Carol Putting in her 2

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Carol Putting in her 2 Worth

• (And, by the way, . . .
• Thanks, Mike, for having me follow Ed Burger.)

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Some People Have Trouble Understanding What We
Are Up To.
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Caricatures of IBL
No books, no outside sources, just you . . .
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Preconceived Notions
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.
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Sometimes we are not so good at explaining what
we are up to. . .
Thats interesting. I just dont see how its
teaching. ---An award-winning Physics Professor
from CalTech
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I think part of the problem is that In what
sense is that teaching? is the wrong
question. Because I believe that education
improves a lot when teachers stop thinking so
much about teaching and start thinking more about
learning.
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First Activity!

• How does learning begin before teaching has
  occurred?
• Class presentations
• What are they for? (And what are they NOT for?)
• Whom do they serve? (And whom do they NOT serve?)
• How do we affect the class dynamic so
  presentations serve the right purposes/people?

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Teacher as Amateur Cognitive Scientist
How do we get our students to think and behave
like mathematicians?
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Teacher as Amateur Cognitive Scientist

• Getting into our students heads.
• How do they learn?
• And (thinking cognitively) what do they need to
  learn?

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Teacher as Amateur Cognitive Scientist
Getting into our own heads How do we operate
as mathematicians?
This is perhaps the crucial question for an IBL
instructor, but, in some weird and subtle ways,
it is harder for us than figuring out how our
students think and learn.
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Lets try an experiment . . .

• Q Who is non-orientable and lives in the ocean?
  
• A Möbius Dick
• Q What is purple and commutes?
• A An Abelian grape
• Q When did Bourbaki stop writing books?
• A When they found out Serge Lang was just one
  person.

Theorem
Mathematics is a Culture
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Culture is, by its very nature, completely
unconscious
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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.
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What does it mean to say that two partially
ordered sets are order isomorphic?
The students first instinct is not going to be
to say that there exists a function between them
that preserves order!
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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?. ???
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Thats 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.
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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. But
sometimes proofs help us understand connections
between mathematical ideas.
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Chasm? What Chasm?
???
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Cultural Elements

• We hold presuppositions and assumptions that may
  not be shared by someone new to mathematical
  culture.
• We have skills and practices that make it easier
  to function in our mathematical culture.
• We know where to focus our attention and what can
  be safely ignored.

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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.

And none of these are even talking about proving
theorems!
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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.
• Whats the problem?

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Charlie came by later. . .

• His problem was similar to Karens.
• But just looking at the definition didnt help
  Charlie as it has Karen.
• He didnt understand what the definition was
  saying, and he had no strategies for improving
  the situation.
• What to do?

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What? . . . Where?
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Sorting out the IssuesEquivalence 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
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There is a lot going on. Most of our students
are completely overwhelmed.
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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.
Relation on A
Collection of subsets of A.
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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) gt 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?
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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.)
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. . . closure under scalar multiplication and
closure under vector addition . . .
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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.....
The argument is the same, but your students are
really having trouble. What's at the root of the
problem? What should you do?
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Scenario 4 A very good student walks into your
office. She has been asked to prove that the
function
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?
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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.
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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
• Other useful ideas e.g. If A, then B or C.

Beyond counterexamples Negating implications!
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Negating Statements
(an ) converges to L if for every ? gt 0, there
exists N ? ? such that for all n gt N, d(an ,
L) lt ?.
(an ) fails to converge provided that for all L
it is not true that for every ? gt 0, there
exists N ? ? such that for all n gt N, d(an , L) lt
?.
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(an ) converges to L if for every ? gt 0, there
exists N ? ? such that for all n gt N, d(an ,
L) lt ?.
(an ) fails to converge provided that for all L
there exists ? gt 0
such that for all N ? ?
there exists n gt N
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For all x, y
f fails to be uniformly continuous provided that
there exists ? gt 0
such that for all ? gt 0
there exist x, y
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Epsilonics
Kabuki dance

• Organizing principle for final proof

Spare and stylized
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Impasse!
What happens when a student gets stuck? What
happens when everyone gets stuck?
How do we avoid THE IMPERMISSIBLE SHORTCUT?
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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
theorems to prove, so they develop intuition for
the definition/new concept. Separate the
elements.
Even if they are not particularly significant!
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Breaking the Impasse
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But all this begs an important question.
Do we want to pre-empt the Impasse?
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Precipitating the ImpasseImpasse 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.
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Precipitating the ImpasseImpasse 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 their own questions.
The intellectual apparatus for understanding
important issues is built in struggling with them.
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. . . 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. Im 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. Im not sure I would
have had that instinctive sense yesterday if I
hadnt been struggling at this line of work for a
while.
Written by David Brooks In one of his NYTimes
conversations with Gail Collins
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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.

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Make Em Care

• Making them care, makes them do it.
• If they do it, they come to care about it.
• When they care about it, they find it easier to
  care next time.

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Assignments that get students guessing and asking
questions. Or
Tree Activity!
What the students think up matters to them a
whole lot more than anything you can say to them.
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Discovering Trees
Consider what happens when you remove edges from
a connected graph (making sure it stays
connected). Your groups 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?
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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.