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?

For Posterity: An Awesome Teaching Day

There are about 180 days in a school year. Multiply that by the number of years you've been teaching. That's a crap-ton of days.

Many of them are good, or just okay. Many of them leave you wondering whether or not you're any good at teaching at all. A few of them turn out to be terrible.

But once in a while, you have a day that makes living through all of those other days completely worth it. For me, yesterday was one of those days.

Last night, I was recognized by our district's Board of Education for "contributions to the school and the community." It was a pretty cool honor.

I was already having an awesome day. I finished grading quizzes that my students had taken this week. They did well overall, but I was particularly impressed by many of my students who had been struggling. Those students have been making a point of coming to me for help, participating more in class, and generally just turning things around. They've been busting their butts, and their work is paying off. It's thrilling to see those students experience success in my class; there have been smiles and high fives all around.

In the evening, I attended the Board meeting to be recognized along with a few other students and staff from our district. When my name was called, I went up to the front to receive my framed certificate:

After my principal said a few things about me, I had a moment to shake the hands of every Board member.

One of them, whose son graduated a couple of years ago (and whom I had taught for two years) stopped me for a moment to tell me: "I want you to know, my son has decided to switch his major to math because of you."

Wow. I mean, just wow. That pretty much describes what I felt at that moment. I didn't really know what to say. I think I almost started crying, or at least got a bit teary. It was just so awesome to hear those words.

That news, that moment, meant a hundred times more to me than any award or certificate. The feels. The feels.

As a teacher, when you have students, you don't always know whether or not you've having any kind of impact on their lives. Most of the time, I feel like I have no idea whether or not I'm having any impact. I guess sometimes you don't find out for sure until after those students graduate and move on. But however long it takes, when you do find out you had a positive impact on a student, it's completely worth it. Nothing else really matters.

Somewhere down the line, I'm going to have a terrible day and I'll need something to lift my spirits; that's why I had to write about my awesome day. I'll always have this to come back to when I need to be reminded why teaching is so rewarding and worth everything.

Two Things From a Tuesday

Or maybe I should title this post "Twosday Things." Because I like portmanteaus.

Thing #1:
Today, I was talking one-on-one with a student about functions. We were talking about the relationship between domain and range, and how to tell if two sets of values make up the domain and range of a function. We talked about how values in the domain are each assigned to one and only one value in the range by the function. I chimed in with the "mailbox analogy" to further explain the relationship: say you're mailing a bunch of letters. The stack of letters is like the domain, and the houses the letters are being mailed to are like the range. You can mail multiple letters to the same house, but you can't mail the same letter to multiple houses. "So you can't mail the same letter to Chicago, New York, and San Francisco simultaneously," I said to the student.

"Unless it's e-mail," the student replied.

HOLY CRAP. That was a really, really good point! I was utterly stunned that I hadn't thought of that. I guess the analogy kind of breaks down in that regard if you throw e-mail into the mix. I'm still pretty sure I got my point across, but it does have me thinking about the analogy I'm using to describe how functions work. Will this be an outdated analogy in the near future?

Either way, I was super-impressed by my student today.


Thing #2:
Some of my students are currently working on compound inequalities. Below is a piece of student work that I found interesting:

The left side of the compound inequality vanished! I've actually been seeing this happen with several students in my class; every time they get one side of a compound inequality equal to zero, they omit it in the rest of their work.

I've been wondering where this is coming from. I imagine it might have something to do with the fact that students are sometimes taught about the existence of an "implied" zero that isn't actually shown. (For example, what is the slope of the line y = 2? There's no x-term, but there's an implied "0x" in the equation; thus, y = 0x + 2, and the line has a slope of 0.)

Maybe it's coming from somewhere else. I don't think it's anything I've done, but I could be wrong.

Anyway, that's two things from a Tuesday. Maybe I'll try to do this weekly, so I'm blogging more often.

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.

Taking a Teaching Mulligan

Sometimes, despite trying to do my best job possible as a teacher, I screw up. I'm pretty sure it's healthy to accept that it happens from time to time.

A few weeks ago, my students took a quiz that pretty much nobody did well on. Like, not even really that close. (I'm not going to go into what the subject matter was or how my lessons were designed or what scaffolding I did -- that doesn't really pertain to the message of my post today.)

Needless to say, this elicited an emotional reaction from me. I actually had to stop grading and walk away for a few minutes because I was feeling a mix of sadness and anger all at once. I reasonably sure that I was uttering curse words under my breath after I came back and continued grading.

I think, unfortunately, there are some teachers who probably would have taken that anger and directed it at their students the next class period. I've seen teachers get absolutely pissed off at their students for doing terribly on a quiz or a test as a whole group.

I'm not one of those teachers. When students don't perform well on an assessment, I blame myself. I blame myself pretty hard, actually. Maybe more than I should. I guess I can't help it.

This happened on a Friday afternoon. I thought about what to do all weekend. I came back to my students on Monday and, in each class, just laid it out for them:

"Guys, nobody did well on this quiz. I'm sorry. I blame myself for that. When nobody does well, that tells me that I probably did something wrong with my teaching. So, I'm not going to include these quizzes in your grade for now. We'll come back to it next week, I'll try to teach differently, and we'll re-take this quiz. Does that sound fair?"

And it sounded fair to everyone.

I imagine part of why my students were amenable to this is because many of them sensed that they hadn't done well. I bet many of them were afraid they'd let me down, or that I was going to be mad at them for failing one silly math quiz. They probably don't know that, when a class bombs an assessment, the first question I always ask myself is, "what did I do wrong?"

Stuff like this is a humbling reminder that, even though I work hard and try my very best as an educator, there will be times where I come up short. I try to keep those instances few and far between, but from time to time it will happen. When it does, I think the right thing is to give my students a second chance -- or, more accurately, ask my students to give me a second chance.