Thoughts on teaching rigor / proofs
Here I discuss *both*:
- how to think about proofs and rigorous thinking
- how to teach them
(i.e., it's both domain and pedagogy)
key part of math
remember, mathematicians know how to *do* rigor,
but *teaching* it is a separate issue
A key issue is:
you can't learn method without content
(you can't learn how to do math without doing actual math)
...so you need math content.
Basic discrete algebra (group and number theory) is the best content
because it is concrete (you can write down the finite sets)
and familiarly computational.
\section{Standard forms of proof}
just as in writing there are various forms,
there are some standard forms of proof
(and remember, work with them *contextually*: give real, meaty examples)
EGs:
- deduction / direct proof (most basic):
to show A -> B, show A -> C_1 -> C_2 -> \dots -> C_n -> B
- contraposition: to show A -> B,
show that ~B -> ~A
- contradiction (but don't overuse):
To show A -> B, show that A \& ~B implies a contradiction
(must use *both* A and ~B; often to construct an impossible object,
like a non-abelian group of order p^2)
- equivalent definitions / characterizations
Given several different definitions, show that they define the same
object.
[NB: to be formal and linear, you would select one as *the*
definition, and some people prefer this, but I prefer "equivalent".]
- counter-example
$\not \forall x A(x)$ because $\exist x_0 \not A(x)$
($\not \forall = \exists \not$)
Some kinds of setups/problems (proofs):
- given just right assumptions to prove result
(if you haven't used all assumptions, your proof is wrong)
These
\section{Active reading}
Some exercises:
- generalize a result
given a prop and proof
that doesn't need the full assumptions,
generalize it (weaken assumptions so they're "just right")
[this helps clarify a *lot* of what happens in analysis]
"put it in its natural general setting"
- find error in a proof
- rewrite a contradiction to use contraposition
- summarize / sketch a proof
Given a long, precise proof,
give a sketch, highlighting main ideas
(for grading this, need to know what main ideas are!)
\section{filling in / how people read and write proofs}
discuss how pros communicate:
- you say the key points and let the listener fill in the details
Why?
- thinking is faster than talking
- (write better)
it's hard for a listener to grasp 20 steps,
[they don't have short-term memory]
but can grasp 3 steps and see how they connect
This *does not work* for novices:
- for them, *everything* is new:
everything seems a key detail
They haven't seen the standard steps (the \epsilon/3 proofs)
I'd suggest:
- introduce standard steps in smaller settings,
so they can be grasped
(and do 'em several times, so they get used to them)
- giving an *outline*,
then filling in the steps