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.

20130311-091546.jpg

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.

hundreds.chart.11

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.

hundreds.chart.9

Example starting with 8.

hundreds.chart.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.

Credit to student Angela Drietz for the complete chart.

Here are the double-digit numbers.

mayan.hundreds.chart.11

;

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

mayan.hundreds.chart.9

;

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.

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