The aim of this article is to advise readers that natural numbers may be introduced as ordinal numbers or cardinal numbers and that there is an ongoing discussion about which come first. In addition, through several examples, the authors demonstrate that in the process of answering the question "How many?" one may, if convenient, use…

Learning the concept of fractions is among the most challenging topics in school mathematics. One of the main sources of difficulties in learning fractions is related to "natural number bias" (Van Hoof, Verschaffel & Van Dooren, 2015). Applying properties of the natural numbers incorrectly in situations involving rational numbers can…

In this article, I argue that the common practice across many school mathematics curricula of using a variety of different representations of number may diminish the coherence of mathematics for students. Instead, I advocate prioritising a single representation of number (the number line) and applying this repeatedly across diverse content areas.…

In this article, I describe a research/teaching experience I undertook with a class of 5-year-old children in Malta. The topic was subtraction on the number line. I interpret the teaching/learning process through a semiotic perspective. In particular, I highlight the role played by the gesture of forming "frog jumps" on the number line.…

Rather than describing the challenges of integer learning in terms of a transition from positive to negative numbers, we have arrived at a different perspective: We view students as inhabiting distinct mathematical worlds consisting of particular types of numbers (as construed by the students). These worlds distinguish and illuminate students'…

Multiplication can be presented to students through different models, each one with its pros and cons. In this contribution we focus on the repeated sum and the array model to investigate the relations between the two models and those between them and multiplication properties. Formal counterparts are presented. Taking both a mathematical and…

In this article, we explore Whitehead's claim that experience is fundamentally affective. We do so in the context of an Italian primary school classroom featuring the use of the multitouch application "TouchCounts." We study the way in which the children and the iPad take each other up and the meanings of number and arithmetic that…

"Where Mathematics Comes From" (Lakoff & Núñez 2000) proposed that mathematical concepts such as arithmetic and counting are constructed cognitively from embodied metaphors of actions on physical objects, and four actions, or 'grounding metaphors' in particular: collecting, stepping, constructing and measuring. This article argues…

How do students cope with and make sense of polysemy in mathematics? Zazkis (1998) tackled these questions in the case of 'divisor' and 'quotient'. When requested to determine the quotient in the division of 12 by 5, some of her pre-service teachers operated in the domain of integers and argued for 2, while others adhered to rational numbers and…

In this article, which was first published in 1991, the late Sandy Dawson, discusses aspects of a Lakatosian approach to mathematics teaching. The ideas are illustrated with examples from three teaching situations: making conjectures about the next number in a sequence; making conjectures about the internal angles in a triangle using Logo; and…

In this article, I present the notion of a set-oriented perspective for solving counting problems that emerged during task-based interviews with postsecondary students. Framing the findings within Harel's "ways of thinking", I argue that students may benefit from this perspective, in which they view attending to sets of outcomes as…

One of the challenges of learning ratio concepts is that it involves intensive quantities, a type of quantity that is more conceptually demanding than those that are evaluated by counting or measuring (extensive quantities). In this paper, we engage in an exploration of the possibility of developing reasoning about intensive quantities during the…

I explore the impact of ambiguous referral to the unit on understanding of decimal and fraction operations during episodes in two different mathematics courses for pre-service teachers (PSTs). In one classroom, the instructor introduces a rectangular area diagram to help the PSTs visualize decimal multiplication. A transcript from this classroom…

Some pedagogical problems in Chinese numeration are described. They involve the teaching and learning of how to speak numerals with fluency in Chinese, using Hindu-Arabic written numbers. An alternative approach which stresses regularity is proposed. (MNS)

Several ways to present two theorems (concerning a square matrix and a property of prime numbers) are demonstrated. One way for each theorem is more stimulating, better setting the stage for the proofs. Several methods of presenting proofs are illustrated, with the outcomes considered from the learner's viewpoint. (MNS)