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This volume is the outcome of ICMI Study 23 (Primary Mathematics Study on Whole Numbers). This editorial introduction outlines the rationale, the launch, the discussion document, the study conference, the content of the study volume, the main implications of the study, merits and limits of the study, concluding remarks, and acknowledgments.

This chapter witnesses the growing importance of understanding the role of the social and cultural context in which the teaching and learning of mathematics is situated, outlining the background to the studies conducted in recent decades. The aim of this chapter is to report on the process that led the IPC to prepare a background information for each submitted paper, for the first time in an ICMI Study and, also, in any major international conference. A short analysis of the 66 completed context forms that were received is provided with elaboration and commentary. These data are a based on these submissions, rather than a representation of the population of mathematics educators as a whole. The information provides an ‘insider’ view based on the applicants’ perceptions, rather than a collection of objective data. This contextual information, however, concerns the participants attending the ICMI23 Macao Conference; hence it is important to understand the perceptions of the contributors involved in writing this volume: the data provide outlines of the pre- and primary level schooling cultures and systems within which particular kinds of findings, interventions and outcomes are achieved.

Language and culture play a common, key role in conveying concepts in mathematics teaching and learning for mathematical thinking development. Linguistic transparency can foster the construction of mathematical meanings and support the understanding that occurs in learning discourse. A cross-cultural examination of languages should thus allow us to understand linguistic supports or limitations that may interfere with students’ learning and teachers’ teaching of mathematics. This chapter examines number naming and structure across languages and language issues related to whole number structure, arithmetic operations and key concepts of place value and equality from a linguistic perspective. It also specifically considers how the Chinese language has been linked with Chinese arithmetic in ancient and present times.

This chapter primarily offers a commentary on Chap. , before moving off at the end into some wider issues. I have organized my comments under four broad headings – links among what is said, written and gestured vis-à-vis number; place value; the (dissolving) distinction and its pertinence between count nouns and mass nouns (and the place of English-language number words in relation to this distinction); and some similarities and differences among the systems of cardinal, ordinal and fractional number words – before concluding with a (very) few summary remarks.

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.

This chapter discusses some matters raised by Chap. , both the general ideas and some specific sections. A main theme is the non-trivial nature, both of the historical development and the teaching, of the base-ten place value system. A second main theme is that the power of the system stems from its consistency with algebraic structure. A sequence of five stages of understanding of the base-ten system is reviewed, with emphasis on the importance of understanding that base-ten notation expresses each number as a sum of pieces of a special type. The value of being able to work with the pieces, for understanding the computational algorithms, and especially for discussing approximation, is reviewed. To facilitate focus on the pieces and their role, a standard name for them is called for. Also advocated is greater emphasis on understanding large numbers.

This chapter focuses on the neuro-cognitive, cognitive and developmental analyses of whole number arithmetic (WNA) learning. It comprises five sections. The first section provides an overview of the working group discussion. Section reviews neuro-cognitive perspectives of learning WNA but looks beyond these to explain the transcoding of numerals to number words. In the third section, children’s early mathematics-related competencies in reasoning about quantitative relations, patterns and structures are explored from new theoretical perspectives. Studies presented and discussed in working group 2 are presented in the following section as exemplars of intervention studies. The final section examines methodologies utilized in neuro-cognitive, cognitive and developmental analyses of children’s whole number learning. It discusses study designs and their potentialities and limitations for understanding how children develop competencies with whole numbers as well as task designs in cognitive neuroscience research pertinent to number learning. The chapter concludes with implications for further research and teaching practice.

The commentary relates to two main points included in Chap. : (a) cardinal and ordinal numbers and (b) patterns and structure. The comments start with mentioning some philosophical approaches to the nature of natural numbers and add the psychological point of view represented by Piaget, that there is no need to differentiate between the cardinal and the ordinal senses of number. Further, the comments support the approach advocated in Chap. about the merits of employing Davydov or Gattegno methods of teaching whole numbers. Several developmental studies about the origins and counting principles are described. Finally, an emphasis is put on the additive structure and the importance of teaching the operation signs as well as the meaning of the equal sign and not merely the numerical outcomes of the operations.

The core of this chapter is the notion of artefact, starting from the discussion of the meaning of the word in the literature and offering a gallery of cultural artefacts from the participants’ reports and the literature. The idea of artefacts is considered in a broad sense, to include also language and texts. The use of cultural artefacts as teaching aids is addressed. A special section is devoted to the artefacts (teaching aids) from technologies (including virtual manipulatives). The issue of tasks is simply skimmed, but it is not possible to discuss about artefacts without considering the way of using artefacts with suitable tasks. Some examples of tasks are reported to elaborate about aspects that may foster learning whole number arithmetic (WNA). Artefacts and tasks appear as an inseparable pair, to be considered within a cultural and institutional context. Some future challenges are outlined concerning the issue of teacher education, in order to cope with this complex map.

The main focus of this chapter is the mathematical preparation of primary school teachers in relation to the teaching of elementary arithmetic. Comments are offered, from the perspective of a mathematician involved in the education of pre-service teachers, on the importance and variety of artefacts (often of a cultural and historical nature) that can be used to support the learning of whole number arithmetic, as well as on the role played by mathematical tasks in fostering the ‘mathematical message’ that may be conveyed through the artefacts. I will discuss concrete examples taken from an arithmetic course created by my department specifically for prospective primary school teachers, intending so doing to illustrate a crucial observation about artefacts and tasks, namely, that they form an inseparable pair in the teaching and learning of mathematics. This chapter addresses the mathematical preparation of primary school teachers. The learning of arithmetic by actual primary school pupils is not an immediate aim of the work we do with our student teachers, but we claim that many of the artefacts and tasks discussed in our arithmetic course (and in this chapter) can be transferred to pupils – but of course with a necessary adaptation to young children new at such notions.