Math Stories

My school year is pretty much over. The eighth graders are going to be practicing for graduation - those that didn't make it are going to get an hour of monotonous lecture on the consequences of failure (in this case, enduring monotonous lectures from me).

Today we had a school wide survey, and when I turned on my projector to provide the intro, the bulb blew.

Our supplies department has no replacement bulbs. In fact, they plan on replacing the whole projector rather than replacing the bulb¹. Of course, they have no projectors available right now. Fortunately it's at the end of the school year - I can get away with not using it for three days. During the discussion of its use, I was questioned on how frequently I used it.

I use it daily. I use it all period long. I go from warm up to cool down with it. I wear out the buttons on the remote flipping through all of the animations I've built. I'm at the point where I can't imagine teaching without it.

So, rather than adjusting my usage patterns, as suggested by the supplies coordinator², I think I'll be using a sizable chunk of my next year's discretionary allowance on buying a backup bulb for this thing.

¹ The reasoning is not entirely unsound - it has to do with how budgets need to be managed. We all know that bureaucracy makes stuff more expensive, right? (back)

² I think they just want us to have technology, but not actually use it. (back)

Dina talks about wanting a report card.

It's a thinly disguised list of goals for next year. But, I like the idea. My review this year was an unending row of "Meets Expectations" checkboxes. I want something more.

So here's what I plan on grading myself on 12 months from now:

General Goals:

• Engage students.
• Make standards accessible.
• Effectively collaborate with peers.
• Extend influence beyond the boundaries of the classroom.
• Effectively reflect on practice.

Specific Goals

• Observe more. Be observed more.
• Do more lesson analysis here.
• Solicit feedback from students.
• Provide students with better tools to evaluate their own learning.
• Develop tools to track student behavior/work habits. Make those accessible to students as well.
• Make sure that, as facilitator, I really facilitate. (IOW, keep my mouth shut).
• As department chair, keep teachers focused on teaching, and teaching effectively.
• As department chair, push the administration to make my teachers' jobs easier. Oppose the administration on anything that makes my teachers' jobs more difficult.

And finally, as a meta, I expect to be able to give myself a break if I don't get 100%.

Last Friday was our 8th grade picnic: during 5th & 6th periods all the eligible 8th graders who bought tickets were allowed to go to the sports field where they had a couple of jumpers, a sumo wrestling setup, pizza and a stand for bottled water.

8th graders being what they are, a few of them decided the ideal use for the bottled water would be to pour it all over each other. I managed to pull a couple of them aside to discuss the wisdom of using $1/bottle water for this purpose.

As I was doing this, another teacher walked up behind me and dumped a bottle of water over my head.

Really. Another teacher. And we hadn't even won anything.

The pythagorean theorem expresses a nonintuitive relationship - that's where a lot of its power comes from.

Unfortunately, the way it is usually taught involves a formula (or two or even three depending on how it's taught) and learning which numbers to plug in where.

And it almost never sticks.

This is the lesson I use to introduce the whole thing:

The kids cut out the squares, and spend the period trying to make right triangles out of the sides of the squares.

On its own, it's a horrible lesson. It's almost impossible to tell if the triangle is really a right triangle or not, resulting in a lot of close but not quite guesses, which can be frustrating for the students. It requires me to filter out the correct solutions (there are 7 of them) for the students.

The pattern in the triplets is obvious if you get it. This year, out of 80 kids, I had one who could find it on their own without help. There isn't enough of an aha moment to make it stick.

I still teach it though.

By the labeling, and subsequent use of the squares and counting of the sides, the students learn to associate a square and it's root. It gets them to think about right triangles and the squares associated with the sides. And it gets them ready for this next sheet:

The first two triangles show off the pythagorean association. The next three allow them to find the third side on their own. (i have them add the two smaller squares while I cover the larger square with my hand. When they get to the third problem, they've forgotten that there isn't anything there.) The last three involve finding one of the smaller squares - during which my most frequently asked question of the students is: "Does it make sense to have a larger number for a smaller square?".

The final problem has a triangle without the squares - the kids get to draw them in on their own. The next couple of days involve variations on the theme, building to word problems, none of which ever see even a hint of a²+b²=c². Everything is done by first drawing squares on the triangle, which eventually disappear to be replaced by just the length of the sides and its square.

I've done the second sheet without the first, and it was a struggle, as were all of the lessons after it. They didn't understand why the squares were stuck to the triangles, or why or how to calculate the areas of the squares. That crappy first lesson with all the struggles and frustrations turns the everything after it into a feast of exploration and aha moments.

I've noticed recently that I've been doing something almost subconsciously, and I want to start doing it some more intent.

I've been doing this long enough that I've got an idea of where kids will get hung up, get frustrated, and quit. I haven't found a way to force them through that in situ, but what I've learned to do is give them a particular tool a couple of days or maybe a week earlier (but never a month before) that gets them through. An early example of this is the two step helper, but even that smacks too much of actual process and math.

Here's one I did a couple of months ago. I know that these kids would have difficulty translating slope and rise and run to graphs, or from graphs. They could tell it's a line, but they wouldn't be able to pick out what makes the rise, or the run.

So, before I even start with two variable equations, I had a day of drawing triangles on graph paper. They got to pick two numbers (which I conveniently decided to call the rise and the run), and had to use those numbers for the rest of the day.

I had them draw repeated triangles.

I had them draw scaled triangles.

And I had them draw flipped over triangles.

At the end of the day, they had a piece of graph paper that had a lot of parallel lines - and in particular parallel lines that were not going the same direction as their neighbors parallel lines. They learned that the rise and the run do indeed determine a unique angle, and more importantly, they learned to identify a triangle shape as being related to rise and run.

On its own, this lesson sucked. The graphs were messy, the kids thought it was stupid, and there was nothing concrete that this would ever be applicable to.

But when I taught them to graph lines a week and a half later, just about every kid could figure out the rise and run of a line drawn on a piece of graph paper, or make a line with the appropriate rise and run.