What we say / what they hear Culture shock in the classroom - PowerPoint PPT Presentation

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What we say / what they hear Culture shock in the classroom

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What we say / what they hear Culture shock in the classroom Mathematical Culture Mathematical Culture Helping our students focus Example: Equivalence Relations And. . . – PowerPoint PPT presentation

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Title: What we say / what they hear Culture shock in the classroom


1
What we say / what they hearCulture shock in
the classroom
2
Mathematical Culture
We hold presuppositions and assumptions that
may not be shared by someone new to mathematical
culture.
3
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.
4
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!
5
For every ? gt 0, there exists a ? gt 0 such that
if...
6
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?. ???
7
Mathematical Culture
We know where to focus our attention for
maximum benefit and we know what can be safely
ignored.
8
What? . . . Where?
9
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
10
There is a lot going on in this theorem. Many of
our students are completely overwhelmed.
11
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
12
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.
13
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?
14
Mathematical Culture
We have skills and practices that make it
easier to function in our mathematical culture.
15
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!
16
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
17
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!
18
Impasse!
What happens when a student gets stuck? What
happens when everyone gets stuck?
How do we avoid THE IMPERMISSIBLE SHORTCUT?
19
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
20
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
21
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!
22
But all this begs an important question.
Do we want to pre-empt the Impasse?
23
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.
24
Changing Gears
25
What they say/what we hearListening to our
students
  • (Sometimes) hearing what they mean instead of
    what they say

26
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.

27
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?

28
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?

29
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.
30
  • 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.
31
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.
32
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.
33
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.

34
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?
35
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.)
36
. . . closure under scalar multiplication and
closure under vector addition . . .
37
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?
38
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?
39
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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