Monday, December 29, 2014

Favorite comments of '14, cont: R. Craigen

On Conversations on the Rifle Range 6: Grant’s Tomb and the Benefits of Boredom

R. Craigen said...
Concerning squaring and "rooting" as inverse functions, it is critical that this sort of thing be familiar to students in some form before they begin calculus because much in that course hinges upon elaborations upon that idea, or complex instances of it.

One needn't have a sophisticated understanding of inverse functions, but an operational comfort with them. Like that of the student Matt here, who thinks about the radical sign disappearing when you square.

That's about the level I need calculus students to understand things when they enter my class. What is a log function? I want them to understand that ln (e^x) = x and e^(ln x) = x when defined. From here we can work another useful version: y = ln x is equivalent to x = e^y, and to get comfortable switching between the two for convenience. For all of this it is best that they arrive in my class with a notion of inverse functions already intact. Unfortunately, few do, so I have to go to first principles with them -- and it's just too much to take in all at once. Things should be done in order, and students need time to master basics before moving up a rung on the ladder of abstraction.

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