Monday, November 1, 2010

Goldbach's Conjecture and the Millennium Prize Problems

I just got out of my 8th grade Algebra I class, and we didn't get through any of the material I had planned on covering. Yet, it was one of the best classroom experiences I have had in my brief teaching career. The class was supposed to begin with some mental math warm-up exercises, then take a brief detour into number theory and Goldbach's Conjecture, and then seamlessly return to our lesson on functional notation so the students would understand their homework. Instead, what was meant to be a 5 minute nugget of number theory, turned into a 45 minute explanation of communication in the 1700s, a reading of one of the Millennium Prize Problems, and Google Image searches of Grigoriy Perelman and Terence Tao.

First, I baited the students with the questions: "Find two prime numbers that add up to 64" and "Find two prime numbers that add up to 90". They saw it as a challenge, and the fact that there are multiple correct answers to each question makes it such that multiple students can chime in with their answers. In essence, almost everyone was invested and involved in solving these math 'puzzles'. Then I presented Goldbach's Conjecture, that any even number greater than 2 can be written as the sum of two primes. I explained how Goldbach proposed it in a letter to his friend, Leonhard Euler, a fellow mathematician. The students loved the idea of two great math minds conversing through letters to present and prove conjectures and theorems. The Goldbach Conjecture is also one of the oldest open problems in math, which led me to talk about the Millennium Prize Problems and their $1,000,000 award. All of a sudden we were Googling, reading the open questions, and looking up Grigoriy Perelman (the mathematician credited for solving 1 of the 7 problems - the Poincare Conjecture) and Terence Tao (the preeminent number theorist in the world). One student asked, "What do they look like?", so we searched for images of them. It was a brief detour into history and current events and images, and while they will never have to know any of this for a math test or any other test for that matter, it piqued their interest to the extent that they left class saying, "I actually had fun!"

Building curiosity in students is far more important than building their knowledge base, so while I didn't get through my lesson plan, I think I achieved more in this class period than I originally could have hoped for. Here's what I learned from this experience:

1. Baiting students is important. If I had just presented the Goldbach Conjecture, which I find inherently interesting because I'm a math teacher, then the lesson could have flopped. Baiting students and getting them interested is critical to a successful class.

2. If a lesson is taking a detour, but you get the sense that the students are learning or sense that their interest is piqued, throw your original plans out the window and go with what is working. To be honest, I think this was the most fun the students had in class and the most fun I've had teaching.

3. Incorporating other disciplines into your class is a great way to build a lesson or a lecture. Too often students come into math class and all they think they will see are numbers and letters, when in reality our education system should focus on interdisciplinary approaches to learning. Even a 5 minute explanation of how Goldbach and Euler communicated brought the lesson to life, and took it out of the realm of pure mathematics and into the realm of real life.

My favorite professor at Williams, Michael Lewis, taught American Art and Architecture and the reason so many people loved his class was because his lectures weren't Art History lectures, they were just Interesting Lectures. He synthesized so much information into his class, from all sorts of disciplines, that it felt like you were just listening to him tell a story and you couldn't help but be engrossed.

One day I hope to be as eloquent, interesting, and knowledgeable as Professor Lewis, and today was the first time I really felt like I had the potential to get there.

1 comment:

  1. Just out of curiosity, did any student ask you why does Goldbach's conjecture work only for even numbers? If none did, do you think asking them and subsequently giving them a simple analytical answer would augment the fun?