Bad statements
- incorrect statements
- statements that are correct but misleading
\section{Incorrect statements}
It's awkward to give *wrong* intuitions or *bad* statements:
- you want to mention 'em b/c students often think them
- but to even mention them gets 'em stuck in peoples' heads
Easiest is if a bad intuition is
when students pick up on some *other* phenomenon,
and conflate it with the one you're talking about.
In that case *naming* this other phenomenon helps keep it separate.
...
For instance, in topology, the terms \emph{open} and \emph{closed}
are confusing, because closed means ``complement is open'', not ``not open'',
as it does in everyday language.
(This conflict between everyday language and mathematical language is quite frequent:
math terminology can be technical (like syzygy) or arbitrary (like functor)
or everyday (like neighborhood); everyday language has the advantage of familiarity,
but the disadvantage of being confusing if it is not precisely analogous.)
``A subset is not a door: it can be open, closed, both or neither.''
...
Counter-examples
An easy case to deal with is when a statement has a good counter-example.
``The limit of rational numbers is *not* necessarily a rational number:
the rational numbers are not a closed subset (of the reals).''
Turn into an exercise:
give a sequence of rational numbers that converges to an irrational number.
...
Often students draw false analogies;
(e.g., cancelling a *summand* in a fraction: (x^2 + 2)/(x + 2) = x^2/x)
...best is to clarify the correct statement that they are misusing,
and
(e.g., non-commutativity of linear operators:
- (similar, familiar setting: composition of functions)
compute composition of x^2 and x+1 both ways
- (basic failures, at every level):
give matrices A,B s.t. AB is defined but BA isn't
give matrices A,B s.t. AB and BA are both defined but are different shapes
give matrices A,B s.t. AB and BA are both defined and the same shape but not equal
)
...
The hard case is when students are simply confused,
and come up with mystic ``rules''.
Rules should "work" iff they reflect correct intuition;
*ask* students to explain their thinking,
and if it's incorrect,
try to come up with a counter-example
(or an example that shows the problem)
***and in future give this example as homework when teaching the topic***
...so that they will not build this incorrect notion into their thinking
(this is pretty Socratic)
Further, *give students all short-cuts and mneumonics*;
if you don't, they'll figure them out themselves, often incorrectly.
*make life easy*, if your goal is to teach a concept
\section{Structure}
\begin{quote}
A good theorem reflects the structure of the phenomenon it describes.
\end{quote}
If you wish to state a result which, while true, does not reflect the structure,
state it as a *corollary* of a good theorem.
[Clarify exactly the statement,
and refine it until it's true.]
Here's a bad statement:
(due to Ani)
\begin{thm}[True but misleading]
An operator $T$ on a finite dimensional vector space space is simple
iff the characteristic poly has no proper factor.
\end{thm}
[check this statement]
This suggests the *incorrect* statement:
\begin{conj}[False]
If the characteristic polynomial factors as $p_T = fg$,
then the $K[T]$-module splits as $V = W \oplus W'$,
(i.e., $T$ preserves a direct sum decomposition)
where
\end{conj}
See, it's confusing simple and indecomposable:
for polynomials (in one variable, over a field), having a factor means factoring,
but not so for modules.