Judy Sayers

Comparative studies of mathematics teaching: does the means of analysis determine the outcome?

This paper addresses four questions concerning the influence of culture on mathematics teachers’ professional practice. Firstly, drawing on categorical data yielded by the application of low inference coding schedule to video recordings of sequences of lessons taught by case study teachers on four common topics in England, Flanders, Hungary and Spain, we undertook an exploratory factor analysis to examine the ways in which such coded variables interact. This process yielded five factors, each of which was interpretable against the literature and highlighted the extent to which dichotomisations of mathematics teaching as reform or traditional are not necessarily helpful, not least because all project teachers exhibited characteristics of both. Secondly, factors scores were analysed by nationality to reveal culturally located practices resonant with the available literature. Thirdly, cluster analyses yielded four well-defined cross-cultural clusters of episodes, each indicative of particular didactical perspectives that appeared to challenge the exclusivity of these culturally located practices. Finally, the key methodological finding was that the manner in which data are analysed influences greatly the outcomes of comparative mathematics research.

The What and Why of Whole Number Arithmetic: Foundational Ideas from History, Language and Societal Changes

Increasing globalization encourages assumptions of universalism in teaching and learning, in which cultural and contextual factors are perceived as nonessential. However, our teaching and learning are unavoidably embedded in history, language and culture, from which we draw to organize our educational systems. Such factors can remain hidden but can also provide us with opportunities to gain a deeper understanding of constraints that are taken for granted. This chapter provides a meta-level analysis and synthesis of the what and why of whole number arithmetic (WNA). The summary provides background for the whole volume, which identifies the historical, cultural and linguistic foundations upon which other aspects of learning, teaching and assessment are based. We begin with a historical survey of the development of pre-numeral and numeral systems. We then explore the epistemological and pedagogical insights and highlight the differences between linguistic practices and their links with the universal decimal features of WNA. We investigate inconsistencies between spoken and written numbers and the incompatibility of numeration and calculation and review a number of teaching interventions. Finally, we report the influence of economics and business, academic mathematics, science and technology and public and private stakeholders on WNA to understand how and why curriculum changes are made, with a focus on the fundamental losses and gains.

The Role of Conceptual Subitising in the Development of Foundational Number Sense

Evidence indicates that children with a well-developed number sense are more likely to experience long-term mathematical success than children without. However, number sense has remained an elusive construct. In this chapter, we summarise the development of an eight-dimensional framework categorising what we have come to call foundational number sense or those non-innate number-related competences typically taught during the first years of schooling. We also show, drawing on grade one lessons from Hungary and Sweden, how focused instruction on conceptual subitising, the teaching of children to identify and use easily recognisable groups of objects to structure children’s understanding of number, facilitates children’s acquisition of a range of foundational number sense-related competences.

HOW DO TEACHERS OF MATHEMATICS TEACH? A FOUR-WAY INTERNATIONAL COMPARISON

This paper reports on a small-scale EU-funded study of the teaching of mathematics in the age range 10-14 in Flanders, England, Hungary and Spain. Based on video-recordings of sequences of lessons taught on standard topics, and exploiting a coding schedule developed from live observations in each country, we examine the didactic strategies teachers employ in the episodes that comprise a lesson, where an episode is that period of a lesson in which a teachers observable intention remains constant. Analyses identified consistent teacher behaviours within countries and considerable variation across them. Some implications for English mathematics education are discussed.

It is quite confusing isn't it?

Research on practice informs us that finding an effective balance between mathematical instruction and mathematical construction is not an easy task when developing young children’s learning of mathematics. Framed against Presmeg’s (2014) dance between instruction with construction as a dance, this chapter examines one teacher’s practice in relation to a centrally imposed curriculum and support materials determining the mathematics curriculum (5–7 years). Drawing from a larger multiple case study of six specialist primary mathematics teachers’ conceptions of the whole class interactive phases of their mathematics lessons, this chapter presents Jane’s introduction of place value and partitioning through her exploitation of particular manipulatives and language. The aim of this chapter is to highlight a conflicting and confusing dance between national professional guidance for teachers, an individual’s interpretations of this guidance, and the creation of meaningful opportunities for young children to construct mathematical knowledge.