TITLE: Mathematics, Problems, and Teaching AUTHOR: Eugene Wallingford DATE: July 17, 2007 8:17 PM DESC: ----- BODY: I'm surprised by how often I encounter the same topic in two different locations on the same day. The sources may be from disparate times, but they show up on my radar nearly coincident. Coincidence, perhaps. Yesterday I ran across a link to an old article at Edge called What Kind of a Thing is a Number?. This is an interview with Reuben Hersh, a mathematician with a "cultural" view of mathematics. More on that notion later, but what caught my eye was Hersh's idea of how to teach math right:
A good math teacher starts with examples. He first asks the question and then gives the answer, instead of giving the answer without mentioning what the question was. He is alert to the body language and eye movements of the class. If they start rolling their eyes or leaning back, he will stop his proof or his calculation and force them somehow to respond, even to say "I don't get it." No math class is totally bad if the students are speaking up. And no math lecture is really good, no matter how beautiful, if it lets the audience become simply passive. Some of this applies to any kind of teaching, but math unfortunately is conducive to bad teaching.
Computer science isn't mathematics, but it, too, seems conducive to a style of teaching in which students are disengaged from the material. Telling students how to write a program is usually not all that enlightening for students; they need to do it to understand. Showing students how to write a program may be a step forward, because at least then they see a process in action and may have the gumption to ask "what?" and "why?" at the moments they don't understand. But I find that students often tune out when I demonstrate for too long how to do something. It's too easy for me to run ahead of what they know and can do, and besides, how I do something may not click in the right way with how they think and do. The key for Hersh is "interaction, communication". But this creates a new sort of demand on instructors: they have to be willing to shut up, maybe for a while. This is uncomfortable for most faculty, who learned in classrooms where professors spoke and students took notes. Hersh tells a story in which he had to wait and wait, and then sit down and wait some more.
It turned out to be a very good class. The key was that I was willing to shut up. The easy thing, which I had done hundreds of times, would have been to say, "Okay, I'll show it to you." That's perhaps the biggest difficulty for most, nearly all, teachers--not to talk so much. Be quiet. Don't think the world's coming to an end if there's silence for two or three minutes.
This strategy presumes that students have something to say, and just need encouragement and time to say it. In a course like Hersh's, on problem solving for teachers, every student has a strategy for solving problems, and if the instructors goal is to bring out into the open different strategies in order to talk about them, that works great. But what about, say, my compilers course? This isn't touchy-feely; it has well-defined techniques to learn and implement. Students have to understand regular expressions and finite-state machines and context-free grammars and automata and register allocation algorithms... Do I have time to explore the students' different approaches, or even care what they are? I agree with Hersh: If I want my students actually to learn how to write a compiler, then yes, I probably want to know how they are thinking, so that I can help them learn what they need to know. How I engage them may be different than sending them to the board to offer their home-brew approach to a problem, but engagement in the problems they face and with the techniques I'd like them to learn is essential. This sort of teaching also places a new demand on students. They have to engage the material before they come to class. They have to read the assigned material and do their homework. Then, they have to come to class prepared to be involved, not just lean against a wall with a Big Gulp in their hands and their eyes on the clock. Fortunately, I have found that most of our students are willing to get involved in their courses and do their part. It may be a cultural shift for them, but they can make it with a little help. And that's part of the instructor's job, yes -- to help students move in the right direction? That was one article. Later the same afternoon, I received ACM's regular mailing on news items and found a link to this article, on an NSF award received by my friend Owen Astrachan to design a new curriculum for CS based on... problems. Owen's proposal echoes Hersh's emphasis on problem-before-solution:
Instead of teaching students a lot of facts and then giving them a problem to solve, this method starts out by giving them a problem.... Then they have to go figure out what facts they need to learn to solve it.
This approach allows students to engage a real problem and learn what they need to know in a context that matters to them, to solve something. In the article, Owen even echoes the new demand made of instructors, being quiet:
With problem-based learning, the faculty person often stands back while you try to figure it out, though the instructor may give you a nudge if you're going the wrong way.
... and the new demand made of students, to actively engage the material:
And [the student] might spend a couple of weeks on a problem outside of class.... So you have to do more work as a student. It's kind of a different way of learning.
The burden on Astrachan and his colleagues on this grant is to find, refine, and present problems that engage students. There are lots of cool problems that might excite us as instructors -- from the sciences, from business, from the social sciences, from gaming and social networking and media, and so on -- but finding something works for a large body of students over a long term is not easy. I think Owen understands this; this is something he has been thinking about for a long time. He and I have discussed it a few times over the years, and his interest in redesigning how we teach undergraduate CS is one of the reasons I asked him to lead a panel at the OOPSLA 2004 Educators' Symposium. Frank Oppenheimer's Exploratorium This is also a topic I've been writing about for at least that long, including entries here on how Problems Are The Thing and before that on Alan Kay's talks ... at OOPSLA 2004! I think that ultimately Kay has the right idea in invoking Frank Oppenheimer's Exploratorium as inspiration: a wide-ranging set of problems that appeal to the wide-ranging interests of our students while at the same time bringing them "face to face with the first simple idea of science: The world is not always as it seems. This is a tall challenge, one better suited to a community working together (if one by one) than to a single researcher or small group alone. The ChiliPLoP project that my colleagues and I have been chipping away slowly on the fringes. I am looking forward to the pedagogical infrastructure and ideas that come from Owen's NSF project. If anyone can lay a proper foundation for problems as the centerpiece of undergrad CS, he and his crew can. Good coincidences. Challenging coincidences. -----