Project Pentagon

Pentagons are taking over my life.

You may have heard the announcement this summer that mathematicians found a new tiling pentagon. Previously, there were 14 known classes of convex pentagons that tile the plane. Now there are 15. Maybe that’s all there is; maybe there is another class, or even infinitely many classes, remaining. No one knows.

My Normandale colleague Kevin Lee brought some samples of this new pentagon to Math On-A-Stick this summer, mere days after the announcement. This led to discussing the nature of sameness of the pentagons with my father, which led to further reading, and so on…

I am now drawing an example of each of pentagon type using Geometer’s Sketchpad and Adobe Illustrator, cutting them out of wood on a laser cutter, and then figuring out how they go together. No phase of this project is simple.

I consider a pentagon “solved” if I have at least once figured out how it tiles.

I have successfully drawn and cut pentagons 1 through 11. I have solved all of these but number 9.

The project is making me think a lot about learning.

For example, tonight I was working on pentagon number 8. I solved it.

These sets of four can continue to go together in a way I see and can describe.

But that’s not the only way to view the solution. Maybe someone else solves it using sets of three.

This is the exact same arrangement—the same solution—organized differently. The threes are meaningful here, whereas the fours were meaningful in the first solution. Which is better? Which is right?

Another solution uses sixes.

With that set of six pentagons, you can tessellate by translation only. The three pentagons at lower right are the beginning of the next set of six. Each of these has the same orientation as its corresponding pentagon above it. Does that make it a better solution?

I’m thinking a lot these days about the kinds of questions I’ve posed here. I’m trying to sort out my answers to a larger question:

What is (or should be) the relationship between informal outside-of-school math, and school math?

I have given a couple versions of a talk that asks four basic questions about people’s mathematical activity that occurs outside of school:

• Is this math?
• Is it school math?
• Do we value it?
• Why or why not?

I invite you to join me on this journey.

I’ll keep you posted on the pentagon project.

Advertisement

An interesting story about research and assumptions

Nature v. nurture. Age-old debate on relative importance. Not gonna settle it here. Not even in the limited context of factors influencing mathematics success.

There is lots of interesting research going on, of course. I want to tell you a quick story about a very small subset of that research.

A few years back, a group of educational psychology researchers published a study that phys.org headlined, “Math ability is inborn“.

The study investigated the ability of 4-year olds to choose the larger of two sets of dots when these sets were viewed briefly (too briefly to allow for counting).

They found that children who were better at this task also knew more about numeration and counting.

A quote from one of the researchers, Melissa Libertus:

“Previous studies testing older children left open the possibility that differences in instructional experience is what caused the difference in their number sense; in other words, that some children tested in middle or high school looked like they had better number sense simply because they had had better math instruction. Unlike those studies, this one shows that the link between ‘number sense’ and math ability is already present before the beginning of formal math instruction.”

Read more at: http://phys.org/news/2011-08-math-ability-inborn.html#jCp

Let’s pause for a moment to think, shall we?

If a child has not had formal instruction in mathematics, is the only remaining possibility that her mathematical performance is due to innate skill?

Of course it isn’t.

There is also the possibility that the child has absorbed some mathematical knowledge from her environment, and that different environments might provide differential input.

Maybe the child who is better at discerning the larger set has more practice doing just that. Maybe that child’s parents have been asking her how many? how much? and which is more? for the last two or three years.

Maybe that child’s parents have been Talking Math with Their Kids.

True confessions

Here are two questions we can ask about educational technology.

The differences between them are important.

1. Is the activity this technology supports more intellectually stimulating than what children would otherwise be doing?
2. Is the activity this technology supports more intellectually stimulating than what children should otherwise be doing?

I will confess, here, now and publicly that I hold (for example) Khan Academy to the latter standard.

And, it seems to me, the typical defense of Khan Academy is that it should only be held to the former.

What made this difference especially salient for me was a recent article in the New York Times, which describes (among other things) a waffle-cutting app on the iPad. (See video at this link.)

Now, it seems to me that the children in the study were telling the researchers that there is something inappropriate about the activity when the 2-year old was trying to taste the waffle, and the 4-year olds needed to be coerced into not tap, tap, tapping everything on the screen.

But if we imagine a perfected version of the app, optimized for the ways 4-year olds interact with electronics, then we can ask those two questions about the idealized waffle-cutting app.

If kids are cutting virtual waffles in a daycare environment that otherwise provides little to no math talk, then perhaps this app would be an improvement. But I cannot really imagine an app that would be better than having children cut real waffles and talk about the nature of their activity with sympathetic adults as they do so.

I cannot imagine a virtual waffle app that is better than what 2—4 year olds should be doing, which is talking math.