Breakthroughs, "A-has", and "Finding the Door"

I highly recommend this article on Yitang "Tom" Zhang's breakthrough on the Twin Primes conjecture. It's readable, inspiring, and full of lessons for elementary and secondary teachers of mathematics.  Don't worry about understanding the math (although it does explain it pretty nicely), just read Tom's story, from unemployed math PhD helping out at Subway, to becoming a professor, to his big breakthrough.

First, our students have these kinds of breakthroughs each and every day.  They don't have to be Archimedes style "eureka" and "jump from the bathtub" breakthroughs.  They aren't Einstein imagining himself on a beam of light, and conceptualizing relativity.  But they are breakthroughs, nonetheless.  Rather than bolts from the blue, these are often subtle and sudden realizations that they are closer to solving a problem than before.  You know, the "a-ha" moments.

These are the moments when things become clear, when they can see their way through a problem. They can begin to articulate a solution through the complex interaction of math processes, content and background knowledge, and classroom culture that we call problem solving.

There is a lovely description in the article of Mr. Zhang walking around and thinking about the problem, until one day, he "found a door".  He realized what tool he needed to solve the problem, and how he would do it.  The unbelievably complex math he was working on began with selecting tools he would use, and finding a way to represent the problem.  Sound familiar?  They should- they're two of our math processes.  If the math processes are the "actions of doing" math, doesn't it make sense that a professional mathematician would use the same processes as our young learners?  I think so- practice, experience, and knowledge background being the main difference.

The growth mindset math learner might say, "I can't solve this problem...yet."  We might change that to "I haven't found the right tool...yet."  Or:  "I don't know how to represent this situation with math...yet."

We could also say, "I haven't found the door...yet."  After you find it, you can walk right through, after all.