I'm in Cleveland at my dad's house for the annual event... the homemade gnocchi and sauce were great, the kids got lots of noisy plastic crap to forget about by tomorrow, and the vegan chocolate-peanut butter cake from Mustard Seed cafe was tasty: I had two, ok three, slices.

After desert, board games, and being shot at with nerf guns (which have gotten scarily high-caliber), I was talking with my 10-year-old half-brother about his school. Specifically, I wanted to see how he is doing in math. He is (not surprisingly) unable to articulate exactly what he is doing in math, so I was asking him specific questions to see what his math is like. Last year, I was surprised to find out he knew square roots already - I explained about cube and higher roots, and he picked it up instantly.

I wanted to see what he knew about fractions as a 4th grader at a typical Cleveland-area public school. I asked him if 1/2 or 3/4 was bigger.. way too easy. I asked him if 2/3 or 3/4 was bigger. He got it right, and quickly, but couldn't really explain why. I then asked him if 3/7 or 3/8 was bigger, and he said 3/7 immediately. I asked him to explain how he knew, and he looked at me like I was stupid, saying, "a seventh is bigger than an eighth, so..". I asked if he had worked with mixed numbers, and he hadn't, so I asked him to figure out what 3 1/4 - 1 1/2 is. He couldn't do it in his head, so I told him to get paper and draw a picture. That's all I said - he drew fractions circles correctly, crossed off a whole, the fourth, and then another fourth from a whole, and came up with 1 3/4.

I've only ever taught at DCP, so I don't have much of a frame of reference for knowing if he is above average or not, but this is the kind of thinking that all students must have to be successful in high school math. This is the kind of numeracy ability I want my students to develop; this way, when they get to a new problem, instead of giving up, they can reason it through and at least make progress. I struggle daily to get them to pay attention, to care, to think, to not give up when a problem is hard, and their mathematical progress is painfully slow. In a couple weeks, when we start reviewing for finals, and half the kids don't even remember what an integer is, it will be painful. But I know that, by the end of the year, most of my students will have improved their math abilities in many ways. Never as much as I want, but it will have to do! I just gave our grade-level equivalency test before break, and the median score has improved by 1.1 grade levels (from 5.9 to 7.0) and the average by 1.65 grade levels (from 5.76 to 7.42) since the summer. If I can squeeze that kind of growth or better out of them during the second semester, most will be in pretty good shape for next year.

What Do We Mean by ‘Understanding’?

1 day ago

## 1 comment:

Hello Dan,

Finally got some times to read your blog. Sounds like your half brother is way above average and has an exceptional talent for math.

I watched your movies which are fun, but they were not so clear and no sounds. I'll try to see them online.

Hadas

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