Towards a more conceptual calculus

I love Paul Foerster. His Calculus textbook, published by Key Curriculum, is excellent. My students today said that it's a little hard to follow sometimes, but not too bad, which, for a calculus textbook being read by underprepared students, is pretty awesome.

Last week, when I started having 8 students in my calculus class, I decided to start plowing ahead and teaching full-on calculus. Those 8 now have textbooks and TI-89 calculators. I've gotten off on a really conceptual method, really focusing on my questioning strategy, getting into some amazing diversions, as the whole class (of 8, mind you, which is probably why it works) goes deep into each student's ideas. The culture of the class is really amazing: very interrogative, with most of the students really willing to share their ideas. I took time today to thank them for that, to say, "Be thinking about the things that you think are the best and worst about this class-- I'm going to want your feedback. And, since I'm talking about it, let me say that I think that the best thing is how you guys all open right up and share your ideas, even if you're not sure that they're right. I'm really proud of you for that."

Not a lot of real reflection in this post-- just reminding myself that THIS is how I want to be teaching.

It's very meta, what I do...

Still scrambling for useful activities for the four calculus students I have-- I'm still waiting for the rest to have their programs changed (I'm told "probably by the end of next week." I'm understandably depressed about losing as much as three weeks from an already challenging situation, but it's like the man says: "Life is what happens while you're making other plans.")

So, today's particular scramble was this: I walked into the room, thinking "we should build something." I opened the cabinet of building supplies, and saw that I have 20 popsicle sticks and a ton of glue. Okay, glue-and-wood structures, with obvious materials limits (5 sticks per person). Too few to make much of a bridge. Let's go for height: "Build the tallest free-standing tower you can build," says I.

"I need some tape," says Coria. "No," says I, "no tape."

"Can I have some scissors?" asks Diana. "Sure, here."

So, tools are allowed, but materials are not. I observe what they're doing, out of one eye, while also checking and replacing calculator batteries. "Are you just making us do this so that you can do the calculators?" Diana again. "No," says I, "or I'd make you do the calculator batteries. My point here is to watch how you develop a plan, how you design and modify that design."


After 20 minutes, the projects are all wet and messy, and nothing's coming together too well: we need some time for glue to dry. I assign a writing piece: "Write about your original plan, and how your plan changed: what caused your plan to change, and how did you integrate that change? Finally, do you expect any further changes in your plan?"

Then I decide to sit down and do the same writing assignment, myself...


So, my plan was to force them into making some sort of plan of attack, on a very open-ended problem. As soon as they started into it, it occurred to me that the teachable moment here probably has to do with metacognition, and getting them to look at how they approach a problem, and how they deal with setbacks in that approach. I started hatching the idea of making them write about it about 10 minutes in, and slipped that in as a nice way of forcing them to take hands off, while they waited for the glue to dry a bit.

From here, the plan is to let them finish, and measure, and do a show-and-tell of their towers, followed by some reading (and re-writing, where needed) of their written pieces, as well as adding a fourth question to the writing: "Compare and contrast your final tower with your very first idea. You may include sketches, and discussions of expected vs. actual height." But it's very possible that the plan will end up leaving that bit out, or the presentations may get cut short, depending on how responsive they are to all of it.

I'm also planning to end by sharing this writing with them. Maybe I'll even start getting students commenting on this blog.


Juliana suggests, "Next time you do this, use marshmallows instead of glue. It works better with marshmallows." Good idea, and less messy.

Status

Lots of differentiation going on. Unfortunately, a lot of it is happening at and below the "red zone". 7th period, in particular, had no one who even attempted any kind of numerical method for the first stab at the Fireworks problem. There was some considerable engagement when I scaffolded the guess-and-check process, though, so that's something.

1st period actually had a lot more success than any of the others, in terms of finding numerical solutions. Possibly this is the right time for me to be modeling appropriate methods and solutions, but I also don't want to set the pattern of me being the expert, doing the work for them (especially as that's a mistake I've made in the past).

I ought to prep for tomorrow, a fallback plan that involves something they can have more success at. Also, if I write a quiz for Friday, tonight, then I'll likely remember to mention it in class tomorrow...

Year begins

Today was the sixth day of classes. So far, it's a pretty good start:

All four days of the first week were spent learning names, taking a pre-test, and teaching concepts of area, especially area of irregular shapes, with an emphasis on converting shapes into rectangular and triangular sub-parts. Largely, I chose to spend this early time on this subject to emphasize to these students (most of whom bounced around between subs for at least part of last year) that they're going to be given adequate time to master ideas, and that I'm not going to move on until they express some mastery of a topic. It was a pretty good trust-building exercise (NOTE TO SELF: develop some trust-building exercises for the classroom!) though the material was all off-the-cuff, and so a bit sketchy.

This week, we're starting in on the fireworks problem and unit. I'm starting to figure out what works in terms of developing understanding of the topic: time spent in class on silent reading, question-asking, and summarization was really crucial. Today, each class was called on to summarize "in as few words as you can" the three main questions in the fireworks problem-- it APPEARED to have really brought them from "I have no idea what's going on" to "yeah, I'm on this homework."

I'm also doing a better job of laying out the daily agenda, and starting to get control of my opener timing (not letting it drag out to 10 minutes!). I need to keep working on wrapping up with a review of what we've done-- I'm doing it, but not 100% of the time. Also, I need to do a better job of ensuring that I know the attendance, before class is over... :)


There are, predictably, serious problems relating to the installation of a huge new student database (like, my calculus class has 5 people in it!) but they'll sort themselves out soon. This does mean that I haven't taught any actual calculus yet, so we're going to be a bit under the gun for time, now... Muss man durch.