Tag Archives: 6 years old

Zero=half

A propos of nothing the other day, Tabitha asked a strange question.

Tabitha (six years old): Why are zero and half the same?

Me: They aren’t.

T: Like seven is one more than six, but zero and half are the same. They’re both nothing.

Me: One half? if you have half of something, that’s more than nothing.

T: But half, the number, that’s the same as the number zero.

Recall that last fall, she was not convinced that one-half was a number at all.

She now accepts that one-half is a number. But she hasn’t really dealt with the idea that there are numbers between other numbers. She is doing a bit of beautiful kindergarten logic here. Her premise is that there is only one number less than 1, namely 0. She has also accepted that one-half is a number less than 1. Therefore, one-half and zero are the same.

And—rightly—she is suspicious of this conclusion. The logic is sound, but it doesn’t make sense.

I go to work on that first premise.

Me: Oh. I see. Well, one-half—the number—is between zero and one.

I draw this picture, which I feel is certain to be totally unconvincing.

I was writing upside down. Forgive the crummy 2's. Note the complex fraction. Take that, Common Core!

I was writing upside down. Forgive the crummy 2’s. Note the complex fraction. Take that, Common Core!

But then again, we hadn’t talked about one-half being a number since October. That last conversation seems to have been fermenting all this time, so maybe this one will do the same.

To be continued, I am sure.

Advertisements

Planting Seeds with Tabitha (or, The Pigeonhole Principle)

We were planting seeds the other day. Indoors. This is Minnesota, after all.

Over the course of many years of gardening I have worked out a system. Yogurt containers, potting soil and these lovely clear IKEA containers.

photo (1)

The IKEA boxes are a recent innovation. They keep soil moisture high (yet have enough volume to allow the plants to breathe), and they let me move plants inside and out according to the ever-varying spring weather (it was 80° on Sunday this week, and it snowed on Wednesday).

Sorry for the digression. Back on task.

We were planting tomato seeds by poking holes into the soil, placing one seed in each hole and covering the seed. We had discussed how deep to make the holes; that the depth corresponding to Tabitha’s first knuckle is not at all the same as the depth corresponding to my own, et cetera.

Tabitha (six years old): How many holes should I put in this one?

Me: Five. Put one in each hole.

I hold out my hand with several seeds for her to take.

T: But there’s more seeds than holes.

Me: So what?

T: So then they’ll be crowded.

This is her line of reasoning, not mine. I had not been at all concerned with trying to offer the precise number of seeds she would need. I had simply shaken some from the pack into the palm of my hand.

But since she started it, I develop a plan. I am going to do my best to get her to state the pigeonhole principle.

Me: But what did you say about the seeds and holes?

T: There are more seeds.

Me: And what are the consequences of that?

T: You said the plants wouldn’t grow as well if there are two in the same hole.

So close! She is using the pigeonhole principle, but I cannot quite get her to state it.

So I do.

I tell her about pigeons and pigeonholes.

TooManyPigeons

We proceed to a lovely (and thoroughly uninformed) discussion of the mechanics of sending messages by carrier pigeon. She wonders, for instance, about how to send a message to your friend, since the carrier pigeon’s unique skill is to fly home from anywhere, but not vice versa. We deduce together that you must need to borrow your friend’s pigeon.

Oh, and those tomato seeds? Brandywine.

Peeps math with Tabitha

After the Peeps photo session last week, I test drove my images with Tabitha (six years old).

Me: Which are there more of in this picture? Purple Peeps or pink?

peep.compare.1

Tabitha: Purple.

Me: How do you know?

T: It goes all the way to the top.

Piaget would be proud. Tabitha’s focus is on one dimension, rather than on overall quantity. So let’s test that hypothesis. Does she really believe that’s all that matters?

Me: What about in this picture?

peep.compare.4

T: Purple.

Me: But they both go to the top in this one.

T: This one (purple) has full rows, and this one (pink) has holes.

Me: Interesting. You know what I see? I see that if you moved that last bunny on the bottom row up to the next row, you’d have two rows of three and an extra bunny, while the purple has three rows of three.

T: Yeah.

Me: OK. One last one. What about this picture?

peep.compare.3

T: Purple.

Me: Because it goes to the top?

T: Yeah.

Me: Look carefully, though.

T: Pink.

Me: Why?

She proceeds to count 9 pink bunnies. I correct her and have her count over. She again counts 9 pink bunnies. I show her that if you move the two top purple bunnies into the second row, you would fill that row. She is uninterested and we move on to other things.

Pumpkin muffins

My wife Rachel made pumpkin muffins last night. Her contribution to the world of baking is the chocolate chip pumpkin muffin. I feel The Honest Toddler would approve. I know that Tabitha does.

Tabitha told me about a dream she had last night. In the dream, there was only one pumpkin muffin left.

Later in the morning, Tabitha counted the muffins in the tin.

photo

She got eight.

Me: Wait. Count those again?

Tabitha (six years old): [Points to her mouth] One. [Points to first muffin in tin] Two, three, …

Et cetera, ending at eight.

Tabitha: So really, we had eight muffins left.

Me: I see, last night when you dreamt there was one, there were really eight.

T: Yes.

Me: That must be reassuring to you.

T: Did you eat one this morning? Then it would be nine.

Place value and language

Timon Piccini writes about a conversation he recently had with his niece.

My niece is in grade 1, and she is adept at adding single digits.  With little hesitation she can do her basic addition.  She even showed me that she could do things like add 100 + 100.  I thought this was really neat so I asked her some questions.

Me: What’s 1+1?
Niece: That’s easy it’s two.

Me: What’s 100+100?
Niece: It’s 200 duh!

Me: What’s 1000 + 1000?
Niece: 2,000 these are easy!

Me: What’s 10 + 10?
Niece: … I don’t know.

This leads Timon (TIM-in) to wonder about how language is related to early numeracy and later mathematical development. Interesting stuff.

I learned after commenting that his blogging platform doesn’t allow html code. So my comment is hard to read. I have reproduced it below. But go read his full post. And read the comments while you’re there.

My comment

This is where my mind has spent the last few years. So lovely to see that I’m not the only one intrigued by this sort of stuff.

I love that conversation with your niece. Just like we need to read aloud to our children, we also need to talk math with them. I don’t think we do any damage when we move to symbols (as you did in this conversation), but I don’t think we have any evidence that it’s really helpful, either. Like teaching a pig to sing, I suppose (wastes your time and annoys the pig).

What does seem to be helpful is that you’re interested in the child’s ideas. This can take many, many forms. One interesting activity for a curious teacher such as yourself is to take a moment to formulate a hypothesis and then a question to test it. Here you noticed that she could do 1+1 and 100+100 but not 10+10. Your hypothesis (which I also believe to be correct) is that this is language based. So ask her, What is 1 ten plus 1 ten? I’m curious whether she would say “two tens” or “two ten”. I can’t tell from your transcript whether she said “two hundreds” or “two hundred”.

Anyway, I think you and I would both be surprised if she had no answer for 1 ten plus 1 ten. When she offers it, follow up with How much is that? How much is two tens?

Many thanks to all the folks who have contributed references. Those will be helpful as I develop my own understanding of this territory. I’ll add my own (a bit self-serving, admittedly). I wrote a paper on relationships between quantity, numeration and number language (my bit starts a few pages into the file). That paper grew out of work I do with future elementary teachers and research from Karen Fuson. It contains several worthy references.

My 6 year old’s definition of algebra

Overheard in a conversation between Griffin, 6 years old, and Tabitha, 4:

Griffin: I hate algebra.

Tabitha: What is algebra?

Griffin: Algebra is a piece of paper with math problems that are very hard.