"The Jigsaw Puzzle of Teaching,"
Vol. VII, No. 4, May/June 1995

One of the best pieces of advice I've heard recently on teaching came from one of our senior faculty members. The two of us were attending a workshop for graduate students who had been chosen to lead recitations in his department the following semester. It was the last day of the three-day workshop, and we had already sat through a dozen or so attempts on the part of the new recruits to present 10 to 15 minutes of their best teaching. As was to be expected with these first tries at flying solo in front of a class, we had seen our share of aborted take-offs, spin outs, and even a couple of crash landings. Finally, on that third day, as we sat through one more explanation of a problem whose purpose and method were not as clear as they might have been, this professor's patience must have worn thin for he gruffly admonished the group as a whole, "Don't turn the lessons you teach into mystery stories."

Mysteries, of course, can be terrifically engaging. They can pull their readers/viewers/listeners into a web, keeping them guessing about what will happen

next, or where the next twist or turn will be. A mystery is mysterious precisely because information is doled out in measured amounts; gaps are purposely created; crucial elements of the story are withheld. Mystery makers challenge their audience to see if they can outrun or second-guess them, fitting the clues together before the surprise at the end is finally revealed. But this, I believe, is not the best model for what should happen in the classroom as we try to teach complex ideas.

In the classes I've watched, often (although unfortunately not always) the instructor starts well by announcing what the subject of the day will be: the harmonic oscillator, Markov chains, two-dimensional momentum transport processes. If the topic gets written on the board as well as being announced, all the better. Here is a good way to begin to inoculate students against the creeping inroads that the mysterious can make.

Then the class launches into the day's work. Very often, this means problems are introduced, examples offered, or proofs presented. Numbers fill the board, equations abound, calculations accumulate. This level of detail is rightly at the heart of what should go on in many MIT classes; it is fundamental to the work of the scientist, mathematician, and engineer. The danger is that both instructor and student can become so absorbed by these details that they lose sight of what I call the "picture on the box."

Sometimes I think teaching a class is akin to working a jigsaw puzzle. In order to put together a jigsaw puzzle, you need to alternate between the details--all those hundreds of pieces of oddly shaped cardboard--and the picture on the box, which provides the overall design for how the pieces go together. As I've said, the individual pieces--the problems, the equations, the calculations--are important in their own right. They are the building blocks of the lessons we teach. But there is a danger that the details can be so delightful in and of themselves, that the larger picture, the "why are we doing this" somehow gets lost. Observing MIT classes, I've sometimes felt as if I were watching a movie in which the director begins with a fabulous panoramic view--perhaps of the skyline of New York or windswept rocks overlooking a vast expanse of ocean--and then gives the audience nothing for the next two hours but close-ups of the leading man's nose. Without the long shot, without some context much is lost.

The good news is that this is a simple problem to remedy. Every once in a while (maybe every 15 minutes), all you need do is pull back the lens and survey the territory. It isn't enough to simply announce the topic at the beginning of class; students, like most of us, have short attention spans. Research into the information processing capabilities of members of an audience shows that attention is greatest at the beginning and end of a presentation with a substantial dip in the middle. (One study found that students' attention begins to wane after ten minutes!) So continually remind your students why you're solving the equation, what the problem exemplifies, how the calculation furthers an understanding of the topic at hand.

Transitional statements help, too. How does subpoint A relate to the theme of the day's lesson? How does subpoint B relate to subpoint A? Does one idea further the next, or is it a contrast? Is subpoint C a consequence of subpoint A and B? Why did you need to talk about A and B before you presented C, anyhow? Provide clear, explicit signposts along the way reminding students where you started, where you are in the process at a particular point in time, and where you expect to end up.

What all this means, of course, is that you need a unifying thread that will weave itself throughout the class. Patrick Winston, in a handout entitled "Lecturing Heuristics," which accompanies his not to be missed IAP presentation, "How to Speak," writes of the "central, exciting question" that forms the basis of any good lecture. The faculty member mentioned at the beginning of this column talks of the need to identify and articulate the "important problem" that must motivate every class period. The question that students need answered for them is, "Why are we learning what we are learning?" What is the key viewpoint, the fundamental insight that propels the rest of the day's material? What does the picture on the box look like, anyway? Studies on what makes lectures successful identify two basic components: a simple organizational plan and a number of germane examples. Both elements are necessary for the effectiveness of the lecture will be weakened if either is missing.

As I've observed classes here over the past year, I have been extraordinarily impressed at the ability of those I've watched to manipulate the symbols that build the examples, the problems, and the proofs. Sometimes I feel as if I am being treated to amazing feats of underwater endurance; as the instructor fills the board from one end to the other, it is as if I'm watching a swimmer glide effortlessly lap after lap on a single breath. But it is important to come up for air regularly, to look around, to check where you've been and where you're headed. I would like to suggest that that is good pedagogy because it helps students figure out how the pieces come together not only so they can duplicate the picture on the box, but ultimately so they will have the confidence to create their own.