Teaching to See like a Mathematician

My goal with the use of visuals in mathematics I am a research geometer, working on applied problems such as: What shapes of built frameworks will stand up or fall over? How can we program geometric constraints for CAD or CAGD? How can we reconstruct 3-D objects from a single picture or several partial views? What parts of proteins are flexible? and What patterns occur in computational origami. It will be no surprise that I make substantial use of static and dynamic diagrams as well as physical models in my research, and in my communication. However, over the last decade I have worked consistently to make the use of visuals more explicit, more precise, and more extensive in my mathematical practice. When I write, I often have a sequence of visuals (or mixed visuals and text, as in an augmented PowerPoint presentation) in mind around which the text is structured. (This is true of the current paper see two samples below. The first is a visually presented mathematics problem.) This point form from that presentation summarizes where I begin these reflections: There has been a difference between my public and private face in mathematics • I do my mathematics visually, in private; • I question, problem solve, analyze, explore, prove, answer, and communicate visually. • Choosing when and how I use visuals has changed the questions I pose; changed the methods I use to solve them changed the answers I give changed my communication and my teaching • I wish the same options for my students.