Imagine a plastic cup lying on the floor. Give the cup a nudge so that it begins to roll. What does the path it takes look like?
So begins the journey the author takes you on in this book. Along the way, he makes the case that the ability to imagine, manipulate, and explain mathematical images and situations is fundamental to all mathematics and particularly important to higher level study. Most importantly, drawing on years of experiments in his own classroom, he shows that mathematical imagining is a skill that can be taught efficiently and effectively. This text describes an original routine that gives students space and time to imagine a mathematical situation and then revise, discuss, and act upon the mental images they create. You can use this creative routine to glimpse into your students' thinking and discover teaching opportunities, while empowering them to create their own mathematics. Inside you'll find the following: an introduction to the routine including the rationale behind it, facilitation guidance, and classroom examples; modifications to implement the routine in your classroom, even with varying time constraints; 37 exercises broken into four categories: constructions, problem-solving, reasoning, and paradoxes; discussions of the mathematics involved in each exercise, including possible follow-up questions; instructions on how to create your own exercises beyond the book. This one-of-a-kind resource is for secondary teachers looking to inspire student creativity and curiosity, deepen their own subject matter knowledge and pedagogical content knowledge, and invite all students to access the power of their own mathematical imaginations.