Tuesday, July 19, 2011

Frank Noschese's TED Talk*

* DISCLAIMER: I'm making this up.

[Dramatic TED music.]


Thank you everyone. We've heard some fascinating talks today from some very knowledgeable people. Not only are the ideas great, but the lectures have been funny, exciting and engaging. And they really need to be. Every speaker here wants everyone to walk out of the room having learned more about their research, their work, or their writing.

I wonder, though: is a lecture the best way to do this?

Full disclosure: I'm a high school physics teacher, and there are often debates between those who think that the future of education is taking short lectures and distributing freely and widely, and those who think that there is something better than lecturing.

So what I thought we'd do today is run a little bit of an experiment. Let's do a quick poll to start things off. Text your answer to this number, and we'll get the poll results immediately and put them up.

[The question is a counter-intuitive implication of a point made by one of the previous speakers. Like, maybe the previous speaker.]

OK, so we've got the results and here they are.

They're not bad. Most folks got the answer correct. A big group didn't. That's not entirely surprising. Research supports the limitations of lecturing, and the research is constantly confirmed by any classroom teacher.

Now, if you're an educator, you might say, well, what can you do? Most of the people listening to the talk got the point, and you'll get the rest through remediation and tutoring. Send them to Khan Academy or keep them after school.

The issue is, that's pretty inefficient. There are other issues as well (etc.)

It would be great if we could do better than this. So that's what I propose we spend the next 15 minutes doing. Let's try something, and you won't have to believe me that it's better than lecturing. You're going to see for yourselves that it is.

First, I need to tell you an important fact about something called the modeling method. At its heart is the modeling cycle, which begins with modeling development, and is followed by modeling deployment. Here's what those terms mean... Now, remember this, because this is important.

Now, let's try something all together different.

[Insert: inquiry-based, modeling learning here. It can be about anything that you like. Here's how you pull it off:
* Start with another text poll to figure out how many folks get the answer
* Recruit a bunch of educators to help you.
* Have the educators pass out materials.
* Educators go around to groups and ask Socratic questions to groups of people in the audience. ]

OK, now let's all come back together. Sir in the front, what did your group's experiment show, please come up and show us.


Now, let's end with two polls. Right before the experiment, I told you an important fact about how modeling instruction works. Let's see how many of you remember that fact, which I told you, and I told you was very important.

[Poll results.]

OK, I'd say just OK. Now, let's ask another question, based on what you learned through inquiry and experiment.

[Show results.]

Let's not do lecturing better -- ladies and gentlemen, let's do better things.

Wednesday, July 13, 2011

Algebra 2: "How do we predict the future?"

I took Kate's Algebra 2 standards and I reorganized the first semester as an answer to the question, "How can we predict the future?"

Here's the results. Nothing earth-shattering, but feedback would be appreciated.

Here's the doc: Algebra 2 Rejiggered: Predicting the future.

A waste of time?

I'm not a curriculum developer. It's insane to think that I can do this. But I'm devoting a lot of time to making sense of my curriculum, writing lessons over the summer, etc. What's the point? There are, out there, better lessons and course-wide structures than I could possibly produce. I'm devoting a lot of time, right now, to thinking through Algebra 2. Why bother?

Here are my reasons:
(1) Thinking through curricular issues will make me more sensitive to student needs. I'll know which questions to ask, which points to emphasize, which ones to let go more easily.
(2) I don't have access to all those resources, and it's not a given that my school can provide me with them. I am, still, on my own.
(3) My school year has about 130 math teaching days, so I'm forced to make curricular decisions. I can't rely on someone else's curriculum.
(4) My students take the Regents exam, so my challenge is right there: I need to make tough curricular decisions while still getting my students to pass NY's Algebra 2 Regents.

Still, it's exhausting and doesn't have much to do with the actual practice of helping kids learn these things. Sigh.

Monday, July 11, 2011

How to put modeling at the center of NY State's Alg2/Trig

Q: What new skill should Algebra 2 students leave with?
A: The ability to model a system mathematically.

NY's A: Yeah, that. Also, how to solve an absolute value equation, how to employ Degree, Minute, Second notation to represent an angle, how to graph an inverse cosine curve...

And that, basically, is the challenge in reorienting the Alg2/Trig course around a single question or theme.

Still, I've been working on reorienting NY's Alg2 curriculum along these lines. This isn't exactly ground-breaking: Kirk Weiler's e-text, for instance, points at such an orientation. He starts by discussing functions, and then introduces different families of functions that end with a regression and modeling unit.

Here's what I would like to do differently in organizing the curriculum:
1. I want to bring the modeling and regression to the beginning of the function unit, to motivate our study of the function family.
2. I want to discuss a concrete example of a function, such as the familiar linear functions, before talking about functions in the abstract.
3. I want all the other stuff -- and boy oh boy is there a lot of other stuff -- to fit into the larger discussion about modeling.

1 and 2 are doable. 3 is hard. Still, there are some things that can be done to integrate the various skills of the course. For instance, much time is spent in Alg2 solving equations. By the end of the year, students should be able to solve absolute value equations, radical equations, quadratic equations, trig equations, exponential equations, log equations and rational equations.

These sort of skills, however, become necessary when you've mathematically modeled a system, found a representative function, and now wish to extrapolate. You're either going to be evaluating an expression, or solving an equation. If you think about the curriculum in this way, you have functions at the center of the curriculum, the functions are there for modeling, and a clear distinction between evaluating an expression and solving an equation will be constantly reinforced.

Ditto for inequalities.

So functions, modeling, and solving equations are taken care of. They fit into the larger framework. What's left over is all the stuff that has to do with manipulating expressions. For instance: simplifying radicals, simplifying complex exponents, simplifying complex fractions, exponent rules, etc. How do these things fit into the larger framework?

The best that I can do now is to say that these are upgrade packages, so to speak. The ability to manipulate expressions will allow us to have an easier time evaluating function expressions for a value, or expressing answers to function equations. So I think what I'm going to do is be explicit that these areas don't directly fit into our modeling narrative -- they're not used to describe or extrapolate based on data -- but they're excurses, upgrade packages that will allow us to model certain relationships more accurately.

In summary: 1) Bring statistics and regression to the foreground, to motivate the study of functions. 2) Put extrapolation at the center of function units. Extrapolation motivates both the evaluation of expressions and the solving of equations. 3) Explicitly bring out all the leftovers into upgrade packages, that will assist us in our next modeling exercise.

My next post will organize Alg2 standards into this framework. The post after that, hopefully, will reflect critically on this and think about what some of the problems of this will be.

Monday, July 4, 2011

Numbers for 9th graders

I was tutoring a kid the other day. I'm introducing her to Algebra2, and we spend the hour talking about relationships between numbers. Once we've got a few of these relationships pinned down, I tell her that they're called functions, and then we talk about some other functions. She asks, "So Algebra2 is pretty much about functions?" Yep, that's right.

She pauses, and thinks. "It's weird. When I took Algebra the first time it was just all these random topics that we needed to know, and I knew them, but they were all different. I thought that Algebra 2 would be the same, but I guess I'm wrong, it's all about functions."

That's satisfying. Because she knows what the question is she knows what's important (general foundational stuff about functions, stuff that relates to the nature of these functions) and what, relatively, isn't. She knows what she's studying, she'll know how to integrate the new knowledge. We'll introduce each new function with a similar "big" question ("When will the missile fall?"; "Are we going to overpopulate Earth?"; "Why does Albany want us to study DMS notation?")

So, what's the question for Algebra 1? At first I thought Algebra 2 was the challenge, but now I'm having trouble constructing a meta-narrative for Algebra 1 and teasing out a question that introduces that narrative. Clearly a lot of the course is driving towards the concept of a function/2-variable equation. But at the beginning of the year we're still doing arithmetic, so how do I describe the endgame early on?

The best I'm doing right now is thinking about the question, "What counts as a number?" I'm imagining this as a mini-arc that develops, with care, the concept of what we're going to treat as a number in Algebra while also giving me a chance to brush up their arithmetic skills. I'd like the answer of this question to involve integers, fractions, properties of real numbers, square roots, expressions and variables. I could add a historical subplot to the story, revealing info about when this stuff was thought up ("People invent math? WTF?").

Here's what I have so far: