The problem of perception – how far we can trust the senses, and how we come to believe what we think we know about the external world – goes back to the ancient Greeks. Naive realism – that we immediately know reality in what we see – is refuted by cases in which appearances are obviously distorted by perceptual artifacts, like the black drop in the case of transits of Venus. Perhaps the most famous case of the difficulty in visual telescopic observations is that of the canals of Mars, which led to a long-running controversy, not only among observers but engaging the wider public as well, about the nature of the actual surface markings of the red planet. In the end, there proved to be no canals in the artifactual sense, and instead the subtleties of perception revealed themselves – which include the nature of the visual system, the way that sense-information is processed in the brain, and the way the brain uses guided perception and prediction to make “sense” of the sense perceptions.

Chapter 16 examines what it means to be a primary mathematics teacher in a professional context. It considers the attributes of effective mathematics educators, the requirements for professional accreditation, and the value of professional learning and engagement with mathematics education networks. You will also reflect on your preparation for the LANTITE and how to continue developing your identity as a confident and capable mathematics educator.

Chapter 13 explores the concept of mathematical identity and how both students and teachers come to see themselves in relation to mathematics. It examines the impact of maths anxiety – particularly in the early years of teaching – and how identity is influenced by community, context, and experience. The chapter highlights the importance of understanding and responding to diverse school settings, including rural, regional, and remote communities. You will also consider how to meaningfully embed the Australian Curriculum cross-curriculum priorities – Aboriginal and Torres Strait Islander histories and cultures, Asia and Australia’s engagement with Asia, and sustainability – within mathematics lessons.

Chapter 15 explores the foundational practices of planning, assessing, and using data in mathematics education. It unpacks the essential components of effective mathematics lessons and outlines strategies for designing learning sequences that are responsive to student needs. The chapter introduces formative and summative assessment practices, the role of feedback, and how student learning data can be used to inform, adapt, and improve teaching and learning.

Chapter 14 focuses on diversity in the primary mathematics classroom and the need to create inclusive and responsive learning environments. It examines strategies to recognise and value students’ diverse cultural, linguistic, and learning needs, and offers practical approaches for differentiation and equitable participation in mathematics. You will explore how inclusive pedagogy supports all learners in developing confidence and capability as mathematical thinkers.

Chapter 5 explores computational thinking and how digital tools and processes – such as decomposition, abstraction, and algorithmic thinking – can support students’ mathematical understanding. You will consider how robotics, apps, and modelling can be used to promote logical reasoning and real-world application.

Chapter 7 introduces the interrelated strands of Number and Algebra (Foundation to Year 2) and explores how young children build informal understandings of number through everyday experiences. The chapter focuses on early numeracy skills such as magnitude, counting, number order, and using numbers in real-world contexts. Thinking and working mathematically is foregrounded through rich tasks that encourage flexible thinking and build foundational knowledge for later learning.

Chapter 9 focuses on how students develop foundational understandings of Measurement and Space in the early years (Foundation to Year 2). It explores how young learners engage with concepts such as length, area, time, and mass through hands-on experiences and everyday contexts. You will consider how to structure learning experiences that build conceptual understanding, support spatial reasoning, and introduce key mathematical vocabulary and thinking processes.

Chapter 3 introduces the concept of Big Ideas in mathematics—such as number sense, algebraic thinking, and spatial reasoning—and how these support deep learning. You will consider ways to develop and connect key mathematical ideas, using strategies like Fermi problems to promote critical and creative thinking.

Chapter 1 introduces the structure and purpose of the Australian Curriculum: Mathematics (Foundation to Year 6), exploring the importance of mathematics in education and everyday life. You will examine what it means to think, reason, and work mathematically, including the roles of inductive and deductive reasoning. This chapter also highlights how reasoning skills develop across the primary years and clarifies the relationship between numeracy and mathematics in the curriculum. These understandings form the basis for designing meaningful learning experiences that connect curriculum content with students’ developing mathematical reasoning.

Chapter 6 delves into experiments, models, and simulations in mathematics, introducing powerful strategies for inquiry and representation. You will learn how to scaffold students’ understanding of mathematical concepts through modelling frameworks and interactive tools, enhancing conceptual understanding and problem-solving.

Chapter 2 focuses on the role and potential of technology in the mathematics classroom. You will be introduced to the TPACK and SAMR frameworks, which support effective planning and teaching with digital tools. This chapter demonstrates how technology can enhance engagement, feedback, personalisation, and collaboration, while providing guidance on selecting high-quality digital resources aligned with the cross-curriculum priorities. The integration of technology is framed to enrich pedagogical practice and support diverse learners.

Chapter 10 builds on earlier learning and explores how students extend their understanding of Measurement and Space in the middle and upper primary years (Years 3 to 6). It focuses on the development of key concepts such as units of measure, angles, transformations, and geometric properties. You will investigate strategies for using precise mathematical language, addressing common misconceptions, and applying spatial and measurement thinking to solve purposeful, real-world problems.

Chapter 1 is mainly devoted to articulating Wittgenstein’s normativism. This label reflects what I consider to be Wittgenstein’s fundamental insight: that mathematical sentences actually serve to formulate norms, or rules; furthermore, as rules, they lack truth-values. (However, in my reading, he also accepts that these sentences can make truth-apt assertions.) The chapter also sketches the two doctrines relevant to the subsequent discussion – platonism and conventionalism – and highlights the eliminativist aspect of normativism while emphasizing its therapeutic consequences: by eliminating problems, one achieves ‘peace of mind.’

The Introduction sets the stage for the project. It situates the idea of the book in the scholarly landscape, by explicating the main directions of the present account and how it relates to other accounts. It briefly introduces some of the central positions appearing in the text (e.g., ‘normativism,’ ‘eliminativism,’ ‘(non)revisionism’), presents the methodological commitments of the work, and gives summaries of the chapters.

This final chapter provides a summary of the book, together with some additional clarifications. Along the way, I will highlight several open questions and difficulties faced by the present account – and, if the account offered here is accurate, by Wittgenstein’s own position.