Chapter 1 will examine the ontological and epistemological questions surrounding music in the knowledge system of the medieval Islamic world by exploring the philosophical system of Ibn Sina and his later followers, all of whose works laid the foundations for scholars of music in the centuries to come. In particular, I will address how mathematics was conceptualized vis-à-vis the cosmology of the falsafa tradition as the discipline that examined the existents whose existence was dependent on physical matter but could be conceptualized without the said matter. Through this conceptualization of music and mathematics, scholars of music were able to broaden their subject matter to cover topics from the melodic modes in vogue in their time to the poetics of music. At the same time, since everything in the universe was connected to one another, music was linked with many other scientific disciplines such as astronomy and medicine.

Chapter 4 considers another major actor in the learning of musical knowledge, besides the patrons: professional scholars. While it is true that musical treatises were for the most part commissioned for the elites, once a text was out in the market, anyone with an interest in the subject and a small amount of money in their pocket could acquire a copy. Professional scholars pursued music as a part of their training in mathematics. I center my discussion around the studies of one such scholar of music at the madrasa of Mustansiriyya, who was a student of al-Urmawi himself. I analyze a rare manuscript that contains marginal notes written by this scholar who studied the subject matter under the master. This rare manuscript grants us a unique perspective into how scholars actually went about learning their subject matter.

Focusing on Menippus’ description of his celestial journey and the great cosmic distances he has travelled, I argue that Icaromenippus is a playful point of reception for mathematical astronomy. Through his acerbic satire, Lucian intervenes in the traditions of cosmology and astronomy to expose how the authority of the most technical of scientific hypotheses can be every bit as precarious as the assertions of philosophy, historiography, or even fiction itself. Provocatively, he draws mathematical astronomy – the work of practitioners such as Archimedes and Aristarchus – into the realm of discourse analysis and pits the authority of science against myth. Icaromenippus therefore warrants a place alongside Plutarch’s On the Face of the Moon and the Aetna poem, other works of the imperial era that explore scientific and mythical explanations in differing ways, and Apuleius’ Apology, which examines the relationship between science and magic. More particularly, Icaromenippus reveals how astronomy could ignite the literary imagination, and how literary works can, in turn, enrich our understanding of scientific thought, inviting us to think about scientific method and communication, the scientific viewpoint, and the role of the body in the domain of perhaps the most incorporeal of the natural sciences, astronomy itself.

We briefly offer the reader a sense of what “logic” is supposed to be: its scope, its goals, and the kind of tools logicians use. We discuss the relationship between logic and the rest of mathematics, outline various conceptions of logic and ways it has been applied, and offer a concrete example of the kind of reasoning one might wish to “formalize” and how this might look.

The economics used by governments is based on ideas from the 1870s, when economists adopted the language of science, but not the method. To make the maths easy to solve, they assumed the economy was simple, predictable, and static. Nobody believes these assumptions are true, but they still shape analysis that informs policy. When the economy is complex, uncertain, and changing, this kind of analysis can lead us to bad decisions.

This chapter is divided into three main sections. The first proposes the Ten Teacher Questions framework. This set of questions is designed to provide you with a generic framework for critical enquiry into all your pedagogical choices, and to connect your pedagogical knowledge to what you have learned in previous chapters. The second section provides the curriculum context structures ‒ that is, the ACARA Cross-curriculum Priorities and General Capabilities, which inform our work. The third section presents Teaching Ideas in Mathematics, The Arts and English. Our key message is not that you must implement every Teaching Idea! Instead, we hope the examples will consolidate a practical approach to harnessing the linguistic diversity of your students. We hope that you will grasp the principles which you can see at work in the Teaching Ideas, and the way that they respond to one or more of the Ten Teacher Questions.

In ‘Early Learning in Plato’s Republic 7’, James Warren provides an analysis of Socrates’ account of the sort of early learning needed to produce philosopher-rulers in Republic 7 (521c–525a), namely a passage describing a very early encounter with questions that provoke thoughts about intelligible objects and stir up concepts in the soul. Warren explains how concepts of number, more specifically the concepts ‘one’, ‘two’, ‘a pair’, and so on, play an essential role in these very early stages of the ascent towards knowledge, and he stresses the continuity between the initial and very basic arithmetical concepts and the concepts involved in more demanding subjects taught in later stages of the educational curriculum. On this account, Socrates is prepared to ascribe to more or less everyone an acquaintance with some, albeit elementary, intelligible objects. This, in turn, can shed some light on broader debates in Platonic epistemology about the extent to which all people – not just those whom Socrates calls philosophers – have some conceptual grasp of intelligibles.

The Introduction provides an overview of the book’s argument about how novels in nineteenth-century Britain (by George Eliot, Wilkie Collins, William Thackeray, and Thomas Hardy) represented modes of thinking, judging, and acting in the face of uncertainty. It also offers a synopsis of key intellectual contexts: (1) the history of probability in logic and mathematics into the Victorian era, the parallel rise of statistics, and the novelistic importance of probability as a dual concept, geared to both the aleatory and the epistemic, to objective frequencies and subjective degrees of belief; (2) the school of thought known as associationism, which was related to mathematical probability and remained influential in the nineteenth century, underwriting the embodied account of mental function and volition in physiological psychology, and representations of deliberation and action in novels; (3) the place of uncertainty in treatises of rhetoric, law, and grammar, where considerations of evidence were inflected by probability’s epistemological transformation; and (4) the resultant shifts in literary probability (and related concepts like mimesis and verisimilitude) from Victorian novel theory to structuralist narratology, where understandings of probability as a dual concept were tacitly incorporated.

Chapter 11 focuses on EC STEM education. It describes what STEM looks like in EC settings and identifies ways in which STEM elements can be incorporated into children’s learning. The chapter describes how STEM-related play can enhance young children’s appreciation of the world and provides a range of examples that have potential for STEM learning. Digital tools and applications for STEM learning are featured in this chapter.

The Tokugawa period saw a transformation in the systematic inquiry into nature. In the seventeenth century scholars were engaging in discrete fields of study, such as astronomy or medicine. But over the course of the next two centuries the fields that initially seemed distant and unrelated gradually converged into one enterprise that we now call “science.” Although Japanese scholars were not isolated from European science, it was not the outside influence that caused this transformation. Rather, the new conceptualization of science came from within, as different scholars came to align themselves along different lines. What brought them together was no longer social status, practical goals, or even their respective disciplines, but the kind of questions they asked, the kind of evidence they considered acceptable, and the sources they deemed authoritative. Together, they now engaged in Science, with a capital S, that was greater than the sum of its parts.

Symbolic tools represent, organize, and transform our knowledge of objects and events. The acquisition and internalization of symbolic tools change the way we think about the world. Different cultural subgroups use different symbolic tools and as a result, they shape their cognitive processes, even those as basic as spatial memory, differently. Moreover, some of the psychological functions that at the first glance should progress developmentally irrespective of the person’s experience actually depend on the acquisition and mastery of specific tools, for example, the graphic representation of objects. Even in societies with formal educational systems, the teaching of symbolic tools as tools is often neglected. Tables, graphs, and formulae appear as a part of the content material instead of being learned as specific tools. Many of the problem-solving mistakes made by students, for example in international exams such as PISA and TIMMS, reflect their poor mastery of symbolic tools rather than a lack of curricular knowledge. Educational interventions aimed at teaching students how to identify and apply the instrumental properties of symbolic tools lead to improved problem-solving in subjects ranging from mathematics to foreign language learning.

This chapter focuses on mathematics and computational thinking. In each chapter, the practice is dissected into distinct and clear learning tasks that serve as process goals for learning the practice. These tasks are then examined within the context of a self-regulated learning cycle and coaching strategies for instruction and assessment are emphasized. The instruction and assessment strategies are contextualized for students in grades 9–12 and focus on conducting an investigation on the factors influencing the period of a pendulum. The data practices for the investigation are infused with computational thinking. The tasks are reassembled into two case studies focused on the heating curve of water– one positive and one negative – to demonstrate how the learning tasks can be used by students and how teachers can support students learning how to plan and carry out investigations.

Thinking encompasses a very wide range of phenomena. Chapter 6 first comes back to a study focused on the pleasure of thinking itself. Pleasure is then examined in three modes of thinking: sense-making, reasoning, and daydreaming. Second, as acts of thinking are always situated in specific activities and anchored in various domains of experience, the chapter distinguishes various domains of knowledge: all are complex semiotic systems, culturally mediated, which can be more or less culturally shared and formalised. Third, the chapter examines trajectories of thinking in many systems of knowledge, formal or informal; starting with daily modes of thinking and their pleasures, it examines the pleasures of thinking in professional thinkers before exploring a specific form of sense-making connected to personal experiences. Altogether, this chapter shows that trajectories of thinking are dynamic and that they intermesh elements from a diversity of knowledge systems, moving along various modalities of pleasure.

Until recently it was thought that the mathematical solution to the formation of nonreplicating patterns that go on to infinity had not been solved before current advances in Western mathematics. But the discovery of such patterns at several Islamic religious sites prompts us to ask why it mattered to the Muslim artisans of that day to solve the mathematics and create such patterns. Just as Western cathedral art represented a cosmological view so, too, we may conjecture that to these Muslim craftsmen the representation of a world that is full of individual elements that relate to one another in unique ways replicates in the visual world what Allah has created for mankind in the world of social relations. By linking the art form and the mathematics to this broader social vision, we may be able to understand why the masters of that age chose to represent on the walls of their religious structures this particular cosmological pattern.

This chapter considers some practical applications of STEM in the primary classroom with a particular emphasis on STEM’s relationship to mathematics outcomes and the integrity of the mathematics as taught in the STEM context. This will extend to an exploration of Education for Sustainability (EfS) in the primary mathematics classroom, and opportunities for STEM tasks that are based on inquiry within the EfS space.