Fundamental Rule of Exponents

For a basic algebra class, given the rudimentary order-of-operations that looks like this:
  1. Parentheses
  2. Exponents & Radicals
  3. Multiplication & Division
  4. Addition & Subtraction
 Then we have:

The Fundamental Rule of Exponents: Operations on same-base powers shift one place down in the order of operations.

(1) Exponents will multiply powers, i.e., (am)n = am∙n. Example: (x6)2 = x12.
(2) Radicals will divide powers, i.e., n√am = am/n. Example: 3√x15 = x5.
(3) Multiplying will add powers, i.e., am∙an = am+n. Example: x4∙x7 = x11.
(4) Division will subtract powers, i.e., am/an = am−n. Example: x9/x2 = x7.

We've discussed this before, but I just recently decided to apply the name shown here to the pattern. It doesn't show up on a Google search yet, so I think it's fair-game to do so. Cheers!


Against Inverted Classrooms

The "inverted classroom" (or as Wikipedia calls it, "Flip Teaching") is the idea that lectures can be watched (say, by online video) prior to class meetings, and then classroom time dedicated to questions and problem sets with the teacher's coaching and assistance in trouble spots. Obviously, it's the reverse of the standard math-class process of lecturing in class and then homework after.

When I heard about this a few months ago, I was really excited and initially felt like it was a great idea. Here's why: I agree that I've independently found the moment where I can help an individual student, working on a specific problem, and identify-fix-correct-clarify the exact location where they're making a mistake, to be the most satisfying and productive use of time for both student and teacher. I try hard to get as much time for those moments of practice and error-catching in my classes as I can. So it sounded like fully devoting class time to that process would be ideal. For my summer courses, I thought very deeply how much I could go in that direction.

So at the end of that analysis, I am now very skeptical that this technique will have legs and work long-term for mathematics education. Here are some of the reasons why I say that:

(1) It could have been done at any earlier time with books, but wasn't. It appears that online video lectures and published books are pretty obviously equivalent (in fact, I think books have the advantage in any way I can think to compare, especially for math). While other disciplines have commonly run classes with assigned reading beforehand, and critical discussion in-class (e.g., literature, history, law, etc.), math seems pretty ironclad in having avoided that in any place or time that I can detect. (Can you think of any counter-examples?) This suggests that there's something about math that demands live presentations in the first place.

(2) Questions still need to be asked during lecture presentations. One reason why the initial presentation has to be live: expert feedback isn't just necessary during individual problem sets, it's also necessary to clear up the initial presentation itself. Almost certainly some different level of detail will have to be presented for different audiences, and there needs to be live back-and-forth questioning in order for that initial lecture to be valuable in the first place -- and this value accrues with interest the more people are watching/present at the time. If a student simply doesn't have access to a particular necessary detail during a recorded video, no amount of rewinding or re-watching will conjure it up, and that time will be for naught. (It's been argued in the past that a fully hyper-linked presentation to arbitrary depth of detail could satisfy this need, but in practice that seems to have failed -- arbitrarily large amount of work on the part of the writer, and similar great workload and discipline needed by the student to follow all the needed links.)

(3) Nonstandard class times are particularly ill-suited for it. The inverted classroom might work a bit better for regular one-hour, once-a-day classes (students need to catch up on one-hour chunks at a time, etc.) But like a lot of methodologies it breaks down in other cases. For example, take my summer statistics courses: they run for 6 weeks, meeting twice a week for 3 hours at a time (other classes might be 4 hour sessions at a time). There needs to be an in-class test about every 3rd class session, which will last about an hour -- note that between one-half and two-thirds of the same meeting will be spent on some other lecture topic. Experience (if not common-sense) shows that students will not have presence of mind for any new topic prior to the test, either in-class or before. So I absolutely must resign myself to presenting new information myself after the test on test days, which themselves are 1/3 of the class meetings. Work out this staggered effect (including the very first class), and I saw that there's essentially no way to make "flip teaching" work in my evening summer courses.

In conclusion: It seems like the general student of any time period hasn't been able to learn math on their own (either from a book or a video) -- that's why they're in a classroom in the first place. It would be nice if there was an expert in the discipline with them at all times, during both initial presentation and homework. But since that's infeasible, the best we can do is some mix of presentation and troubleshooting together in the limited classroom time.