The Word Problem Problem

Recently I was faced with the question of whether to use some of the same books in two different courses. Isn’t each course supposed to have distinctive content? But what about the unity of knowledge? My internal debate eventually led me to conclude that when it comes to curriculum design, most university faculty — at least at the undergraduate level — are like failed restaurateurs. Everyone who likes cooking in the kitchen thinks they can run a restaurant. Every faculty member thinks designing a curriculum is the same as constructing a syllabus.

We often preach about the importance of critical thinking and claim that our courses and curricula help develop this skill. Yet students are often unable to transfer analytical  reasoning techniques from one context to another. The classic example of this is the word problem from elementary school:

  • A Japanese train with seven cars travels between Osaka and Tokyo at a constant speed of 200 km/hour. How long does the trip take for someone riding in the second car?
  • A Japanese train with four cars travels between Osaka and Tokyo at a constant speed of 200 km/hour and someone accidentally drops a book from the window of the second car. How long does the book take to hit the ground?

Students who have learned how to solve the first problem will frequently fail to solve the second one, even though they are essentially identical.* Why does this happen?

As pointed out by the cognitive psychologist Daniel Willingham, the human mind is lazy — it is predisposed to look for familiar patterns, and when it thinks it has found one, its decision-making ability is often negatively influenced. To get around this habit, a person must learn to differentiate between a problem’s superficial aspects and what Willingham calls its deep structure:

When one is very familiar with a problem’s deep-structure, knowledge about how to solve it transfers well. That familiarity can come from long-term, repeated experience with one problem, or with various manifestations of one type of problem (i.e., many problems that have different surface structures, but the same deep structure).

How does this relate to the readings I assign in my courses? If I assign the same text in multiple courses, students get repeated exposure to the same problem, and their ability to solve it is reinforced, or I can use the text to demonstrate how problems with the same deep structure can manifest themselves differently in different environments. Students also get more training in the knowledge that they should look for a problem’s deep structure instead of erroneously focusing on surface aspects (like the number of cars in a train or the train being in Japan).

How does this relate to curriculum design? Many curricula follow a checklist paradigm — students must take one Western civilization history course, two math courses, one religious studies course, ten courses in a major, etc. We like to claim that problem-solving techniques learned in an economics class can be applied to situations presented in an anthropology class, and vice versa, but the checklist sends the opposite message — that knowledge resides in discrete boxes. The way we design our curricula makes it less likely that students will ever learn how to see past surface structure and become effective problem-solvers.

*The solution to both requires knowing distance; in the former, the distance between Osaka and Tokyo, and in the latter between the window and the ground.