The study investigates the knowledge profiles of elementary pre-service teachers (PST) concerning numbers and operations before their professional classroom practice. By validating an instrument through exploratory and confirmatory factor analyses, it identifies and categorizes the PST' performance into distinct profiles based on the mathematical…

Children learn the cardinalities of the first numbers one, two, three and four before they learn how counting tracks cardinality for all numbers. It may be that when children start to understand counting, they also discover how numbers relate to one another in a structured number system. Do children who understand that the cardinality of a set is…

Early mathematics education presents middle-school students with the challenge of adding and subtracting negative integers. This paper reports on results from the experimental implementation of a proposed educational design for integer arithmetic that utilized the number-line (NL) form as a resource for students to enact simple addition and…

To explain children's difficulties learning fraction arithmetic, Braithwaite et al. (2017) proposed FARRA, a theory of fraction arithmetic implemented as a computational model. The present study tested predictions of the theory in a new domain, decimal arithmetic, and investigated children's use of conceptual knowledge in that domain. Sixth and…

Equal representation as a social issue is about the participation of one social group, in a particular context, in proportion to the numbers in the group within the total population. The proportion of women in parliament, and of the participation rate of students of lower socioeconomic status in higher education, are examples. The aims of this…

Early mathematics plays an important role in introducing foundational concepts for number sense in children. One of the critical areas of learning is the establishment of a linear view of numbers. It is essential to create opportunities for young children to understand that numbers are equally spaced on the number line and that they increase in…

To understand misconceptions with rational numbers (i.e., fractions, decimals, and percentages), we administered an assessment of rational numbers to 331 undergraduate students from a 4-year university. The assessment included 41 items categorized as measuring foundational understanding, calculations, or word problems. We coded each student's…

Rational numbers are represented by multiple notations: fractions, decimals, and percentages. Whereas previous studies have investigated affordances of these notations for representing different types of information (DeWolf et al., 2015; Tian et al., 2020), the present study investigated their affordances for solving different types of arithmetic…

Integer arithmetic is difficult for students worldwide. Although students' integer thinking has frequently been studied, little is known about typical instructional practice for this difficult topic. Thus, to investigate what resources teachers use, we surveyed U.S. middle-grade teachers who teach negative numbers. About half the teachers said…

Rational numbers, such as fractions and decimals, are harder to understand than natural numbers. Moreover, individuals struggle with fractions more than with decimals. The present study sought to disentangle the extent to which two potential sources of difficulty affect secondary-school students' numerical magnitude understanding: number type…

This study sought to understand how students activate number sense in determining the position of fractions on a number line and identify how the natural number bias and number sense influences students' thinking processes. The study utilized the Cognitive Task Analysis (CTA), involving four fifth-grade elementary students as the research…

In this paper, we first focus on the sum of powers of the first n positive odd integers, T[subscript k](n)=1[superscript k]+3[superscript k]+5[superscript k]+...+(2n-1)[superscript k], and derive in an elementary way a polynomial formula for T[subscript k](n) in terms of a specific type of generalized Stirling numbers. Then we consider the sum of…

Over the last years, there is a growing interest in studying students' difficulties with rational numbers from a cognitive/developmental perspective, focusing on the role of prior knowledge in students' understanding of rational numbers. The present study tests the effect of the "whole" or "natural number bias" (i.e., the…

Skills and understanding of operations with negative numbers, which are typically taught in middle school, are crucial aspects of numerical competence necessary for all subsequent mathematics. To more swiftly and coherently develop the field's understanding of how to foster this critical competence, we need shared measures that allow us to compare…

Although many U.S. children can count sets by 4 years, it is not until 5½--6 years that they understand how counting relates to number--that is, that adding 1 to a set necessitates counting up one number. This study examined two knowledge sources that 3½- to 6-year-olds (N = 136) may leverage to acquire this "successor function": (a)…