I was thinking about the bit in the Grace Hopper video I linked to the other day, in which she complains about the mental challenge when she did her initial navy training: she had forgotten how to memorize, and there was a lot of memorization involved. As she put it, you can’t derive the organization of the navy.
In physics, however, you can derive a lot of things. I don’t recall exactly when I realized it, but somewhere along the way I realized that I didn’t have to waste time memorizing page after page of equations, because from a few basic ones, many others can be derived. This is clear right off the bat in physics, because the first topic taught is usually kinematics, and all of the equations derive from the mathematical definitions of acceleration and velocity being derivatives. Doing the proper integral recreates a whole bunch of equations. Applying them properly (i.e. adding in some trig and algebra) yield a whole host more that many students memorize (like several related to projectile motion).
I had trouble convincing most students of this when I was teaching. Invariably, they would blanch in horror at the suggestion that they derive equations, but these were typically not the physics majors who were resisting me, so perhaps that’s one of the kinds of thought processes that separate us from other other kinds of students, even within STEM topics. (though even physics majors are not totally immune to the “you’re not going to actually make me apply the math I learned in math class” attitude.) So it was nice to hear RDML Hopper say that.
Being a survivor of innumerable training sessions, I have thought along these same lines considerably. I have found that one of the most effective approaches to explain the idea of deriving solutions to individuals is to start with a discussion of the idea of a kernel e.g. the idea of a fundamental solution to an idealized situation one can understand.
After you have your kernel one can then do several things, but mostly one can add or remove constraints to the problem and think about how it effects the solution (you perturb the kernel).
When discussing it in these terms you find that most people understand what they are doing fairly quickly.
I think sometimes though most students end up being trained not to think creatively, either due to instructor inability or inflexibility in course material. So the mindset of students simply becomes focused on getting the “right answer” and not on actually thinking about the problem. This is understandable since it is often more time efficient to memorize than to derive, especially when most teachers are focused on students following a particular method (especially when they are paid to make sure students abide to some particular method).
This brings up a good point about how we do not generally teach students to be efficient problem solvers.
I love nothing more than deriving an equation for a scenario. Anyone can plug and chug. Then again I doubt that many of my fellow chemistry folks share this sentiment regarding kinematics equations.
All true but starting from the Peano axioms to get to the correct structure can be …. time-consuming.
Derivation is a powerful attribute, but it won’t save you if postulates are defective. Eventually somebody must look at assumptions. That is often politically noisome.
Any Boolean algebra is defined by not(not(p or q) or not(p or not(q)) = p. Organic chemistry is instead interesting for its structural aspects, and tar. A molecule that does not default to a planar noncrossing Schlegel diagram is rare and wonderful. Be humble before low yield, then kick its butt.
I dislike memorization — This right here is one of the reasons I majored in physics :).
High school physics students would rather memorize. It is a strategy that has been successful for them since at least middle school.
Their optimal course would look like this: On Monday I show them a problem and an algorithm for solving it. For the rest of the class, they practice a number of problems. However, the problems all have the same words, just different numbers. On Tuesday and Wednesday, the process is repeated with a new problem each day. On Thursday, there is review of the three problems. On Friday is a test. The test has three problems, phrased exactly the same as the Monday, Tuesday, and Wednesday problems, just with different numbers. After the test, they can forget everything they’ve done that week.
The next week the process repeats with three new problems.