Back in September I wrote a long post about a recent book written by Rafe Esquith. My second paragraph began with this observation:
It wasn’t clear to me at first whether this book would have anything to offer a math teacher in an academically superior high school in a wealthy suburb. Esquith teaches fifth grade in the Los Angeles barrio: it’s elementary school, it’s all subjects, and it’s low-income kids. All of that makes his experience enormously different from mine.
It turned out, of course, that the book did have a lot to offer me. Many ideas in it have stuck with me, but none more than the idea of having a year-long project:
In my high-school courses even a month-long project is hard to achieve but Esquith convinces me that the level of student commitment required to carry out a year-long project is well worth the investment in time and effort. In his case he has the kids make hooked rugs that are works of art, and he has a majority of them involved in a major professional-type Shakespeare production each year. Wow! Can I learn something from that in high-school math? I think so.
So I needed to mull over this idea and discuss it with others. After a lot of mulling — including conversations with six colleagues — I came up with a proposal at the beginning of the second quarter. It was too late for a year-long project. But one that lasted three quarters of a year would still make a good pilot, so that’s what I proposed. This would be a small pilot, involving just 13 students. Think of it as a little investigation, not a study. (As an aside, note that one of the things that bug me about so many educational initiatives is that they so often involve massive commitments instead of starting small. This problem can be seen in everything from imposing Bring Your Own Technology on an entire grade of one school to the huge Common Core.) The proposal was to take my one section of Algebra II and construct a long cryptography project that would incorporate materials from all three remaining quarters of the year. We picked my section for several reasons: it has very few students, so it would be easy to make adjustments on the fly; it’s the only section of this course where the teacher doesn’t also have a second such section, thereby preventing the undesirable consequences of doubling the number of lesson plans; and I would be implementing my own proposal, rather than imposing it on others. We picked cryptography for several reasons: we study it in fourth quarter anyway; it’s a high-interest topic, with many applications and controversies that are currently in the news; and it can readily tie into other algebraic topics.
Because of recent current events, I constructed the following (slightly realistic) scenario:
You, the student, have just accepted an offer of a summer internship working for the NSA. You will research what the NSA does, study and apply the basic principles of cryptology, learn the ropes of enciphering and deciphering messages, and write reports along the way. You will keep track of everything you’re doing and everything you learn, assembling it all into a final product at the end of the internship. (Quarters 2–4 of the school year are, of course, a stand-in for the summer here.)
My hope is that this will accomplish most of the objectives of Esquith’s year-long project. At the very least it will require a commitment of many months, it gives each student a tangible product of their own creation to take home at the end of the year, and it encourages connections among ideas. I have felt all along that most math students (especially in Algebra 2) see mathematics as a disconnected set of tiny, arbitrary, barely related skills. We as teachers know that there are Big Ideas behind what we’re doing, but the students don’t see those Big Ideas. For most of them it’s all atomized and purely skill-based. A high-interest crypto project should go a small way toward fighting this problem. Cryptology is an amazing branch of applied mathematics, particularly as it incorporates inputs and outputs, deep understanding of functions and their inverses, some statistics, software use, representations, topics in number theory (modular arithmetic, prime numbers, bases, etc.), some history, privacy issues on the Internet, Unicode, alphabets in foreign languages, and other topics I’m just not thinking of right now. So it’s a great subject for an over-arching project.
“But doesn’t Algebra 2 already have projects?” you ask. Yes, for about ten years we’ve had two projects in this course: early in the third quarter (i.e., right now) there has been one on exponential and linear functions, and late in the fourth quarter (near the end of the school year) that has been one on cryptology. In my class we’re incorporating both of these as components of our “year-long” project. The second one is easy to incorporate, since it’s already the right subject matter.
The first one required more mulling before I figured out how to incorporate it into the summer internship scenario. This year the story line that the other sections are following is that Mr. Montgomery Burns wants to dump nuclear waste (from the Springfield Nuclear Power Plant, of course) in Weston, in return for which he will pay the town a whole lot of money. The nuclear waste decays exponentially, the payments linearly. Students have to create equations, tables, graphs, and a well-reasoned argument that either supports Mr. Burns or opposes his proposal. Each student accepts a consulting contract, either from Mr. Burns or from Protect the Earth, an environmental non-profit. That’s for the other sections.
In my section a context is wrapped lightly around this scenario: the NSA has intercepted an encrypted message from Mr. Smithers (assistant to Mr. Burns, as you surely know), who is secretly working as an industrial spy for Protect the Earth while ostensibly working for Mr. Burns. Otherwise it’s pretty much the same story and the same set of requirements. So some crypto is included, all within a unified story line.
I’ll let you know how the rest of it goes. So far, so good.