Whole number arithmetic is the foundation of higher mathematics and a core part of elementary mathematics. Awareness of pattern and underlying problem structure promote the learning of whole number arithmetic. A growing consensus has emerged on the necessity to provide students with the opportunity to engage in algebraic reasoning earlier in their…

This article begins with a theoretical discussion of the characteristics that a task should feature to be regarded as a mathematics problem suitable for pre-primary students. Those considerations are followed by a report of a classroom experience in which three problems involving quotative or partitive division were posed to pre-primary school…

This is the second of a two-part article that presents a theory of unit construction and coordination that underlies radical constructivist empirical studies of student learning ranging from young students' counting strategies to high school students' algebraic reasoning. In Part I, I discussed the formation of arithmetical units and composite…

A first-grade teacher has students use their hands and fingers to engage in and develop understanding of counting, to combine groups to facilitate counting by fives and tens, and to describe their findings using words and equations.

There is a renewed debate about whether educated adults solve simple addition problems (e.g., 2 + 3) by direct fact retrieval or by fast, automatic counting-based procedures. Recent research testing adults' simple addition and multiplication showed that a 150-ms preview of the operator (+ or ×) facilitated addition, but not multiplication,…

Assessment is a notorious source of preoccupation for faculty and university governing bodies, especially when an institution initiates curricular reforms which shift the programme learning outcomes for knowledge to competencies. One obstacle to acceptance arises from a culture of quantitative assessment (often represented by a single mark), which…

This paper presents an experiment that attempts to mobilise an arithmetic-algebraic way of thinking in order to articulate between arithmetic thinking and the early algebraic thinking, which is considered a prelude to algebraic thinking. In the process of building this latter way of thinking, researchers analysed pupils' spontaneous production…

The ability for students to understand numbers and other mathematical symbols is a crucial part of success in mathematics. Accordingly, it is important for researchers to understand the nature of symbolic number processing -- the connections between a symbol or collection of symbols that convey numerical information (e.g., Arabic digits,…

This study investigated a highly accomplished third-grade teacher's noticing of students' mathematical thinking as she taught multiplication and division. Through an innovative method, which allowed for documenting in-the-moment teacher noticing, the author was able to explore teacher noticing and reflective practices in the context of classroom…

Progressing from additive to multiplicative thinking is critical for the development of middle school students' proportional reasoning abilities. Yet, many middle school mathematics teachers lack a thorough understanding of additive versus multiplicative situations. This article describes a sequence of instructional activities used to develop the…

There is an ongoing debate over the psychophysical functions that best fit human data from numerical estimation tasks. To test whether one psychophysical function could account for data across diverse tasks, we examined 40 kindergartners, 38 first graders, 40 second graders and 40 adults' estimates using two fully crossed 2 × 2 designs, crossing…

In this 3-year longitudinal study, we examined the interplay of children's motivational orientations, spontaneous focusing on numerosity (SFON) and number sequence and arithmetical skills from kindergarten (at age 6) to grade 1 (at age 7). We also examined the direct and indirect contributions of motivation and SFON to the acquisition of…

The Department of Basic Education in South Africa has identified factorisation as a problem area in Mathematics teaching. Learners in earlier grades are exposed to mathematical tasks, involving concepts, such as factors of integers, fractions, equivalent fractions and prime numbers, that are easier to solve when the divisibility rules are applied.…

Digit comparison is strongly related to individual differences in children's arithmetic ability. Why this is the case, however, remains unclear to date. Therefore, we investigated the relative contribution of three possible cognitive mechanisms in first and second graders' digit comparison performance: digit identification, digit--number word…

Research has demonstrated that providing labels helps children notice key features of examples. Much less is known about how different labels impact children's ability to make inferences about the structure underlying mathematical notation. We tested the impact of labeling decimals such as 0.34 using formal place-value labels ("3 tenths and 4…