"Place value," which underlies the meanings of multidigits, encompasses the principle of position and base-10 rules. To understand 65, one needs to know that the digits 6 and 5 occupy different positions and thus represent ordered values of different magnitudes (i.e., the "principle of position") and that the value of each…

This qualitative case study aims to address the ways that pre-service mathematics teachers (PSTs) used a rope in a daylong research meeting for cognitive development of children's understanding of counting numbers in Tanzanian elementary schools. Three university mathematics pre-service teachers volunteered participating in this study. Collective…

"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…

Although the early development of children's number sense is a strong predictor of their later mathematics achievements, it has been overlooked in primary schools in Malaysia. Mainly attributable to underdeveloped number sense of Malaysian primary and secondary school children, their inability to handle simple mathematics tasks, which require the…

Conceptualizing numbers as discursive constructs generated in, and for the sake of, communication, we investigated the development of the numerical discourse of Milo, a boy who was 2 years and 8 months old when we first met him and whom we then followed for 18 months. Our analyses of the child's evolving responses to the question "Where is…

Understanding place value is a critical and foundational competency for elementary mathematics. Classroom teachers who endeavor to promote place-value development adopt a variety of established practices to varying degrees of effectiveness. In parallel, researchers have validated models of how young children acquire place-value understanding.…

There are many continuous quantitative dimensions in the physical world. Philosophical, psychological and neural work has focused mostly on space and number. However, there are other important continuous dimensions (e.g., time, mass). Moreover, space can be broken down into more specific dimensions (e.g., length, area, density) and number can be…

An essential part of understanding number words (e.g., "eight") is understanding that all number words refer to the dimension of experience we call numerosity. Knowledge of this general principle may be separable from knowledge of individual number word meanings. That is, children may learn the meanings of at least a few individual number words…

Many students encounter difficulty in their transition to advanced mathematical thinking. Such difficulty may be explained by a lack of understanding of many concepts taught in early school years, especially multiplicative reasoning. Advanced mathematical thinking depends on the development of multiplicative reasoning. The purpose of this study…

In this article, the author discusses her research into the development of counting in pre-school children. Her research involved a small-scale case study with a personal flavour--the children in the study were her twin daughters (called Emily and Alice for the purpose of the research). Observations of their mathematical development began at 18…

In this study, we use children's prior knowledge to support their development of cardinality understanding, based on Bermejo's (1996) model of cardinality understanding and on the distinction between cardinality (the number of items in a set is represented by a number-word) and the cardinality principle (the last number-word used in counting…

Explored whether a system between written place-value system and base-10 manipulatives helped children understand place-value. Found evidence that the intermediate system helped children differentiate between face values and complete values of digits in multidigit place-value number representations, and to grasp that the sum of the digits'…