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Not a regular figure, but just any one that came to hand, using the usual technique of following the diagonals of an imaginary pentagon.\u00a0 He then asked for the sum of the angles at the points of the star.<\/p>\nI am inspired, by several previous blog entries, to write about my own mathematical awakening, and what I\u2019ve learned from reflecting on it.<\/p>\n
I went to New York City Public Schools, in the Bronx.\u00a0 I always enjoyed arithmetic and mastered it easily.\u00a0 I remember not knowing what \u2018fractions\u2019 were, but don\u2019t remember learning about them, any more than I remember learning to read.\u00a0\u00a0 The understanding came to me naturally, and I hardly noticed the process. \u00a0Even first year algebra didn\u2019t seem like a learning process, more like a set of exercises.\u00a0 So I had mastered a lot of mathematics (well, a lot of algorithms) before I really understood what it was I was learning.<\/p>\n
A revelation came in ninth grade, when I was 13.\u00a0 Ms. Blanche Funke, a good math teacher in JHS 135, took some of us during lunch and organized us as a math team, to compete against other local Junior High Schools.\u00a0 Now this is work I have since spent decades doing, and I know now what could have been done.\u00a0 But Ms. Funke didn\u2019t quite.\u00a0 Her idea was to give us advanced training in textbook algebra\u2014not to find ways to make us think differently about that same algebra.<\/p>\n
So she gave us the definition of an arithmetic progression, and the standard formulas.\u00a0 And a problem something like: \u201cInsert 3 arithmetic means between 8 and 20.\u201d\u00a0 I loved this work.\u00a0 Plug into one formula, get the common difference, then plug into another formula and get the three required numbers.\u00a0 I could see what I needed to do and took joy in starting the work.<\/p>\n
But next to me was my friend David Dolinko, and he was busy drawing something in his notebook\u2014some diagram of a molecule in chemistry.\u00a0 (Professor Dolinko has lately retired from the UCLA School of Law).\u00a0 I poked David.\u00a0\u00a0 \u201cC\u2019mon.\u00a0 Let\u2019s do this problem.\u00a0 It\u2019s fun!\u201d<\/p>\n
David looked at me, as if annoyed at the interruption:\u00a0 \u201cOh, I did that already.\u00a0 Eight, eleven, fourteen, seventeen, twenty.\u201d\u00a0 And went back to his drawing.<\/p>\n
That moment changed my world.\u00a0 Suddenly I realized that these formulas had meaning, were trying to express something.\u00a0 They were expressing that the numbers were \u2018equally spaced\u2019.\u00a0 So David could just pick them out\u2014the numbers were small\u2014and didn\u2019t have to bother with the algebra.\u00a0 Algebra has meaning.\u00a0 And if you know its meaning you can use it more effectively.\u00a0\u00a0 Suddenly, instead of black and white, I saw the world of algebra in color.<\/p>\n
I thought about this a long time.\u00a0 The colors attracted me more and more.\u00a0 I wasn\u2019t just good at mathematics.\u00a0 I enjoyed it, and enjoyed being good at it.<\/p>\n
Well, the next year I was still sitting next to my friend David, in the last seat, last row of a classroom in the Bronx High School of Science.\u00a0 We were taking geometry, the classic neo-Euclidean syllabus, taught by one Dr. Louis Cohen.\u00a0 He was a somewhat impersonal teacher, or so we thought, but a master of his discipline.\u00a0 And of teaching it.\u00a0 So one day he had covered (I don\u2019t remember how) the theorem that the angles of a triangle sum to 180 degrees.\u00a0 The lesson had gone quickly, so he filled the time with some \u2018honors\u2019 problems: the sum of the angles of a quadrilateral, some problems with exterior angles, and so on.\u00a0 And to cap it off, he drew a five-pointed star on the board:<\/p>\n
Not a regular figure, but just any one that came to hand, using the usual technique of following the diagonals of an imaginary pentagon.\u00a0 He then asked for the sum of the angles at the points of the star.<\/p>\n
My hand shot up, seemingly of its own accord.\u00a0 \u201c180 degrees,\u201d I said, without quite knowing why.\u00a0 And to my horror, Dr. Cohen strode calmly down the aisle to my desk, with a piece of chalk in his hand, handed me the chalk, and asked me to explain to the class how I had figured this out.\u00a0 But I didn\u2019t know how I had figured it out.\u00a0 I just saw it, with intuitive clarity.\u00a0 What was I going to do?<\/p>\n
I was lucky that we sat in the back of the room.\u00a0 As I saw him coming towards me, I began to analyze my own thoughts.\u00a0 And as I walked to the front, I figured out what to say.\u00a0 To this day I remember my hand trembling and my voice shaking as I pointed out certain triangles, certain exterior angles, and got the angle measures all to \u2018live\u2019 in the same triangle.\u00a0 Dr. Cohen praised me, then gave a slicker version of the proof that must have clarified it for the other students.\u00a0\u00a0 Of course, there are better ways even than his to prove this statement.\u00a0 If the reader can\u2019t think of a nice proof offhand, take a look (for example) at\u00a0\u00a0 https:\/\/www.khanacademy.org\/math\/geometry\/hs-geo-foundations\/hs-geo-polygons\/v\/sum-of-the-exterior-angles-of-convex-polygon (accessed 6\/2018).\u00a0 \u00a0The argument can be adjusted to cover non-convex polygons.<\/p>\n
Why is this important?\u00a0 Well, it is important for us to understand that the language of mathematics is a language of thought.\u00a0 And that thought is synonymous with intuitive thought.\u00a0 We sometimes get caught up in the expression of our intuitions, and fail to go back and make clear, even to ourselves, what we are talking about.\u00a0 This phenomenon has deep implications for teaching.\u00a0 How we do this, how we know it has happened, how we integrate it into the teaching of mathematics as a forma language, are all questions we must struggle with.\u00a0 But they are not questions that we can beg.\u00a0 We must somehow be sure that students can eventually understand our results on an intuitive level, whether or not we communicate with them on this level directly.\u00a0\u00a0 Without that, we are teaching algorithms\u2014even algorithms of proof\u2014and not mathematics.<\/p>\n
I invite readers to contribute their ideas to this blog about how to make mathematics accessible on an intuitive level.<\/p>\n
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