math education

If You Need Help, THEN TAKE IT!

I started something new last week. After I finish the lesson portion of class and it’s time to start on the homework, I have the kids move around. Those who feel like they’ve totally got it, ready to rock head to the back and work quietly. Those who are still feeling a little (or a lot) fuzzy come to the front, and I work with that smaller group on a few select problems from the homework.

The first day I did it was interesting. My A1 class had several takers who were like, “Dude, yes, help!” Most other classes, I had to twist some arms to get anyone to join in.

Second time around, though, more people joined in. I think some kids were like, “Uh, yeah, that actually looks helpful. Might be a good idea.”

It’s nice, because in those smaller groups, the struggling kids are more likely to ask questions, stop me when they don’t understand. I’m liking it. I think I’ll stick with it.

Still, some kids who I know really ought to join in are heading to the back and working with their friends instead. That’d be fine if their friends were helping them understand, but based on the daily quiz results and homework scores, it’s more likely their friends are breezing through the assignment and distracting them with random chatter instead.

It makes me mad at the struggling kids for not prioritizing. It makes me mad at their friends for not recognizing how much harder they’re making it.

I mean, I get it. Social pressure and all … not wanting to “look stupid.” I wish they’d notice that several popular kids are joining the extra-help group. Then again, an outward self-confidence often coincides with teen popularity. (Comes with its own problems, often under the surface, but that’s another post.)

I’ve only been through it two times with each class so far. I could force it, telling specific kids they have to come to the front. I’d rather not. For now, I give a strongly worded suggestion that if they didn’t get the homework done, struggled on the daily quiz, or got a bad grade last quarter, they really ought to join us.

Hopefully the more we do it, the less stigmatized kids will feel.

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Tales of a Tutor

The past couple of weeks, I’ve been helping a friend’s daughters with a college math course they’re taking over the summer. I’m geeky enough that this is fun, and getting paid is a nice bonus.

While doing so, certain things have struck me more than they might while working with my own students. So I figure, why not share?

Even math teachers don’t remember all the math, all the time. Conic sections … I’ve never actually taught them as a whole topic. I’m fine with circles and parabolas, because those come up regularly on their own. Ellipses and hyperbolas, however, not so much. I remember some general things about them, but not how to find the coordinates of the foci, or how to rewrite an equation to the proper form. Fortunately, all it takes is twenty seconds glancing at the right material in the book.

Math teachers don’t always agree. When tutoring, I almost always come up against something where the way the teacher showed them is bonkers (in my opinion). I try to determine if there’s any good reason to do it that way. If there is, I go along with it. If there isn’t, I try to determine whether the teacher will know or care if the students do it a different way. If not, I’ll show the kids my way, explain how it relates to the teacher’s way, and tell them they can choose whichever they like better.

Math teachers don’t always act rationally. Often these college courses don’t allow the use of calculators. I understand the idea—with some calculators these days, you could solve every problem on the test without engaging more than a couple of your own neurons. But it’s kind of ridiculous when the long division to reduce a fraction takes longer than applying the math concept that’s actually being tested.

And the thing is, I’m sure I’m guilty of all of the above in my own math classes. Somewhere out there a math tutor is saying, “Miss Lewis said that? Is she nuts?”

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The Make-or-Break Teachers

I’m getting ready to start a new school year. As always, there’s a thought that lingers over all my preparations.

I hope I don’t screw up any of the kids too much.

To be fair, I’m pretty sure I haven’t screwed up any kids yet. There have been a few I wish I could’ve done more for, but I think my track record’s pretty solid. There’s a little extra anxiety this year since I’m starting at a new school—or rather, my old school after several years away.

It’s an interesting situation, because it’s the school I went to as a teenager, along with being where I launched my teaching career. My family and I are rooted in this area, so a lot of the neighbors know I’m returning to teach there. Several of them are hoping to transfer their child into my class if at all possible.

No pressure, ha-ha.

Seriously, though, one thing I’ve heard from parents in the last several weeks (and indeed the past several years) is how important they feel it is that their child gets the right math teacher. A good math teacher can take a student from hating math to at least tolerating it, if not better. A bad math teacher can bring a skilled student’s progress to a grinding halt. Often that damage is never recovered.

Is it the same in other disciplines? Probably, to a degree, anyway. But it seems like the near-irreparability is more severe in math. I had English classes that I hated, but they couldn’t kill my love for reading and writing (obviously). Then again, if I’d been a struggling reader in elementary school, and a bad teacher only reinforced and exacerbated my struggles, that could’ve set me back for the rest of my life.

Once past the learning-to-read stage, moving on to reading-to-learn, it seems the make-or-break power of teachers lessens somewhat. (I hope so, considering teens I’ve known with English teachers of … questionable quality.) Math works a little differently, always with a new skill, a new principle to learn.

That makes my job potentially dangerous.

Maybe a different approach is in order. Maybe if I keep the focus on helping kids develop their ability to think, to reason, to problem-solve—and I don’t mean “A Train leaves Station A at 6:45 am” kind of problems, I mean real “Here’s a situation and we need a solution” problems—maybe that means I won’t have to worry so much about breaking anyone.

Because you know what? There’s something else underlying this whole line of thought. To have the power to break, I have to keep a monopoly on the power to build.

The students need to be allowed to build themselves. Maybe they’ll suffer minor breakages along the way, too, but maybe that’s what I’m really there for …

… to provide the super-glue when they need to mend their own breaks.

Have you had experiences with teachers (math or otherwise) who had that make-or-break position in your life? What made the good ones good, and the bad ones terrors?

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Math Rant: Screwy Stats

I have to say, I love me some statistics. Have I collected student scores and done a little analysis? Why, yes, I have. Have I collected and graphed data related to my writing? Oh, wait, you already know I have.

The thing is, I also know the limitations of statistics—what it takes for them to be meaningful, how far you can or can’t take the results. That data I analyze from my students? I use it to give me some direction as a teacher, figuring whether things are improving, whether a particular concept fell through the cracks, etc. Not much more than that.

As we all know, of course, statistics on education can get used for a lot more. I get the need for assessment (in some form) and accountability (in some form), but often when I see articles reporting school success/failure, I wonder if the people involved have the first clue about statistics.

Case in point: I recently saw an online report about the 50 best and 50 worst schools in the state, in reference to percentage of students achieving proficiency on the state’s high-stake testing. It reported results for Language Arts, Science, and Math.

The first thing that struck me was that whether looking at the 50 best or 50 worst, the percent passing math was WAY lower than the other two the majority of the time. That made me scratch my head, so I glanced down at the comments.

Several people noted that AP students didn’t take the state test.

I haven’t had a chance to dig into it yet, but if true, it makes those reported percentages almost meaningless. “We want to see how your school measures up … but we’re not going to count the top students.”

This is why when I see statistics reported, I have next-to-no reaction. Not until I know more about where the numbers are coming from. In broader situations, I ask myself questions like, who was included in the sample? How was the sample selected? How were questions worded?

Be careful when reporting statistics as part of an argument. They may or may not back you up as much as you think. Dig a little deeper to find the whole story.

ETA: Did my own digging-a-little-deeper, and it’s actually worse than I thought. The last math courses to participate in the state test are Algebra I and Geometry. The website was reporting on the results of high schools, many of which around here are only grades 10-12. By that time, even the “average” students are past those levels. So the published results only showed the proficiency of the lowest group. No wonder the math percentages were so much worse than the other subjects (which I believe test higher numbers of students in high school).

Have you run into questionable statistics? Any pet peeves on how you see them reported? Do you find yourself completely confuzzled when facing the numbers?

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Mistakes vs. Incompetence

As some of you know, I’m entering a transition in the day job. This involves a lot of interviews, some where I’m the interviewee and some where I’m on the panel of interviewers. It makes for an interesting dual perspective.

My current school includes something extra in the interview process—candidates have to teach a brief mock-lesson. For me, that’s the make-or-break portion of the interview. I can forgive a few weak answers on the standard interview questions, but if the math teaching isn’t up to snuff, I’m not recommending.

Since we’re nearing the end of the school year, I’m also leading my classes through reviews to prepare for their final exams. This includes going through problems we haven’t discussed in-depth since last fall. Most of the time, it’s fine. But a couple of times last week (in calculus, naturally), I had some ridiculous cerebral failures.

That’s fine, too. I make a point of emphasizing to my students early on that I can make mistakes, and if they catch me at it, good for them. Seeing me make mistakes without falling apart seems to help them be more willing to take risks even though they might be wrong.

I got to thinking about the two situations. Where’s the line between “Oops, the teacher’s human and makes mistakes” and, “No, this interviewee doesn’t have what it takes”?

My guess is that the line is in awareness. When I screwed up in calculus, I either knew right away or within moments. I immediately ‘fessed up to the students and set about figuring out what I’d done wrong. With interviewees who aren’t cutting it, they generally seem to think what they’re doing is fine. Top interviewees often have more to criticize about themselves. There’s a question in the interview about what they think they need to work on most. It’s always interesting to correlate their answer to this question with their performance in the mock-lesson.

So, everyone, let’s aspire to make mistakes. Own them, learn from them. But never let it cross into incompetence. If we are incompetent in an area, let’s be aware of it, and work to correct it.

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The Makings of Mathematical Mistakes

In my years of teaching math, I’ve amused myself by taking note of the types of mistakes students make. (Yeah, okay. I’m easily amused.) You can pretty much figure out the type of mistake by watching my reaction.

The “Fell Through the Cracks” Mistake

This usually happens in complex, multi-step problems. The student does all the hard stuff right but overlooks something. More often than not, it’s losing a negative or mistyping something in the calculator.

R.C.’s Reaction: I just point wordlessly at the paper or calculator and wait while the student looks, ponders, then says, “Oh! Oops.”

The “You Know Better” Mistake

Another “careless” variety of mistake. Can I tell you how many times I’ve asked what four-squared is only to hear, “Eight. No—wait! Sorry. Sixteen.”

R.C.’s Reaction: Students often catch those without any help from me. When they don’t, they get my ‘Did you seriously just say that?’ look. If that’s not enough, they get a verbal, “Really?”

The “You’re Still Learning” Mistake

This happens when students are mostly getting a new concept but aren’t quite there yet. OR … when they have to apply something they learned previously that hasn’t quite solidified.

R.C.’s Reaction: Usually I ask them to explain their thinking first, then ask some follow-up questions until they see the wrong turn. Sometimes a neighboring student will try to tease the other about the mistake, at which point I remind them that they made the exact same mistake two minutes ago when I was helping them.

The “Someone in Your Past Failed Both of Us” Mistake

I teach high school math, which naturally relies on concepts learned over several years before arriving in my class. Sometimes we’re working on some complicated algebraic thing and I realize some/all of the students have a problem with an underlying principle. (Fractions, anyone? Or measurement conversions?)

R.C.’s Reaction: What can I do? Go off to the side of our work and make up a simplified example (i.e., non-algebraic addition of fractions), quickly refresh the kids’ memories on that, and parallel it to the problem at hand.

The “Back the Truck Up” Mistake

These mistakes on the part of the student tell me that I screwed something up as the teacher. Didn’t explain clearly, allowed for a massive misconception to take root, etc. Sometimes I even did something just plain wrong.

R.C.’s Reaction: Confess to the class that I made a boo-boo, very clearly indicate where we went wrong, and emphasize the proper way to move forward.

Some people might say it’s a teacher’s job to eliminate mistakes and a student’s job to avoid them. I don’t agree with that. Mistakes are great! They’re how we learn. (Well, so they’re great as long as we learn from them.) And one thing to keep in mind is that I have extremely small classes, and I’ve taught most of my students for more than one year, some for up to five straight. My reactions to the “Fell Through the Cracks” and “You Know Better” varieties are done in an environment where the students and I are able to laugh off mistakes without embarrassment. (And where I’ll accept “It’s calculus on a Monday morning,” as an excuse for the careless mistakes as long as they keep trying.)

I know some people who were always terrified to volunteer information in class, certain they’d make a mistake. I was one of them. Now, I’m okay with making mistakes in the classroom. Still working on being okay with it in the rest of my life.

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