Transcript static slide - Kenyon College

What is the Definition of Definition?
And other mathematical cultural conundrums
2010-2011 Project NExT Workshop
Mathfest, 2010
Pittsburgh, PA
Carol S. Schumacher
[email protected]
Kenyon College
My teaching improved a lot when I stopped
thinking so much about teaching and started
thinking more about learning.
Teacher as Amateur
Cognitive Scientist
How do we get our
students to think and
behave like
mathematicians?
Teacher as Amateur
Cognitive Scientist
• Getting into our students’ heads.
– How do they learn?
– And (thinking cognitively) what do they need to
learn?
Teacher as Amateur
Cognitive Scientist
Getting into our own heads:
How do we operate as mathematicians?
A great deal of 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.
These are different skills that have to be learned.
A great deal of versatility is required....
And we aren’t even talking
about proving theorems yet!
Let’s try an experiment . . .
• Q: Who is non-orientable and lives in the ocean?
A: Möbius Dick
• Q: Why is the contour integral around Western
Europe zero?
A: Because all the Poles are in Eastern Europe!
• Q: When did Bourbaki stop writing books?
A: When they found out Serge Lang was just one
person.
Mathematics is a Culture
Chasm?
What Chasm?
???
Culture is, by its very nature,
completely unconscious
Cultural Elements
• We hold presuppositions and assumptions that
are unlikely to be shared by a student who is
new to mathematical culture.
• We have skills and practices that make it easier
to function in our mathematical culture.
• We know where to focus of our attention and
what can be safely ignored.
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 bijection 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=.
???
“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, they will readily agree that it is
“obvious” that the sequence 1, 0, 1, 0, 1, 0, . . . fails to
converge.
We must be sensitive to some students’ (natural)
reaction that it is a waste of time to put any work into
proving such a thing.
The Purpose of Proof
Our students (and most of the rest of the world!)
think that the sole purpose of proof is to establish
the truth of something.
I Stipulate Two Things
• First: people don’t begin by proving deep theorems.
They have to start by proving straightforward facts.
•Second: this is a sort of ‘test’ of the definition. It is so
fundamental, that if the definition did not allow us to
prove it, we would have to change the definition.
That’s Obvious, too!
If I give my students the field axioms, and 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.”
The Purpose of Proof
Sometimes proofs help us understand connections
between mathematical ideas. If our students see this
they have taken a cultural step toward becoming
mathematicians.
What? . . . Where?
Our
students!
There is a lot going on.
Most of our students are
completely overwhelmed.
Us
Karen came to my office one day….
• She was stuck on a proof that
required only a simple application
of a definition.
• I asked Karen to read the definition
aloud.
• Then I asked if she saw any
connections.
• She immediately saw how to prove
the theorem.
What’s the problem?
Charlie came by later. . .
• His problem was similar to Karen’s.
• But just looking at the definition
didn’t help Charlie as it did Karen.
• He didn’t understand what the
definition was saying, and he had no
strategies for improving the situation.
What to do?
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 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.
Sorting out the Issues
Vector Subspaces
scalar
multiplication
. . . closure under
and closure under
vector addition
...
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.
Partitions
Equivalence
Relations
Are we directing our students’ attention in the
wrong direction?
The usual practice is to
define an equivalence
relation first and only then to
talk about partitions.
Chasm?
What Chasm?
???
Collection
of subsets
of A.
Relation
on
A
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.
Exploratory Exercises
Suppose that f : A B and g: B C are functions. For each of the
following statements decide whether the statement is true (if so,
give a proof) or false (if so, give a counterexample). In the cases
where the statement is false, decide what additional hypothesis
will make the conclusion hold:
1. If g
f is one-to-one, then f is one-to-one.
2. If g
f is one-to-one, then g is one-to-one.
3. If g f is onto, then f is onto.
4. If g
f is onto, then g is onto.
Discovering Trees
Consider what happens when you remove edges from a
connected graph (making sure it stays connected).
Your group’s task is to look at example graphs and
remove edges until you have a graph
Group A: “with no circuits.”
Group B: “ that is minimal in the sense that if you remove
any more edges you disconnect the graph.”
Group C: “in which there is a unique simple chain
connecting every pair of vertices.”
Can it always be done? What happens if you take the same
graph and remove edges in a different order?
.
Impasse!
What happens when a student gets stuck?
What happens when everyone gets stuck?
How do we avoid
THE IMPERMISSIBLE SHORTCUT?
Breaking the Impasse
In beginning real analysis, we typically begin with
sequence convergence:
Definition: an  L means that for every  > 0,
there exists N 
such that for all n > N, d(an , L) < .
Don’t just stand there!
Do something.
•
•
•
•
an  L means that for all  > 0 there exists n  ℕ such
that d(an , L) <  .
an  L means that for all  > 0 there exists N  ℕ such
that for some n > N, d(an , L) <  .
an  L means that for all N  ℕ, there exists  > 0
such that for all n > N, d(an , L) <  .
an  L means that for all N  ℕ and for all  > 0, there
exists n > N such that d(an , L) <  .
Students are asked to think of these as “alternatives” to the
definition. 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.
Pre-empting the Impasse
Teach them to construct examples. If necessary throw
the right example(s) in their way.
Look at an enlightening special case before considering
a more general situation.
When you introduce a tricky new concept, give them
easy problems to solve, so they develop intuition for the
definition/new concept.
Even if they are
Separate the elements.
not particularly
significant!
But all this begs an important question.
Do we want to pre-empt the Impasse?
Precipitating the Impasse
Impasse as tool
Why precipitate the impasse?
The impasse generates questions!
Students care about the answers to their own
questions much more than they care about the
answers to your questions!
When the answers come, they are answers to
questions the student has actually asked.
Precipitating the Impasse
Impasse as tool
Why precipitate the impasse?
The impasse generates questions!
At least as importantly, when students generate their own
questions, they understand the import of the questions.
The intellectual apparatus for understanding
important issues is built in struggling with them.
. . . the theory of 10,000 hours: The idea is that it takes 10,000
hours to get really good at anything, whether it is playing tennis
or playing the violin or writing journalism.
I’m actually a big believer in that idea, because it underlines the
way I think we learn, by subconsciously absorbing situations in
our heads and melding them, again, below the level of awareness,
into templates of reality.
At about 4 p.m. yesterday, I was working on an entirely different
column when it struck me somehow that it was a total
embarrassment. So I switched gears and wrote the one I
published. I have no idea why I thought the first one was so bad
— I was too close to it to have an objective view. But I reread it
today and I was right. It was garbage. I’m not sure I would have
had that instinctive sense yesterday if I hadn’t been struggling at
this line of work for a while.
Written by David Brooks
In one of his NYTimes
“conversations with Gail
Collins
Morale: “Healthy frustration” vs. “cancerous
frustration”
•Give frequent encouragement.
•Firmly convey the impression that you know they can do it.
•Students need the habit and expectation of success--“productive challenges.”
•Encouragement must be reality based: (e.g. looking back
at past successes and accomplishments)
•Know your students as individuals.
•Build trust between yourself and the students and between
the students.