This is a quick summary of My Favorite from TMC.
I spent three weeks before Twitter Math Camp backpacking the John Muir Trail in the Sierras of California. I spent a lot of time in really beautiful places, like this one:
Of course, hiking alone for a few weeks, I got bored from time to time, and I brought some math along with me (Five Triangles and Paul Lockhart’s Measurement were great sources of fun problems). One in particular that I was working on is this problem from Five Triangles, finding the proportion of the area that is shaded of the three congruent equilateral triangles shown below.
It had driven me nuts for awhile before my trip, and I spent some more time digging into it.
While sitting in my tent at the spot I took the picture above, I ended up finding a long, circuitous solution. It’s pretty ugly, and I won’t share it here so you can enjoy playing with it. My solution involved solving several equations that looked similar to this one:
Of the four equations I needed to solve, I got three of them wrong the first time. I managed to make mistakes on the above problem three different ways before I got it right.
At first this really frustrated me. I know how to solve that equation (I promise, I do!). But it was just one step in a long, frustrating process. And I was sitting in a tent at 11,500 feet, in the middle of a backpacking trip, wondering why I was hiking alone for three weeks. Worried that the noise I just heard was a bear, or a mountain lion. Dreading eating oatmeal squares for the 15th time for breakfast the next morning.
And I think that’s the way too many of our students experience math. Hard math problems — the problems that we, implicitly or explicitly, are the ones that we communicate to students are the most important to be able to solve, are ones with lots of steps. And problems with lots of stops are the toughest one for teenagers in particular, but any human, to solve. They’re thinking about whether that girl two tables over likes them, or whether their hair looks good, or whether they made the basketball team. In particular for the students who struggle the most, we push these problems as what they need to be able to practice until it makes perfect. But there’s no reason that every math student in the country needs to be able to solve that equation perfectly every time. I have a degree in mathematics, and when I was a bit distracted, I failed miserable at solving those equations.
This is a call for something that I think all teachers understand, but is too easy to forget. Deep problems, problems that are worth solving, are not defined by the number of steps needed to solve them, but by the quality mathematical insight students need to reach a solution.
This was my favorite mathematical experience of the year, and it is a reminder for me, as the school year starts, that I send explicit and implicit messages to my students about what quality math looks like. I want to pay particular attention to what I hold up as high-quality math, because all of my students, but in particular my students who struggle the most, will be best served if they come into math class trying to look at a problem, or a topic, or the world in a new way that helps them learn something — and not trying to get all of the steps in the right order so they get a check mark on a quiz or a test.
I’m glad you enjoyed your hike, not getting bitten by snakes, scorpions etc.
The triangle problem is delightful. I get 5/12. It seems to work for any triangle. Hint: join the tops of the first two triangles and figure out where the dividing line crosses the left side of the second triangle. DO NOT USE ALGEBRA OR COORDINATES!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
Addendum: That is, 5/12 of two of the triangles.
Dylan, thanks for sharing this at TMC. I’m still trying to think of a simple solution, but no luck yet. To be honest, I didn’t read this particular post because I’m worried you put a solution in there.
Thanks, Nathan. No solution in the post, so feel free to read away. If you solve it or get sick of it, here’s another problem I spent a long time on — http://fivetriangles.blogspot.com/2013/11/115-november-nonagon.html
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