Archives for the month of: June, 2011

The great comments about Algebra 2 have got me thinking about relationships, descriptions of “having to do with”.

Every pair of people that have anything to do with each other is a different case, yet each relationship is pretty unambiguously father-son, or business partners, or best friends, or lovers, or whatever it is. We all relate to each other in a small number of very common ways. The examples of each category contain a dazzling, unending variety–but in spite of that, every variation on grandmother-granddaughter is still an example of grandmother-granddaughter. The grandmother and the granddaughter have something to do with one another, and it’s a specific something.

Variables–anything we can count or measure; any quantity that changes, or could–can be related, or not. They can have something to do with each other, or not. The period and length of a pendulum have something to do with each other. The height of a projectile and time have something to do with each other. We need not be applied, we can stay pure: x^2, the product of a number with itself, certainly has something to do with how big x is. 2^x also has something to do with how big x is, but it’s a different something. There are a few, very common ways things can be related (along with lots more less common ways). Is the goal of Algebra 2 to find out what those common ways are?

What are the different kinds of relationships among variables? What can we infer about one if we know something about the other? If Mike is my father, then I must be Mike’s … what? If y=10^x, then x must be … what? What is the only number that has the following complicated relationship to 3? What makes a relationship quadratic? If two people are seen at the movies kissing, what can you infer about their relationship? What can’t you? If two variables take on values together in the following pairs, what kind of relationship is it? What does that tell you about the other ways those variables will be seen–excuse me, observed–together?

This might work.

I think the biggest thing I’ve learned this year is this: if students aren’t paying attention and haven’t prepared a space in their brain for new knowledge to fit into–or in other words, if they themselves haven’t asked the question that is about to be answered–then they can move through a discovery exercise and not take anything away from it just as easily as they can hear a lecture and not take anything away from that.

Modeling is in the air. Thursday a colleague told me he was looking at Modeling Instruction for next year. Then yesterday, Shawn committed. So I started reading up. I was a full-time modeler before teaching, so I’m biased, but I love it. In Modeling Instruction physics, students organize their year by making and testing hypotheses about observable aspects of a few archetypical physical systems. The hypotheses arise from curiosity: we see the pendulum moving, what are things we could measure? What predicts their values? Here is the world, you are already fluent in it, go make sense of it. What the students construct in response is a solid understanding of the core content of introductory physics.

I could make Algebra 2 into a modeling course. But modeling is using math to describe what you are studying; I would like to be studying the math. I’m looking for a destination such that the study of each of these functions is a step towards something pretty deep and beautiful.

These functions are the structure of everything. Yes, you can model radioactive decay with an exponential. And yes, you can model compound interest with an exponential. But the exponential, the idea of the exponential, is the abstraction of what interest and radioactivity have in common: what you get is proportional to what you have. Yes, the height of a ball over time is quadratic. Yes, as you fence off your rectangular llama pasture, making it squarer and squarer, the area is quadratic. The parabola is the abstraction of what constant acceleration and constrained rectangular area have in common: the unceasing influence of the second difference, patiently turning things its way one bit at a time, until the system is inevitably flying in its direction.

So, what? “Algebra 2: The structure of everything”? It’s grandiose, not compelling. It isn’t even a question. It certainly isn’t “Would You Go to Mars?” (via Dan).

These objects, the stuff of Algebra 2, are too fascinating and too pervasive not to lead to some summit worth attempting.

If you want students to master something–call it R–and R is a means to S, then work on S; R will slip unobtrusively in under the radar….” -Bob and Ellen Kaplan, Out of the Labyrinth, p. 87

I need an overarching theme, question, or mission for Algebra 2 that transcends and motivates the required skills.

The content goals of Algebra 2 are to invert, transform, solve, and apply graphical, tabular, and analytic representations of linear, quadratic, exponential, sinusoidal, and rational equations. That’s R.

What’s S?

We’ve just plotted some data on school enrollment vs. year (and discussed the meaning of “enrollment” as a number).

Question: “What does the slope represent?”

Response: half-open mouths, half-shrugs, half-smiles, total silence. Their looks say: What are you talking about?

“Ok, try this: What does the slope tell you about the school?

Backs straighten (a bit). Eyes narrow in concentration. Conversation sputters once, twice, then takes off. I sit and listen. After some false starts, a few people put it together, and after another minute they’ve all got it.

They turn to me and now their looks say, Why would you ask us something so easy?


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