Professor Tanton speaks about the ceiling tile pattern that inspires his career in mathematics; he unlocks the mystery of those tiles, demonstrating the power of visual thinking; how similar patterns hold the key to feats of mental calculation.

Professor Tanton's tips on cutting through the confusing details about groups and objects, particularly when ratios and proportions are involved; learning through handy visual devices, which include blocks, paper strips and poker chips.

Seeing visual proofs for computing the areas of rectangles, parallelograms, triangles, polygons in general and circles; proving that for two polygons of the same area, one can dissect one polygon into pieces that can be rearranged to form the other.

Probing the computational miracle of place value, where a digit's position in a number determines its value; using this idea to create a dots-and-boxes machine capable of performing any arithmetical operation in any base system.