How do children acquire exact meanings for number words like three or forty-seven? In recent years, a lively debate has probed the cognitive systems that support learning, with some arguing that an evolutionarily ancient "approximate number system" drives early number word meanings, and others arguing that learning is supported chiefly…

A pivotal information about the structure of students' comprehension underlying from which the knowledge is gained can be identified through the mental model. This study aimed at describing the six levels of students' mental model in comprehending the concept of the integer. The subject of this research was 40 students consisting of 20 students…

It is widely documented that the density property of rational numbers is challenging for students. The framework theory approach to conceptual change places this observation in the more general frame of problems faced by learners in the transition from natural to rational numbers. As students enrich, but do not restructure, their natural number…

Inservice (n=36) and preservice (n=67) mathematics teachers were asked for a commutative, nonassociative binary operation. Responses were analyzed for correctness, productiveness, mathematical content, and underlying difficulties. Both groups exhibited a weak concept by failing to produce an example and using a limited content search space.…

Data from (n=4) students in grades three through six showed a consistent parallel between the types of units constructed in a geometric setting with those in a numeric context. Students who constructed abstract composite units in tiling the plane also did so in adding and subtracting whole numbers. (Author/MKR)

Adapted Keil's predictability method to examine adults' and preschoolers' conceptions of numbers, focusing on the ontological distinction between numbers and sets of objects. Found that children, like adults, attribute spatial-arrangement properties to collections much more than to numbers, although both are considered to have quantitative…

Interviews with 107 children, ages 4-7, about uncounted quantities, counted quantities, and numerical equations showed that the ability to predict changes to counted quantities increased with age. Only 7-year olds were able to use covariance and compensation in the purely numerical context of derived equations. (Author/MKR)

This paper theoretically characterizes number sense as a set of cognitive capabilities for constructing and reasoning within mental models. This perspective provides support for viewing various aspects of number sense as features of students' general condition of knowing about numbers and magnitude, rather than as skills needing specific…

In the present article, we argue that the conceptual change approach to learning can apply in the case of mathematics, taking into consideration the particular nature of mathematical knowledge and the neurobiological bases of mathematical cognition. In the empirical study that is reported in this article, we investigated ninth graders'…

Investigated the continuous fraction concepts of (n=59) students in grades four, five, and six. Students were confident and accurate when performing sharing tasks, but were much less successful on continuous quantity tasks involving formal fraction language and symbol manipulation fraction tasks. (14 references) (Author/MKR)

Compares the cognitive representation of number of 24 American, 25 Chinese, 24 Japanese, and 40 Korean first-graders, and 20 Korean kindergartners. Chinese, Japanese, and Korean children preferred to use a construction of tens and ones to show numbers, whereas English-speaking children preferred to use a collection of units. (RJC)

Based on a two-tiered epistemological model (preliminary physical concepts and emerging mathematical concepts), studied the construction of number among 91 kindergarten students in France, Canada, and the United States. Subjects' cognitive numerical structures were surprisingly similar despite differences in language and culture. (SV)

Attempted to verify knowledge regarding decimal and rational numbers in children ages 10-14. Discusses how pupils can receive and assimilate extensions of the number system from natural numbers to decimals and fractions and later can integrate this extension into a single and coherent numerical structure. (Author/MKR)

Kindergarten (n=30) and grade 1 (n=15) students of contrasting socioeconomic backgrounds were interviewed to assess their knowledge of number word sequences and spatial patterns and their ability to recognize numerals and use counting to establish numerosity of visible and screened collections. Findings indicate greater variation in abilities…

Validates a framework for assessing children's thinking in multidigit number situations and uses the framework to evaluate instructional programs. Key constructs--counting, partitioning, grouping, and number relationships--appear stable within each of 5 levels across 12 case studies, suggesting a possible hierarchy of thinking. (DF)