That’s the name of one of the classes I’m teaching this term. We have trimesters. So each term is 12 weeks long and we have a week of “intersession” in between the terms. Except that this first term is not quite 12 weeks. The expansion area of the building wasn’t quite finished when we started school, so we had some alternative programming called “Baxter Foundations.” It included stuff like my Intro to Spreadsheets workshop. Classes started this week. And one of my classes is called Problem Solving with Algebra. I came up with that name, and I honestly don’t know exactly what it means. I have a rough idea, but it could go in a lot of different directions. Mostly, I want my students (and all the students taking this course) to think and puzzle and use algebra and solve problems.

Then I got an email that Jo Boaler has published a short paper called The Mathematics of Hope. In it she discusses the capacity of the human brain to change, rewire, and grow in a really short time based on challenging learning experiences. We’re not talking about learning experiences that are so challenging that they’re not attainable, but productive struggle. Challenging learning experiences that produce some struggle, but are achievable. The ones that make you feel really good when you solve them. You know the ones I mean.

So I decided to start this class with a bunch of patterns from Fawn Ngyuen‘s website visualpatterns.org. The kids are amazing. They jumped right in. Okay, so I taught most of them last year and they know me and what to expect from me, but seriously. Come up with some kind of formula to represent this pattern. Kinda vague, don’t you think? And I’m pushing them to come up with as many different formulas as they can, and connect those formulas to the visual representation. For example, an observation that each stage adds two cubes to the previous stage would result in a recursive formula like: C(n) = C(n-1) + 2 when C(1) = 1 (which is a recursive formula for pattern #1).

On Tuesday, different groups of students were assigned different patterns. Wednesday, each group presented what they were able to figure out. Some had really great explicit formulas, while others had really great recursive formulas. A few had both. Most were stumped at creating an explicit formula for pattern #5, pattern #7, and pattern #8.

Tuesday night, I received this email from Sam, a student:

“After staring at the problem for 2 hours, (5:45 to 7:45) and scribbling across the paper as well as two of my notebook pages, I am still unable to find a explicit equation. Then, reading the directions, I realized that the way they are worded allows the possibility of no explicit equation, as well as the fact that I only had to come up with equations as I can find. So after 2 hours, several google searches, lots of experimentation and angry muttering, I decided I have all that I can muster, and must ask you in the morning.”

I left them with the challenge to find an explicit formula related to one of these patterns. Their choice. Just put some thought into it before we meet again on Monday. Wednesday night, Sam sent me this followup email:

“After another hour at work, I found the explicit formula. I realized that the equation was quadratic, not exponential, and youtubed a how-to for quadratic formulas from tables. I kid you not, the man said the word “rectangle” and from that, I solved the problem. Then I watched the video through and took quick notes for future reference.”

Then, Dan Meyer posts this: Real work vs. Real world. Makes me think – as always. What am I asking of my students? This is real work – they are engaged and they are thinking. Sam, and the others, were not going to be defeated by a visual pattern. The fact that they are working in a “fake world” doesn’t matter.

Found you via Dan Meyer

This desire to find real-world situations with real-math opportunities is really only part of the purpose of math in schools. I look back on the hours spent puzzling out difficult integrations an realize that the point of it was to continue the development of pattern recognition. This is exactly what you are doing, very explicitly in your case, and to me it is the main point of math, as only a tiny percentage of students is ever going to “use the math in the world of work”. Why else on earth do algebra students have to simplify

3x^2 + 5x^2y + 7x^3y – 8x^2 + 11y -17xy

This is pure pattern recognition, and also incredibly boring and pointless.

Geometry is a fountain of possibilities for pattern recognition.

I had a long computer conversation with Elaine Watson about one of Fawn Nuygen’s dot pattern examples in which it was possible to continue the pattern in several different ways, leading to different equations.