Twosday Things: Hearts, Stars, Messy Numbers

Time again for Twosday Things!

Taking a cue from last Tuesday's post, I'll discuss two teaching-related things (however big or small) that happened over the past week. I'm trying to post about two things every Tuesday throughout the school year (hence the title, "Twosday Things"). This makes the second week in a row; so far, so good.

Thing #1:
Something I've noticed that happens A LOT in my class:

• Student is working through a (typically algebraic) problem.
• Student gets a non-integer answer (i.e. a "decimal answer").
• Student immediately assumes they must be wrong. Often accompanied by asking the teacher, "am I supposed to get a decimal for my answer?"

This is a near-daily occurrence in my class, despite my frequent insistence that "decimals are numbers, too!" ("Fractions are numbers, too!" is similarly used often.) I cannot even count the number of times this happens in a school year.

How does this happen? How do our students reach the point where they automatically assume that "decimal answers" must be wrong? How do we let them get to high school with this assumption cemented into their mathematical psyche?

Yesterday, I took this question to my Twitter feed:

Some super-awesome math-types from the Twittersphere chimed in with their thoughts on the topic:

"Give them messiness." I love that. I feel like our students need more practice and earlier exposure to "messy numbers," because real-world math is messy and complex. Students need to learn that decimals, fractions, irrationals, etc. are all numbers, too.

At the same time, I don't think it's inherently bad that students question their answers every time they get something "messy." Sometimes (often, in fact), their answer actually is the result of a mathematical mistake, and they need to be able to figure out where the mistake was made.

I can see some potentially good habits here: stopping to think about whether the answer makes sense in the context of the problem; double-checking work for mathematical mistakes; and so forth. I just don't think that "getting a messy answer" should be the sole reason a student thinks they did something wrong. If anything, students should be trained to question "messy" answers and "clean" answers. Students should be in the habit of doubling back and re-checking their work to make sure their reasoning makes sense.

Maybe the mistrust in "messy" numbers can be a good thing; but if it is, it needs to be applied to all numbers. Equal opportunity, darn it!


Thing #2:
Today in class, I had a few students who asked for help with the following problem:

We discussed the fact that the problem mentioned "two numbers." We had no idea what those two numbers were, offhand. But, we had enough information to be able to set up a couple of equations. We just needed to pick two variables to represent the numbers first.

"We can call these two numbers anything we want," I said. "We can call them x and y. We can call them a and b, or c and d. We could even call them stuff like, 'dollar sign' and smiley face.' What do you want to call these two numbers?"

One of my students said, "heart and star."

Math, learning, and hilarity ensued:

I had a terrible time keeping a straight face, especially when I said things like, "so what expression do we plug in for heart?" or "yep, we have to simplify by combining our star terms, so star plus eight equals twenty-four," or "there we go, star equals sixteen and heart equals forty."

It was a fun little way to talk about the concept of representing unknown values with variables. Why settle for boring old x and y when you can have a bit of fun?