Sunday, September 9, 2012

What's the Big Idea with Algebra 2?

Lately, I've been following some of the conversation around the big ideas in an advanced algebra/pre-calculus course. The Global Math Department* hosted an interesting panel discussion around this topic a couple of weeks ago. I appreciated the thoughtfulness and complementary ideas of the presenters (John BurkDan Goldner, Michael Pershan, and Paul Salomon), and especially the thoughts behind proof and 'the well-crafted solution.' Without entirely reaching a consensus, the focus of the discussions tended to lean towards prediction as the overarching theme for algebra ii. The reasoning was thoughtful and grounded, but this theme did not satisfy me. While I can certainly see it, I also think that prediction is the theme for statistics. Can Algebra 2 and Statistics have the same theme? They can, I suppose, but it is not satisfying enough.

Some of the new bloggers from the New Blogger Initiative also tackled this topic last week.
gooberspeaks got me thinking about the focus on families of functions and David Price included ideas about varying ways of representing functions and modifying their behavior. Kyle Eck has a strong bent towards applications which resonates with the GMD theme of predictions. And all these ideas muddled around in my brain for a long time before emerging as a single construct that currently satiates my desire for deeper inspection.

Algebra 2 is all about: generalizing patterns of behavior in bivariate relationships.

But that's my academic's definition. In the UbD-influenced language of a high school classroom, I'd say that Algebra 2 asks these questions:

  • How can we communicate the behavior of a relationship between two ideas?
  • Are there rules of behavior that apply to all relationships?
  • Why is it important to be able to generalize patterns of behavior?

Functions certainly play a large role here, because it's easier to generalize patterns when there are overt rules of behavior to follow. But just as importantly, we also look at conic sections and the elusive inverses of even polynomials and periodic functions, because these ideas give us essential insight about the comforting nature of functions that are both one-to-one and onto, and about the obstacles presented by relationships that are not.

Graphing also plays a large role, because it is a most excellent tool for alternate representations of bivariate relationships. Seeing patterns emerge in the shape of coordinate graphs can be enlightening long before symbolic manipulation clears a path through the brain... and I thank the math gods for that! I am wary though of too much graphical emphasis, for our well-loved coordinate system has obvious limits as our brains allow us to consider relationships with more variables.

And applications clearly play an important role too, especially in the attempt to answer that third question. But I hesitate to put applications at the forefront of an advanced algebra theme. I think that is perhaps better handled by a physics class. In algebra we are attempting to represent scenarios with a generalized pattern of behavior, and manipulate this generalization to highlight useful information. I think I agree with Paul Salomon in that proof and 'well-crafted solutions' may trump (but certainly not replace) applications in the hierarchy of an overarching theme in algebra.

To end, I'll just say that my desire to ask (and attempt to answer) the big questions is never entirely satiated, but I do so enjoy the conversations that emerge from them. I welcome your thoughts, criticisms, and further insights. The discourse is what makes being a mathematician so much fun.

*Megan Hayes-Golding, where have you been all my life? What a terrific thing the GMD is, and one of these Tuesday nights, I will not have bedtime routines or NBI deadlines to worry about and will be able to attend a session while it is actually happening! Thanks to you and all others who are making this happen.

3 comments:

  1. I really enjoyed the thought behind your post and the conclusion(s) you came up with. I particularly like the three everyday-language questions you pose here.

    I have taught Algebra 1 more than any other subject, and I wonder what the Algebra 1 Big Idea would be. I talk about relationships when discussing this with my students, but I tend to refer to relationships between people. I am interested in thinking further on whether I could expand my discussion to include ideas.

    I am curious if in your framing of the Big Idea you think it could refer to relationships between two people? (Or other types of relationships perhaps, like relationships between species in Biology or some such thing. I'm just thinking out loud here.)

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  2. Steve, I like where you are going with these questions. To me, I think that algebra is about logic and communication of patterns. We observe the way quantities relate to each other and use abstraction to communicate patterns of behavior. We quickly discover that some relationships are easier to describe than others. Why is that?

    Human behavior, for example, while it does follow some predictable patterns, is not in and of itself predictable. Some numerical relationships, on the other hand, are especially easy to describe: like the relationship where two quantities are always the same (y = x). I think it is no mistake that in Algebra I, we mention numerical 'relations' only briefly in order to define the truly utilitarian 'function.' We start our students off with simple, highly-predictable, injective relationships between two quantities... and then take them as far as they will let us.

    I think statistics shows us a wonderful marriage of the power of algebra combined with the less uniform patterns we observe in other life relationships around us. We observe, gather data, and attempt to fit a uniform pattern of behavior to our model. How close is close enough? Let the statistics decide.

    Thank you for providing me with more food for thought. It's this kind of stuff that keeps me up at night!

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  3. Very impressive and knowledgeable blog.Algebra is a major component of math that is used to unify mathematic concepts. Algebra is built on experiences with numbers and operations, along with geometry and data analysis. Some students think that algebra is like learning another language. This is true to a small extent, algebra is a simple language used to solve problems that can not be solved by numbers alone. It models real-world situations by using symbols, such as the letters x, y, and z to represent numbers.

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