Transcript Document 7262334

What we say / what they hear
Culture shock in the classroom
Mathematical Culture
We hold
presuppositions and assumptions
that may not be shared by someone new to
mathematical culture.
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
our students, it is a phrase or sentence that is
used to help understand what a word means.
What does it mean to say
that two partially ordered
sets are order isomorphic?
Most students’ first instinct is not going
to be to say that there exists an orderpreserving bijection between them!
For every  > 0, there
exists a  > 0 such that if...
?
?
?
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=.
???
Mathematical Culture
We know where to
focus our attention
for maximum benefit
and we know
what can be safely ignored.
What? . . . Where?
Helping our students focus
Example: Equivalence Relations
We want our students to
understand the duality
between partitions and
equivalence relations.
We want them to prove
that every equivalence
relation naturally leads
to a partitioning of the
set, and vice versa.
Partitions
Equivalence
Relations
Our
students!
There is a lot going on in
this theorem.
Many of our students are
completely overwhelmed.
Us
Sorting out the Issues
Partition of
A
Equivalence
Relation on
A
Every partition of a set A
generates an equivalence
relation on A.
&
Every equivalence on A
relation generates a
partition of A.
Sorting out the Issues
Collection
of subsets
of A.
Relation
on
A
Every collection of
subsets of A generates a
relation on A.
&
Every relation on A
generates a collection of
subsets of A.
And. . .It’s not just about logical
connections
The usual practice is to define an equivalence relation
first and only then to talk about partitions.
Motivation for
Are we directing our
students’
attention in the wrong direction?
defining
equivalence
relation?
Mathematical Culture
We have
skills and practices
that make it easier to function in our
mathematical culture.
A great deal of versatility is required....
•We have to be able to take an intuitive statement and write
it in precise mathematical terms.
And none of
•Conversely, we have to be able to take a (sometimes
these are
even
abstruse) mathematical statement
and “reconstruct”
the
intuitive idea that it is trying
to capture.
talking
about
•We have to be able to take a definition
provingand see how it
applies to an example or the hypothesis of a theorem we are
theorems!
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.
First Day Out
• Ed Burger
• Mike Starbird
• Carol Schumacher--- “the thought experiment”
What are the goals of each instructor?
Are there common elements/goals?
The impermissible
shortcut
Logical Structures and Proof
• Proving a statement that is written in the form
“If A, then B.”
• Disproving a statement that is written in the
form “If A, then B.”
• Existence and Uniqueness theorems
Beyond
counterexamples:
• Other useful
ideas:
e.g. “If A, then B or C.”
Negating implications!
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 define the convergence of
a sequence:
Given any tolerance
Definition: an  L
there exists N 
there is
some fixed
position
means that for every  > 0,
such that for all n > N, d(an , L) <  .
beyond which
an is within that
tolerance of L
Don’t just stand there!
Do something.
• an  L
means that   > 0  n  ℕ  d(an , L) <  .
• an  L
means that   > 0  N  ℕ  for some
n > N, d(an , L) <  .
• an  L
means that  N  ℕ,   > 0   n > N,
d(an , L) <  .
• an  L
means that  N  ℕ and   > 0,  n > N 
d(an , L) <  .
Make it “real”
Pre-empting the Impasse
Teach them to construct examples. If necessary throw
the right example(s) in their way.
even if they are
Look at an enlightening special case before considering
not
particularly
a more general situation.
significant!
When you introduce a tricky new concept, give them
easy theorems to prove, so they develop intuition for the
definition/new concept.
Separate the elements.
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.
More importantly, students understand the import of their
own questions.
The intellectual apparatus for understanding important
issues is built in struggling with them.
What they say/what we hear
Listening to our students
(Sometimes) hearing what they mean
instead of what they say
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.
And these are different skills that have to be learned.
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 has Karen.
• He didn’t understand what the
definition was saying, and he had no
strategies for improving the situation.
What to do?
“That’s obvious”
To a mathematician this 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
for all x  F , x  0  0
they are very likely to wonder why I am asking them to
prove this, since it is “obvious.”
I find that it is helpful to stipulate two things:
• First: students don’t begin by proving the deep theorems.
They have to start by proving straightforward facts.
•Second: 0x = 0 is a sort of ‘test’ for the axioms. It is so
fundamental, that if the axioms did not allows us to prove
it, we would have to add it to our list of assumptions.
Then we make an amazing observation: The field
axioms discuss only additive properties of 0, but because
addition and multiplication are assumed to interact in a
certain way (distributive property), this multiplicative
property of 0 is obtained “for free.”
0x = 0 holds because nothing else is possible.
Our students (along with the rest of the world) think that
the sole purpose of proof is to establish something as
true. And while this is the case, sometimes proofs can
help us understand deep connections between
mathematical ideas.
If our students see this they have taken a cultural step
toward becoming mathematicians.
In what sense is that
teaching?
Basically, Susie doesn’t get it.
Susie is a pretty good
student.
But “work on these
problems and we will
talk about them next
time” is a little
nebulous for her.
•She thinks student presentations are a
waste of everybody’s time.
•(She may secretly believe that I don’t
lecture because I’m lazy or unprepared.)
•She is conditioned to respond to what I
say and she doesn’t believe that her
work starts (or even can start) before
she understands the material.
•She is wondering when I will get
around to actually teaching her
something!
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.
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 ℝ2 and ℝ3 . 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.