Kathryn C. Irwin

Professional Conversations and Professional Growth

A professional development program for 18 teachers was conducted over a two-year period. The participating teachers taught in intermediate schools (students aged 11–13) and secondary schools. The teachers worked collaboratively to improve their mathematics teaching, with encouragement to reflect on their practice but with minimal instruction from the researchers. Results, as defined by change in teaching practices, beliefs, and reflections, and student achievement, indicated that the collaborative program was particularly useful for experienced secondary school teachers but less useful for intermediate school teachers. We concluded that this type of professional development was most useful for teachers who had sufficient knowledge of mathematics; these teachers were able to focus on pedagogy and to draw connections between aspects of the mathematics they taught, without recourse to a specialists advice.

Algebraic Thinking with and without Algebraic Representation: A Pathway for Learning

The origins of algebraic thinking precede understanding of arithmetic, as shown in a study of children aged 4–7. A mathematics curriculum introduced in some New Zealand schools in 1999, The New Zealand Numeracy Project, now encourages this algebraic thinking within arithmetic. The underlying framework for this curriculum is described, with examples of the type of thinking encouraged. The effect of this emphasis on the algebra underpinning arithmetic operations was examined in two further studies. One of these involved students in their final year of elementary and intermediate school, at age 12. This study showed that on a test that focused on students’ awareness of the underlying algebraic structure of arithmetic, those students who had been included in the new curriculum in its early stages outperformed those who had received a traditional curriculum. A later study followed a cohort of students who received the new curriculum through their two intermediate school years (aged 11–12) and into their first year of high school at age 13, when traditional algebra is introduced. The results of this study showed that students who had developed their understanding of the interrelationship of mathematical relationships for additive, multiplicative and proportional operations could display this understanding algebraically. The ramifications of these findings for further teaching algebraic thinking with or without algebraic representation led to a proposal for a ‘pathway for algebraic thinking’ accessible to all students.

Identification of Mathematically Gifted Children in New Zealand

A popular way of identifying mathematically gifted children in New Zealand is to select those ranked at or above the 90th percentile on a mathematics multiple-choice test, the Progressive Achievement Test (PAT). Giftedness in mathematics can properly be defined as the ability to solve difficult mathematical problems. We therefore compared the performance of sixty-six children on a published problem-solving test and the PAT. Receiver Operating Characteristic analysis showed that, independent of any chosen percentile, the PATs accuracy at identifying mathematically gifted children was 78 per cent. Such a degree of accuracy will lead to many mathematically gifted children being overlooked, or many being mistakenly identified as gifted. Use of the PAT for this purpose is therefore not recommended.

Understanding linear measurement: A comparison of filipino and new zealand children

An understanding of linear measurement depends on principles that include standard unit size, iteration of units, numbering of a unit at its end, and partial units for measuring continuous length. Children may learn these principles at school, for example through experience with informal measurement, or they may learn them through use of measurement in society. This study compared the application of these principles by children aged 8 and 9 from the Philippines and New Zealand. These countries were selected because they have quite different curricula, societal influences and economies. Ninety-one children were interviewed individually on a common set of unusual tasks that were designed to tap underlying principles. Results showed many similarities and some differences between countries. Most tasks requiring visualisation and informal units were done more accurately by New Zealand children. Some tasks involving the use of a conventional ruler were done more accurately by Filipino children. These differences appear to be related to differences in curricula and possibly to differences in societal use of measurement. We suggest that these results, like those of other writers cited, demonstrate the need for extensive work on the underlying concepts in measurement through work on informal measurement and a careful transition between informal and formal measurement.

Young Children’s Formation of Numerical Concepts: Or 8 = 9 + 7

The ways in which young children develo mathematical concepts have been of interest to many writers. Theories related to children’s de of mathemati cal concepts have been developed by such diverse writers as Case (1985). Fuson (1989), Gelman and Gallistel (1978). Inhelder and Piaget, (e.g., 1958), Resnick (e.g.,l989), and Steffe. Cobb, and von Glaserfeld(1988). All of these authors work within a broadly constructivist framework which examines the ways in which children build their own understanding. This view of learning is put well by Lindfors when writing about children’s language (Lindfors, 1987, page x). She says, ‘I see children as shapers of their own knowing....I see children’s learning in all areas as an active sense-making process—necessary and inevitable and individual.’

The Benefits And Challenges For Social Justice In International Exchanges In Mathematics And Science Education

International exchanges in higher education are influenced by globalised economies. Whether globalization is seen as a relabelling of capitalism or as the homogenization of cultures, social justice must not be an alien concept in academic endeavours because of the many inequalities that exist both within and between nations and peoples. This chapter presents an analysis of an exchange between two universities by examining certain aspects that impinge on issues in social justice such as fairness, equality, and equity. While the exchange benefited both universities in various ways, the authors who were the principal actors in the exchange admit that many lessons in social justice have been learned. Thus, the chapter lays out some reflections on how future academic exchanges might be conceived to ensure fairness and equity for all parties involved