# From concrete to abstract part 2

Started a new chapter today in our Prealgbra book and the first section was about the Pythagorean theorem. My son had certainly heard about the Pythagorean theorem in our various talks as well as from this fantastic piece from Numberphile:

One of the examples in the section was the proof of the theorem that students usually learn first. We worked through that proof with numbers initially and followed up with symbols. At the end of the morning I asked my son to go talk through both approaches.

Prior to these videos, the algebraic approach gave him a little difficulty, so I was interested to hear him talk through that part on his own in the second video.

With numbers:

With symbols (sorry if the thumbnail preview for this video isn’t working – still having that odd problem occasionally) :

I was definitely happy that we was able to work through both of these examples mostly on his own. After we finished up the last video we rewatched the old Numberphile video for some additional fun!

# A deep dive into a problem from Grant Wiggins / Patrick Honner

Almost two years ago Grant Wiggins and Patrick Honner had a conversation about rigor:

A Conversation About Rigor, with Grant Wiggins — Part 1 <target=”_blank”>

One of the problems (from an old TIMMS exam) referenced by Grant Wiggins caught my attention:

At the time I used it for a fun talk with my kids:

In talking through this problem with them back then the intention was just to show them a problem with a fun solution. At the time they were 9 and 7 and I didn’t see any value in diving into any of the details, but thought that they would like the idea of unrolling a cylinder.

The main reason this problem caught my attention wasn’t the fun solution, though, rather it was Wiggins’s reference to the number of correct solutions to this problem. Quoting a NYT report:

“It also turned out to be one of the hardest questions on the test. The international average of advanced mathematics students [in 12th grade] who got at least part of the question correct was only 12 percent (10 percent solved it completely). But the average for the United States was even worse: just 4 percent for a complete solution (there were no significant partial solutions).”

I’ll be the first one to admit that I’m a terrible judge of the difficulty for problems like this one. On one hand I have virtually no experience teaching math to kids and thus close to zero experience seeing what kinds of problems are hard for kids and what kinds of problems are easy. On the other hand I grew up in math contests and also have a PhD in math, so this problem is a type of problem I’ve probably seen thousands of times – solving it is almost automatic.

The desire to find out a bit more about why this is a difficult problem never went away, though the opportunity to explore the problem more deeply never really came around . . . until today. Today my older son and I arrived at the section in our Introduction to Geometry book that talks about cylinders. Finally 🙂

The series of videos below present how a kid seeing cylinders for the first time (formally) approaches the problem referenced by Grant Wiggins. It takes quite a while to get to the solution and the process of getting to the solution is not even remotely a straight line. Having gone through this exercise today I feel that I have a much better understanding of why the problem is difficult.

[ Before starting with the videos, I’ve been having a mystery problem with video thumbnails on Youtube lately. All of the videos should play properly even if the thumbnail is all grey. I haven’t figure out what’s causing this problem or any good way to fix it. Really all I know about the problem is that yelling at my computer doesn’t seem to help. It usually goes away after the videos have been up for a few days. ]

Part 1: Introducing the problem and getting a few ideas. He has an interesting idea right away about chopping the string up into 6 lengths. There’s a little confusion his mind about these lengths, though, and that confusion relates to the lengths in 3 dimensions vs how they are drawn in 2 dimensions on the board. It will take a few more videos – and looking at a square version of the cylinder – until that confusion gets sorted out.

The bit of this first conversation that really surprised me is that he saw that unrolling the cylinder might be a good idea. Seeing how the string would appear in the unrolled picture was hard for him, though.

We ended the last video with my son thinking that the side of the cylinder would be chopped into 4 equal slices by the string. I was caught off guard by this idea but realized that working through how to think about the lengths was going to be important if we want to solve the problem. The confusion about the 3d and 2d settings is still giving him problems, and it takes him a while to see that one of the “lines” he’s seeing on the board goes half way around the cylinder.

It was really hard for me to sit on my hands here.

Now that we have a little bit better understanding of the height of the string at various points on the cylinder we can return to the unrolled cylinder. The main difficulty now is translating the heights we see in the cylinder into heights on the rectangle. At the end of this video he suggests one way around this difficulty is looking at a rectangular prism. This was an interesting suggestion – especially since our Zometool set was laying on the ground right in front of us making it easy to build a prism.

So, with the camera off we built a rectangular prism out of our Zometool set. With the camera on we began by wrapping a string around it 3 times. At first he struggles to see the connection between the prism and the cylinder, but then the new model does help him understand the geometry better.

Once he makes the connection that going half way around the cylinder moves you up 6 inches, he’s able to draw a new picture on the board that nearly shows the solution. There’s still a little residual 3d vs 2d confusion, but eventually he’s able to get the picture (almost) right.

One not super surprising error is some confusion about what happens along the border where you cut a cylinder. I didn’t want to dwell on this point for now since I didn’t think it was a super important part of this problem. Even though one of the pieces of string is drawn going the wrong way, the length of the string is still correct.

At this point we’ve walked all the way to the end of the problem – the only thing left is the final calculation.

For the last part of the project, I asked him to walk through the solution from start to finish. I hoped that this final talk would allow him to pull all of the ideas together and give him the feeling of walking away having solved a really challenging problem:

So, quite an interesting study for me. It was fascinating to see the struggle my son had translating the 3D geometry to the 2D picture on the board. That struggle makes me think we would have struggled to get to a solution if we didn’t have our large cylinder and Zometool set to help. As I said above, I have a much better understanding of why this is a hard problem having gone through it with my son today.