Although numbers are universal, there are great differences between languages and cultures in terms of how they are represented. Numerical notation can influence number processing. Two well-known types of notational systems are sign-value, such as the Roman numeral system, and place-value systems, such as the Indo-Arabic numeral system. What is…

Understanding the concept of completeness for an ordered field is known to be difficult for many university mathematics students. We hypothesise that the variety of possible axioms of completeness for the set of real numbers is one of the sources of difficulties as is the lack of understanding of the "raison d'être" of these axioms. In…

Using the basic properties of the base-b representation of rational numbers, we will give an elementary proof of Gauss's lemma: "Every real root of a monic polynomial with integer coefficients is either an integer or irrational." The paper offers a new perspective in understanding the meaning of 'irrational numbers' from a deeper…

Humans count to indefinitely large numbers by recycling words from a finite list, and combining them using rules--for example, combining sixty with unit labels to generate sixty-one, sixty-two, and so on. Past experimental research has focused on children learning base-10 systems, and has reported that this rule learning process is highly…

Initial acquisition of the first symbolic numbers is measured with the Give a Number (GaN) task. According to the classic method, it is assumed that children who know only 1, 2, 3, or 4 in the GaN task, (termed separately one-, two-, three-, and four-knowers, or collectively subset-knowers) have only a limited conceptual understanding of numbers.…

The purpose of this study was to examine the ways advanced mathematics students define "number" and the degree to which their definitions extend to different number domains. Of particular interest for this study are learners' fundamental conceptions of number and the implications for learners' interpretations of complex numbers (a + bi).…

Sixty (35 girls and 25 boys) 9th-grade students' conceptual understanding of the number line was qualitatively assessed through verbal explanations and visual representations. The assessment included an open-ended question focused on students' number line descriptions and the explanations coalesced around six features: sequential ordering (i.e.,…

Understanding numerical magnitude is critical to the development of mathematics proficiency, and math interventions targeting magnitude knowledge have been shown to improve outcomes for students with math learning difficulties across grade levels. While recent studies have found that growth in magnitude knowledge mediates fractions intervention…

Understanding numerical magnitude is critical to the development of mathematics proficiency. Math interventions targeting magnitude knowledge have been shown to improve outcomes for students with math learning difficulties across grade levels. However, while recent studies have found that growth in magnitude knowledge mediates fractions…

Supporting children's understanding of the everyday, cultural use of written numerals is highly significant, as it is this understanding that gives meaning to classroom conversations on the purposes of written numbers. This paper presents findings from a phenomenographic study of the qualitatively different ways in which 3-5-year-old children…

Math performance is negatively related to math anxiety (MA), though MA may impact certain math skills more than others. We investigated whether the relation between MA and math performance is affected by task features, such as number type (e.g., fractions, whole numbers, percentages), number format (symbolic vs. nonsymbolic), and ratio component…

Acquiring robust semantic representations of numbers is crucial for math achievement. However, the learning stage where magnitude becomes automatically elicited by number symbols (i.e., digits from 1 to 9) remains unknown due to the difficulty to measure automatic semantic processing. We used a frequency-tagging EEG paradigm targeting automatic…

This paper contains an analysis of some early thinking of 94 young children aged 5 years 7 months to 6 years 5 months. These children were interviewed as part of a larger study of the multiplicative thinking of children who were midway through their first year of school in Australia. They had not been formally taught multiplication or division at…

Magnitude information, for instance, regarding weight, distance, or velocity, is crucial for planning goal-directed interactions. Accordingly, magnitude information, including numerical magnitude, can affect actions: Responses to small numbers are faster with the left hand than the right and vice versa (hand-based SNARC effect). Previous…

On the surface, the number line may seem like a basic tool with obvious applications. However, using a number line is not always intuitive for students. Students may not recognize significant features such as the size of the unit, how units are represented by iterated equal lengths, or that the accumulation of iterated units is a magnitude of…