In this article, the authors provide some insight into how teachers can potentially facilitate among students' representations and activities on mathematical tasks: between or among -- (1) student representations and the context; (2) multiple forms of student representations; and (3) processes in student representations. The authors use the…

In most mathematics textbooks, each lesson is followed by a set of homework problems in one of two ways -- blocked practice and mixed practice. Additionally, most mathematics textbooks rely on a common learning strategy called overlearning, that is, mastering a skill and continuing to practice this same skill. This study investigated the effects…

In undergraduate mathematics, deductive reasoning plays important roles in teaching and learning various ideas, and is primarily characterized by the concept of logical implication. This comes up whenever conditional statements are applied, i.e., one checks if a statement's hypotheses are satisfied and then makes inferences. In calculus, students…

The purpose of this study was firstly to investigate the mathematical metacognitive characteristics of high-efficiency students by comparing high-efficiency and low-efficiency students. Secondly, the aim was to explore the pathways of mathematical metacognition of efficient mathematics learners in mathematics achievement. In this study,…

The aim of the study reported in this article was to reveal how instruction, enriched with reasoning about students' thought processes and strategies, regulates prospective teacher's noticing skills in the context of whole number multiplication. In addition, the study examined the evidence of the prospective teacher's (Matt) knowledge that…

Possessing the mathematical competence to encounter the challenges under globalisation and the ubiquity of digital technology is an important goal of education. Herein, we propose a broad conceptual framework for understanding and developing mathematical competence. The framework comprises four non-overlapping but interlinked cores:…

Retrieval practice is a learning strategy that has repeatedly been found to have positive effects on memory and learning. However, studies indicate that students rarely use retrieval practice on a voluntary basis. The objective of the present study was to examine students' self-regulated use of retrieval practice, and to determine whether sex and…

Generalisation is a skill that enables learners to acquire knowledge in general, and mathematical knowledge in particular. It is a core aspect of algebraic thinking and, in particular, of functional thinking, as a type of algebraic thinking. Introducing primary school children to functional thinking fosters their ability to generalise, explain and…

In this paper, we explore how students' algebraic noticing's and explanations changed across a two-year period with the introduction of designed instructional material. The data in this report is drawn from n=53 Year 7-8 students' responses to a free-response assessment task across two different years. Analysis focused on how students noticed and…

A solid foundation in mathematics is beneficial not only for individuals' concurrent and later academic achievement, but also for future career success and financial stability. Rational number knowledge plays a pivotal role in mathematical success, yet it is often regarded as a challenging area to grasp comprehensively. The dissertation aims to…

Mathematics instruction can work to open or constrain children's opportunities to learn, and the busy, interactive nature of instruction makes it challenging for researchers to identify patterns in high-quality opportunities to learn. While researchers have documented how whole-group discussions and small-group work can contribute to such…

The development and use of learning trajectories is a body of research that has made enormous contributions to the field of mathematics education, offering insight into the teaching and learning of topics at all levels. Simultaneously, the work of building learning trajectories can benefit from explicitly adopting an anti-deficit stance,…

The current study used microanalytic interviews and behavioral traces to examine sequential phase relations among forethought, performance, and self-reflection processes, and to investigate the predictive influence of these processes on mathematics performance of 96 eighth grade students. Consistent with expectations, students' microanlaytic goals…

Examples are an indispensable part of mathematical thinking through analogy, learning and the teaching process. In this study, the strategies that high school students used in example generation activities in mathematics course; operational and conceptual knowledge levels were determined and their effect on example generation ability was examined.…

In the present study, we tested the hypotheses that: a) there are individual differences in secondary students' conceptual and procedural fraction knowledge, and b) these differences are predicted by students' approach (deep vs. surface) to mathematics learning. We used two instruments developed and evaluated for the purposes of the study which…