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Note to the instructor

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This is an “exploratory math” outreach activity, specifically aimed at dispelling the thrall of math phobia. There are two parts to that phobia - thinking that math is impossible to learn and that it is dull. Everybody agrees mathematics is useful, but less commonly appreciated, mathematics is something you can learn like anything else (nobody is “just bad at math”, but it seems a lot of people have wrongly internalized that rote memorization is all there is to math). Moreover, something people miss when talking to kids is that mathematics is something that can be pretty. Here we use the prettiness to trick kids into playing with math. We avoid the “scary” algebra by using purely numerical work and a lot of guided exploration. To facilitate the exploration, we use a programming language as a calculating and plotting tool.

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The final goal is to have the students be able to plot and explore natural fractals on their own. We will start by presenting a simple language for calculations and plotting (the Julia programming language). We learn the basics of manipulating lists of numbers, repeating operations until a condition is met, and plotting lines with given coordinates through the first hour or so. We introduce the notion of dynamical laws (using the example of population growth and avoiding any mathematical jargon) in the first half of the second hour. We introduce Mandelbrot’s “dynamical law” and then we let the students explore its convergence properties (by giving them the code to do that) for another half an hour. In the third hour we explain and rebuild that code from scratch.

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From the very beginning be very clear with the students that this activity is exploring a very particular mathematical object that happens to be beautiful - we are not teaching general mathematical or computer science techniques and what we show them will _not_ be directly useful in class, _nor_ will it be particularly related to any mathematical techniques employed in computer generated art in general. We are exploring one pretty object in and endless see of mathematical beauty.

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Directions to how to approach the students is left in similar panels.

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 See “A Mathematician's Lament” by Paul Lockhart, freely available on the Internet.

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The activity is designed at Yale for SWE's Engineering Day.

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