My post last night on teaching engineering thinking had a coincidental resonance with another blog post that came out yesterday: Calculus and formal reasoning in intro physics, by Bruce Sherwood, one author of the textbook I’m using with my son for us both to learn calculus-based physics: Matter and Interactions. I’ve had several e-mail conversations with Bruce Sherwood, and I’ve been reasonably pleased with the textbook. He is an infrequent blogger, but he generally has something interesting to say when he does post to his blog.
In the Calculus and formal reasoning in intro physics post, Bruce says
… there is a tendency for older faculty to deplore what they perceive to be a big decline in the mathematical abilities of their students, but my experience is that the students are adequately capable of algebraic manipulation and even calculus manipulation (e.g. they know the evaluation formulas for many cases of derivatives and integrals). What IS however a serious problem, and is perhaps new, is that many students ascribe no meaning to mathematical manipulations.
We are convinced that an alarmingly large fraction of engineering and science students ascribe no meaning to mathematical expressions. For these students, algebra and calculus are all syntax and no semantics.
This problem with formal reasoning may show up most vividly in the Matter & Interactions curriculum, where we want students to carry out analyses by starting from fundamental principles rather than grabbing some secondary or tertiary formula. We can’t help wondering whether the traditional course has come to be formula-based rather than principle-based because faculty recognized a growing inability of students to carry out long chains of reasoning using formal procedures, so the curriculum slowly came to depend more on having students learn lots of formulas and the ability to see which formula to use.
I had not thought of the problem along those, lines, but that is fairly consistent with what I’m seeing in the circuits class. They can do algebra and calculus when the problems are set up for them, but they have difficulty turning word problems and electronics problems into the corresponding algebraic problems. They also rarely do “sanity checks” to make sure that they haven’t reversed the sign or inverted the value. If they see math as consisting purely of formal manipulation of symbols, with no attached semantics, or only a purely formal mathematical semantics, with no connection to the real world phenomena they are modeling, then the difficulty with setting up equations and the lack of sanity checks becomes much more explicable.
Unfortunately, being able to explain their behavior doesn’t immediately give me a handle on how to modify it—if this explanation is even the right one (or, more correctly, if this model of their behavior is a good enough one to be useful). I will be continuing to give problems that require thinking rather than applying a random formula, trying to get them to reason from a few basic principles rather than from memorized formulas. My “basic principles” are not as basic as Sherwood’s, of course, since this is an electronics course, not a physics course. Throughout Chapter 19 (which my son and I just finished), Sherwood prohibits the use of Ohm’s Law, which has not yet been derived in the book, insisting on more fundamental, but more difficult, reasoning based on fields, electron mobility, and density of mobile charges. In my class, I’m happy to take Ohm’s Law as a fundamental principle in electronics, without going deeper into the physics.
I made a one-page study sheet for the students and handed it out today, listing the small number of formulas I expect them to have instantly available in their memory. There are only 15 formulas so far, and most are pretty trivial. I suspect that we’ll only add 4 or 5 more for the rest of the quarter. I’ll be using all the impedance formulas in Monday’s quiz, but not much else. But it will be a closed-notes quiz—I want them thinking, not randomly scrambling through their notes looking for a formula that isn’t there. Of course, the small number of basic formulas doesn’t mean they’ll find the quiz trivial, because I’m not going to ask them to recite the formulas, derive them, or apply them to problems they’ve already done, but to apply them to problems they’ve not seen before. They’ll be needing to figure out how to apply these few formulas to the design and analysis questions I’ll ask.