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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.

Chapter and the different contributions in the proceedings of working group 4 have explored different aspects of the teaching and assessing of whole number arithmetic. We often underestimate teacher’s required knowledge, not only for selecting challenging and dense tasks but also, within a determined task, for assessing the diverse needs of individuals and respect to these needs. It is certainly not a little challenge for mathematics education research to describe the knowledge at stake: even within the field of WNA and even if we take a single lesson. In this chapter, I will focus on the difficulty to make coherent choices in teaching and assessing whole number arithmetic and the various pieces of knowledge that are necessary to do so. Whole number arithmetic is not a homogeneous domain in this chapter, so I will discuss some aspects: memorizing numerical facts, writing numbers and numerical sentences and additive structure’s field. These considerations will lead to a commentary about Macao’s lesson.

In this chapter, we attend to presentation/discussion of structure and structuring activities as two key routes through which whole number arithmetic can be connected to other mathematical content areas and to central mathematical processes and products like defining/definitions and generalizing/generalization. In the body of the chapter, we use literature to distinguish between approaches focused more on the presentation of structure and those oriented towards structuring activities, before presenting an overview and discussion of studies geared more towards one or other of these approaches. We incorporate studies that have been directed towards both students’ mathematical learning and mathematical (and pedagogical) teacher learning and conclude with commentary on biases towards structure-based or structuring activity-based approaches across these contexts. Our argument is that both approaches show promise for building towards stronger connections between whole number arithmetic and other mathematical areas, with a number of examples in each category included. Given the evidence of difficulties for so many children in many parts of the world in moving beyond the terrain of whole number, our findings suggest that attention to structure and structuring can provide important routes for bridging this chasm.

Expressing generality, by learners, is seen as the heart and soul of school mathematics. To engage successfully requires shifts of attention from recognizing structural relationships in a particular situation to perceiving properties as being instantiated. Some examples of tasks which involve learners in encountering mathematical structure are provided as an extension of Chap. , with observations about a learning trajectory concerning the structuring of such tasks. It is proposed that arithmetic is best seen as the study of the structural relations between numbers rather than as a process of getting answers to calculations, which is a sideline. When it becomes the focus, learners merely mimic calculators, badly. The summary from Chap. provides a basis for further suggestions on important directions for further work to bring structural awareness and structural relationships into the core of mathematics teaching.

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.

This chapter provides an overview of the ICME 23 Study panel on special needs in research and instruction in whole number arithmetic. It starts with a general introduction by Verschaffel about the state of affairs in and the major issues and challenges for research and educational practice in the field of mathematical learning difficulties (MLD). Afterwards these issues and challenges are explored and discussed from four different angles by four scholars with complementary specializations in the domain of children with MLD and/or other special needs in the curricular domain of whole number arithmetic, namely, Anna Baccaglini-Frank, Joanne Mulligan, Marja van den Heuvel-Panhuizen and Yan Ping Xin. Finally, Brian Butterworth discusses these four contributions, particularly with respect to the questions ‘what constitutes the “mathematics” that MLD research and practice should address’ and ‘what can be considered as appropriate interventions for children with special needs’

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.

In 1999, presented readers with a puzzle. Four of its six chapters concerned interviews of US and Chinese primary teachers. (Note that the two groups of teachers were not chosen to be representative or similar in status.) The teachers described how they would respond to four classroom situations: teaching subtraction with regrouping; addressing a mistake in multi-digit multiplication; creating a word problem for division by fractions; and addressing an incorrect student conjecture about area and perimeter. The puzzle: Why did the US teachers, all of whom had tertiary degrees and were enrolled in intensive post-graduate programs, show less knowledge of school mathematics than the Chinese teachers, who had so much less education?

An important part of the answer may be that school arithmetic in the two countries was profoundly different. (We write ‘was’ because expectations have changed in both countries. For example, influenced by the 1989 US , significant changes in school arithmetic appear in the 2001 Chinese .) The knowledge displayed by the Chinese teachers was supported by the solid substance of school arithmetic in China. But, such knowledge was not readily available to their US counterparts. In this article, we give an account of this ‘solid substance’ for whole numbers and end with some examples of its connection with the teachers’ responses.

This paper describes an approach to developing concepts of number using general notions of quantity and their measurement. This approach, most prominently articulated by Davydov and his colleagues, is discussed, and some arguments favouring this approach are offered. First is that it provides a coherent development of both whole numbers and fractions. Second, it makes the present from the start of the school curriculum as a useful mathematical object and concept into which real numbers can eventually take up residence. Third, in the Davydov approach, there are some significant opportunities for some early algebraic thinking. I further present an instructional context and approach to place value that simulates a hypothetical invention of a place value system of number representation.

It is widely agreed that humans inherit a numerical competence, though the exact nature of this competence is disputed. I argue that it is the inherited competence with whole numbers (the ‘number module’) that is foundational for arithmetical development. This is clear from a longitudinal study of learners from kindergarten to year 5. Recent research has identified a brain network that underlies our capacity for numbers and arithmetic, with whole number processing a core region of this network. A twin study shows a strong heritable component in whole number competence, its link to arithmetical development and to the brain region. These findings have implications for improving numeracy skills especially among low-attaining learners.