Plane Geometry - Bruce E. Shapiro

2 SECTION 1. PREFACEare enrolled in integrated programs that teach content simultaneously withpedagogy (but in different classes), or already have teaching experience andwant a refresher in the content area. To these students what I have to sayhere is probably nothing new.Integrating Content with Pedagogy. Our focus is on mathematicalcontent. But can content be separated from pedagogy? Should it be? Thisis not an easy question. Some would say that as mathematicians we should“leave sociology to the sociologists.” This has a grain of truth to it - wecan best teach what we know best - and instructors might do more harmthan good by straying outside their own specialized disciplines. It seemsnearly everybody has an opinion on the matter. But to leave pedagogy outof the equation completely would be you, as a future teacher, a disservice.This is not a class in pedagogy. I am not going to tell you how to teach.I couldn’t even if I wanted to. But what I want you to do as you progressthrough the semester is to think about how you can communicate yourknowledge to others. Start with your colleagues in this class. If you reallyunderstand something you should be able to explain it to people with backgroundssimilar to yours. Ask for their criticism and listen to what theysay. Take their suggestions into account the next time you work together.Of course being able to explain your proof of the Pythagorean theorem tothe student sitting next to you does not mean you’ll be able to explain itto your ninth grade students. But it will get you thinking about explainingthings in your own words. Listen to what you say. Explain things the wayyou would want them explained to you. Think about what questions youmight have, and answer them to yourself.As a teacher in training, you should continually be asking yourself “howam I ever going to teach this?” whenever you are presented with newmathematical content. Yes, you should certainly master the material. Yes,you should learn the “big picture” - how does this new content mesh withwhat you already know about math? Where else can we go from here?What other theorems, results, and methods can we obtain? How can thiscontent be applied to other applications outside pure math?Topics and Presentation. This class will give you the big picture.You’ll get “down and dirty” in other classes that are more focused on specificgrade level requirements. How to present this big picture has a longand well storied history.You could say it all started with Euclid – but all he really did was write« CC BY-NC-ND 3.0. Revised: 18 Nov 2012