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?

Twosday Things: Solution Methods and Student Initiative

It's Tuesday, which means it's once again time for Twosday Things!

In my dauntless endeavor to blog regularly, I am continuing to write about two (big or small, mostly small) things that happened in my teaching world over the previous week. This makes the third week in a row. Not bad.

Before you read on, be sure to open up Geoff Krall's awesome PrBL starter kit in a new tab; this should be your next bit of reading after you're done here. You're welcome!


Thing #1:
One of my students (we'll call her Susie) was having trouble working through the following problem:

"Find the value of two numbers if half the larger number plus two equals the smaller number and their sum is 44."

Setting up a system of equations to represent the problem wasn't terribly much of an issue; Susie was able to do this on her own with a bit of questioning from me to prompt her thinking.

After we had set up the system, Susie seemed stuck on what to do next (though we had been working on systems of equations all week and I'd seen her succeed in completing similar problems).

Before I even said anything, another student (let's call her Nadia) offered to help explain what to do next, and I gladly obliged. Nadia used the elimination method to solve the problem while explaining her steps to Susie:

After Nadia finished, Susie seemed confused. She understood that 28 and 16 had to be the numbers described in the problem, but she wasn't clear on the elimination method that Nadia had used. "I actually thought you were supposed to plug the equation for b into the second equation," she said. Susie proceeded to solve the problem using the substitution method:

I found what happened next to be interesting: Nadia seemed confused about the method that Susie had used to get her answer, even though they both came up with the same thing! I said to them, "so Susie, it sounds like you were confused when Nadia used elimination to solve this problem, and Nadia, it sounds like you were confused when Susie used substitution." We talked about it, and both girls said the methods they each used just made more sense to them. I stressed to them that it was important to understand both of these methods (as well as solving by graphing), but also that it was great to see that each of them had their own way of figuring out this problem. As has happened in my class before, students are seeing there can be more than one path to a solution.


Thing #2:
The above situation touches on something that has been developing among my students in my classes over the past few weeks: they're starting to regularly help each other out on their own.

Stuff like the above has been happening with greater regularity in my classes. It's happened faster among my advanced students, but my other students are starting to do it as well.

A lot of classroom time is spent allowing the students to work through problems at their own pace, with me providing one-on-one or small-group assistance as needed. This can be challenging to manage, particularly with having my "regular" and "advanced" classes in my room at the same time every period.

That's part of the reason why I love it when students start to take the initiative and help each other out. I also love this kind of initiative because it's so important for students to be able to take agency of their own learning. It's an important life skill (at least I think so).

One of my classes has really figured this out. Every day, they come in and they all get with their usual groups (I did no grouping; they formed these groups on their own and they work really well). They figure out what their assignments are. If I don't have a new workshop or learning module for them, they get to work and ask me for help whenever they have questions. They're also getting very good at helping each other out, checking each other's work, and asking each other questions (Thing #1, above, happened with this group).

When students figure out how to take charge of their own learning, good things happen. One student (let's dub this one Marie) had been struggling all year with math. Last week, the small group of friends Marie regularly works with really focused on helping her understand how to solve systems of equations. They were able to give her a greater amount of attention and assistance than I was able to by myself. After a while, Marie started to be able to solve systems of equations on her own; she even got so excited about getting a problem right, that she wanted to do it on the board! AWESOME!

It's not like this every day, and it's not like this in every class. But it's starting to happen more, and it's great to have one class that's really taken off with helping each other out.