"The lecture method is a process by which the lecture notes of the professor get to the notebooks of the students without passing through the brains of either."
Part I. Teaching Physics
The video (Confessions of a Converted Lecturer) is over an hour; it had crappy sound quality on my laptop (not so bad on my desktop); it's mostly just one person talking and showing a few slides of his data analysis; and it's not even about math. But it was totally worth it.
Eric Mazur is a physics professor at Harvard. He teaches physics to pre-med students. They don't have an interest in physics, but they are very good students. Eric was, by conventional standards, a very good teacher. But then he got his hands on the Force Concept Inventory (FCI), which has conceptually-based physics questions with no calculations, and it changed him. His students were doing well when measured by conventional physics questions, but could not answer questions that physicists see as much easier. His students could recognize which formulas to use for which problems, and could then work their way through long calculations, but they still didn't understand some of the most basic principles of physics.
As he described this process students use, of learning just the procedures, he said:
"Some students told me that physics is boring, and I could never imagine it. But, imagine that it's been reduced to following a recipe, that you don't even understand. Yes, then it would be boring."He said that most students come in, even after getting a 5 in an AP physics course, as Aristotelian thinkers. (Aristotle codified some of the ideas the ancients had about physics. Those ideas were often wrong.) After a conventional Harvard physics course, and after scoring well on a conventional exam, most of the students are still 'Aristotelian thinkers' - "they haven't understood the material in week two, on which everything else depends."
I remember one thing like this from my college physics course. I remember a question about something rolling off a table, and what its path would be. It's hard for me to believe I didn't always imagine a parabolic path. But I do remember being surprised. I think my previous conception was that the object would go straight down. It sounds so wrong to me now... Apparently, there are lots of basic concepts in physics like that, where people have firmly entrenched wrong ideas, until they've really understood the principles involved. Then they have trouble believing they ever had the wrong picture in their heads before. We don't remember not knowing...
Is math like this too? Can you imagine not knowing that 3+11 is the same as 11+3? Kids don't know that at first, and when they get it, it's a powerful tool, since counting up 3 from 11 is way easier than counting up 11 from 3. Are there misconceptions that we all had? I can't think of any right now. (Anyone who works with little kids, are there wrong ideas that come to us naturally?)
When Mazur gave a pre-test to his Harvard students, using the Force Concept Inventory, and told them how low their initial scores were, they wanted to spend time going over each question to understand it. There wasn't time during class, so he arranged to meet with interested students in the evenings. In this voluntary, ungraded environment, perhaps they felt more comfortable expressing their confusion. One evening, he gave what he thought was a good explanation of why the forces a light car and a heavy truck exert on one another in a crash are equal (as they must be, if one understands Newtonian mechanics), but what he got was a roomful of confused looks. He tried again, adding acceleration into the picture, and saw he'd only increased the confusion. He knew that about 40% of them already understood this idea, and, giving up, asked them to explain it to each other. Out of the chaos was born Mazur's version of "peer instruction".
Nowadays, when he teaches, he explains for just a few minutes, and then asks a concept question. In a lecture hall with hundreds of students, they all begin to talk. Here's the process:
- He shows the question and explains it,
- Students silently think for 1 to 1 1/2 minutes,
- Individuals answer (using clickers, which allow immediate tallying of anonymously given answers),
- Discussion ensues among small groups of students (he doesn't try to control the groupings),
- Students answer again, after their group has come to some consensus,
- He explains the answer (and goes on to repeat the process for the next small bit of lecture).
There's more to savor in the video. If you can find the time to watch it, I think you'll be glad you did.
Part II. Teaching Math
So how do we use this for math? There is some work being done along similar lines. Jerome Epstein has developed a Calculus Concept Inventory, and found similar results - that conventional courses don't help the students to truly understand the concepts.* And Cornell has the Good Questions Project (which I mentioned before, here).
Good questions hook in to a student's personal experiences. This one may work at Cornell (the author's students really liked it), but would definitely not work at my college:
An article in the Wall Street Journal's Heard on the Street column Money and Investment (August 1, 2001) reported that investors often look at the change in the rate of change to help them get into the market before any big rallies. Your stock broker alerts you that the rate of change in a stock's price is increasing. As a result:My students definitely do not have stock brokers, nor do most of their teachers. As I glance through the Good Questions file I found here, I don't see others like this one. Perhaps someone has thought about the issues of privilege already. Here are a few questions from the 53-page list:
a. you can conclude the stock's price is increasing
b.you cannot determine whether the stock's price is increasing or decreasing
c. you can conclude the stock's price is decreasing
True or False. At some time since you were born your weight in pounds equaled your height in inches.
True or False. As x increases to 100, f (x) = 1/x gets closer and closer to 0, so the limit as x goes to 100 of f (x) is 0.
Your mother says “If you eat your dinner, you can have dessert.” You know this means, “If you don’t eat your dinner, you cannot have dessert.” Your calculus teacher says, “If f is differentiable at x, f is continuous at x.” You know this meansI'm hoping all these good questions will help me to teach a much better section of calc II this fall. (It sure would help if I could get a 'smart classroom' to teach in for the fall, so I could have students use those clickers...)
(a) if f is not continuous at x, f is not differentiable at x.
(b) if f is not differentiable at x, f is not continuous at x.
(c) knowing f is not continuous at x does not give us enough information to deduce anything about whether the derivative of f exists at x.
I'd like to think about how to use these same ideas at the algebra level, too. How do we ask the questions that will get students thinking about concepts rather than just focusing on the procedures? There's lots of good work being done by the bloggers I read. Jason Dyer, at Number Warrior, has a Fractions Concepts test that I'll use when I review fractions at the start of the term with my beginning algebra students. Hmm, maybe we could all work together to create an Algebra Concept Inventory...
[Hat tip once again to Dan, whose blog, Math for Love, has been very inspiring.]
*This is not available online, since they'd like to keep the content of the questions private, in order to retain their validity as a test instrument. I have emailed the author, asking for a copy to use in my calculus course in the fall.