Exercising With Learnstream.page

I want to analyze (once again) the idea of using the underlying SRS-ed components with exercises.

First, there is a danger of not even training the right components: Because many exercises are multiple-choice, I may have used something other than applying each of the concepts to figure out which was the right answer. More generally, some problems have more than one solution (e.g. I may not have known "When is energy conserved?" if I took a kinematics approach.)

Let's assume though that the components are actually perfectly assigned. The question now is, is the SRS sufficient, or should there be more to it? Where I'm concerned is where something may be considered a "skill" rather than something to be memorized. Take for example, the component "What is the derivative of f(g(x))?" (i.e. chain rule). Pretty much any course is going to drill this in a wide variety of applications, whereas our approach is going to be giving one problem now and then repeat every once in a while. Which of the following conclusions would you bet on? (Assume the student is currently "strong" on any obvious other components for relevant problems, like the derivative for polynomials and trig functions)

1. SRS is better: there's actually no measurable benefit to doing the drilling.
2. Drilling is better: Learning something like the chain rule is not memorizing a fact but a different "rewiring" process that takes more training.
3. SRS is better BUT there should actually be components for different applications of the chain rule. (Explain.)
4. Looking at the chain rule in isolation is worthless; the cognitive load of everything else going on is too significant. 
5. Other (Explain.)

So, I just managed to screw up one of the calculus problems: Minimum on [-1,2] of y = 4x^3 - 6x^2 - 9x

I *know* the rules, but I still screwed it up. (And right now I can't even figure out how I screwed it up---maybe looking at the second derivative when I should have been looking at the first.) It shows there is more to even a simple problem like this than just applying the (easily defined) rules. Algebra manipulations, keeping track of variables, keeping track of which derivative I've just computed, keeping track of which things I'm writing or thinking about are what, not get confused with other things that are going on in my head. So basically I'm favoring #4 right now. So once you break it up, you have to look at the individual pieces and how they fit together. I think the chain rule itself is SRS-able, but I'm not sure how well we can actually understand the results when it gets combined with other things. (I.e. not just reproducing it in a pure form.) Like even if we get to the point where you can apply the components to solve the problem robotically with zero ambiguity, the students will never be robots. Does that mean we should try to consider metacognitive things too, or do we just teach the facts as best we can and assume it will be good enough if the students are being sufficiently focused and organized with their thoughts?

Neal:

I'm tending to agree with you on #4, ryan.

To take an example from Big E&M, often times a problem is done by solving laplace's (or poisson's) equation in some volume, given certain boundary conditions at the surface, and then computing the electric field by differentiating the potential. Conceptually, it is all very simple and very elegant-- the boundary conditions are well defined, and it's one of the simpler PDE's you could run across. However, actually getting the right answer is not trivial-- there are too many steps involved, and although it is clear what you're supposed to do conceptually, the mathematical skills are hard to do properly. In other words, the concept is simple, but the cognitive load to get an answer is huge (especially working in 3-dimensions, or in spherical coordinates, or something like that).

So maybe we need a middle-ground approach...? We SRS the "big ideas" and make sure the students are fluent with them/fluent with what tools they need (ie, given a problem, the SRS-able task is "what approach would you use to solve this?" or maybe going so far as to define relevant parameters/variables, etc.) Then, the only way to get proficient at the actual process is to drill the problems.

On a semi-related note, I was thinking about how SRS is set up in contrast to a traditional class. If you think about the number of problems that are actually assigned, it's not like students are getting a whole lot of drilling in the HW. But, they also aren't getting the advantage of SRS (unless they are overly-diligent in reviewing past material). For the mudd course, I think they assign about 75-80 problems over the course of a semester (about 5 a week, give or take). What does it mean, then, that the problem-to-component ratio isn't very high? Some of the problems do build on past material, but it's still a far cry from drilling the chain rule 100 times, either.  Usually in AE I've told people before tests that if they can go back through and do every recitation problem, they should be fine on the exam. But students should be doing those recitation problems as they are forgetting the material, not the night before the exam when it may be too late.

So I think there is an advantage to SRS, but I'm betting our algorithm is too simplistic. But, on the flip side, it is surely better than the status quo, I believe.