Carol Putting in her 2¢ Worth

Carol Putting in her 2¢ Worth

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

SOME PEOPLE HAVE TROUBLE UNDERSTANDING WHAT WE ARE UP TO.

No books, no outside sources, just you . . .

Caricatures of IBL

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.

“That’s interesting. I just don’t see how it’s teaching.”

---An award-winning Physics Professor from CalTech

Sometimes we are not so good at explaining what we are up to. . .

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.

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?

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?

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.

Let’s 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.

Mathematics is a CultureMathematics is a Culture

Culture is, by its very nature, completely unconscious

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, thereexists 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 goingto be to say that there exists a functionbetween them that preserves order!

As if this were not bad enough, we mathematicianssometimes 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.

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.

The Purpose of Proof

???Chasm?What Chasm?

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.

•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.

A great deal of versatility is required....

And none of these are even talking about proving theorems!

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?

What? . . . Where?What? . . . Where?

Sorting out the IssuesEquivalence Relations

EquivalenceRelationsPartitions

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.

There is a lot going on.

Most of our students are completely overwhelmed.

Us

Our student

s!

RelationonA

Collectionof subsets

of A.

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?

???Chasm?What Chasm?

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:

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.”)

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.

. . . closure under scalar multiplication 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.....

The argument is the same, but your students are really having trouble. What's at the root of the problem? What should you do?

and C C

C CA A A A

Scenario 4: A very good student walks into your office. She has been asked to prove that the function

f x xx

( ) 1

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.

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!

(an ) converges to L if

for every > 0, there exists N such that

for all n > N, d(an , L) < .

(an ) fails to converge provided that

for all L it is not true that “for every > 0, there

exists N such that for all n > N, d(an , L) < .”

Negating Statements

(an ) converges to L if

for every > 0, there exists N such that

for all n > N, d(an , L) < .

(an ) fails to converge provided that

for all L

such that d(an , L) .there exists > 0 such that for all N

there exists n > N

f fails to be uniformly continuous provided that

f is uniformly continuous if

For all > 0, there exists > 0 such that

if d(x,y) < , then d(f (x), f (y)) < .

For all x, y

such that d(x,y) < and d(f (x), f (y)) .there exists > 0 such that for all > 0

there exist x, y

“Epsilonics”

“Organizing principle” for final proof

Spare and stylized

if for every > 0, there exists > 0

such that if , then .

2

23

9lim

1 10x

x

x

0 3x

2

2

9

1 10

x

x

= Kabuki dance

Impasse!

What happens when a student gets stuck?

What happens when everyone gets stuck?

How do we avoid THE IMPERMISSIBLE SHORTCUT?

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!

Breaking the Impasse

But all this begs an important question.

Do we want to pre-empt the Impasse?

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.

Precipitating the ImpasseImpasse as tool

Why precipitate the impasse? The impasse generates questions!

The intellectual apparatus for understanding important issues is built in struggling with them.

At least as importantly, when students generate their own questions, they understand the import

of their own questions.

. . . 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 BrooksIn 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.

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.

Assignments that get students guessing and asking questions.

Or

What the students think up matters to them a whole lot more than anything you can say to them.

Tree Activity!

Discovering TreesConsider 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?

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.