Sunday, March 20, 2016

Teaching teenagers that math is more than what they learn in school

I found slides for a presentation by Ben Brubaker titled "Exploring great mysteries about prime numbers".

On his first slide, he lists some big questions:

  • What is mathematics? 
  • What kinds of questions do mathematicians try to solve?
  • What do mathematicians do all day? 
  • How does mathematical proof resemble a poem or a painting? 
  • Why should we learn mathematics? 
  • How can we become better at mathematics?
These are great questions, and I think that all math outreach should aim to help students answer them. 

He also talks about two mistaken impressions about math that high school perpetuates: 

1. Mathematics is a single path leading to an endpoint: calculus. 
2. All mathematical proofs are like the two column proofs you do in geometry. 

Here's a wikipedia article which talks about all the different areas of math - many of which do not use ANY calculus.  It would be possible to design a high school mathematics curriculum which is leading towards some other pinnacle - like elementary number theory or probability and statistics!  A mathematical proof is an argument that shows why your statement is true.  It often takes the form of a paragraph, and the details you need to include include depends on the audience that you're trying to convince.  

I've been thinking a lot about having high school students explore the prime numbers by writing programs to get lots of data, looking for patterns, and trying to explain them. I think that it's a great topic - prime numbers are extremely concrete and fundamental, but students can quickly make super deep conjectures and interact with the forefront of mathematical knowledge.  It's important to me to give students the experience of exploring, creating, and doing mathematics - looking at examples, asking questions, making conjectures, and trying to understand why they're true.  While there's not a lot that high school students can prove about primes (very quickly, the proofs get complicated and require a lot of background), they can get a good sense of what's going on, and I think that this is an extremely important but untaught skill. 

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