This is another story of impulsive teaching. I'm not recommending that, but we got to a good place, so I want to tell someone.
In my preservice elementary course, we were headed towards decimals, passing through place value, so it was time for the base 10 blocks. A wise elementary teacher taught me that new manipulatives should always start with play time. (If you can't, tell them when they will be able to play. Chris's other lesson was to use each new manipulative as a chance for the students to tell you the rules about using them. Pro tip.) The wooden Base 10 blocks we have are particularly good for building. But playtime always ends with: 'so what did you notice about these?'
They found the 10 fit into the next one pattern, and noticed irregularity in these old, hand-cut materials.
I love when manipulatives are used for a purpose or a problem rather than a set of exercises. So I asked how many blocks were in each tub, in terms of the small cubes as the unit. 3510, 4873, 4508, 4508, 3377. Hey! That's not very fair. Ooh, I have an idea: what if each group gave the next group half of their blocks? 4191, 4691, 4003, 4508, 3443. Is that any better? Some say yes, some no. Let's give half again! 4441, 4500, 3741, 4256, 3817. Still disagreement about what's happening.
Okay. Let's settle this like mathematicians. Make a display of the data that proves your point. We collected one more round of give half away.
I was really impressed at the diversity of displays by happenstance. It made for a great discussion of results. As we often do, I asked for each group to get feedback from the other students: one specific thing you like about their work, and one thing that would make it stronger.
I missed one of the graphs, but here are the other three.
|The first graph shown, people liked how it made the visual comparison of the round by round numbers. Convinced people that the numbers each round were getting closer.|
|This display charted each group's round by round count compared to the mean.|
|The classic lineplot follows each group's total round by round. People agreed that this showed convergence to the mean the most strongly.|
Our individual block distribution is out of wack. Two groups don't even have enough to compose the next unit! We had to make some trades to get a better balance. We could play...
We went table by table with people proposing trades. The whole class decided if a trade was fair. It was the most fun I've ever seen composing and decomposing by place value. Trading was heavy and fast paced. But occasionally we had to stop to check fairness. We even had one crazy three way trade. There was lots of interesting reasoning about the quantities and how they got out of whack even while the totals converged. Final count - not bad.
The idea of social relevance in math class has always been an interest. My colleague Georgi Klein made great use of Marilyn Frankenstein's algebra work. And we almost got a chance to hire Mathew Felton who looks at the political aspect of math learning. So I closed with an observation that with so many math problems about maximizing or candy, it might be nice to address big issues, and disparity is something that's going to be an issue. I got a little preachy, really. But it felt like a good day, with some real values in our place value.
Transitioning to decimals, after work with a fixed unit, we traditionally do something like the top part of this next activity. (Probably originated with Jan Shroyer.) It starts the idea of shifting the unit for different situations. Pretty effective. This time around I added the problems at the bottom as puzzles. They were very interesting for the students to think about, and seemed to push consolidation of their decimal strategies. It really requires a lot of reunitizing. I'd love to know how middle school students thought about them. Each group made up a puzzle of their own to swap, and that also seemed beneficial.
I'd love to hear your thoughts about political values in math class, block market trading for place value, or the representation puzzles.