Tag Archives: numbers

Number and numeration gone wrong

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

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



Zero=half revisited

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: 4\frac{1}{2} is a number.

Me: Oh? 4\frac{1}{2} 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 4\frac{1}{2}.

Me: Oh. So when we count count, we skip over 4\frac{1}{2}?

T: Yeah.

We are both silent for a few moments, thinking.

T: Zero, too. People don’t count starting at zero. They say 1, 2, 3…

Me: Yeah. Isn’t that funny?

T: It should go half, zero, 1, 2, 3…

It seems clear that has indeed been thinking about that conversation. She is struggling with the betweenness of \frac{1}{2}; that it expresses a number between 0 and 1.

What does five mean?

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.

Top ten numbers of 2011

In numerical order:

  1. \frac {1}{4}.
    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: \frac {\frac {1}{2}} {\frac {1}{4}} miles per hour. Sean and Bill were pro; I was con.
  2. \frac {1}{1}.
    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.
  3. 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. 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.
  5. 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.
  6. 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?
  7. 39.
    Tabitha counted perfectly to 39 yesterday. She’s in public pre-K.
  8. 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.
  9. @!!!.
    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. 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.