Xu Hua Sun

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 context form 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 convenience sampling 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.

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 goal of the chapter is to explore and discuss teacher education in different parts of the world and to emphasize the commonalities and differences not only in the panellists’ countries, but in a broad perspective. By looking at differences in the parts and processes of different educational systems, we can learn from each other and develop a more integrated perspective on teacher education. Most research studies in the field of primary mathematics teacher education at the international level focus on curricula within teacher education and on the knowledge a primary teacher needs for teaching well. WNA provides a context for developing understandings and constructing arguments that adhere to the practices and norms of more advanced mathematics. Two key issues frame the chapter: ways to increasing and deepening teachers’ mathematical understandings and developing tools that support their mathematics teaching. Examples from seven countries are accompanied by brief information about the organization of primary teacher education in each of them.

The main topics discussed by the panel and the resulting questions to be answered are introduced along with some bibliographic references. The main topics of discussion concern the relationships between tradition and the verbal and non-verbal representations of numbers, numbers and artefacts of arithmetic and the role of technological devices in emulating traditional abaci and allowing direct interaction with the screens of multitouch devices in counting activities. Another crucial issue concerns the different languages that can be present in a classroom for historical and cultural reasons. This represents a challenge for teachers, who must cope with the ways in which words can shape the specific connotations of the meanings of numbers. Although all of these facets of numbers need to be coordinated with the standard mathematical concepts, they also appear in the multimodal representations that are used to teach them, such as words, textbooks, notes and teachers’ and students’ gestures. All of these factors intertwine and sometimes conflict with the richness of the representations and practices that children encounter outside school in their everyday lives.

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 focusses on the diverse theoretical and methodical frameworks that capture the complex relationship between whole number arithmetic (WNA) learning, teaching and assessment. Its aim is to bring these diverse perspectives into conversation. It comprises seven sections. The introduction is followed by a narrative of a Macao primary school lesson on addition calculations with two-digit numbers, and this sets the context for the subsequent three sections that focus on the development of students’ mathematical and metacognitive strategies during their learning of WNA. Apart from examining the impact of teachers’ knowledge of pedagogy, learning trajectories, mathematics and students on children’s learning of WNA, learning theories are also drawn on to interpret the lesson in the Macao Primary School. Two interpretations of the variation theory (VT), an indigenous one and a Western perspective, provide much needed lenses for readers to make sense of the lesson. In addition, the theory of didactical situations (TDS) is also applied to the lesson. The chapter also includes a reflection on possible classroom assessment and the role of textbooks, both of which were less apparent in the lesson, for the teaching and learning of WNA.

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.