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Threshold concepts are those which are characteristic of the discipline, and without which students are able to make only limited progress in developing disciplinary ways of thinking. They are characterised by being: transformative, in that once understood, they result in a significant shift in the perception of a subject; irreversible, in that the change of perspective is unlikely to be forgotten; integrative; in that they expose the previously hidden relatedness of ideas; bounded, in that they open up new conceptual spaces, and; potentially troublesome, in that they are hard to acquire. This chapter examines curriculum resources from the Primary Connections and reSolve: Maths by Inquiry projects and proposes some threshold concepts in primary mathematics and science. The identification of threshold concepts is important for the research community, for educational designers and for teachers as explicit attention to the concepts may help to address the misconceptions that often lie at the heart of the troublesomeness of concepts encountered in higher education.

This concluding chapter draws on the work of previous chapters to draw out three dimensions of connection. The first dimension is the connection between mathematical ideas that is characteristic of deep understanding and the distinctive epistemological basis of the discipline of mathematics. The second is the connection between mathematics and culture, in which the joy and delight of playing with and discovering mathematical ideas permeate learning. The third is the deep connection between mathematics and the world, in which mathematics both arises from a consideration of real-world problems and models those problems. Such connections take time and are often undervalued in an era of accountability and easily measured standards. The chapter challenges three dominant metaphors of education—those of education as factory, clinic, or race—and proposes a metaphor of slow maths, borrowing from the slow food movement in which connections to people, to culture, to history, to knowledge, to learning, to the world, and to self, are paramount.

This chapter examines the philosophical underpinnings of Australasian mathematics education research between 2012 and 2015. It takes a hermeneutic approach, seeking to uncover often hidden assumptions about ontology, epistemology , aesthetics, ethics and logic. The first part of the chapter explains the approach taken and outlines the set of papers considered. The chapter then examines the set of keywords used in Australasian mathematics education research papers published in major international journals, seeking to identify broad themes or omissions. It then takes a more detailed look at a purposive selection of 26 papers, chosen to reflect the themes in this Review. Finally it examines papers that explicitly discuss epistemology to identify researchers’ underlying assumptions about the nature of knowledge and its acquisition. The chapter points to some important tensions within the research and suggests that such tensions can be used as a creative force to enable mathematics education researchers to better identify and question the philosophical assumptions that underpin their research.