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.

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More on the language of place value

Read aloud the following number:

182,356

Now mentally answer this question: What is the value of the 8 in this number?

I see two correct ways of stating this:

Eighty thousand, and

Eight ten-thousands

And I’m trying to decide whether I care about the difference between these. I’m not sure that I do.

So now I do what I always do to test ideas. I ask, What if? Specifically, What if we looked to the right of the decimal point? What would this question look like there?

So consider the number 0.0008.

If you said Zero-point-zero-zero-zero-eight, then I’ve got a lot more work to do with you.

Place value language matters in math classrooms.

And the sentence in the picture below is confusing.

No, I’m guessing we would all agree that this is eight ten-thousandths, which is not at all the same as eighty-thousandths (although it is the same as eighty hundred-thousandths).

So now I see that my What if? question has muddied the waters, rather than clarified them.

What can we conclude?

I suppose that the major conclusion is this:

We need to stop pretending that decimal place value (i.e. to the right of the decimal point) behaves exactly like whole-number place value (i.e. to the left of the decimal point).

In the abstract, this is certainly true. But composing units is not conceptually (or linguistically) equivalent to partitioning them.

Like I was saying here:

“and”

How many dalmatians were in that movie again?

It was one-hundred-and-one, right?

Have I angered you yet? If you teach math, I probably have.

See, we math teachers are precise people. We like things to be just right. Part of being just right is using language correctly. In English number language, and is reserved for separating the whole number part of a number from the fractional part.

One and one-half

One and seven tenths

That sort of thing.

So when we hear one hundred and one, we freak out.

But separating the whole number part from the fractional part? That’s just one perspective. Sure, it’s the grammatically correct one.

But here’s the thing about grammar rules…they are arbitrary. Totally and completely arbitrary. Name a grammar rule and there’s a language somewhere that violates it completely.

Yet we English-speaking math teachers act as though the use of and were a signifier of mathematical understanding or competence. Which it is not.

Here’s another interpretation of and in English number language. Maybe and signifies a change of unit. In one and seven tenths, one is counting the original unit, seven is counting tenths. The and helps the listener to follow along; it signifies this shift.

In that case, one hundred and one is the same way. The first one counts a unit-hundreds. The second one (following the and) counts a different unit (which, awkwardly, we call either ones or units).

If I’m right about this, then you will have heard native speakers of English say something such as,

Three hundred and four thousand and twelve.

Three hundred and four counts thousands. Twelve counts ones. Then within the three hundred and four part, there are two different units also. Hence the and.

If I’m right, then you will probably not have heard native speakers of English say something such as,

Thirty and four.

Both of those words count the same units.

So I say let’s give up on this little obsession we have about and. Let’s not let it get in the way of effective, efficient communication in mathematics classrooms.

Let’s save our wrath for this:

Three point twelve.

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.

 

Theoretical models and all that

Let’s say you had a four-year old daughter and she was learning to count.

Let’s further say that you had a deep interest in number language, had read widely on the matter and thought very, very hard about it. Maybe you had even published an academic paper on relationships among number language, quantity and numeration (pdf).

Let’s say that you (among many other people before you) had noticed that while the English teens have a pattern, it’s a fairly obscure one.

Let’s also say that you notice that (unlike some other languages such as Japanese) there is also a relatively obscure pattern for the names of the decades: twenty, thirty, forty, fifty, etc.

Finally, let’s imagine that you noticed that we start counting at one, but we don’t start the twenties at twenty-one nor the thirties at thirty-one.

Don’t you think you’d put all of this together to build a theoretical model of learning to count that includes (1) trouble in the teens, (2) skipping twenty in favor of twenty-one, (3) more success in the twenties than in the teens, and (4) ending the count at or about twenty-nine?

And then, when you saw that theoretical model play out in great detail, don’t you think you’d want to capture it on video?

I know I would.