Introduction to the Special Issue on Personal Mathematical Knowledge in the Work of Teaching/Introduction au numéro spécial sur les connaissances mathématiques personnelles dans le travail des enseignants

Anne was on a crowded bus looking over the shoulder of a young student who was trying to solve a geometry problem: to select, from several choices, the ratio of the radii of the inner and outer circles of a regular hexagon. The student was clearly stuck and going along irrelevant directions. When Anne asked, “How are you tackling this?” the young woman looked surprised and then took out her earphones and said she had let the radius of the inner circle be 1. Anne said, “Good choice” (while thinking it would have been much more use to label the outer radius), and asked if that had been helpful. A discussion of possible approaches ensued, with Anne trying not to give any direct clues. One of the possible answers had √3 in it and Anne asked if that suggested anything. The student talked about angles of 60 degrees, and Anne said, “You have told me everything you need to know to resolve this.” Before she left the bus, Anne said, “Whenever you label anything in a geometry diagram, choose a label that applies to as many aspects as possible.”