It is widely assumed within developmental psychology that spontaneously-arising conceptualizations of natural number--those that develop without explicit mathematics instruction--match the formal characterization of natural number given in the Dedekind-Peano Axioms (e.g., Carey, 2004; Leslie et al., 2008; Rips et al., 2008). Specifically,…

Processing rational numbers is difficult for many children. The natural number bias is one possible explanation for why children struggle with rational numbers. It refers to the tendency to overgeneralize the properties of natural numbers. In this study, it is argued that in order to be successful in rational number tasks, individuals need to…

Humans perceptually extract quantity information from our environments, be it from simple stimuli in isolation, or from relational magnitudes formed by taking ratios of pairs of simple stimuli. Some have proposed that these two types of magnitude are processed by a common system, whereas others have proposed separate systems. To test these…

Background: While it has been widely demonstrated that children's and adolescents' understanding of rational number plays an important role in their mathematics achievement, we have limited knowledge about the cognitive correlates of this understanding. Aims: The current study aimed at examining whether children's non-symbolic ratio processing and…

This case study examines the salient features of two individuals' reasoning when confronted with a task concerning the cardinality and associated cardinal number of equinumerous infinite sets. The APOS Theory was used as a framework to interpret their efforts to resolve the "infinite balls paradox" and one of its variants. These cases…

Unlike old perceptions of algebra as a subject that needs to be taught in the upper grade levels of schooling, much of the recent research reports that young students are capable of reasoning algebraically. It is important to note that the recommendation to include algebraic experiences in the early grades is not made simply to introduce typical…

This study examined how four domain-specific skills (arithmetic procedural skills, number fact retrieval, place value concept, and number sense) and two domain-general processing skills (working memory and processing speed) may account for Chinese children's mathematics learning difficulties. Children with mathematics difficulties (MD) of two age…

Numerical fractions are commonly used to express ratios and proportions (i.e., real numbers), but little is known about how they are mentally represented and processed by skilled adults. Four experiments employed comparison tasks to investigate the distance effect and the effect of the spatial numerical association of response codes (SNARC) for…

Because of the educational importance of mathematics and science, IEA's (International Association for the Evaluation of Educational Achievement) Trends in International Mathematics and Science Study, widely known as TIMSS, is dedicated to providing countries with information to improve teaching and learning in these curriculum areas. Conducted…

A brief survey of models in dealing with various types of understanding is given. Then a hybrid model, which proved inadequate for describing understanding, is outlined. Finally, four levels of understanding are discussed: intuitive, procedural, abstract, and formal. The concept of number is used to illustrate these levels. (MNS)

The name of Zoltan P. Dienes (1916- ) stands with those of Jean Piaget, Jerome Bruner, Edward Begle, and Robert Davis as a legendary figure whose work left a lasting impression on the field of mathematics education. Dienes' name is synonymous with the multibase blocks that he invented for the teaching of place value. Among numerous other things,…

A mathematics teacher who avoided students' questions about zero, notes that children are able to interpret this concept and any system of numeration in a language that they have made up and can understand. (AM)

This report, written in Finnish with a 10-page summary in English, concerns research on the interrelationship between Piagetian operations, various aspects of information processing capacity and mathematical thought processes and strategies related to number and measurement. In a theoretical discussion, Piaget's research on mathematico-logical…

Children from grades 2, 3, 4, 6, and 8 made speeded, bimanual parity (odd/even) judgments of numerals. Response times indicated that from fourth grade on, parity information was retrieved directly from memory. By grade 3, children represented magnitude information as a left-to-right number line which was accessed obligatorily and, by grades 6 and…

Decimal numbers have become an increasingly important topic of the elementary and junior high school mathematics curriculum. However, national and state education assessments indicate that students have incomplete and distorted conceptions of decimal numbers. This paper reports initial data from a two-year project designed to elicit and describe…