Class today was an unholy mess.

Truly a mess.

You know that feeling you had as a kid on the playground? When you thought you were jumping from the first level and then remembered that you had actually started on the second? Or the first time you jumped off the swing and you realized only after doing so that your landing wouldn’t be like falling from the height of the swing, but like falling from quite a bit higher because you were headed UP when you left the swing?

I had that feeling in class today.

But you’re never gonna know how high you can jump from if you don’t go ahead and jump.

I wrote last week about my composed units assignment in the math for future elementary teachers course:

In this exercise, we had already dealt with two of three major components of the number concept, which are (with a shout out to Karen Fuson):

- Quantity (how many things there are)
- Numeration (how we write how many things there are), and
- Number language (how we
*say *how many things there are).

In particular, we dealt with 1 and 3. A *pair* of socks is number language that indicates we have 2 socks. A *dozen eggs* is number language that indicates we have 12 eggs.

And we had already done other activities to deal with quantity, numeration and number language. But I wanted to bring the composed unit activity full circle by considering numeration.

So I returned to this example:

And I asked, *How many eggs are there?*

There are 12.

But imagine a world in which our number system weren’t based on making groups of ten. Imagine that it were based on making groups of a dozen.

Then we could say that there is *one group of eggs*, and that there are no ungrouped eggs.

In that alternate world, we might write the number of eggs this way:

This means that we have 1 group, no leftovers and that the size of our groups is a dozen. We will agree to read this as “1 dozen”, not as “one-zero-base-dozen” although the latter may be necessary to facilitate communication later on, just it is sometimes necessary to spell out my last name…”de-eh-en-i-ee-el-ess-OH-en”. And we probably shouldn’t read it as “ten, base dozen” because we’ll get confused.

If we have two dozen, we write this:

Next, we’ll write an equation to show some different ways we can notate a dozen.

We talked about the conventions of notation. The little *dozen* on the 10 there indicates to us the size of the groups we are making in this particular numeration example. We call that the *base*. The middle part of the equation is a typical, well-formed English phrase, and we have added the little *ten* as a subscript on 12 in order to make clear the size of the groups we’re making. But that 12 on the right should feel like home-that’s how we write how many there are in our usual number system.

We did another example together.

How many tires? Well, let’s call the group of tires a *set*. Then there are:

Then I turned them loose to consider what *How many* question to ask, and how we should notate it in a number system built especially for counting things according the composed unit in the picture.

Interesting examples to consider include the ones that were interesting the first time around:

How many shoes?

And this one:

How many Pop-Tarts?

Considering only the silver pack lying on the table, we have:

But there are 4 packs in a box. How should we notate that? Here are three possibilities for your consideration:

Each is justifiable, and we worked through the first two. That discussion took a lot of mental effort on everyone’s part. One student asked very apologetically, “This might be a question that doesn’t go anywhere, but writing one-zero-zero doesn’t seem right, because aren’t we making groups of 2, not of 4?”

After insisting that she retract the apology, I observed what a smart question it was. After all, don’t we make groups of ten, and then ten groups of ten, and then ten groups of ten groups of ten, etc.? Wouldn’t it be a crummy number system if the size of the group kept changing on you? (Side note-the Mayan number system did this: first grouping is of 20, second is of 18, then 20′s the rest of the way.)

So why *might* we want to make the third place worth four times the second place? Because that’s how Pop-Tarts come (according to the photograph). The original unit is a Pop-Tart. These are composed into packs of 2. These packs are composed into boxes of 4.

We tried one more thing…what if we write (for the sake of moving forward):

And then someone comes along, opens a pack and eats one Pop-Tart. How should we write how many remain?

Yup…I jumped off that swing today. Didn’t really have a solid plan for sticking the landing. But I learned an awful lot by scraping my elbow when I landed.

And my students made some lovely connections. After class, a student wanted to check his thinking on the gallon of milk. How many ounces are there? He wrote:

See a previous post for information on our alternate base system that led to the expression at the far right.