I read a lot of teacher blogs these days.

(Incidentally, I turned MTBoS_Blogbot into an RSS feed, which was my reason for begging Lusto to make it in the first place.)

Anyway, I read a lot of teacher blogs. I see your beautiful activities, clever games and meaningful conversations. I wish I had an ounce of the teacherly creativity that Sarah Carter has, but really I don’t. It’s not what I do.

So, what exactly is it that I do?

In 8th Grade we’re going to study exponential functions. Class began with a lovely Desmos activity. They worked with randomly assigned partners.

After thinking through these questions, I thought kids could begin learning about equations for exponential functions, and towards this it would be helpful to contrast linear table/equations with exponential ones.

In years past, I would have aimed to elicit these ideas out of a conversation. I’ve lost faith in this move, though. While it’s nice to get kids to share ideas, their explanations are often muddy and don’t do much for kids who don’t already see the point. (Just because a kid *can *say something doesn’t mean that they should.) This, at least, is what I suspect.

Better, I’ve come to believe, to follow-up an activity like this one with briefly and directly presenting students with the new idea. I worry more about visual presentation than I used to. Here is what I planned to write on the board, from my planning notebook:

I put this on the board, so that it would be ready after the kids finished the Desmos activity: what could the equations of each of these relationships be? boom, here they are:

During planning I hadn’t fully thought through what I was going to ask kids to do with this visual. At first, I stumbled. I gave an explanation along with the visual, but I got vibes that kids weren’t really thinking carefully about the equations yet. So I asked them to talk to their partners for a minute to make sure they both understood where the exponential equation came from.

You can tell when a question like that falls flat. There wasn’t that pleasant hum of hard-thinking in the classroom, and the conversations I overheard were superficial.

Remembering the way Algebra by Example (via CLT) uses example/problem pairs, I quickly put a new question on the board. I posted an exponentially growing table and asked students to find an equation that could fit this relationship.

There we were! This question got that nice hum of thinking going.

While eavesdropping on kids, I heard that L had a correct equation. I thought it would be good to ask L to present her response, as she isn’t one of the “regular customers.”

Her explanation, I thought, gave a great glimpse of how learning works. She shared her equation but immediately doubted it — she wasn’t sure if it worked for (0,5). After some encouragement from classmates she realized that it would work. Turns out that her thought process went like this: 10, 100, 1000, that’s powers of 10 and this looks a lot like that. But how can I get those 5s to show up in there too…ah! The example involved multiplication so this one can too.

(Of course, she didn’t say this in so many words. After class I complimented her on the explanation and she put herself down: *I don’t know how to explain things*. I told her that learning new stuff is like that — your mind outpaces your mouth — but I thought I had understood her, and confirmed that I got her process.)

With the example properly studied, I went on to another activity. Following my text, the next twist was to bring up compound interest. I worried, though, that my students would hardly understand the compound interest scenario well enough to learn something from attacking a particular problem.

While thinking about this during planning, I thought about Brian’s numberless word problems. (My understanding of numberless problems is, in turn, influenced by my understanding of goal-free problems in CLT.)

I took the example problem from my text (*$600 investment, 7% interest/year, how much money do you have in 10 years?*), erased the numbers and put the variables on the board.

Then, I asked kids (again with the partners) to come up with some numbers, and a question. If you come up with a question, try to answer it. (A kid asking *But I** can’t think of a question *is why this activity was worth it. And with some more thought, they could.)

I collected their work from this numberless interest problem, and I have it in front of me now. I see some interesting things I didn’t catch during class. Like the kid who asked ‘*How much $ does someone lose from interest after 5 years?’* (And why would an 8th Grader know what interest is, anyway?) Or the kids who thought a 10% interest rate would take $100 to $180 over 8 years.

No indications from this work that anyone uses multiplication by 1.10 or 1.08 or whatever to find interest. Not surprising, but I had forgotten that this would be a big deal for this group.

For a moment I’m tempted to give my class feedback on their work…but then I remember that I can also just design a short whole-group learning activity instead, so why bother with the written feedback at all.

I’m not exactly sure what ideas in the student work would be good to pick up on. I should probably advance their ability to use decimals to talk about percent increase, but then again there was also that kid who wasn’t sure what interest was.

My mind goes to mental math. I could create a string of problems that use the new, exponential structure with decimals:

- 600 x 1.5
- 600 x 1.5 x 1.5
- 1000 x 1.5^3
- 50% interest on a $200 investment

That’s awfully sloppy, but it’s just a first draft.

Or maybe the way to go is a Connecting Representations activity that asks kids to match exponential expressions with interest word problems.

I’m not sure, but all this is definitely a good example of what I do. It’s what I’m learning how to do better in teaching, at the moment. It’s not fancy or flashy, and no one’s lining up to give me 20k for it, but it’s definitely representative of where I am now.

I’m not sure at all how to generalize or describe what it is this is an example *of*, though. Is it the particular flow of the 45-minute session that I’m learning to manage? Or is it the particular activity structures that I happen to have gathered in my repertoire?

None of those are satisfying answers. Maybe, instead, this is just an example of me basically doing *what I think I should be doing.* My reading is piling up, and I’m getting some coherent ideas about how learning and teaching can work. This lesson is a good example of how those principles more-or-less look in action. It might not be right (and it sure isn’t at the upper limits of what math class can be) but I’ve got a decent reason for most of the decisions I made in this session.

I think what I have to share, then, is how what I’m reading connects to how I’m teaching. This episode is an example of that.

Michael, this is a great post. I love reading about what you’re doing and thinking. I just wanted to say Thanks!! for taking the time.

I also loved your post on Beyond Explanations…it had so many great topics, what really hit me was about how students relate and remember stories. So thanks for all of that too!!

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Michael, really appreciate the look you give into your “teaching thinking”. Your “operation arrows” drawn on the board after the desmos activity are very impactful to students trying to figure out how the equations are “built”. My constant question is how I can “make it stick” and your push to get kids to explain their thinking is in the same vein. Also loved how you started the unit on exponential that you described in this linked post https://problemproblems.wordpress.com/2016/04/13/cognitive-load-theory-explains-answer-getting/

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I read what you’re doing as working hard to inspire children to engage deeply with mathematics in ways that make sense mathematically and in ways that make sense to them. You’re wrestling with thinking about what creates a “pleasant hum of hard-thinking in the classroom” — love this phrase — and creating a community in which students explore mathematics and make progress on understanding more. I’m also very drawn to this phrase: “your mind outpaces your mouth” — YES, that’s JUST how learning works. And you’re understanding more about children’s thinking and about mathematics from doing this work… through sharing with us, we can learn from you.

That’s my rough draft thinking about what I got out of your post. Thanks for writing it.

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Did you ever read how I introduced exponential functions? It’a always with word problems, first thing. https://educationrealist.wordpress.com/2013/05/05/modeling-exponential-growthdecay-interspersed-with-a-reform-rant/ The kids get it really well, and quickly see the pattern. It’s changed a bit since I’ve written this up, but the basics are the same.

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Nice!

One thing that I realized this year — in this lesson — is the possible connections to scientific notation are really interesting. I’m excited to teach this stuff next year again. Like you say in that post, exponentials are a lot of fun.

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