Tag Archives: definitions

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.


Classifying hexagons

In the spring, my prospective elementary teachers move on to the second course in our math content sequence. This one focuses on geometry and measurement.

In my never-ending quest to problematize the routine, I’ve been brainstorming ways to help them with one of our goals for the course: Distinguishing between properties and defining characteristics of a shape.

Several things make this difficult:

  1. The standard domain for examining this is pretty tired-the hierarchy of quadrilaterals.
  2. Students bring a lot of cultural and educational baggage to this hierarchy.
  3. Depending on how we choose to define a shape, its defining characteristics and properties change. For example, we can define a square as an equilateral, equiangular quadrilateral. In that case, having right angles is a property. Alternatively, we can define a square as a rhombus with a right angle. Then being equiangular is a property instead of a defining characteris

To rid ourselves of this baggage, I propose to have my students develop a classification of hexagons. Many details to be worked out.

But consider the hexagons below as examples.

These two hexagons have in common the fact that each side is parallel to another side of the hexagon. Sort of like parallelograms. Perhaps we end up calling these parallelogons. And then we’ll have to decide whether we want to subdivide this category further. Do we want to distinguish between these two? Based on what characteristics? And can we agree whether all regular hexagons are parallelogons? And can we distinguish between that claim and this one: all parallelogons are regular?


I’ll have to think all this through.