What we say / what they hearCulture 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 order-preserving

bijection between them!

For every ? gt 0, there exists a ? gt 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 focusExample 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.

Equivalence Relations

Partitions

There is a lot going on in this theorem. Many of

our students are completely overwhelmed.

Sorting out the Issues

Every partition of a set A generates an

equivalence relation on A. Every

equivalence on A relation generates a partition

of A.

Equivalence Relation on A

Partition of A

Sorting out the Issues

Every collection of subsets of A generates a

relation on A. Every relation on A

generates a collection of subsets of A.

Relation on A

Collection of subsets of A.

And. . .Its not just about logical connections

The usual practice is to define an equivalence

relation first and only then to talk about

partitions.

Motivation for defining equivalence relation?

Are we directing our students attention in the

wrong direction?

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. - 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!

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
- Other useful ideas e.g. If A, then B or C.

Beyond counterexamples 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 means that for every ? gt 0,

there exists N ?N such that for all n gt N,

d(an , L) lt ? .

an is within that tolerance of L

there is some fixed position

beyond which

Dont just stand there!Do something.

- an ? L means that ? ? gt 0 ? n ? N ? d(an , L) lt

? . - an ? L means that ? ? gt 0 ? N ? N ? for some

n gt N, d(an , L) lt ? . - an ? L means that ? N ? N, ? ? gt 0 ? ? n gt

N, d(an , L) lt ? . - an ? L means that ? N ? N and ? ? gt 0, ? n gt N ?

d(an , L) lt ? .

Make it real

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!

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.

More importantly, students understand the import

of their own questions. The intellectual

apparatus for understanding important issues is

built in struggling with them.

Changing Gears

What they say/what we hearListening 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.
- Whats the problem?

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?

Thats 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 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 dont 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 doesnt get it.

- She thinks student presentations are a waste of

everybodys time. - (She may secretly believe that I dont lecture

because Im lazy or unprepared.) - She is conditioned to respond to what I say and

she doesnt 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!

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.

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) 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?

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.)

. . . 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?

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?

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.