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,…

Despite the prominence of analogies in mathematics, little attention has been given to exploring students' processes of analogical reasoning, and even less research exists on revealing how students might be empowered to independently and productively reason by analogy to establish new (to them) mathematics. I argue that the lack of a cohesive…

This study aims at elaborating a well-established theoretical framework that distinguishes three modes of thinking in linear algebra: the analytic-arithmetic, the synthetic-geometric, and the analytic-structural mode. It describes and analyzes the bundle of signs produced by an engineering student during an interview, where she was asked to recall…

This paper aims to shed light on an overlooked but essential aspect of informal reasoning and its radical implication to mathematics education research: Decentralising mathematics. We start to problematise that previous studies on informal reasoning implicitly overfocus on what students infer. Based on Walton's distinction between reasoning and…

Since statistical inference is a probabilistic generalization about a population analyzed on the basis of a sample, inferential reasoning demands producing reasons ("statistical" and "contextual") to substantiate and validate generalizations. To convey an understanding of students' inferential reasoning, we present a…

This paper investigates how mathematics teachers describe their ethical decision making related to instructional practices, drawing on frameworks that incorporate both rational and non-rational approaches. We employed a multiple-case study method, selecting three teachers as cases through criterion sampling. Data were collected via four…

Definitions are an integral aspect of mathematics. In particular, they form the backbone of deductive reasoning and facilitate precision in mathematical communication. However, when multiple non-equivalent definitions for the same term exist, their ability to serve these purposes can be called into question. While ambiguity can be productive, the…

Findings on children's proportional reasoning abilities strongly vary across studies. This might be due to the different contexts that can be used in proportional problems: fair-sharing, mixtures, and probability. A review of the scientific literature suggests that the context of proportional problems may not only impact the difficulty of the…

It is well-known that a key to promoting students' mathematics learning is to provide opportunities for problem solving and reasoning, but also that maintaining such opportunities in student-teacher interaction is challenging for teachers. In particular, teachers need support for identifying students' specific difficulties, in order to select…

Past studies have documented some pre-service teachers' (PSTs) difficulties in reasoning about sampling variability. This study adds to the body of literature by investigating the ideas that PSTs employ in reasoning about sampling variability, and by conjecturing what is behind the difficulties especially during the contextuality episodes. This…

Teachers' questioning plays an essential role in shaping collective argumentative discourse. This paper demonstrated that rationality dimensions in teacher questions can be assessed by adapting Habermas' three components of rationality. By coordinating Habermas' construct with Toulmin's model for argumentation, this paper investigated how two…

We investigate how students make sense of irrational exponents. The data comprise 32 interviews with university students, which revolved around a task designed to examine students' sensemaking processes involved in predicting and subsequently interpreting the shape of the graph of f(x)=x[superscript square root of 2]. The task design and data…

Our study aims to determine how Habermas' construct of rationality can serve to identify and interpret the difficulties experienced by university students in the mathematical problem-solving process. To this end, a problem which required modelling and solving a differential equation was used. The problem-solving processes of university students…

Binary operations are one of the fundamental structures underlying our number and algebraic systems. Yet, researchers have often left their role implicit as they model student understanding of abstract structures. In this paper, we directly analyze students' perceptions of the general binary operation via a two-phase study consisting of task-based…

Sampling distributions are fundamental for statistical inference, yet their abstract nature poses challenges for students. This research investigates the development of high school students' conceptions of sampling distribution through informal significance tests with the aid of digital technology. The study focuses on how technological tools…