Sunday, December 16, 2012

Algebra 2 & relationships

We've been working on our algebra 2 curriculum over the past year, both to align it with the Common Core and make it seems less like a random collection of function things. Last year we had the idea to make it very data-based; the idea was to have students gather a ton of data, and then base our study of functions on those data sets.

There are a few problems with this approach. For one, our district decided that prep periods are a luxury that cannot be afforded, so we don't have the time to prepare such a radical departure from our current curriculum.

Second, and more problematically, I think this would put way too much emphasis on the applied part of mathematics, and would make it difficult to meaningfully teach any aspects that aren't as practical. If we're motivating the study of functions based on answering questions about real data, I'm not sure how I'm going to help students understand removable discontinuities. Or motivate them to solve difficult equations algebraically when a graphical method yields a solution quickly. And so on.

Mathematics is not only about answering practical questions about the real world and data. It is certainly useful for that, but it's also much more. Thus a solely data-based approach to teaching algebra 2 would be a mistake, in my opinion.

So we need a more flexible approach if we're going to be true to all of mathematics. Thinking about the essence of our algebra 2 course, and really algebra in general, what we're dealing with is relationships. Linear, polynomial, rational, radical, exponential, logarithmic, and trigonometric, to be specific. We learn about these relationships by looking them in different lights - algebraically, graphically, and numerically.

Currently, a major problem with algebra 2 is that, to students, it feels like a ridiculously long laundry list of things to learn. I get the feeling that they don't understand that the equations, graphs, and tables that we work with represent the same relationships. I try to teach the connections, but they're mostly lost on students. And it's not surprising - the focus of the course isn't on understanding the relationships, it's on the set of techniques and facts about them. Without understanding the relationships themselves.

So, my idea: refocus the course on the relationships. One way to do that might be to start numerically. For example, we might start with the following:
(1, -3), (10, 1019), (-1, -4.5), (?, 59), (2, -1), (3, 3), (0, -4), (0.5, ?) }
I think this would really put the focus on the relationship itself. We're dealing with two quantities, and there is some way to predict one from the other. However, it is difficult to figure that out simply with numbers, so translating the problem into different forms - equations and graphs - takes on more meaning. 

Our new semester starts in about a month; I'll let you know how it goes. Any wisdom you have to offer would be appreciated.

Sunday, December 9, 2012

Rekindling why I teach mathematics

I haven't blogged for a long time, and I'll explain that in relatively short time. But in the mean time, I have a brief thought to share.

I've really, really been struggling lately to find my passion for teaching pure mathematics. I'm very passionate about teaching computational thinking skills and statistics - ideas and methods that clearly are relevant to developing an informed citizenry - but I've been questioning the point of teaching apathetic teenagers rather abstract mathematics.

Well, I've found the beginning of a cure in, of all places, a graphic novel. This weekend I've been reading Logicomix, a story about the life and ideas of Bertrand Russell.

For those of who don't know, or have forgotten, Russell's main role in the story of mathematics was his attempt to create a solid foundation of all of mathematics in his "Principia Mathematica", co-authored with Alfred Whitehead. This graphic novel does a beautiful job of connecting the ideas in mathematics to the questions that make our existence so wonderfully interesting.

I need to remember, I'm teaching ideas. And these ideas have a context; they are part of the story of real people attempting to more fully understand our place in the universe.

I haven't been teaching that at all.