The learning and teaching of number. Paths less travelled through well-trodden terrain

As suggested by its title, this book takes the reader along paths in the “well-trodden” terrain of the learning and teaching of number, successfully guiding her through some less traveled paths. Right from the start, the authors point out the complexity of such a terrain, but they do so in a graceful and non-threatening way; they assign the reader a first task: “Give an example of number. And another, and another. Give an example of a number that is different from all the previous examples.” (p. 3). The authors then hint at the struggle of agreeing upon a definition of number. From there, they rapidly turn to the comforting and well-known solution in Mathematics of “defining by properties” (p. 4) and to the ways in which numbers are used: as labels, as count, as measure, as location, as multicultural artifact, as action, as abstract mathematical construct. The first chapter closes with a section on operations on numbers. Once the mathematical scene is set, the authors continue to prepare the reader for her journey with a second chapter, “Getting ready to act,” in which the authors act as the reader’s teachers, but also as teacher educators, imagining that she might find herself in the position of teaching students of her own about number. Hence, the authors explain many of the pedagogical moves that they, as educators, are prepared to make as their classes explore the terrain along the paths presented in the subsequent nine snapshots. Indeed, the tasks in each snapshot, as well as the comments and invitations to act between tasks, are built around the aspects introduced in this second chapter: mathematical themes (reversibility, invariance in the midst of change, freedom and constraint, presentation and re-representation), mathematical actions (discerning difference and seeking sameness, stressing and ignoring, specializing and generalizing, imagining and expressing, conjecturing and convincing, organizing and characterizing), actions to initiate and support activity (doing, talking and recording, say what you see, watch what you do, manipulating, “ – getting-a-sense-of – ” articulating, constructing personal narratives and explanations, scaffolding, and fading). Needless to say, they also provide advice on using tasks with learners. Instead of summarizing the content of each snapshot, I will touch on some of them in the following sections, where I highlight what I see as two great strengths of this book: it is a beautiful application of research to practice, and it offers pathways to inclusive approaches to the teaching of number for a broad spectrum of students.