[Edu-sig] explaining functions [Possibly OT]

[Kirby Urner]

Brian van den Broek:
> I'd agree with you if experience hadn't taught me otherwise. I do see
> the CS and math notions of functions are tightly related (with the
> difference that math functions don't have side effects).

However, I think mathematicians should accept the CS "side effects" as just
more of the function's result, wherever and however it shows up.

One thing I like about computer programming is it quickly gets you into
operating with inputs/outputs other than numbers, strings for example.  The
early math exposure kids get is too exclusively number-focused IMO.  Books
like 'Godel Escher Bach' are a good antidote.

My suggestion that a factory machine which twists lids onto milk bottles or
the like, is a good model of a function, would be designed to break this
fixation on numbers, always numbers.

In a hybrid techie-math class, we'd wire sensors to chips, so that inputs
were things like temperature, outputs various GUI gizmos -- more like the
cockpit of an airplane (instrumentation readings = output, environmental
conditions, like altitude, wind speed = input).

The mathematicians' *notation* 'f : x -> y' or whatever they use, may well
be about turtles moving, toasters popping, cranes swiveling, cells dividing.
The abstraction is about taking inputs to outputs, stimuli to responses,
causes to effects or whatever.

In the case of random.randint(5), it needs to be explained that just as
outputs may be other than what's explicitly returned (so-called
side-effects), inputs may be coming from places other than the argument list
-- in this case, there's a sequence of pseudo-random numbers being computed
algorithmically, with state saved between calls (like in a generator).
 
I think lots of examples of simple causes -> effects drawn from ordinary
life helps phase in the new function semantics in a way that's very glued to
everyday experience.

In the case of propositional calculus in particular, I favor using logic
gates at some point in the discussion.  Little diagrams of AND OR NAND NOR
XOR.  http://encyclozine.com/Connective

You can use these to assemble the behavior of p->q i.e. p,q are inputs to a
logic gate setup, and the value of p->q is the output.

p  q  p->q
T  T    T
T  F    F
F  T    T
F  F    T

In Python, you can use Booleans, e.g. the above is equivalent to 

  >>> def ifthen(p,q):
	    return (not p) or q

i.e. (~p | q) <-> (p -> q)

  >>> combos = ((True,True),(True,False),(False,True),(False,False))

  >>> for p,q in combos:  
           print "Inputs:  %s\t%s\tOutput: %s" % (p, q, ifthen(p,q) )

  Inputs:  True	  True	Output: True
  Inputs:  True	  False	Output: False
  Inputs:  False	  True	Output: True
  Inputs:  False	  False	Output: True


Kirby