Current reforms in undergraduate mathematics education devoted to proof and counterexample highlight its importance in teaching and learning mathematics. While undergraduate students who have taken a number of upper-level mathematics courses are expected to have mastered the skills required to verify statements as well as to read and write proofs…

This paper summarises part of a longitudinal study to investigate the possibility of establishing an Index that would indicate the appropriate starting point for instruction for students entering high school and reports on the first group of students to sit the Index tests in 2005, comparing the predictive nature of the tests with other state and…

This paper reports on a study to examine mental computation strategies used by Year 1, Year 2, and Year 3 students to solve addition and subtraction problems. The participants in this study were twenty five 7 to 9 year-old students identified as excellent, good and satisfactory in their mathematics performance from a school in Penang, Malaysia.…

As a final year Bachelor of Education student at the University of Tasmania, this author recently participated in an innovative program for improving mathematics learning outcomes for primary school students and the mathematics teaching development of pre-service teachers. As a participant in the program, she was required to plan, teach, and…

The purpose of this study was to investigate the relation between early mathematical skills and cognitive processing abilities for two samples of children in Italy (N = 40) and the Netherlands (N = 59) who completed both a cognitive test that measures Planning, Attention, Simultaneous, and Successive (PASS) processing and an early mathematical…

According to reform documents, teachers are expected to teach proofs and proving in school mathematics. Research results indicate that high school students prefer verbal proofs to other formats. We found it interesting and important to examine the position of secondary school teachers with regard to verbal proofs. Fifty high school teachers were…

We report the development of a multiple-choice instrument that measures the mathematical knowledge needed for teaching arithmetic with fractions, decimals, and proportions. In particular, the instrument emphasizes the knowledge needed to reason about such arithmetic when numbers are embedded in problem situations. We administered our instrument to…

This review evaluates the role of language--specifically, the Chinese-based system of number words and the simplicity of Chinese mathematical terms--in explaining the relatively superior performance of Chinese and other East Asian students in cross-national studies of mathematics achievement. Relevant research is critically reviewed focusing on…

This report focuses on prospective secondary mathematics teachers' understanding of irrational numbers. Various dimensions of participants' knowledge regarding the relation between the two sets, rational and irrational, are examined. Three issues are addressed: richness and density of numbers, the fitting of rational and irrational numbers on the…

Children's natural curiosity about numbers, big and small can lead to exploring place-value ideas. But how can these abstract concepts be experienced more concretely? This article presents some practical approaches for conceptualising very small numbers using linear models, area models, volume models, and diagrams.

"S"pontaneous "a"ttention to "n"umber (SAN) is the tendency to notice the relatively abstract attribute of number despite the presence of other attributes. According to nativists, an innate concept of one to three directs young children's attention to these "intuitive numbers" in everyday situations--even before they acquire language. According to…

"The Number Race" is an adaptive game designed to improve number sense. We tested its effectiveness using a cross-over design in 53 low socioeconomic status kindergarteners in France. Children showed improvements in tasks traditionally used to assess number sense (numerical comparison of digits and words). However, there was no…

Over the last century many individuals with exceptional memory have been studied and tested in the laboratory. This article studies Chao Lu, who set a Guinness World Record by memorizing 67,890 decimals of pi. Chao Lu's superior self-paced memorization of digits is shown through analyses of study times and verbal reports to be mediated by mnemonic…

Deficits in basic numerical processing have been identified as a central and potentially causal problem in developmental dyscalculia; however, so far not much is known about the typical and atypical development of such skills. This study assessed basic number skills cross-sectionally in 262 typically developing and 51 dyscalculic children in…

In teaching primary teacher trainees, an awareness of the characteristic features, especially commutativity and associativity of basic operations play an important role. Owing to a deeply set automatism rooted in their primary and secondary education, teacher trainees think that such characteristics of addition are so trivial that they do not need…