Entries from TipsAndTricks

The kids pick the number of digits (we usually start with 5). They put 5 dashes on their paper. I turn over 5 cards in a deck one by one. They have to decide where to put the numbers. Then each kid reads their number to me while I put it on the white board. The kids with the highest number wins.

For some reason, they love this game. On the next round, we go up one digit. Today, we went all the way up to 100 million.

It's a great game.

• They gain familiarity with large numbers. They get a lot of practice with reading large numbers out loud and hearing large numbers read out loud while it is being written on the board.

• They have to use strategy. In some games, we have a lot of high numbers at first which every kid puts in the same place. Then, they winner is the determined by the numbers in the ones and tens place. Conversely, sometimes we have a lot of low numbers in the beginning. Then the winner is determined by the highest digits. Much more interesting is when we have medium and low cards. Then, they have to do a lot more thinking about where the cards go.

• There are very concrete results from this game that allow us to explore numbers even further. In one game, 5 out of 8 kids had the same highest number. So we talk about why and when does this happen? In one game, we had one winner that was a lot higher than anyone else. When does this happen?

We have a gang of kids that run semi-wild in our neighborhood in the summer. They are very mixed in age (ranging from 7 through 11). I have thought about corralling the whole lot of them and bringing them in to teach them all some math together; it would do them all some good to work on it over the summer, and Ben would enjoy his math sessions more if he shared them. I'm a little stumped, though, about how to teach a wide range of ages and interest levels simultaneously.

I'd love to collect some more math games that are as simple and elegant as this one is, especially games that might appeal to a broad range of ages, and (like this one) start a math session off on the right foot.

There are two ways that a teacher might see what looks like evidence for modality theory in the classroom. First, a teacher who believes the theory may interpret ambiguous situations as support for the theory. For example, a teacher might verbally explain to a student - several times - the idea of borrowing in subtraction without success. Then the teacher draws a diagram that more explicitly represents that the 3 in the tens place really represents 30. Suddenly, the concept clicks for the student. The teacher thinks "Aha. He's a visual learner. Once I drew the diagram, he understood."

But the more likely explanation is that the diagram would have helped any student because it is a good way to represent a difficult concept. The teacher interprets the student's success in terms of modality theory because she has been told the theory is correct and because it seems to explain her experience.

Willingham offers the following suggestion: teach to the best modality for representing the idea, not to the student's best modality.

But what if there are multiple modalities to choose from, for an idea? More generally, what if there are a whole host of different ways to represent an idea, and the kid's not getting any of them?

I ran into that situation recently, when teaching Ben how to do simple problems by adding and subtracting constants on both sides of an equation. Actually, trying to help Ben get the hang of this has taken quite a bit of effort this week, and I don't think it's a hard idea. I've got kinesthetic, visual, and auditory ways of teaching it, too. I could even sing it, though that's getting a bit ridiculous.

For the kinesthetic learner, you could get out a balancing scale or use Bornstein manipulatives. You could draw pictures of pan balances for a visual learner. You can explain verbally, as I did repeatedly, that what you're doing to solve the problem x + 4 = 13 is to 'undo the addition' of the 4 on the left hand side of the equation. If none of this works, what do you do then?

Try each modality over again, I suppose. Round 2: in case he was a kinesthetic learner, I had him copy each step I made in his own handwriting (laugh, if you will, but it works for me when I do it). In case he was visual, I drew pan balances again, next to the equivalent equation: no dice. "Subtracting the 4 is applying the inverse operation to get the x by itself," I said, auditory-like, but that didn't help either.

All this time, of course, he was able to do the problems by repeating the steps I made; he is a fabulous rote learner (is 'rote' a modality? If not, it should be). But I could tell he wasn't really getting the gist of it. Finally, in exasperation, I said, "Look, Ben, what's the opposite of adding 4?

"Subtracting 4."

"Good! And what's the opposite of subtracting 13?"

"Adding 13."

"Good. All you're doing to get the x by itself is doing the opposite of adding or subtracting the number that's with it," I said, but I didn't even get it all out before he said, "OH! I get it!"

And that's the sound I love to hear.

So, knowing Ben's best learning modality didn't help, and wouldn't have helped. I wish teaching, and learning, were so predictable that all you needed to do to teach a whole class reliably was to know what each kid's best learning style was. But I think that learning is inherently unpredictable. The trick is to be able to hit the teaching problem from a bunch of different angles, and you need to know lots of different ways to present the information. The more, the better (by the way, this is a major part of what Liping Ma's Chinese elementary math experts do with their release time; sit around together, thinking up new ways to teach problems to tough cases).

As an aside, I have never been able to figure out Ben's best learning modality (aside from 'rote'. His raw memory is unbelievable). As a person on the autism spectrum, he's supposed to be a visual learner; this is accepted theory to such a degree that teachers will assume he needs to learn visually, but it's not always the right approach.

What Ben really is, is an unpredictable learner. You never know what's going to be easy, where he'll get stuck, and what will unstick him. He's the kind of kid who keeps a teacher on her toes.

Things I note about her teaching approach:

• It's actually multimodal! She's talking, she's getting the kid to write AND draw his own pictures, she's doing whatever it takes to get the kid on track.
• It's feedback-driven: she switches gears in midstream if the kid isn't getting it.

Carolyn's post:

SteveH is correct. Whatever you do, you must bring all students to UNDERSTANDING. And this is what students really do want. "Understanding" or "not understanding" are the reasons students "hate" or "love" math.

A good math teacher learns how to "approach" a student having difficulty. The teacher has all of these ideas (hopefully) stored away back there some place, ready to be pulled out when needed. But a teacher's most important job is determining where the student's understanding fell apart, identifying where there might be gaps in reasoning, and knowing how to bridge those gaps. This is where choosing the right approach comes in. It might involve reteaching, reviewing a step that is being omitted, or helping a student reason through a difficult story problem.

So a hand goes up, and a student says, "I need help."

(Those are my favorite times of the math hour because it means I get to find the puzzle piece that is needed to make this all fit together in his/her head and give understanding to what I've just taught or to what's needed to solve this problem.)

So I have some choices, but I always look to see what the student has already done or tried. That tells me what to do next.

I then start by having the student read the problem to me (if it is a word problem).

Then I make a choice:

I might say, "OK, draw Bill's house. Now write 'B' on it for Bill. Now, draw the schoolhouse; now write 'S' on it. Now, draw the road from the house to the school. Now, look at the problem again to see how far it is to the school (and the student answers outloud 4 1/2 miles). OK, write that number on the map you've just drawn."

I could have drawn that little map for the student, and might do it under certain conditions, but having a student draw the map involves his sight and his movement (and mouth from speaking and ears from hearing his own voice) and it involves more importantly his BRAIN. (I've got to make sure his BRAIN is working and focused on the problem so he can "understand".)

So I would continue, "Start at Bill's house with your pencil; now walk to school. OK how far did Bill just walk? ('4 1/2 miles') OK, write that down. Now, he's at school, but he wants to come home, so have Bill walk back home. How far did Bill walk to get home? OK, write that down under the first number. How would you know how much he walked to school and back on that one day? ('add the two numbers together.') Good, do that. OK, but that is just one day. Now, let's read the problem one more time and let's see what the questiion was. (How far does Bill walk in a week going to school and back?) OK, now how could we figure that out?"

Many times the problem just works itself out in the student's brain as they begin to draw out a picture of the problem.

Or, if I've checked over his work and seen that he's added 4 1/2 miles 5 times for the 5 days of the week, I can see that he's overlooked a part of the question. So I have the student reread the question. Many times, the student will catch his own mistake when he hears his voice repeat the words "to school AND BACK". If not, I have him read just that part again.

Something really important: for some students it's just a matter of not knowing "how or where to get started". There are gaps in processing the information and gaps in understanding.

Not only does he need to know where to start, but he needs to know that where he is starting will get him going in the right direction and will help him get the right answer. This is very important to a student's confidence. If a student doesn't know "where to start" or isn't sure "if he's going about solving it properly", a teacher's trying to find the right modality isn't necessarily the answer.

This is where an "constructivist" approach is so devastating to the student. That student wants to be able to KNOW what to do to get the right answer.

It's terribly upsetting and deflating to a student not to know "where to start" and "if they're taking proper steps to solve the problem correctly." To leave this student to come up with his own idea isn't helping him.

Hopefully, though, a teacher will NOT just repeat the instructions that were given initially (if any were given). If a student didn't get it the first time, at least try a different approach.

One of my former students said of another teacher, "Why should I ask her for help? She always just repeats the same instructions that didn't make sense the first time." This student, smart as a cookie, just wanted to understand the entire process and to know how to work to get the right answer.