Mathematical Mondays

Student-Centered, Math-Anchored

There’s something in education you may or may not have heard of—the student-centered approach. Here’s what some people think it looks like:

Students doing whatever they want, however they want, as long as it has some tenuous connection to the subject at hand. There are no wrong answers. Math facts are left by the wayside.

Basically chaos, with very little education happening.

I imagine some teachers actually carry it out that way, but that’s not the philosophy as I understand it. When we hear the term “student-centered,” I think we tend to have ideas of, “Let the student lead the way. Let the student determine everything.” So I try not to think about student-centered without including something else.

Math-anchored.

I envision students out at sea, paddling around in the water, exploring to their hearts’ content. The earlier illustration would end there, but when I think of it, each student also has a tether. How much rope they have might vary, but all the lines are connected to a stable post. They’ll reach that post from different sides and at different rates—of course, as the teacher, I’m there giving gentle tugs to each rope to urge them in my general direction—but they’ll all get to the same endpoint.

That destination is the core principle, the major mathematical idea that’s the reason we’re doing the activity or exploration at all. Students are empowered to delve into the thick of it, really engage their brains to make sense out of a situation. They see the different approaches their classmates took and discuss whether they’re equally valid.

Most importantly, they come away with an understanding of that root concept.

Like most things, easier said than done. Even if I set up a great lesson, it can be awfully tempting to forget those “gentle tugs” and just haul each student in by their tether. It’s also easy to run out of time before they reach the destination, and then forget the next day that I’ve left them adrift.

Before we even get started, then, I’d better make sure they all have life preservers. In other words, setting up a classroom environment where they know the expectations and what to do when they’re left without the mathematical understanding we’re looking for.

Can you tell the first day of school is coming up?

Wish me luck.

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Why I Won’t Tell You What to Do

This is one of my biggest guiding principles in teaching: I won’t tell my students what to do.

Okay, I will sometimes. Like when I tell them to clear their desks before a test, to get out a piece of paper, to work with their partner, or to stop playing games on their calculator when they’re supposed to be working (and I know they’re playing because no one uses their thumbs that much when they’re calculating).

But that’s not what I’m talking about.

I’m talking about when a student asks, “How do I solve this problem?” Sometimes I slip, but more often than not, I answer that question with a question. “What do you know about the problem already?” “What are we trying to find?” “How is this similar to/different from this other problem?”

Yup, I’m one of those teachers.

Even when I do “tell” a little more, it’s often with options. “What are the tools we’ve been using? Tables, graphs, and equations. You could try using any of those.”

It’s easier just to tell students how to solve the problem. Really, it is. (That’s why I slip once in a while.) So why don’t I just do it that way?

Because it’s not about what’s easy … especially not what’s easy FOR ME.

It’s about getting the student to the point of doing mathematics independently. And before anyone says most people never use anything from algebra or above in “real life,” that’s not what doing mathematics is truly about. It’s about thinking and reasoning and working out what makes sense.

Like so many things from my teaching life, it carries over into my writing life. People ask for feedback, critique, suggestions. In that case it’s peer-to-peer, but that makes me even less likely to say, “Do it this way.” I try to focus on giving my reaction as a reader, what worked and didn’t, leaving it to the writer to figure out how to best resolve any problem areas—if they even agree that the area is a problem.

Some people give feedback by saying, “What if you did it like this?” and proceed to rewrite a whole paragraph or query letter. I can’t say it’s wrong and no one should do that. Maybe that works for some people. Just me, personally … it makes me cringe. Once in a while I throw in a “such as” and give a possible sentence to illustrate my point, but I try to keep that really limited with a tone of “but in your own way.”

That’s the thing. When I tell someone how to solve a problem, they’re not really doing mathematics. When someone is writing, feedback is critical. Taking in that feedback, processing it, and deciding what to do about it (if anything) is a necessary skill. It needs to be their work, their writing, their voice. We can suggest and spitball and yea-or-nay ideas, but when it’s our writing, we must do the heavy lifting.

And yes, sometimes I slip in that department, too. But I try. I just want to make people think.

But if I said my way of giving feedback is the only way, that would be telling you what to do.

Have you seen or experienced benefits of the direct-instruction approach? Have you seen downsides to being left to puzzle it out, picking and choosing from more general bits of advice?

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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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Talking Basically About Bases

You may or may not know that we operate in a base-10 number system, which is a beautiful thing. It spares us from the agony of Roman numerals, where years looked something like this:

Instead of like this:

Remember all that talk about place value in elementary school? Ones, tens, hundreds, thousands, and so on? That’s the base-10 idea. Multiply 10 by itself successively, and you get the next place value.

Ten isn’t the only number to base a system on, though. Convenient with our ten-fingered anatomy, but it’s not even the only base we use on a regular basis. Time notoriously operates on non-ten bases. (Makes figuring elapsed time tricky for some students.) And since this particular country refuses to go metric, most of our measurements avoid the ease of base-10.

There are plenty of practical applications for other bases, but when I first learned about them in school, I remember just thinking it was cool to write a number that meant something other than what it looked like. Sort of a mathematical code.

For example, take base-8. We have to reassign all the place values. The ones place is still the ones place. The next place to the left is now the eights place. And the next is the eight-squareds (or sixty-fours) place, followed by the eight-cubeds (or five-hundred-twelves) place. So earlier I mentioned the year 1988. In base-8, that’d be

3 five-hundred-twelves
7 sixty-fours
0 eights
4 ones

So the year 1988 converts to 3704.

If someone gave me a worksheet of numbers to convert to different bases right now, I’d probably be a happy camper working through it.

And that concludes our Yes-I-Am-A-Geek moment for this week.

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Just the Facts, Ma’am

One of the fun little debates in math education is over the importance of “knowing your math facts.” By this, people generally mean having your times tables memorized, that kind of thing.

How important is this? I admit, it’s a little frustrating when I’m trying to get a student to understand a complex higher-math problem (algebra, maybe), and they get slowed down trying to remember what nine-times-six equals.

On the other hand, I find it more worrisome when a student has their multiplication facts down pat, but can’t problem-solve enough to figure out that multiplying is what they’re supposed to do in the first place.

Then there’s my favorite situation: Students who know their multiplication facts, but have to count it out to add or subtract.

Instead of memorizing math facts, I’m more a fan of developing math fluency. When I was in elementary school, I had most of my times tables down, but struggled with the twelves. It didn’t matter, though, because I knew I could just multiply by eleven then add the number I wanted to multiply by twelve. I could do it quickly enough that my teachers never knew I hadn’t memorized those facts.

And it didn’t matter.

That’s math fluency. It requires having some math facts under your belt, but more importantly, a fundamental understanding of operations and how they work.

What do you think? What are the benefits of memorizing math facts? How did you handle learning those facts in school? Would you do it differently if you could go back?

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Time May Be Relative, But You Can Control It

Are you guys familiar with Einstein’s big idea about time being relative? It’s one of my favorite topics in physics, but I know some people aren’t as crazy about it. (In my mom’s words, it makes her mind go “blinky.”)

Here’s the basic idea, and it has to do with the speed of light in a vacuum being constant no matter the speed of the source. Suppose one person stays on Earth while another flies away on a spaceship going a significant fraction of the speed of light. Also suppose that they magically have a way of keeping an eye on each other instantaneously as the one travels.

To the guy on Earth, a day passes, but his monitor of the spaceship shows only a handful of minutes has passed there.

To the guy on the spaceship, a day passes, but his monitor of Earth shows years have passed there.

(The exact ratios depend on what fraction of the speed of light the spaceship is going, but you get the idea, I hope.)

Does that seem really bizarre and out there? It shouldn’t. We run across the same thing all the time in our writing efforts.

(Yes, I just segued from Einstein to writing fiction.)

Sometimes you read a scene that’s supposed to happen in a matter of seconds, but you feel like it takes hours. Or significant time is supposed to pass, but it feels like the blink of an eye.

Time passage mismatch = PROBLEM

The most obvious solution may be a matter of real estate on the page. Something that’s supposed to happen quickly takes only a line or two. The passage of more time gets several paragraphs.

That might work in certain situations, but what if the details of that super-quick scene are significant? What if nothing of note happened in the passage of three months?

In the first situation, how do you get those details in there without getting that feeling of sluggishly trudging along? I’ve had some success with shorter, snappier sentences, particularly in fight scenes and the like.

In the second situation, how do you get across that passage of time without just a blink-and-you’ll-miss-it statement of “Three months later…”? I think that’s a matter of transitions. Those three words may be too little. Several paragraphs about a lot of irrelevant nothing happening during those months is too much. But a few carefully worded sentences in the transition can give that weight, get the reader in that feeling of time passing.

Those are the first solutions that came to my mind, but I’m sure there are more. What tips do you have for controlling your readers’ perception of time?

And for those of you interested, here’s one of my favorite clips explaining that whole time dilation concept.

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