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?

Dot-to-dot

Another one from the archives.

About a year ago, Tabitha was doing a dot-to-dot drawing.

This is not the one she was doing. It is an example. Not a very good one, either.

Things were going well. She got to 11.

Tabitha (five years old at the time): I don’t know what twelve looks like.

Me: It’s a 1, then a 2.

With this tip, she cruised through the teens and got to 20.

T: What does twenty-one look like?

Me: A 2 and then a 1.

T: And twenty-two?

Me: A 2 and then a 2.

T: So twenty-three is probably a 2 and then a 3.

This got her going on to 29.

T: I don’t know what 30 looks like.

There are a lot of interesting things going on in this exchange. Among them:

Sometimes in mathematics, we need to live with new notation before picking its meaning apart too carefully. See also fractions, functions and derivatives.

Numeration and number language do not develop hand-in-hand. Tabitha knows the number language; she can count past thirty. She has not learned how to read or write numbers that high.

Patterns are powerful tools in mathematics. Tabitha’s experience in the teens gave her powerful intuitions for the twenties.

What is ten?

Consider the seemingly simple question What is ten?

Quantity. This refers to how many things there are. If ten is a quantity, then it refers to this many things: ***** *****

Numeration. This refers to how we write how many things there are. If ten is a set of symbols, then it refers to this: 10.

Number language. This refers to how we say how many things there are. If ten is a word, then it refers to this word: ten.

To illustrate the difference, ask a French person to read this number: 10. Then ask a fifth grader what this Roman numeral stands for: X. Finally ask a computer programmer what number this refers to in binary: 10.

In order, the French person’s dix illustrates that we can use different number language for the same numeration and quantity. The fifth grader’s ten illustrates that we can use different numeration for the same number language and quantity. And the programmer’s two illustrates that we can use the same symbols to represent different quantities.

If I weren’t so lazy, I’d link to a Karen Fuson reference for the research details. Maybe I’ll get to that sometime. But she’s the go-to person on this.

 

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.

Composed units, continued…

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):

1. Quantity (how many things there are)
2. Numeration (how we write how many things there are), and
3. 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.