Bad mental models

I believe (as noted under "teaching") that pointing out -bad- mental
models is essential for teaching a subject, esp. math.

That is, there are some -very common- incorrect mental models;
pointing them out and showing -why- they're wrong is deeply valuable.

Here's some in calculus:
- "next real"
  This shows up many places: students often think there's an
  -immediate successor- in real numbers.

  One place where this shows up is when some students argue that
  1+1/2+1/4+... is -just less- than 2.
  Rephrase this as: so the lim of 1/2,1/4,1/8,... is "just more than 0"?
  ....which is silly.

  There's lots more: lims do -not- preserve > (as the above shows) -- but
  they do preserve >=.

  In decimal, the number:
  0.0...01 (infinitely many 0s) -doesn't make sense-
  (it's less than every positive real, so it's zero)

  More hard-core: there -is- a notion of "hyperreals", where you
  actually have stuff like epsilon=(1,1/2,1/3,...) etc., but
  -that's not what we're studying-
  (viz, hyperreals are an interesting concept, and perhaps -should- be
  taught, but if they're -not- being taught, one should at most point
  'em out (to show that the concept isn't incoherent, but is just -not
  our object of study-))

  Here's the stock example:
  (and do it slowly, pointing out the steps, else it goes by too fast)
  - what's the least positive number?
    (aka, the number right after '0')
    There isn't one!
    (given any positive number, take half of it and there's a smaller)
    This is really about non-closed sets: they needn't have a minimal
    element. So (0,infty) doesn't have a minimal element!
  Pose it as a question:
  "What's the next number after 0?
   What's the -least- positive number?"
  "It must be a rather special number, right?"
  (lots of other ways to see this:
   if there's a next number, then there's a next one after that, etc.:
   so how many steps 'til you reach 1?
   Doesn't make sense! There's infinitely many numbers between 0 and
   1! And between 0 and 1/10,000 -- oops.)

- monotone convergence
  this is discussed elsewhere: too often students' only model of
  converge is -monotone- convergence;
  not all convergence is monotone: I prefer the "black hole" model:
  you eventually approach within epsilon -and never escape-

- "intermediate state"
  Students -persistently- believe in some mythic state between a
  sequence and its limit.
  They say:
  ``$1 + 1/2 + 1/4 + \dots$ gets close to but never reaches $2$''
  Well, -duh-.
  But the -limit- equals 2. That's what a limit is!
  Do you mean that we should define the limit -differently-,
  that we should say the limit of that sequence is
  ``something less than $2$''?

  A limit is a way to get a -real number- out of a -sequence-.
  Do you have any other real number to suggest?
  (this gets back to ``infinitely near numbers'')
  
  Think of it like sum: ``sum'' takes in 2 numbers and gives you 1 out;
  ``inverse'' takes 1 number in and gives you 1 out (unless it's zero)
  ``limit'' takes in infinitely many numbers and gives you 1 out,
    (unless it diverges)

  Maybe you don't think it's valuable...

A key objection is: ``but you -can't- sum infinitely many numbers!''
To which the response is:
1: I just did
2: What you mean (or -should mean-) is:
   "We haven't defined infinite sums. It doesn't follow from finite
   sums."
   So what we're doing is giving -a- way of defining infinite sums.
   For it to be a "good" definition, it should satisfy some properties
   -- EG, it should agree with finite sums on finite sums.
   (which this does) etc. -- these properties are AKA -axioms-.
