In Numeracy, we have so far been working on two concepts: solving word problems with the bar model method and adding integers with and without manipulatives.

The bar model work has been quite interesting, and I'll post more on how it's going later. Adding integers has gone pretty well, as it is not that difficult of a topic for most students. The hard part, as always, is breaking students of their deeply ingrained habits of wanting the "rule" or the "shortcut" that will let them solve the problem faster. They can't seem to figure out that they have learned these rules again and again over the years, and that they haven't stuck yet. And, even though they may think they know the rule, they might not. Several times today I heard "a negative minus a negative is a positive". But I digress...

Today, I began integer subtraction in two of my classes; subtraction is, of course, much more difficult for students to master. In my first class, there was a lot of buy-in. First, I showed them how to do problems where the second number is smaller in magnitude than the first number (8 - 5, -6 - -4), which is easy to show with unit cubes and an integer mat. I like showing how the second example is no more difficult than the first when you understand what you are actually doing. Then, things got really interesting when we moved to problems like 6 - -4, 4 - 7, and -4 - -9. I showed them why and how we add zero pairs to be able to subtract. After I went through it once, a couple eyes lit up. After the next problem, a couple more. And after the third, a few more. I could actually witness students engaged in the act of finally learning a concept. This is one of the joys of teaching basic math to older students. One of my repeating students raised his hand and said, "I don't get it. Why is this so easy? Last year this made no sense, and now it's easy." I think I was able to convince him that the fact that he was paying close attention throughout the lesson was the answer to his question (I didn't teach him last year, but I know he almost never engaged in his class). I'm not sure if this meta-knowledge will stick, but if it does, I think he may now be set to finally learn some math and pass algebra. For sure, when he does lose focus in the future, I'll remind him about what he discovered today.

But with all successes come setbacks (I didn't say failure! I must be getting less cynical). In the next class, the lesson did not go over so well. A couple kids showed me the bright-eyed look of victory, but most were just playing with their cubes. I think I need to invest in unit cubes that do not lock like legos... Some of the students know the "rule", and though they don't know why it works, they wanted to keep using it and not try the blocks. I wouldn't mind it so much (for the few who really do know how to use the rule), except that it prevents students who don't know the rule yet from seeing the value in using the manipulatives. It's like creating a short-circuit. I have two more classes to go on this lesson, so we'll see how the others react. I am still getting a feel for the different character of my different periods, but certain patterns are already surfacing.

What Do We Mean by ‘Understanding’?

1 day ago

## 5 comments:

Dan -- your posts are inspirational, informative, and reassuring. I teach Algebra for students who failed it the first time, and your experiences really resonate with me. Thanks! Karyn

Thanks, Karyn. How long have you been teaching? What kinds of numeracy support do your students get? What is the school like?

Please feel free to email with any questions you have, or any good strategies that you use.

"I don't get it. Why is this so easy? Last year this made no sense, and now it's easy."

wow.

you can't hear stuff like that enough.

sometimes you could almost believe

teaching is worth the trouble ...

mad owen t.

It is great. The trick is to get a kid to express a sentiment like that more than just once, or once in a while. I wonder how many discrete moments it takes for a student until their learning process becomes a bit more continuous.

I like my addition: Basketball scores. Plus is us, minus is them.

We scored 5, we scored 2

+5 + +2 Who's better? us (+) by how much? 7 (+7)

They scored 3, they scored 7

-3 + -7 Who's better? them (-) by how much? 10 (-10)

They scored 9, we scored 8.

-9 + 8. Who's better? them (-) by how much? 1 (-1)

I don't use blocks much, but when I do, I call them "toys." I expect that many kids will end up playing with the toys, and I recall from other years making playing a reward for finishing work. I mean, I couldn't stop it, at least let me get out in front.

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