Transcript L - Kenyon College

Teaching Students to Prove Theorems
Session for 2007-2008
Project NExT Fellows
Presenter:
Carol S. Schumacher
[email protected]
Kenyon College
and
The Legacy of R. L. Moore Project
Mathfest, 2007
San Jose, CA
Some Things I Learned
the Hard Way
“That’s obvious.”
To a mathematician it means “this can easily be
deduced from previously established facts.”
Many of my students will say that something they
already “know” is “obvious.”
For instance, if I give them the field axioms, and then
ask them to prove that  x  F, 0  x  0,
they are very likely to wonder why I am asking them
to prove this, since it is “obvious.”
What is a definition?
To a mathematician, it is the tool that is used to make
an intuitive idea subject to rigorous analysis.
To anyone else in the world, including most of your
students, it is a phrase or sentence that is used to help
understand what a word means.
For every >0, there
exists a >0 such that if...
?
?
?
What does it mean to say
that two partially ordered
sets are order isomorphic?
The student’s first instinct is not going
to be to say that there exists a function
between them that preserves order!
As if this were not bad enough. . .
. . . we mathematicians sometimes do some very weird
things with definitions.
Definition: Let  be a collection of non-empty sets.
We say that the elements of  are pairwise disjoint if
given A, B in , either A  B= or A = B.
WHY NOT.... ???
Definition: Let  be a collection of non-empty sets.
We say that the elements of  are pairwise disjoint if
given any two distinct elements A, B in , A  B=.
Great Versatility is Required
•We have to be able to take an intuitive statement and write
it in precise mathematical terms.
•Conversely, we have to be able to take a (sometimes
abstruse) mathematical statement and “reconstruct” the
intuitive idea that it is trying to capture.
•We have to be able to take a definition and see how it
applies to an example or the hypothesis of a theorem we are
trying to prove.
•We have to be able to take an abstract definition and use it
to construct concrete examples.
And these are different skills that have to be learned.
Cultural Elements
• We have skills and practices that make it
easier to function in our mathematical
culture.
• We hold presuppositions and assumptions
that may not be shared by someone new to
mathematical culture.
• We know where to focus of our attention
and what can be safely ignored.
What? . . . Where?
Total Immersion
It is not what I do, but what
happens to them that is
important.
Whenever possible,
I substitute something that the students do
for something that I do.
There are many ways to do this
Small scale stuff to enhance lectures.
•Activities that help students become familiar with a
with a new definition.
•Students work on proving theorems in small groups
while you circulate around the classroom helping out.
•Have students present results of their work to each
other.
Projects and Activities instead of lectures.
•GAP projects in abstract algebra: conjugating
permutations. (Google: Judy Holdener Gap)
Sorting out the Issues
Equivalence Relations
We want our students to
understand the duality
between partitions and
equivalence relations.
We may want them to
prove, say, that every
equivalence relation
naturally leads to a
partitioning of the set,
and vice versa.
Equivalence
Relations
Partitions
The usual practice is to define an equivalence relation
first and only then to talk about partitions.
Are we directing our students’ attention in the wrong direction?
Furthermore, there is a lot
going on. Many students
are overwhelmed.
They don’t know how to
focus their attention on
one piece at a time.
Collection
of subsets
of A.
Relation
on
A
• an  L means that   > 0  n  
d(an , L) < .
• an  L means that   > 0  N   for
some n > N, d(an , L) <  .
• an  L means that  N  ,   > 0  
n > N, d(an , L) <  .
• an  L means that  N   and   > 0 
n > N  d(an , L) <  .
Students are asked to think of these as “alternatives” to the definition of sequence
convergence. Then they are challenged to come up with examples of real number
sequences and limits that satisfy the “alternate” definitions but for which an  L is false. I
usually have the students work on this exercise in class, perhaps with a partner.
Scenario 1: You are teaching a real analysis class and have
just defined continuity. Your students have been assigned the
following problem:
Problem: K is a fixed real number, x is a fixed
element of the metric space X and f: X R is
a continuous function. Prove that if f(x) > K,
then there exists an open ball about x such that f
maps every element of the open ball to some
number greater than K.
One of your students comes into your office saying that he has
"tried everything" but cannot make any headway on this
problem. When you ask him what exactly he has tried, he
simply reiterates that he has tried "everything." What do you
do?
Scenario 2: You have just defined subspace (of a vector
space) in your linear algebra class:
Definition: Let V be a vector space. A
subset S of V is called a subspace of V if S
is closed under vector addition and scalar
multiplication.
The obvious thing to do is to try to see what the definition
means in 2 and 3 . You could show your students, but you
would rather let them play with the definition and discover
the ideas themselves. Design a class activity that will help
the students classify the linear subspaces of 2 and 3
dimensional Euclidean space. (You might think about
"separating out the distinct issues.”)
scalar
multiplication
. . . closure under
and closure under
vector addition
...
Scenario 3: Your students are studying some basic set
theory. They have already proved De Morgan's laws for two
sets. (And they really didn't have too much trouble with
them.) You now want to generalize the proof to an arbitrary
collection of sets. That is.....
C
A  
A 
  
 
C
and
C
A  
A 
  
 
C
The argument is the same, but your students are really
having trouble. What's at the root of the problem? What
should you do?
Scenario 4: A very good student walks into your office.
She has been asked to prove that the function
f ( x) x
1 x
is one to one on the interval (-1,). She says that she has
tried, but can't do the problem. This baffles you because
you know that just the other day she gave a lovely
presentation in class showing that the composition of two
one-to-one functions is one-to-one. What is going on?
What should you do?
Scenario 5: Your students are studying partially ordered
sets. You have just introduced the following definitions:
Definitions: Let (A, ) be a partially ordered set.
Let x be an element of A. We say that x is a
maximal element of A if there is no y in A such
that y x. We say that x is the greatest element
of A if x y for all y in A.
Anecdotal evidence suggests that about 71.8% of students
think these definitions say the same thing. (Why do you think
this is?) Design a class activity that will help the students
differentiate between the two concepts. While you are at it,
build in a way for them to see why we use “a” when defining
maximal elements and “the” when defining greatest elements.