Psuedoteaching with a Purpose?

Frank Noschese over at Action-Reaction has been stirring up trouble again - this time, it's about the neologism "pseudoteaching."  This follows Dan Meyer's huge series on "psuedocontext," which is about those obviously phony textbook-type problems where the material is shoehorned dubiously into a faux-"real world" situation.

Psuedoteaching, then, is teaching that looks great, but doesn't really help the students learn.  It frequently even looks great to the students - they think that they're learning a lot, but... they aren't.

One of the big targets has been flashy derivations in class, especially as performed by the teacher.

The main thrust of the collective argument is good, but I think that I might just disagree on some of the finer points here.  To wit, a recent lesson:

Content: Elastic Collisions

Background: We've studied and rocked inelastic collisions, demonstrating how conservation of momentum leads to the inescapable conclusion that kinetic energy is lost during these collisions.  The less elastic the collision is, the more KE is lost.  When the objects stick together, we've lost as much as we're gonna lose.  I save the center of mass frame interpretation of this (as a super-easy facepalm moment) until next year (AP C: Mechanics).  I used to do CM frame in the first-year course, but, well, you know how it goes.

Setup: The natural question arises: "What if we don't lose any energy in the collsion?"  Some dialogue: "How could that happen?", "How would the motion be different than if we did lose some KE?", etc.  We define an elastic collision to be one in which no KE is lost, so now... we've won - in principle.  Having 2 equations (conservation of KE and conservation of p) with two unknowns (the two final velocities) is victory. ...right?

The Homework: OK, smarty pants: if you have 2 equations and 2 unknowns, go home and solve for the two final velocities.  I'll even cut your work in half.  Look how symmetrical the system of equations is:

If you solve for one of them, just switch all of the subscripts around, and you'll have the other one.

The Walk of Shame:  Well, generally nobody figures out how to do it.  There's squaring of trinomials, ginormous quadratic formula use, and sometimes somebody will take that brute-force approach through to the ugliest answer you ever saw.

The Deus ex Machina:  Here's where somebody's going to cry pseudoteaching.  I set up the system of equations again, but then group all of the terms with the first mass on one side and all of the terms with the second mass on the other side.  Nobody among them would ever think to do this.

I prod: "This is screaming something at you... a big alarm bell is going off!  What does it say?"  Eventually, somebody recognizes the difference of squares, we factor and cancel (they generally haven't thought about canceling factors that look different, though it again makes total sense after they see it), and now we have two linear equations to solve, which we can do:

Thus are born the elastic collision equations:

We can now look back and recognize that the second one is really just the "speed of approach equaling the speed of retreat" phenomenon that we noticed way back at the beginning when they were modeling collisions using air tracks and no knowledge of momentum.  They noticed that was only true for the rubber-band bumpers, and... here's why!

Down to it: Yes, I could've evoked the approach/retreat to get them to generate this on their own.  Since nobody thought of that before, though, why would that be any different than me giving this giant algebra trick hint?  Yes, it's a bit of a magic trick with the algebra.  Yes, I'm the one that did it, while they all went "oooooh."  Will they all remember how to do this derivation in the future?  Absolutely not.  A few will, and in the AP class next year, about half will recall it when we get back to elastic collisions.  These are the psuedoteaching targets on my back.

Total recall and mastery, however, is not necessarily my goal here.  Communicating that not everything involves algebraic grinding, that there's elegance in quirky thinking, that the trip's as important as the journey, etc. is an important thing to emphasize, and I try to hit it a couple of times a year in a dramatic fashion like this.  I want something that's not technically above their heads (that is, showing them some calculus isn't the same thing - I'm looking for something that they have the knowledge and skills to actually do), but demands more creativity or conceptual understanding than they have.  I want them to know that there's more than formulas and drudgery.  Higher level math does not equal even more algorithms.  I say that all the time, but this is a kick in the pants. 

That's the goal for most of them.  I do also want to give those few that can file these tricks away in their heads and recall them later... another trick.  I also gave them the "square both sides" of an equation trick earlier in the year, when deriving one of the kinematic equations:

They derived the rest, either from graphical modeling or algebraically from their previous models.  This one just wouldn't occur to them, even though they see why you can square both sides, they wouldn't have thought of it themselves.

Why do you know so many tricks?  I'm guessing that you didn't learn most of them by inquiry or create them yourself.  I look for binomial approximations because one day in high school, a teacher said: "Hey, do you see this crazy equation?  I can make it simpler if...".  The same goes for the small-angle approximation and a million other tricks.  Even the symmetry idea about switching the subscripts is subtle, powerful, and useful for a kid with a math-a-riffic future.  There's nothing wrong with equipping those that are ready to receive them with these tools.  It's not a majority of the kids, but those kids deserve stuff that will help them out later.

Everyone deserves a little wonder and surprise, algebraically speaking.  I also don't have a problem giving a calculator to a herd of mice, because there will be some sharp rats out there that could use it, even if the others just think that it's shiny.