Tuesday, November 12, 2013

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?


  1. I absolutely LOVE when I have those moments of "This is what I WOULD have done but this is what I'm doing now."

    It really shows me how much I've changed over the years and it's very refreshing.

    1. This is exactly why I'm glad I'm doing a better job of blogging regularly this year. I'm doing a LOT of reflecting and it's helping me change my practice for the better. I don't think I'd be noticing these things quite as easily if I wasn't blogging.

    2. This really looks great, this will be a fun educational blog, and i really admire your thought's and looking forward your post in future. i am student of a mathematics subject and i feel little bit difficult in a Math Addition , but my favorite portion of a math is Geometrical section.

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