition of mathematical literacy. Therefore, the OECD’s discourses do not only serve to reproduce the type of mathematical instruction implied in the organization’s services concerning education, but also concomitantly provide a potentially most effective educational technology through which the demand of these very services may be reproduced.

Rational numbers, such as fractions and decimals, are harder to understand than natural numbers. Moreover, individuals struggle with fractions more than with decimals. The present study sought to disentangle the extent to which two potential sources of difficulty affect secondary-school students’ numerical magnitude understanding: number type (natural vs. rational) and structure of the notation system (place-value-based vs. non-place-value-based). To do so, a 2 (number type) × 2 (structure of the notation system) within-subjects design was created in which 61 secondary-school students estimated the position of four notations on a number line: natural numbers (e.g., 214 on a 0–1000 number line), decimals (e.g., 0.214 on a 0–1 number line), fractions (e.g., 3/14 on a 0–1 number line), and separated fractions (3 on a 0–14 number line). In addition to response times and error rates, eye tracking captured students’ on-line solution process. Students had slower response times and higher error rates for fractions than the other notations. Eye tracking revealed that participants encoded fractions longer than the other notations. Also, the structure of the notation system influenced participants’ eye movement behavior in the endpoint of the number line more than number type. Overall, our findings suggest that when a notation contains both sources of difficulty (i.e., rational and non-place-value-based, like fractions), this contributes to a worse understanding of its numerical magnitude than when it contains only one (i.e., natural but non-place-value-based, like separated fractions, or place-value-based but rational, like decimals) or neither (i.e., natural and place-value-based, like natural numbers) of these sources of difficulty.

This article aims to connect two research areas by using positioning theory to review the literature on talk moves, teacher interactions, and discourse patterns in mathematics education. First, a conceptual review identified 44 articles with 94 concepts describing interactions and discourse patterns. Similar concepts were grouped in a process that developed five categories, each describing one teacher position (a teacher who tells, a teacher who supports, a teacher who uses students’ ideas to create learning, a teacher who orchestrates, and a teacher who participates). Related to each position, we describe rights, duties, and communication acts. We suggest that these five teacher positions represent three transcendent storylines (teachers are providers of insight, teachers are facilitators of learning, and teachers are participants in learning). Using positioning theory enables us to understand the underlying powers that shape the classroom in relation to transcendent storylines, rights, and duties. We use this to explore what the implications are of these storylines and positions for equity and access to important mathematical ideas. This article contributes to our understanding of the complexity of classroom interactions and how transcendent storylines might play a role in subverting or promoting particular classroom communication patterns.

Abstract

In this research, our objective is to characterize the problem-solving procedures of primary and lower secondary students when they solve problems in real class conditions. To do so, we rely first on the concept of heuristics. As this term is very polysemic, we exploit the definition proposed by Rott (2014) to develop a coding manual and thus analyze students’ procedures. Then, we interpret the results of these analyses in a qualitative way by mobilizing the concept of semantic space (Poitrenaud, 1998). This detailed analysis of students’ procedures is made possible by collecting audiovisual data as close as possible to the students’ work using an action camera mounted on the students’ heads. We thus succeed in highlighting three different investigation profiles that we have named explorer, butterfly, and prospector. Our first results tend to show a correlation with these profiles and the success in problem-solving, yet this would need more investigation.

Using a mixed methods approach, we explore a relationship between students’ graph reasoning and graph selection via a fully online assessment. Our population includes 673 students enrolled in college algebra, an introductory undergraduate mathematics course, across four U.S. postsecondary institutions. The assessment is accessible on computers, tablets, and mobile phones. There are six items; for each, students are to view a video animation of a dynamic situation (e.g., a toy car moving along a square track), declare their understanding of the situation, select a Cartesian graph to represent a relationship between given attributes in the situation, and enter text to explain their graph choice. To theorize students’ graph reasoning, we draw on Thompson’s theory of quantitative reasoning, which explains students’ conceptions of attributes as being possible to measure. To code students’ written responses, we appeal to Johnson and colleagues’ graph reasoning framework, which distinguishes students’ quantitative reasoning about one or more attributes capable of varying (Covariation, Variation) from students’ reasoning about observable elements in a situation (Motion, Iconic). Quantitizing those qualitative codes, we examine connections between the latent variables of students’ graph reasoning and graph selection. Using structural equation modeling, we report a significant finding: Students’ graph reasoning explains 40% of the variance in their graph selection (standardized regression weight is 0.64, p < 0.001). Furthermore, our results demonstrate that students’ quantitative forms of graph reasoning (i.e., variational and covariational reasoning) influence the accuracy of their graph selection.

This paper has two main goals: first, to analyze current research related with transition to identify the foci being produced and reproduced by research through a literature review and, second, to map the connections between circulating discourses to unfold the discursive network that supports them via an intensive reading. Specifically, we are interested in how social and economic factors, educational background, and cognitive development impact the transition process and the documented potential challenges students may face. In this fashion, the questions that guide the development of the paper are as follows: What are the dominant narratives in mathematics education research about the transition from school to university mathematics? How do dominant narratives entangle particular rationalities to configure a discursive network about the transition from school to university mathematics? As a result, we identify three dominant narratives entangled, which shape a system of reason that regulates what is possible to do, act, and think.

Instructors manage several tensions as they engage students in defining, conjecturing, and proving, including building on students’ contributions while maintaining the integrity of certain mathematical norms. This paper presents a case study of a teacher-researcher who was particularly skilled in balancing these tensions in a laboratory setting. We introduce sociomathematical scaffolding, which refers to the scaffolding of normative aspects for mathematical discourse. We found that the teacher-researcher’s sociomathematical scaffolding entailed inquiring into the students’ intended meaning of their draft and then supporting students in revising their draft to adhere to mathematical norms. We illustrate this pattern in three episodes in which the teacher-researcher supported a pair of students to revise their drafted (1) definition of unbounded above sequences, (2) conjecture of the Archimedean Property, and (3) proof by contraction of the Archimedean Property.