Conference Material

Engaging Parents and the Public

One of my favorite drills that we actually do with parents in the session is a game called Tic Tac Toe Products. How many of you have actually done this before? Experienced this? Only a few. Well, we're going to take just a couple of minutes to do this. I'm going to divide you right down here with you folks being x's and you folks being the o's. And in this game, you are trying to capture four numbers in a row for your team -- either horizontally, vertically, or diagonally. Four in a row. And the way you capture a number is you tell me which two factors you want me to put the paper clips on.

So if you told me four and six, I would say what's four times six? And if you told me 24, I'd mark the 24 with your x or your o. The next team can only move one of those paper clips. You can move it anyplace you want, but you can only move one paper clip. Okay? So let's play it for just a couple of minutes and you'll see how it's played and then we'll come back out of that experience. By the way, those of you who have a need for closure may be deeply disturbed by the end of this session.

So x's, you won the toss. You get to go first. I need two factors. Where do you want me to put the paperclips?

Q: Do you have to put them both down?

Parker: Yes.

Q: Four and six.

Parker: O's, I'm only going to give them this little bit of help because we're not going to have very long to play. But had we had longer to play, students would have been saying, "I don't think so." Because you boxed yourself in and you won't be able to build.

Q: Nine and two.

Parker: How do you argue with that kind of authority? Nine and two. Nine and two, is that okay? No? Yes? Two times nine? 18. O's, you can move one of those paper clips any place you want, but you can only move one. Move the nine to the one? To an eight? Move the two to the three? Two to the five? Nine to the six? Now you can move one paper clip any place you want, but you can only move one paper clip. If you kids would only listen to me when I give directions. Move the two to the three. Three times nine? Move the nine to the three? Nine to the three? Okay? Three times three is? Wait a minute, it's their turn now. O's. Move the three to the nine? Move the three to the one? Just do it, they're chanting in the back of the room.

One times three. Ahh, now that raises an interesting question. Is blocking them, even if it doesn't serve you an efficient move? Or might there be another way to block? I helped you too, so I'm helping them this time. You want to move a three to a five? Three to one? One times three. No, no, no, no. Three to five. It's only a game. Three times five? 15. X's? Move the five to the one? Three times one is? We're going for four in a row.

Are you doing drill? Yes. In fact, you're doing quite a few drills to determine every move. But as a teacher, for some obvious reasons, this is not a great way to play the game. Some of you are wanting to follow a strategy and I'm taking whoever's loudest, so you're frustrated. Also I know that some of you are doing drill, I don't know that all of you are. And it's not okay with me for a sixth of my children to be tuned out in the classroom. So what I would do in the classroom is I would have children seated in groups of four and each group would get a sheet like this. And I would have two children play two children. I'd partner them up for two reasons. First of all, some children don't know their multiplication facts and it helps to talk those over as part of learning them. And some children don't see the strategies. I don't know if we played long enough for you to see that the game gets very sophisticated with regard to strategies. Especially when you realize you not only want to pay attention to what you want, but where you don't want to leave your opponent and where you might want to be positioned next time. Once children catch onto that, the game takes a long time to play. It sometimes takes 45 minutes or more. And all I do as a teacher is I smile inwardly and I say, "They're all practicing their multiplication facts for 45 minutes and nobody's complaining."

Now I also send this kind of thing home for homework. But I've learned that when I send it home for homework, I send it home with a letter that says something like this, "We're working on our multiplication facts this year. It's going to be very important to everything we're doing. We'll be doing lots of drill in the process. Please notice the drill that you're doing while you're playing the game and have fun playing the game." And I've started sending the letters home because I sent the games home for years and I had parents saying, "Ruth, we're having so much fun with math at home as a family. Thanks for sending it home and keep it coming." And then I would have a back to school night and almost always one of the first questions I'd get asked is why don't we see drill coming home this year? It took me a long time to realize that if we'd always thought of drill as looking like forty of these on a page, we don't necessarily see that children might be doing 400 of these in the context of playing a game that also asks them to think and reason and strategize and behave in some mathematically important ways.

So one of the things I think we have to get a lot better at is helping ourselves, helping our children, helping our administrators, helping our parents see that drill is very solidly embedded in the new programs that we have available to us today, but it looks very different from what we're used to seeing.

Okay, now we're going to pull out of that parent piece again and we're going to shift direction. And the third myth I want to look at with you is the myth that the new National Science Foundation funded mathematics programs are fuzzy, devoid of substantive content, and lacking in the basics. I want to share with you the work that my 5th grade students did from a unit that was written by mathematics educators at TERC who developed the series Investigations in Number, Data, and Space. This unit is now embedded in that K-5 program. I don't know where you are in your adoption process, but if you are adopting and if you like what you see in the student work, I'd encourage you to take a careful look at this program.

The unit starts with children using a geoboard. And I'm assuming we've all seen a geoboard. Parents haven't necessarily all seen one. And the geoboard in this case is divided into square regions. Now one of the things that my students have learned before the work that you're about to see is they've learned to find the area of a rectangle. So what would the area of this rectangle be? Two is not an area I would say to my children. Two square units. We need units attached for something to be an area. And they have learned that a diagonal will divide a rectangle in half.

So if this were an actual diagonal, what would the area of the triangle be? One square unit. The unit starts by asking children to divide a geoboard in half in as many ways as they can. And initially, they do it all the predictable ways. But they quickly learn halves don't have to be congruent. They don't have to be the same shape, they have to be the same size area wise. And before the work that you're about to see, they have divided their geoboard in halves, in fourths, and in eighths in a variety of different ways.

This lesson gives children a plain six inch unmarked square. And the assignment says divide the square in a way that shows your understanding of fractions and use at least three different fractions. And I remember one of my children who came to me, having really struggled in mathematics, worked hard to measure and to say something like there's two halves, there's two fourths, there's two eighths. Matthew, age 10, turned in this. Now I don't know if you can see this. It said and write the number sentences that tell us what you did.

The teacher I was working with, Cathy Young, is a teacher who is wonderful philosophically aligned and very math phobic. I think of her as math injured somewhere along the way. So when Matthew turned this paper in, she came up to me with a look of terror in her eyes and said, "Fine, Ruth, what do we do now?" And I said, "Well, I can think of three things that you could do. First of all, you could take it home and have a fair amount of homework to do yourself tonight, Cathy, to know how to write a response to Matthew. Or you could just think about writing, "Nice work, Matthew" not really knowing if it's nice with regard to fractions, but it looks pretty good." Or we could make a black line master of it and we could give it back to the class and we could ask students, "What about this one? Does it make sense?" knowing we've given them a pretty good problem to solve. So with those three choices, do you have any idea what she did? No, she didn't say nice work, Matthew.

We gave it back to the students. Now you're not going to have time to look at the whole thing, but would you just for a couple of moments take a look at the yellow and see if you can come up with a mathematically convincing argument for whether the yellow is or is not one-sixteenth of Matthew's square?

That's a big if on that one. Okay. So what do you think? Is it one-sixteenth or not? Now I gave students the whole class period and I gave you less than two minutes. So this is not really fair. Some of you are saying yes. Can you convince me mathematically that it is? What method would you use? Anybody willing to share what you would do? Ahh, you would use Pic's Theorem. Something we all know and love dearly, I am sure. But actually, those of us who have investigated area on a geoboard might use that method, yes.

Q: Well, (...inaudible) the whole thing would be a square. Parker: He's divided into a four by four. So there are 16 squares.

Q: (...inaudible) is I've got to figure out if the yellow is one whole square.

Parker: Can you hear her all over, by the way? Yes? Okay.

Q: Okay, so take the column on the far right. (...inaudible) square there.

Parker: You're saying take this column over here?

Q: Right. Okay, now left of it, the yellow part to the left of it is a half of a square.

Parker: Okay. So you're saying that this yellow would be half of that square because we have a diagonal that divides it in half. So we've got a half a square there.

Q: The question now is (...inaudible)

Parker: Is what's this odd thing here.

Q: The whole thing to the right is one square. And so the right side of the yellow squidgy is the diagonal of the four squares.

Parker: By the way, this is what my students finally did too. They said here's the diagonal that divides those four squares in half. So this is two square units. Okay.

Q: Okay. So now you take the first three squares down.

Parker: You're talking about this rectangle right now?

Q: And now the left side of the squidgy is a diagonal of that.

Parker: The left side of the squidgy is the diagonal. So you're saying that this is a diagonal that cuts that rectangle in half?

Q: Right. So that orange piece there must be one and a half.

Parker: So this is going to be one and a half?

Q: So if we had four minus--

Parker: We had four. We took off the two.

Q: Minus one and a half.

Parker: We took off the one and a half. That left us with the half. And a half and a half is a square. So that's one-sixteenth. That is precisely what my students did to prove it. After recycling Matthew's, though, Cathy and I looked at each other and said, "Let's not go on. Let's give them another square and ask them to divide it in a way that shows their understanding of fractions and use at least three different fractions." And a very interesting thing happened in the classroom after we recycled Matthew's work.