This paper discusses variation in reasoning strategies among expert mathematicians, with a particular focus on the degree to which they use examples to reason about general conjectures. We first discuss literature on the use of examples in understanding and reasoning about abstract mathematics, relating this to a conceptualisation of syntactic and…

This article is the third in a series of articles describing a research project entitled "Entering into Symbols" (EIS) on the use of mathematical applets at key stages 2 and 3. The first two articles, in "MT200" and "MT203", described applets designed to teach place value ("Tuckshop subtraction") and basic algebra ("Matchbox algebra"). In this…

Here, we investigate what loci are produced when a square of side-length one is allowed to rotate around a square of side-length n, where n is a whole number. We find that if i = 1, 2, 3 or 4 (mod 4), the loci obtained for n [congruent to] i (mod 4) all have the same symmetry and we show how the perimeter of each class can be determined. We also…

Previous work has investigated adults' knowledge of principles for arithmetic with positive numbers (Dixon, Deets, & Bangert, 2001). The current study extends this past work to address adults' knowledge of principles of arithmetic with a negative number, and also investigates links between knowledge of principles and problem representation.…

Theoretical analyses of the development of numerical representations suggest that playing linear number board games should enhance young children's numerical knowledge. Consistent with this prediction, playing such a game for roughly 1 hr increased low-income preschoolers' (mean age = 5.4 years) proficiency on 4 diverse numerical tasks: numerical…

Catalan numbers are defined by combinatorial properties, in numerous situations. Here, an integral representation for these Catalan numbers is computed.

An interesting class of purely pandiagonal, i.e. non-magic, whole number (integer) squares of orders (row/column dimension) of the powers of two which are related to Gray codes and square Karnaugh maps has been identified. Treated as matrices these squares possess just two non-zero eigenvalues. The construction of these squares has been automated…

A sense of number is now generally recognised as a central factor in learning, and later applying, mathematics. Consequently, number sense is increasingly emphasised in curriculum documentation related to mathematics. The "Primary School Curriculum: Mathematics," published by the Government of Ireland in 1999, is no exception. It…

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,…

This qualitative study explored the integration of religion and spirituality in counseling among six African American Christian college students ranging from age 26-48. Specifically, attention was given to the circumstances that either facilitated or discouraged the student's willingness to include their faith system in counseling. Thematic…

Here is the only reference book you will ever need for teaching primary school mathematics and statistics. It is full of exciting and engaging snapshots of excellent classroom practice relevant to "The New Zealand Curriculum" and national mathematics standards. There are many fascinating examples of investigative learning experiences,…

This paper describes the PD program that was delivered during the first year of the study. The main goal of the intervention was to increase teachers' capability to teach positive rational number topics effectively. The program included a 3-day summer institute (18 hours per teacher), five 1-day seminars held during the school year (30 hours per…

Described is an activity in which students develop their own number system. This activity allow students to examine the structure and history of number systems and to discover the power and respectable efficiency of the number system used today. (KR)

Four hundred and three 3rd- and 5th-grade Chinese students took the Multiplication Estimation Test or participated in the interview on it, designed to assess their computational estimation performance on whole-number multiplication. Students perform better when tasks are presented visually than orally. Third graders tend to use rounding based…

The goal of this study is to uncover the successes and challenges that preservice teachers are likely to experience as they unpack lesson-level mathematical learning goals (i.e., identify the subconcepts and subskills that feed into target learning goals). Unpacking learning goals is a form of specialized mathematical knowledge for teaching, an…