In a recent e-mail list discussion, being a math major was justified by the transferability of problem-solving skills from one domain (math) to others (banking, sales, and other jobs). This justification for studying math is a popular one with mathematicians and math teachers. One of the primary justifications for requiring geometry, for example, is that it teaches students how to prove things rigorously. The same case for transferable problem solving can be (and has been) made, perhaps even more strongly, for computer science and for engineering fields that do a lot of design work.
I was a math major (through and MS) and I got my PhD in computer science, and I certainly believed that the constant practice at problem solving made me better at solving certain classes of problems—ones with clear rules, not social problems or biological ones.
Education researchers have tried to measure this transfer effect, but so far have come up empty, with almost no indication of transfer except between very, very close domains. I don’t know whether the problem is with the measurement techniques that the education researchers use, or whether (as they claim) transferability is mainly an illusion. Perhaps it is just because I’m good at problem solving of a certain sort that I went into math and computer science, and that the learning I did there had no effect on my problem-solving skill, other than tuning it to particular domains (that is, perhaps the transferable skill was innate, at the learning reduced transfer, by focusing the skills in a specialized domain).
Two of the popular memes of education researchers, “transferability is an illusion” and “the growth mindset”, are almost in direct opposition, and I don’t know how to reconcile them.
One possibility is that few students actually attempt to learn the general problem-solving skills that math, CS, and engineering design are rich domains for. Most are content to learn one tiny skill at a time, in complete isolation from other skills and ideas. Students who are particularly good at memory work often choose this route, memorizing pages of trigonometric identities, for example, rather than learning how to derive them at need from a few basics. If students don’t make an attempt to learn transferable skills, then they probably won’t. This is roughly equivalent to claiming that most students have a fixed mindset with respect to transferable skills, and suggests that transferability is possible, even if it is not currently being learned.
Teaching and testing techniques are often designed to foster an isolation of ideas, focusing on one idea at a time to reduce student confusion. Unfortunately, transferable learning comes not from practice of ideas in isolation, but from learning to retrieve and combine ideas—from doing multi-step problems that are not scaffolded by the teacher.
“Scaffolding” is the process of providing the outline of a multi-step solution, on which students fill in the details—the theory is that showing them the big picture helps them find out how to do multi-step solutions themselves. The big problem with this approach is that students can provide what looks like excellent work, without ever having done anything other than single-step work. De-scaffolding is essential, so that students have to do multi-step work themselves, but often gets omitted (either by the teacher, or by students cheating a little on the assignments that remove the scaffolding and getting “hints”).
I find myself gradually increasing the scaffolding of the material in my textbook, so that a greater proportion of the students can do the work, but I worry that in doing so I’m not really helping them learn—just providing a crutch that keeps them from learning what I really want them to learn. I don’t think I’ve gone too far in that direction yet, but it is a constant risk.
I’ve already seen students copying material from this blog as an “answer” to one of the problems, without understanding what they are doing—not being able to identify what the variables mean, for example. (I used different notation in class than I used in the corresponding blog post—a trivial change in the name of one variable.) I’m trying to wean students off of “answer-getting” to finding methods of solution—the entire process of breaking problems into subproblems, defining the interfaces between subproblems, and solving the subproblems while respecting the interfaces.
I do require that the students put together a description of the entire solution to their main assignments—a design report that not only describes the final design, but how the various design decisions were made (what optimizations were done, what constraints dictated what part choices, and so forth). This synthesis of the multi-step solution at least has the student aware of the scaffold, unlike the fill-in-the-blank sorts of lab report which makes the scaffold as invisible as possible to the student.
I also try very hard for each design problem to have multiple “correct” solutions, though some solutions are aesthetically more appealing than others. This reduces the focus on “the right answer” and redirects students to finding out how to test their designs and justify their design decisions.
I have been encouraged by signs of problem-solving skills in several students in the course (both this year and in previous classes). Often it is in areas where I had not set up the problem for the students. One year, a student came up with a good method for keeping his resistor assortment organized and quickly accessible, for example. This year, one pair of students used their wire strippers and blue tape as an impromptu lab stand for their thermometer and thermistor, to save the trouble of holding them.
The problems students set themselves often lead to more creative solutions than the ones set for the class as a whole—but how do you set up situations in which students are routinely identifying and solving problems that no one has presented to them? I believe that the students who identify problems that no one has pointed out to them are the ones who become good engineers, but that attempts to teach others to have this skill are doomed by the very attempt to teach. Capstone engineering classes are one attempt to get students the desired experience, but I think that in many cases they are too little, too late.