Modeling is an effective tool to help students access mathematical concepts. Finding a math teacher who has not drawn a fraction bar or pie chart on the board would be difficult, as would finding students who have not been asked to draw models and represent numbers in different ways. In this article, the authors will discuss: (1) the properties of…

Although the early development of children's number sense is a strong predictor of their later mathematics achievements, it has been overlooked in primary schools in Malaysia. Mainly attributable to underdeveloped number sense of Malaysian primary and secondary school children, their inability to handle simple mathematics tasks, which require the…

In their ongoing professional development project, the authors had an opportunity to work with a group of teachers who were particularly insightful. These teachers shared written reflections about their thinking, prior experiences, and changing self-awareness of their mathematics practice. In one session, teachers were provided the…

I propose that students pay more attention to the approximations that lead to integral formulas when they study applications of the definite integral. The goal is to focus students on the underlying concepts and the definition of the definite integral--and to steer them away from memorizing formulas. To address this goal, I have written some…

In this article, we revisit the concept of strong differentiability of real functions of one variable, underlying the concept of differentiability. Our discussion is guided by the Straddle Lemma, which plays a key role in this context. The proofs of the results presented are designed to meet a young audience in mathematics, typical of students in…

Background: Teacher knowledge continues to be a topic of debate in Australasia and in other parts of the world. There have been many attempts by mathematics educators and researchers to define the knowledge needed by teachers to teach mathematics effectively. A plethora of terms, such as mathematical content knowledge, pedagogical content…

A proof is a connected sequence of assertions that includes a set of accepted statements, forms of reasoning and modes of representing arguments. Assuming reasoning to be central to proving and aiming to develop knowledge about how teacher actions may promote students' mathematical reasoning, we conduct design research where whole-class…

This study investigates 3 hypotheses about proof-based mathematics instruction: (a) that lectures include informal content (ways of thinking and reasoning about advanced mathematics that are not captured by formal symbolic statements), (b) that informal content is usually presented orally but not written on the board, and (c) that students do not…

This paper studies the cognitive obstacles related to one aspect of mathematization in physics problem-solving, namely, what might be called "structuring for mathematization," where the problem situation is structured in such a way that a translation to a mathematical universe can be done. We report the results of an analysis of four…

The purpose of this article is to draw the attention of mathematics education researchers to a relatively new semantic theory called inferentialism, as developed by the philosopher Robert Brandom. Inferentialism is a semantic theory which explains concept formation in terms of the inferences individuals make in the context of an intersubjective…

Negative numbers are among the first formalizations students encounter in their mathematics learning that clearly differ from out-of-school experiences. What has not sufficiently been addressed in previous research is the question of how students draw on their prior experiences when reasoning on negative numbers and how they infer from these…

In this article, Jason Taff shares an approach that he presented to advanced seventh-grade prealgebra students. He begins by summarizing some of the shortcomings of equating the order of operations concept with the PEMDAS (often rendered mnemonically as "Please Excuse My Dear Aunt Sally") procedure with the hope of helping teachers at…

This study focuses on how students in different postsecondary mathematics courses perform on domain and range tasks regarding graphs of functions. Students often seem to focus on notable aspects of a graph and fail to see the graph in its entirety. Many students struggled with piecewise functions, especially those involving horizontal segments.…

English schoolchildren do not fare well in international comparisons of mathematical, particularly algebraic, ability. I believe that this is at least partly due to algebra being traditionally taught using only symbols, ignoring the enactive-iconic-symbolic trajectory described by Jerome Bruner, which says that experiencing the way in which…

The study aims to analyze, in terms of content and process, the math talks between the mother and child during the shared book reading of the illustrated storybooks with math contents. The study group under the research is comprised of nine pairs of mother and child. The process of the study 18 videos which were recorded when two storybooks with…