The problem of adverse effects of prior knowledge in mathematics learning has been amply documented and theorized by mathematics educators as well as cognitive/developmental psychologists. This problem emerges when students' prior knowledge about a mathematical notion comes in contrast with new information coming from instruction, giving rise to…

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 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…

It is frequently surprising to new teachers (and even those with experience) when they find that not only do some children need to recount the group they have just counted to be assured of the total but also that this need seems to be resistant to intervention. Although moving from "counting-all" to "count-on" is sometimes…

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…

Reciprocal triangular numbers have appeared in series since the very first infinite series were summed. Here we attack a number of subseries of the reciprocal triangular numbers by methodically expressing them as integrals.

The purpose of this article is to share a new approach for introducing students to the definition and standard examples of Abelian groups. The definition of an Abelian group is revised to include six axioms. A bullseye provides a way to visualize elementary examples and non-examples of Abelian groups. An activity based on the game of Jenga is used…

We use the hyperbolic cotangent function to deduce another proof of Euler's formula for ?(2n).

This paper was originally presented at the 25th Biennial Conference of AAMT, 'Capital Maths', and published in the Conference Proceedings. The author discusses the development and beginning implementation of a detailed sequence of concepts, with the intention of supporting teachers to support students to understand place value.

The effectiveness of an individual, conceptual instruction-based tuition program delivered by personal videoconferencing (PVC), compared to face-to-face (FtF) delivery for 30 upper primary and middle school students with mathematical learning difficulties (MLD), is described in this article. The experimental intervention targeted number sense,…

Adult neuroimaging studies have demonstrated dissociable neural activation patterns in the visual cortex in response to letters (Latin alphabet) and numbers (Arabic numerals), which suggest a strong experiential influence of reading and mathematics on the human visual system. Here, developmental trajectories in the event-related potential (ERP)…

Humans and other animals are able to make rough estimations of quantities using what has been termed the "approximate number system" (ANS). Much evidence suggests that sensitivity to numerosity correlates with symbolic math capacity, leading to the suggestion that the ANS may serve as a start-up tool to develop symbolic math. Many…

Background: Several studies have shown the influence of mathematical knowledge on both individual opportunities and chances for a self-determined and prosperous life as well as the welfare of nations. Against this background, the contents of maths education in the foundation phase as well as the way in which it is conveyed gain importance. While…

Curiosity about student thinking lies at the heart of great mathematics teaching. When teachers are curious about and grounded in their students' thinking, their proficiency at eliciting, interpreting, and responding to student ideas increases (NCTM 2014). Coaching conversations that focus on students' thinking can support teachers in developing…

The study reported on in this chapter describes the justifications that elementary and middle school prospective teachers (PTs) made as they examined the temperature story that a Grade 5 student posed for an integer subtraction number sentence. The ways that the PTs made sense of the student's story that used integer subtraction as distance are…