Let us leave stats and talk about the first-year class in high school.
And let’s start with the practical “presenting problem” we face at the high-expectations independent school where I teach: every year, we have to assess a new flock of eighth graders in order to place them in freshman math. Will they take Algebra I? Inductive Geometry? Deductive Geometry? Or Algebra II? (This year, 20% of the incoming class asked to be tested for Algebra II.) This decision has all sorts of unpleasant implications, among them:
- We’re tracking. Every kid knows on Day One who is a “smart kid” and who is not. We try really hard to break the labels, but at some level we can’t. The “smart” kids feel the burden of having to “get it” quickly. At the other end of the scale (we dare not say “the dumb kids”), it’s easy to be “not good at math” and let that follow you for four years.
- We lose the benefits of heterogeneity. I will not debate this here; let’s just admit that homogeneous classrooms limit what can happen in a class. This is part of the whole point—it makes it easier to plan—but it is limiting.
- It impacts scheduling in the rest of the school, forcing the same kids to be in the same sections in English and other non-tracked subjects, further stratifying the society.
- Placement causes heartache for the department chair, who has to defend every decision to parents who desperately want their cherubs to finish calculus as soon as possible.
- We inevitably get some placements wrong. Of course we can adjust as the new semester begins, but a few days in a class is no substitute for getting to know a student and the student’s work over the course of a semester or longer.
Which brings us to a radical suggestion:
Let’s create a course that every ninth-grader takes: Math Nine.
Its topics and tasks would have to be accessible to the least-experienced students, but still challenging to all. It would have to be rich enough that more students would have a chance to shine, feel successful, and be seen as “good at math.” It would also have to give all students the background they need to do well in subsequent math courses.
The hard part is making it fit with what has to be covered during high school. I could argue about how horrible it is that we teach algebra in eighth grade or how we mistakenly trudge towards to calculus and nowhere else. But current reality intervenes; I accept that seniors will be learning the chain rule and juniors will learn trig identities and all the rest. So whatever goes in Math Nine has to prepare them for that.
So I’ve been making a list of what I thought could go into such a course, asking questions like “what did I really need to know to do well in calculus?” and “what did I really wish my students knew in precalc?” Looking at this list, two principles emerge:
- Instead of focusing on skills and procedures, focus on habits of mind.
- Focus on modeling.
And then, this morning, what actually propelled me to write this post, an epiphany. Bearing in mind that one morning’s epiphany is another’s air ball:
Math Nine is a chance for students to be mathematical naturalists, boarding their HMS Beagle and sailing the globe, drawing the finches and watching the behavior of penguins and tortoises. They dip their nets into the sea and classify what comes up. They measure and catalog and categorize, and develop the tools to do that well.
They are not yet scientists. At this stage, they don’t write Origin of Species. They focus on what, where, and how much—not why.
That’s enough for now. I’ll put the current version of that list in another post.
A Best-Case Scenario 
I am actually trying to do this now for my Algebra II, but I suppose it can be extended to Algebra I. After the CSTs are over, I feel “unshackled” to teach what anfd how I think Algebra ought to be taught. The basic unifying theme of Algebra II is functions: linear, quadratic, exponential, etc. I start by saying this is the theme and then I start with linear functions. However, instead of teh traditional slope and intercept I use videos and other aids to establish some basic principles: you need two quantities to define a linear function, the meaning of a point on a line, an intersection point is a solution (and what does solution mean?), what happens when y “disappears” (x-intercept) and what does that mean. The idea (hope) is to back into slope intercept problems from more general principles.
I do something else as well. Since I have a 59 minutes class, I reserve the last 15-20 minutes for procedures. No general math principles here, but how do you work with proportions, how do you factor, how do you transform an equation with all variables to solve for one specific variable. We’ll see how it works.
If you have access to Fathom or some other help-you-plot-data-and-functions technology—and have the bandwidth at this time of the year to look at something ELSE—you might enjoy some draft materials I’ve put together,called “EGADS”.
See http://www.eeps.com/pdfs/EGADs.April2008.pdf
It’s all about taking measurements that you can model with those very functions.
Thank you – EGADS bears your mark Tim: thoughtful and interesting. Unfortunately, even though I have Fathom and sufficient computers it comes a bit late in the term. Next year..
I forgot to add – that I just started this – I have a month to see how it plays
Dean