Change and invariance: a textbook on algebraic insight into numbers and shapes

Review "This book is a great addition to my bookshelf. In fact, it may spend little time on my shelf as it will be pulled out frequently, as I prepare my teaching in problem solving courses, as well as in mathematics and didactics (methods) courses for teachers .... It is much more than a textbook. It is a resource book for any mathematics educator and for a great variety of courses." --Rina Zazkis, Professor, Mathematics Education, Faculty of Education, Simon Fraser University "This book is ideal for anyone who sees themselves as a learner of mathematics or epistemology. One of the dangers in focusing on levels and characteristics of teacher knowledge is that the knowledge can be seen as static. But we are all learners. The various chapters introduce many ideas that can prompt new insights by educators into apparently basic topics like number operations, analysis of geometric shapes and measurements of perimeter and area. What makes this book outstanding is the literally hundreds of wonderful examples of questions and prompts that pose potential classroom explorations for students supported by mathematical and pedagogical analysis, focusing on the way that change and invariance contribute to an appreciation of the opportunities in both. I have been surprised that in many of the celebrated texts in mathematics education the examples given are not only few in number but mundane is substance. That is not the case here. I highly commend this book to experienced educators with an interest in learning experiences for students, to leaders of teacher professional learning of prospective and practising teachers, and indeed to prospective teachers themselves. (. . . ) I emphasise that the authors pose many similar investigations which are subsequently followed by both mathematical and pedagogical analyses and which can be used by readers to create purposeful investigations for themselves and their students. This is one of those books that will repay the investment (of time and money) many times over. "--Peter Sullivan, Emeritus Professor, Monash University Product Description

What is the connection between finding the amount of acid needed to reach the desired concentration of a chemical solution, checking divisibility by a two-digit prime number, and maintaining the perimeter of a polygon while reducing its area? The simple answer is the title of this book.

The world is an interplay of variation and constancy – a medley of differences and similarities – and this change and invariance is, largely, a language of science and mathematics. This book proposes a unique approach for developing mathematical insight through the perspective of change and invariance as it applies to the properties of numbers and shapes.

After a short introductory chapter, each of the following chapters presents a series of evolving activities for students that focus on a specific aspect of interplay between change and invariance. Each activity is accompanied by detailed mathematical explanations and a didactic discussion. The assignments start with tasks familiar from the school curriculum, but progress beyond the menial to lead to sophisticated generalizations. Further activities are suggested to augment the chapter’s theme. Some examples: “How to represent all the integers from zero to 1000 using ten fingers?”, “How to win at the game of Nim?”, “Why do different square lattice polygons with the same area often have the same perimeter?”

This book can be used as a textbook for pre-service mathematics teachers and is primarily intended for their academic instructors. Essentially, students, teachers and anyone interested in elementary mathematics will enjoy the elegant solutions provided for the plethora of problems in elementary mathematics through the systematic approach of invariance and change.