Happy Pi Day!

Before we begin, you really must watch this video if you haven’t seen it already (and, actually, you should watch it again if you have – it’s that awesome):

Now down to business. There are some kids who just love everything you ask them to do. Some of them love it because they love, love, love math in all its forms. Some of them love it because they love you so very much. (This is true in 3^{rd} grade, at least – it’s less true after that, sadly.) But some kids have a lot of trouble digging in. The things you say, the activities you plan, the enthusiasm you bring – none of it grabs them and compels them to make a niche in the mathematics you’re studying, to make math their own.

The third grader whose conversations with me are the topic of this post is one of those hard-to-grab students. She LOVES to draw, to make up characters, and to create cartoon stories. She HATES math. Or so she told me today. Here’s conversation number one (clearly not verbatim – I don’t have that great a memory. But it’s close enough.):

*Note: I’m calling her Olive, but that isn’t her name.*

OLIVE: Can I draw my ballerinas instead of doing math today?

ME: I think not. We have lots of things to do!

OLIVE: What are we doing?

ME: We’re going to learn about division! *(Exclamation points indicate me communicating enthusiasm!)*

OLIVE: I don’t want to do division. I don’t like division. I hate math.

ME: What about math do you dislike?

OLIVE: I don’t know. I just don’t like it.

ME: It’ll help me to understand what you’re thinking and what we can do about it if you can give me some specifics. Do you think math is boring? Do you think it’s too hard? Do you not like certain parts of math? What do you think?

OLIVE: Math is too hard. It’s too hard because I don’t know the basics.

ME: *(Taken aback a little by response. Not what I expected to hear.)* Can you explain what you mean by that? What basics do you feel like you don’t know?

OLIVE: I can do some things, but I don’t know the basics.

ME: Ok. Here’s a basic – do you know how to add?

OLIVE: I can add two-digit numbers.

ME: Great! You sure can! I’ve seen you do it! How do you add two-digit numbers?

OLIVE: Well, you put one number here *(gesturing) *and the other number here *(gesturing below the first number)* and you line up the digits and you add them.

ME: Awesome! Can you add three-digit numbers?

OLIVE: Yes, I can. You do the same thing. But I can’t add one-digit numbers.

ME: *(Pause.)* No?

OLIVE: Adding one-digit numbers is harder.

ME: Huh. Why don’t you show me how you’ll add two-digit numbers. *(Grabbing paper and pencil.) *Ok – what’s 99 plus 99?

OLIVE: *(Writes numbers stacked digit-wise.) *9 plus 9 is 18, put the 8 here, carry the 1. 1 plus 9 plus 9 is 19. So it’s 198.

ME: Super! Ok, so what’s 999 plus 999.

OLIVE: *(Adds.)* It’s 1998.

ME: Alrighty. 9999 plus 9999?

OLIVE: That’s easy, I’ve already done that. Just stick another 9 on the inside. Ooo, that’s an interesting pattern.

ME: Yeah, it is. Olive, look – *(pointing to the nines in the ones column)* – you just did one-digit addition here.

OLIVE: *(Pause.)* I know. But it’s harder when it’s just one digit.

ME: What about it is harder?

OLIVE: There’s less there. There’s just one digit. *(Doing some gesturing indicating lining numbers up.)*

ME: *(Really trying to understand what’s going on.) *Because there’s nothing to line up? Because there’s more riding on that little addition, because it’s the whole answer?

OLIVE: No, kinda, no. It’s just harder.

ME: Ok. Well, what’s 4 plus 4?

OLIVE: 8!

And the conversation continues in this way for several more minutes, until class begins.

Now for conversation number two:

*Olive is sitting on the rug with another girl. Second girl is working on division sheet. Olive is doodling.*

ME: Hey ladies. What’s going on?

OLIVE: Can I draw my ballerinas instead of doing the sheet?

ME: That’s a great ballerina. But how about you try this problem?

OLIVE: No, I don’t want to! I don’t like this. I don’t like math.

ME: I remember you said earlier that you didn’t like math because it was too hard, because you thought you didn’t know the basics. Well, this kind of division is one of the basics. It’s a good thing to learn if you want to know more basics.

OLIVE: No, that isn’t why I don’t like math. I don’t know why I don’t like math. I just don’t like it.

ME: Well, what are some things that you like that math doesn’t have?

OLIVE: Huh?

ME: What are some things that you like to do that you don’t do in math?

OLIVE: I like things involving pictures.

ME: I have an idea – why don’t you make a picture for this division problem? *(Not really expecting her to go for this.)*

OLIVE: *(Pause.) *That’s a really good idea! 8 divided by 2…

*Starts drawing an 8 spliced into two pieces, top circle and bottom circle. 8 has arms and legs. It is being menaced by a 2 – fanged, with a giant sword.*

ME: I really like the teeth on the 2. But what goes in the eight-halves?

OLIVE: What?

ME: Well, what is each eight-half? What goes in each circle?

OLIVE: Ummmm. 4?

ME: I think so! What can you do with that?

OLIVE: They’re boogie-ing!

*Draws a little dancing 4 in each half of an eight. Moves onto next problem – 14 divided by 7.*

OLIVE: How many pieces should I break the 14 into?

ME: Well, what’s it being divided by?

OLIVE: 7? That’s going to be hard to draw.

ME: And what’s boogie-ing?

OLIVE: No! That’s too hard! Can you tell me, pleeeeeeeease?

ME: Olive! It’s your artwork! I can’t make part of it for you!

OLIVE: Ok, ok! *(Really intensely focused expression crosses face.) *2, 4, 6, 8, 10, 12, 14. The twos are boogie-ing.

Olive and I spend the next 45 minutes doing seven very basic division problems this way.

Pros: She dug in. She was so very into her artwork that we worked almost a half-hour beyond math class on this. This is one of the only times this year that she’s chosen to do math over other things.

Cons: We didn’t do much dividing. We did a lot of personifying numbers and the act of division, which is important for her understanding of division. But we didn’t get much hands-on experience with numbers and division.

She loves to draw, and I like that we found a way for her to incorporate drawing into math. But I’m not sure we did enough math. She wants to learn math – in her own words, she recognizes that her skills are weak, and she wants them to be stronger. But she isn’t compelled to work on those skills. This doodling division was great for today, but I need better ways to get her invested in doing mathematics – for the long haul.

John Burk

said:I suppose she’s way to young for Vi Hart, right? But there’s a strong connection between doodling and math for her.

Anna Weltman

said:Oh boy – she might be too young to entirely understand, but she’s probably not too young to get something out of it. I’ll give it a try. What a great idea! Thanks!

Anna Weltman

said:So, I showed my 3rd graders several of Vi Hart’s videos today – and they loved them! They immediately started drawing like crazy – fractals, Appolonian gaskets, knots, bi-colorable graphs… It was lots of fun, and everyone was into it. We’d already done some colorings of Pascal’s Triangle, so the fractals were familiar and exciting to see in a different context. Thanks again for the great idea!

woodshopcowboy

said:I’ve got a crew of kids who challenge my teaching ability the same way – they have a dislike for the math they don’t “get”. It’s tough showing them the same concept over and over and over again. My current get is a RPG-like game (probability dungeon crawl) which I’m hoping turns into something a little more math, little less chop-the-goblin’s-head-off.

When you figure out how to master that redirect, tell the rest of us, will you?

Anna Weltman

said:I’ll certainly let you all know what happens, at least, regardless of how masterful it is. That dungeon crawl game sounds like fun. It’s really hard to find activities that students will be genuinely invested in for the math and to figure out how to take things they’re interested in that aren’t quite mathematical and bring out the math.

Your post made me think about good fraction games, which reminded me of a fraction game that another teacher at my school made up last year, that my kids really liked and was very mathematically interesting. It’s a lot like war, but the cards have fractions on them, and you get to design your deck. You can use the numbers 1-10, each number exactly once, to make your fractions. (So you should each have 5 cards.) To play, each player flips over a card. The player with the larger card wins the round, and the cards are removed from play. Short and sweet game, allowing for many rounds and lots of strategy-development!

The first tricky step for the kids was to realize that they can make fractions greater than 1. It’s really awesome when the one kid who thought of that from the start creams everyone else – and then they get to remake their decks. The second tricky, and more nuanced step, is deciding how to distribute “largeness” in your fractions. For instance, 10/1 is very large. But the next largest card you can make is then 9/2, which is significantly smaller. Perhaps the better strategy is not to make a top-heavy deck, but to try to make all of your fractions about the same size… or is it good to have a card that always wins, at the expense of a throw-away card?

The teacher who made this game up designed a Mathematica program to try to find the deck that always wins – and it turns out there isn’t one. Interesting, no?

woodshopcowboy

said:You have a smarter teacher than me, I would tell them keep trying.

I would assume that game’s like poker, right? You don’t play the card, but the kid. I might try it. My probability game is here: http://wp.me/p1ew1W-iw

Take what you can from it. I would imagine combat could work similar to my spinners idea…hmmm….

Anna Weltman

said:Here’s an update on how things are going with darling Olive and the rest of the 3rdies:

On Monday, we gave them a sheet full of different types of division problems: a multiplication table with some of the outside portions left out, as well as a large chunk of the middle; some, “What times 3 is 27?” type questions; some, “How many 20s are in 240?” type questions; and some straight-up division. During class discussion time before we gave them the sheet, we tried to emphasize that division can be asked in different ways – as the, “What times…” question or the, “How many in…” question.

They didn’t finish the sheet, so we had them return to it today. Before the started, however, we gave them an additional assignment. As they worked on the sheet, they were to pick a division question they liked and draw it. We had Olive come to the board and demonstrate how she draws division. (She was very proud to do so – but, interestingly, told the whole class that the drawing was my idea. I said that I certainly did suggest to her that she draw division, but that the way of drawing it was entirely her idea – which was the truth.)

Most of the kids got really into the drawing! Their drawings mostly followed Olive’s style, but they also got creative with remainders. And, after doing some drawing, Olive picked up her sheet and started working on it – of her own accord. She didn’t do very much… but it’s a start?

Ben Blum-Smith

said:Anna, this comment is mad late, but do you know the book

You Can Count on Monsters?Olive’s division drawing style is just reminding me of it very strongly…