Twosday Things: Ingenious Responses. Also Fish.

Time again for Twosday Things!

Thing #1:
The other day, I stepped out of my classroom for a moment. When I came back, one of my students had drawn this on the board:

I took one look and figured, "what the hell, I'll tweet it." So I did:

One reply stated that this was probably a reference to Fairly Oddparents, which given the age of my current students wouldn't surprise me.

However, the prize for Most Brilliantly Mathematical Response definitely went to Gregory Taylor (@mathtans on Twitter):

I feel like if I'd gotten that kind of response from a student, I'd have just given them an A for the semester right then and there. (Okay, maybe not. But I'd be impressed.)


Thing #2:
One thing I've noticed about my teaching practice this year is that I've become more open-minded with how students respond to questions and problems.

Here's an example of what I mean. One of my students came to me today with the following solution to a problem:

Two disclaimers: (1) The student obviously took some "mathematical liberties" when drawing this diagram. (2) The student did much of their work without a calculator, but explained to me in person what was done: he used the distance formula to calculate the length of each side, then used the Pythagorean Theorem to see whether the three sides formed the sides of a right triangle.

Out of context, this seems like a perfectly reasonable way to solve to problem.

However, this actually came from a problem set focused on parallel and perpendicular lines. The solution path I was "looking for" was to calculate the slope between each pair of vertices and determine if there were two sides that were perpendicular to each other.

What's my point here?

A year or two ago, this is probably how I would have responded to the student's work: "Um... well, that's ONE way to solve it I guess, but I was really looking for [insert what I was looking for]."

But today, this is how I responded: "Whoa, that's brilliant! I hadn't actually thought of solving the problem that way, but that makes a lot of sense! This is genius!" And I followed that up with an explanation of how most other students were solving the problem by calculating slopes as I described above; but the student's mathematical reasoning was both valid and awesome.

This is a great example of how I've changed as a teacher this year. I've always been okay with students coming up with different solution paths to problems; however, I often tried to steer them toward particular solution paths, even if what my students were doing was perfectly reasonable.

Insisting on particular solutions paths isn't, in and of itself, a bad thing. There are situations where it's good to train students on solving a problem a particular way; doing so adds to their "mathematical toolbox," equipping them with a variety of skills for solving problems.

But there are times, I think, when we as math teachers need to be okay with students solving problems in unexpected ways. I think this instance was one of those times. This was a student who had been struggling with math at times this year, but today he came to me with a brilliant solution that I wasn't expecting to see. That deserved praise and recognition.

As I said, a year or two ago, I would have been "just okay" with the method my student used to solve the problem, but not all that enthusiastic because he hadn't done it the way I was trying to teach.

I shudder to think that, just a year or two ago, I wouldn't have embraced his work as enthusiastically as I did today. If I had responded with, "Well, that's one way to do it, but...", I probably would have done harm to the student's mathematical confidence. He applied previously-learned mathematical knowledge to a different type of problem. How could I have any problem with that?