This came from a workbook bought by the kids’ grandparents.

Can someone please explain the purpose of the jars of bugs here?

The mathematics I encounter in classrooms

This came from a workbook bought by the kids’ grandparents.

Can someone please explain the purpose of the jars of bugs here?

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Posted in Diagrams and decorations

Tagged insects, jars, numbers, numeration, place value

A few weeks back, Tabitha asked *Why are zero and half the same?* I was curious to know whether that conversation had affected her thinking in any way. So I asked.

Me:Tabitha, do you still think zero and half are the same? Or have you not thought about that in a while?

Tabitha(six years old): I think…Half isn’t a number. I mean, it’s made of numbers put together, but it’s not a number.

Me:What is a number?

I love this question. How people answer it can be revealing. I asked a version of it of Griffin when he was in Kindergarten.

T: is a number.

Me:Oh? is a number, but not one-half?

T: Yeah. But it doesn’t really get used.

Me:What do you mean by that?

T: Well, people say,1, 2, 3, 4, 5, 6, but not .

Me:Oh. So when we count count, we skip over ?

T: Yeah.

We are both silent for a few moments, thinking.

T: Zero, too. People don’t count starting at zero. They say1, 2, 3…

Me:Yeah. Isn’t that funny?

T: It should gohalf, zero, 1, 2, 3…

It seems clear that has indeed been thinking about that conversation. She is struggling with the betweenness of ; that it expresses a number between 0 and 1.

Posted in Talking math with your kids

Tagged 6 years old, counting, fractions, half, numbers, Tabitha

A few years back, one of my elementary licensure students was trying to understand a conversation she been part of. In this conversation, an early elementary-aged child could not give a substantive answer to the question, *What does five mean?*

I suggested to my student that this was a pretty abstract question and I wondered what exactly would constitute a good answer to it.

And then I decided to show her what I meant by this. I made a video (embedded below) in which Griffin (who was five at the time) and I had the following conversation.

Me:Can you count out five blocks?

Griffin(five years old): [He does so flawlessly while mugging for the camera] What’s next?

Me:Is five a number?

G: Yes.

Me:What does five mean?

G: I don’t know. Do you?

There you have it. He knows what five *is*, but cannot articulate what five *means*.

There is more fun to be had here, though.

Me: Is one a number?

G: Yeah.

Me: Is zero a number?

G: Mmmmm…No!

Me: Why do you say that?

G: Because it’s…not necessarily a number. It’s not big…I don’t know. It’s just not a number.

Me: How about uh…

G: It’s not like any other number.

This is very common. Another math (and physics) teaching dad, Casey Rutherford, discovered this recently when discussing what *10 takeaway 10 *is with his five-year old daughter. She said *zero*. I asked him to pursue the question of whether zero is a number with her.

Having settled the status of zero, I ask Griffin about one half.

Me: Do you know what one-half means?

G: No.

Me: Do you know what it means to have half of a cookie?

G: Yeah…sort of.

Me: Is one-half a number?

G: No.

Me: Why do you say that?

G: No…because….No. It is not a number

Me: What is a number?

G: A number is like 2, 4, 6, 8, 9, 1…stuff like that.One hundred. A billion. Three hundred and ninety nine. Those are numbers.

Me: But not zero.

G: No.

Me: And not one half.

G: No.

Examples. Griffin tells me what a number is by offering me examples of numbers. Maybe this is because he has no other way of talking about the meaning of words? Maybe he *always* offers examples as the sole means of explaining what a category of objects is.

I investigate this hypothesis too.

Me: So I’m gonna ask you one more silly question.

G: OK.

Me: You’re playing with blocks there, right?

G: Yeah.

Me:What is a block?

G: A block is something that’s made of wood and it can be colorful or just plain. And you can build stuff with them. And it’s a toy that has…

There it is—the abstract nature of numbers.

*What is a number?* All he can give is examples.

*What is a block?* He is full of characteristics of blocks, uses for blocks, categories into which blocks fit, etc. He has a robust and explicit scheme for sorting blocks from non-blocks. He has no such thing for numbers.

Posted in Talking math with your kids

Tagged 5 years old, blocks, counting, definitions, Griffin, numbers, one half, zero

In numerical order:

- .

I argued with reader Sean and with Common Core writer Bill McCallum this year about whether it made any sense at all to think of 0.5 miles in 15 minutes as a complex fractional unit rate: miles per hour. Sean and Bill were pro; I was con. - .

Not “one over one”…”one oneth”. I had a really nice office hour conversation with a future elementary teacher about her always wondering where the oneths place is, and never really getting a satisfactory answer. She got that answer in office hours and I meant to get to it in class as well. Never did. *e*.

Having worked very, very hard on the topic of logarithms this year, I have convinced myself that*e*has no business being in the curriculum prior to Calculus-or that if it does, it needs to be accompanied by some more serious intuitive work on limits at infinity than is presently standard.- 4.

In the words of the great Mariachi El Bronx, I currently have “four different lovers”. My college, Connected Math, a regional MSP grant and a local charter school.*With four different lovers and forty-eight roses, I need a confessional that never closes.*Seriously, check out Mariachi El Bronx. - 13.

My daughter, Tabitha (4), skipped right over this all autumn long.

[vimeo http://vimeo.com/23543459]

On a daily basis she would ask me, “Want to see how high I can count?” Perfect to 12 every time; skipped right to 15, then a random assortment of teens before going straight to 21. - 31.

Minnesota state standards call for Kindergarteners to count to 31. I have never understood this. If you can get to 31, surely you can get to 39, right? - 39.

Tabitha counted perfectly to 39 yesterday. She’s in public pre-K. - 100.

Common Core State Standards expect Kindergarteners to count to 100. This strikes me as a more demanding, but equally peculiar milestone to Minnesota’s 31. One-hundred ten would be more meaningful; far enough to establish patterns past 100. - @!!!.

In my math content courses for future elementary and special ed. teachers, we have finally named our number system’s first four-digit number. We have agreed that it is*flap*. Long story. Read my place value posts and shoot me a comment if you are at all interested. If you must know, @!!!=625. But @!!! is*like*1,000. - 10,000.

After meeting Dan Meyer in February and having him link to my site the next week, I started to gain a readership. Ten-thousand page views felt like a remarkable milestone, which I reached in May.

Posted in Opinion

Tagged minnesota state standards, numbers, random assortment, retrospective, Tabitha, top ten