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?

Multiple Solutions (A follow-up to "When Is the Right Answer the Right Answer?")

A couple of weeks ago, I wrote this post about how I wanted my students to determine equations of lines, given certain information. The broader point, I think, was realizing that my students had more than one option for determining answers to the problems they were working on, and being okay with that. (Why wouldn't I be?)

I had another "when is the right answer the right answer?" moment in class yesterday that I thought was really super-cool.

Two students were working together on the same problem. They came up with what they thought were different answers, so they were wondering who was correct. Their work is shown below:

So both students used point-slope form for their equations, and came up with two answers that looked different. This peculiarity made them wonder who was right and who was wrong. (Which, in turn, makes me realize that I still have a lot of work to do with teaching them about making sense versus being right.) They called me over to ask me who had the correct equation.

I must have been really busy at that moment and not really thinking, because I looked at their answers and said, "actually, you're both right." Not that I was wrong in saying so; but I regret that I didn't recognize the teachable moment that had presented itself. This would have been a great opportunity to ask each of them what they thought about their equations, how they came up with them, why they thought their answers made sense, why the other person got something different, and whether or not it made a difference which point they used for point-slope form. Still, it was a really cool moment: two students have a spirited debate over who had the "right" equation, when really they were both right. It was my favorite moment of class from yesterday.

Fortunately, the same thing happened today, on the same problem, with the same work as shown above, between a different pair of students. Grateful for a second chance, I was able to stop and facilitate an awesome math discussion between the two of them.

One student was adamant that the "first" point, (-4, 3), had to be plugged in for point-slope form instead of the "second" point, "because they're X₁ and Y₁," she reasoned. She said this because she had labeled the coordinates as such when using the slope formula to determine the slope:

And point-slope form was written on the board as Y - Y₁ = (X - X₁). So I could see where she was coming from.

I asked her, "so, how would you label these points if the order was swapped?" In other words, what if the problem listed the points "(6, 1) and (-4, 3)" instead of the order they were given? She responded that she would have labeled (6, 1) as (X₁, Y₁) and (-4, 3) as (X₂, Y₂).

My next question was, "So would that change things? Would you get a different slope, for instance?" The student initially thought that yes, she would get a different slope. The other student, who was working with her, said that the slope should be the same. I had both of them determine the slope of the line with the different designations for the coordinates; naturally, the slopes turned out to be the same as in their original work.

I asked, "how did changing the order of the points affect the slope?" The student replied that the order of the points didn't change the slope at all. "Cool," I said. "So what about the two different equations you guys came up with? What difference does choosing one point over the other [when plugging a point into point-slope form] make?" The first student still wasn't quite convinced that it didn't matter what point she chose; her partner said it didn't matter what point was chosen for the point-slope form of the equation.

We decided to have each of them solve their equations for y, so they'd both be in slope-intercept form. When they did so, they came up with the same equation, and the first student was finally convinced that it didn't matter which of the two points she chose. Both students were convinced that they'd both determined correct equations for the line described in the problem. "Why doesn't it matter which point you choose?" I asked. The first student wasn't quite sure. The second student guessed, "because both points are on the same line?" I replied, "that sounds like it makes sense."

I love when students find different (yet equally valid) solutions to problems like this. It makes for some great discussion. I need to keep myself aware that it's more important to ask my students to make sense of their work instead of telling them that they're right; I missed out on having a great conversation with two students yesterday, but I'm glad I had another chance at it today.