How many tens?

Here is one from the archives.

Nearly a year ago, Griffin was seven years old and I was doing some thinking about the number course I teach for future elementary teachers. I decided to see how Griffin was thinking about place value.

Me: How many tens are in 32?

Griffin (seven years old at the time): Three, and then two leftover.

Me: How do you know that?

G: Thirty—that’s three tens, and then the zero means no ones.

Me: How many tens in 268?

G: [long thoughtful pause] Twenty-six, and then there would be 8 left over.

Me: What would you say to someone who thought there were six tens in 268?

G: I’d say there are 20 more than that.

That’s my boy.

A place value thought experiment

The hundreds chart is a fixture of elementary classrooms. Such a fixture that most of us probably don’t stop to think about it.

That’s where I come in.

The trouble with place value is that it is too easy.

Q: Why does the hundreds chart have 10 columns?

A: Because our number system is base-10.

Q: Why does the hundreds chart have 10 rows?

A: Because our number system is base-10.

These answers are so simple that they mask the conceptual complexity underlying place value.

The hundreds chart is rich with patterns.

The double-digit numbers lie diagonally.

If you start with a number in the top row and read diagonally down and to the left, the digits of the numbers sum to the number in the top row.

Example starting with 9.

Example starting with 8.

And on and on.

Why do these patterns exist? Because the structure of the hundreds chart matches the structure of the number system.

But there is something unsatisfying about this answer.

So here’s a thought experiment.

The Mayans had a quasi-base-20 place value number system. Quasi-base-20 because the third place was not worth 20 twenties, but only 18 twenties. All other places have value 20 times the previous.

Imagine stepping into a second-grade classroom in a modern society that used the Mayan numeration system. What chart would they have on their walls instead of our hundreds chart?

What would a Mayan “hundreds” chart look like?

I have used this question as one of two parts of an A assignment in my math content course for future elementary and special ed teachers for several years*.

A common answer in students’ first drafts is the following (this image from wikipedia):

This is no good.

We want to represent place value in the hundreds chart and this chart does not do that. All of these are single digit numbers as far as Mayan place value is concerned.

That chart above is the equivalent of one that goes 0—9 in our decimal place value system.

Another common example has literally 100 cells, in 10 rows of 10.

Also no good. That chart is based on the structure of our number system, not the structure of the Mayan number system.

No, we want a Mayan “hundreds” chart that has patterns equivalent to those we find in our hundreds chart. Patterns such as the ones highlighted above.

Here is what we need.

Credit to student Angela Drietz for the complete chart.

Here are the double-digit numbers.

;

And if you look closely, you can add the “digits” on the leftward-running diagonal to get the number in the top row.

;

That, my friends, is the beauty of place value. It’s not 10, the quantity, that is special. It’s the set of symbols. It’s the 1 and the 0.

Dig.

—

* The other part is to create number language to reflect the place value structure of the Mayan number system. How might the Mayans have read these numbers aloud? As far I know, no one knows the answer to how they did read them aloud, so the task allows for structured creativity.

Decimal place value v. whole number place value

As the rest of my household heads back to school, I am getting down to the business of planning for my new semester. One instructional problem I am trying to solve is that of pushing my future elementary teachers to understand decimals more deeply. Whole number place value I have nailed down. But decimal place value is another story. The rules are so deeply embedded in them that it is very difficult to get them to question these rules, or to seek a better understanding of them.

I have collected substantial data over recent semesters (not research quality data, but consistent formative and summative assessment results) to demonstrate that my students operate on decimals using a combination of known rules and whole-number place value principles. They can switch back and forth between these fluidly and generate right answers to nearly any decimal task.

Which leaves me seeking questions that they cannot answer without digging more deeply.

I have a candidate question in mind: Why can we put zeroes after a decimal number without changing the value, but not after a whole number without changing its value?

 

It occurred to me last evening that I knew what my answer to this was, but that I lacked a wide repertoire of correct explanations. So I asked Twitter. Lots of good stuff followed. Read the full conversation on Storify.

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:

What century is this?

At breakfast one morning, Griffin (7 years old) asked me,

Griffin: What century do you think this is? The 20th or the 21st?

Me: It’s the 21st century.

G: Oh darn. I was hoping you wouldn’t say that. I’ll have to ask Tabitha and Mommy and a bunch of other people this question.

At this point, I was thinking that he was feeling pretty smart for knowing that the year 2012 is the 21st century, even though there are only 20 sets of 100 in 2012. I was wrong. He was making the opposite argument-we should call the present century the 20th.

G: What about when you were born, in 1970? Was that the 20th century or the 19th century?

Me: It was the 20th century.

G: See, I don’t think that’s right. It should be the 19th century.

Me: Because it starts with a 19?

G: Yeah.

Me: I see. Well, what about the year 50? Not 1950, just the year 50? What century do you think that was?

G: Zero.

Me: Right. Well, we agreed that we would start counting with the first century instead of the zeroth century. So the year 50 was in the first century, and the year 150 would have been the second century.

G: Well, it shouldn’t be that way. I want to start counting at zero. So I’ll keep asking people and find people who agree with me.

A couple minutes later

G: So, have there only been people for 2000 years?

Griffin is an independent, contrarian thinker. If there is a way to think about something differently, or even a way to perform some physical deed differently, he’s all in. A critical thinker in the extreme, this boy never accepts “Because I said so” as an answer. This will serve him well in some areas of life and poorly in others. It makes parenting him a unique challenge.

From a mathematical perspective, it doesn’t matter at all whether Griffin thinks of this as the 20th or 21st century. Sooner or later he’ll give in to convention so that he can communicate with the rest of the population of the first world. The important mathematical thing is to explore the basis and the consequences of the argument he is making.

Is the basis of the argument purely the fact that 2012 begins with 20? Or is there an attention to place value? In other words, is he thinking about 2000 as 20 groups of 100, or just as beginning with 20? My question about the year 50 was intended to get at that. There is no beginning with in this case. He had no trouble, which suggests that he is thinking about groups of hundreds-there are no full groups of 100 in 50, so it should be the zeroth century according to his rule.

This is the consequence of his argument. If you’re going to argue that 2012 is part of the 20th century, you need to be ready to accept the idea of a 0th century.