It took 358 years to crack Fermat’s Last Theorem. And, prior to its proof in 1994 by Andrew Wiles, it held The Guinness World Book of Records for having the most number of unsuccessful proofs. It is kind of deliriously maddening to think that something like the Pythagorean Theorem, that is almost universally known by every kid by their teens, had an “innocent” idea about exponent solutions other than two become entrenched in the lore of mathematics for 4 centuries.
It has even made several guest cameos on The Simpsons.
The entire history of the problem, while woven with many advances and mathematicians, is really a story about one of the most important and overlooked ideas of mathematics — especially in the world of speed and competitiveness found in K to 12 math education — time and inertia.
Some of the most brilliant minds in mathematics were stuck on this problem,and were outlived by its final resolution. But, there is something hidden here which speaks to what binds strategic thinking and a solution — that is…nothing.
In my IGNITE Talk at the Annual NCTM Meeting in San Diego, my opening two slides were blank, and I didn’t speak for the first 5 seconds or so.
I was trying to pay symbolic homage to the most beautiful and unheralded gift of mathematics — stillness. We have all been given that gift. You know, when you have that feeling that this problem has gotten the best of you. No more thinking. No more strategies left. Pondering and mulling coming to a stop. The only think left in the wake of all this problem solving is some blank staring that is a mix of mild fatigue, daydreaming, and quiet reconciliation. That the problem in front of you is outside of your capabilities — or at least, for now.
And, that is more than okay. That is the world of mathematical thinking.
I had hardly any of those moments prior to high school. The best courses and the best teachers I had offered plenty of moments to hit that mathematical wall, and yield to stillness. And, of course, university was filled with those moments — more than I should be admitting!
Those days — sadly — are almost gone. I don’t have as much time to be beaten by time. Most of the mathematics I encounter and promote is in the wheelhouse of my understanding. I can only talk about mathematical struggle from a nostalgic lens. So many days I wish I could sit with a problem and struggle with it, yield to it, and share that story.
We have been asked to slow down by Carl Honore for almost this entire century.
Mathematics goes one better. It asks us to stop. Well, “asks” is being polite. It demands that we stop. That stopping is not a sign of weakness, ineptitude, or failure. That stopping is a pause button is absent from our lives. And, while we should aim for rich understanding and illumination of mathematical ideas for our students and ourselves, we are doing a great disservice to the learning of mathematics and to the general wellness of how we think if we are obsessed with just showing the colors of mathematics — without the needed white space.
Just remember, most of the mathematical canvas is blank. It is not to be filled in. It is to valued for exactly what it is — nothing.
Promoting productive struggle is wonderful. But, we should also be promoting “letting go”, stopping, and maybe doing something else — like go for a walk(a James Tanton strategy). If a solution comes back to you. Great. If it does not. Great. Try again, or try another problem. Appreciate those blotches of vivid color you find, but be respectful, mindful, and equally appreciative of all that is not.
It is just as important.