The present study aimed to examine the opinions of in-and preservice middle school mathematics teachers about teaching four operations with integers through "number line" and/or "counters". Participants were 42 middle school mathematics teachers and 51 pre-service mathematics teachers. The data were collected through open-ended…

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…

Visualising positive and negative numbers on a number line is helpful for exploring problems involving operations with positive and negative numbers. This is because number lines lend themselves to exploring problems involving continuous linear contexts such as travelling distances and temperature. Teachers in a professional development program…

This study explored the relationships between task beliefs about the empty number line (ENL), mathematical ability, gender, and voluntary ENL use in multi-digit subtraction and addition. One hundred twenty-three Dutch third-grade students and nine teachers from six schools participated in this study. The multilevel path analysis showed that task…

The main goal of this study was to analyze the contribution of predictors of both domain-general (working memory, processing speed, and receptive vocabulary) and domain-specific variable (estimation and magnitude comparison) processes to informal mathematical performance (numbering, comparison, calculation, and understanding of concepts) in…

In this paper, differences in the implementation of a number activity called the snake game are studied. Nine Swedish preschool teachers worked in collaboration with a research team, enacting the same activity with their groups of 5-year-old children over a 3-month period. Variation theory forms the basis for the analysis of 67 videorecorded…

Difficulties in mathematics are argued to stem from impairments of a specialized system of numerical magnitude representation. This study investigates whether different measures of numerical magnitude understanding in kindergarten uniquely predict mathematical achievement concurrently and 6 months later, and also examines the relative explanatory…

Mental rotation is positively related to arithmetic ability; however, the mechanism underlying this relationship remains unclear. The possible roles of working memory, place-value concept, and number line estimation in the correlation between mental rotation and whole-number computation were investigated. One hundred and fifty-five first-grade…

The aim of this research is to reveal the understanding of dimensions and sub-dimensions of the place value in natural numbers of gifted and non-gifted students attending fourth grade. A mixed approach was adopted in the research. The research is quantitative in terms of revealing the levels of place value, counting, representing, naming,…

Numbers (i.e. 45) are symbols that are common in reading texts in various forms such as dates, percentages, and quantity expressions. Each digit corresponds to a word, unlike words in which each letter represents a phoneme. Despite their high frequency in reading texts, eye movement research on numeral processing is still rare. Numerals are easy…

This study explored whether kindergarteners who had yet to learn about multi-digit numbers at school could automatically process the underlying magnitudes, i.e., place-values, represented by the digits in a multi-digit number. A place-value Stroop task showed a pair of price tags in each trial. Each price tag contained a three-digit number, of…

Infants and toddlers experience the world in interaction with others. Likewise, social interactions are important in learning about math--concepts of number ("one," "two," "three"), space ("on top," "upside-down," "round"), and magnitude ("more," "big,"…

The current study assessed whether adding worked examples with self-explanation prompts focused on making connections between mathematical principles, procedures, and concepts of rational numbers to a curriculum focused on invented strategies improves pre-algebra students' fraction number line acuity, rational number concepts and procedures.…

In mathematics, three integer numbers or triples have been shown to govern a specific geometrical balance between triangles and squares. The first to study triples were probably the Babylonians, followed by Pythagoras some 1500 years later (Friberg, 1981). This geometrical balance relates parent triples to child triples via the central square…

This article describes some of the essential mathematics that underpins the use of algorithms through a series of learning pathways. To begin, a graphic depicting the mathematical ideas and concepts that underpin the learning of algorithms for multiplication and division is provided. The understanding and use of algorithms is informed by two…