Friday, January 30, 2015

Teaching Logarithms Using Suspense (okay and GeoGebra)

Every year, when I start the logarithms unit, I brace myself. I know I'm going to lose a lot of kids. I've tried all kinds of ways to make logs clear. I've written about it here, in fact, and each time I think "This is gonna be great!" but then it's not. But this year, I think I hit on something. I really mean it this time. No really!

And guess why!?!? It's because of the geogebras I've been getting them to create. (If you have no idea what I mean by that, this will give you some idea.) Another unexpected benefit that just kind of fell into my lap. I'm using their exponential geogebras to create suspense about logarithms!

Where we are now

We're currently studying the exponential function, and this time I had them start creating their geogebra explorers very early in the unit, so that they could use them in parallel to the lessons, for pinning things down, validation, exploring, whatever. (Note to self: Do that next year for every function. It's too overwhelming for them to do it all at the end of the unit.)

Anyway, I knew that they'd be able to get their sliders, asymptote, domain etc etc all done, but that when they'd get to the zero, specifically figuring out a formula for it, they'd be at a loss. When I first realized this, I thought, ooh, that might be frustrating or confusing, but then I realized that it would be an opportunity to motivate the need for logs. Just the act of asking me "How do we solve for the x in this?", ie an equation like   indicates that they are aware that this is a thing. A new thing. Which let's face it, logs are.

The plot thickens

They haven't done logs, so this kind of makes it suspenseful for them! I'm hoping that in a week or so, when I reveal logs to them for the first time, instead of the usual confusion and horror and OMG THAT'S IT I'M GOING INTO ART, I'll get "Oh! So that's how you solve that equation!" or something like that.

To add to the suspense, I had them spend some time struggling with the question: What can I do to both sides of this equation: , that is the opposite? I wanted to use that kind of language because that's what they are used to - again to motivate that this calls for something totally different. Trying to find that opposite operation, and failing to do so (which they did spectacularly), points them away from that old familiar safe language, which is a good thing. Kids, you're so not in Kansas anymore.

Questionable pedagogy?

It's not completely true that they haven't seen logs, actually, because I have shown them that to solve , they can get the value of x by either trial and error, or by using the log button on their calculator and punching in:

But they have no idea why that works, or what the log button does yet. Normally I don't encourage my students to do something without understanding it, but logs are different. I find the word itself is intimidating and doesn't sound at all like what it is - an exponent. So I get them used to hearing it for a while before actually explaining it. I'm not sure about the pedagogical appropriateness of that....but emotionally I think it helps. And it also lets them know that logs have something to do with solving exponential equations.

My evil plan's results so far 
Dr. Eeeevillllll!!!

So far, it has all worked exactly according to plan, at least for a few students. Several students got everything done in their geogebra except for the zero, and asked me to help them figure out the formula for the zero. I of course refused. Nicely though!

Another asked "Are we allowed to use logs to do the zero in our geogebra?" I said sure, in fact, you'll have to, there's no other way!  Now I think I should have just batted my eyes, all innocently, and said, "Well, sure, if you think it'll help..."

So the suspense is building, for them and for me! If only one kid figures out how a formula for their zero, all by themselves, I will be thrilled. That'll be better than all preceding years.

In about a week, maybe two, I'll write an update. Watch this space!

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