Teaching mathematics bass-ackwards
I have now taught the "master method" for solving recurrences twice. The first time I managed to muddle through and present the gist of the idea without screwing up too much. But when I revisited it this semester, I was able to develop a pretty complete intuition for why it works and what is going on.
And to realize that one of the world's leading algorithm books presents this in a completely backwards fashion. The student is presented with page after page of formalism, and left to work out what the hell is going on on their own.
To present it properly, show how historically the theorem developed out of solving recurrences with recursion trees. Solve a few from each type of case (three) the master method contains. Lead the students to see that the trees keep falling into one of these three categories. Show how which category they fall into depends on the values of a, b, and x in the equation:
T(n) = aT(n / b) + nx
Then suggest, "Hey, perhaps we can state a rule that shows how the closed form solution to the recurrence depends upon a, b, and x. Then we could just solve recurrences using that rule!" Have the students try formulating the rule themselves if you have time.
Then show them the master method.
At this point, most of them will grasp what is going on well enough that the rule will make good sense to them, and they should be able to use the master method fairly easily.
The backwards approach is very typical of mathematical textbooks. It is as though they're teaching students at Hogwarts, who will just have to accept whatever incantation the professor feeds them.
A friend of mine with a PhD in mathematics explains this by saying, "Many mathematicians are sadists."
And to realize that one of the world's leading algorithm books presents this in a completely backwards fashion. The student is presented with page after page of formalism, and left to work out what the hell is going on on their own.
To present it properly, show how historically the theorem developed out of solving recurrences with recursion trees. Solve a few from each type of case (three) the master method contains. Lead the students to see that the trees keep falling into one of these three categories. Show how which category they fall into depends on the values of a, b, and x in the equation:
T(n) = aT(n / b) + nx
Then suggest, "Hey, perhaps we can state a rule that shows how the closed form solution to the recurrence depends upon a, b, and x. Then we could just solve recurrences using that rule!" Have the students try formulating the rule themselves if you have time.
Then show them the master method.
At this point, most of them will grasp what is going on well enough that the rule will make good sense to them, and they should be able to use the master method fairly easily.
The backwards approach is very typical of mathematical textbooks. It is as though they're teaching students at Hogwarts, who will just have to accept whatever incantation the professor feeds them.
A friend of mine with a PhD in mathematics explains this by saying, "Many mathematicians are sadists."
Similar: I am currently a measurement engineer, with a lot of specialized knowledge in flow measurement. I can explain how Venturis work to people and write down most of the equations from memory.
ReplyDeleteIn college, I did terribly in the fluids class where these flow equations were taught. The classes were taught, not by engineers, but by mathematicians. Most of the class was spent memorizing the derivations of useful equations from less useful ones. e.g. a test question might be "derive Bernoulli's law from the Navier-Stokes equations."
Not only is this backwards (Bernoulli was dead before Navier or Stokes was born,) the Navier-Stokes equations are utterly useless for practical engineering, unless you're creating a CFD toolset. The derivation is difficult, probably took a year of someone's life the first time it was done, but we can memorize and regurgitate the steps for a test.
Yes, I think a lot of this derives from what you have pointed out many times before -- the dominance of the Analytical Thought Police.
ReplyDeleteAnalytical thought is great, but it isn't everything. Just because it often does really well at things like justifying, 'proving,' and defending (some) beliefs, it isn't the only sort of thought there is. It generally isn't the way most new knowledge is initially discovered (intuitively) or mastered (through deep concrete familiarity, as Andy describes above).
To use the methods most suited for 'proving,' and justifying for teaching is just as you say - backwards. The classroom isn't supposed to be a courtroom.
This is common in mathematics. Instead of showing the path that led to the idea that led to the result you present the most elegant, minimal final form of the proof. My advisor decades ago commented that papers in computer science seemed to be written to be understood, unlike math papers.
ReplyDeleteHey Andy,
ReplyDeleteAre there any good books on fluid dynamics? Ie not one with Navier Stokes and derivations therefrom, but something that actually explains the concepts?
Ken,
DeleteI learned most of the stuff I use from papers (ASME, I think) published in the 1960s through 1980s. There's one on subsonic obstruction meters, and one on critical flow Venturis.
Ok, Ken, this is not fluid dynamics in general, but more geared towards measurement, which is what I do. The best general reference I have for that is a copy of:
DeleteFluid Meters
Their Theory and Application
Report of the ASME Research Committee on Fluid Meters
Sixth Edition
1971
Edited by Howard S. Bean
I'm sure there is a more recent version.
Funny that I'm using a reference book from the year I was born. My colleague gave me this when he retired a couple of years ago.
Delete