Transcript Trainer Power Point

Proof and Reasoning in
Grades 9-12
NCTM Proof and Reasoning Standard
Instructional programs from prekindergarten through
grade 12 should enable all students to—
•Recognize reasoning and proof as fundamental
aspects of mathematics;
•Make and investigate mathematical conjectures;
•Develop and evaluate mathematical arguments and
proofs;
•Select and use various types of reasoning and
methods of proof.
Mathematical Proof
Today’s Agenda

What is Mathematical Proof?

Different Kinds of Proof

Mathematical Language
What is Mathematical
Proof?
Initial Discussion
With your neighbors at your table,
discuss the idea of “proof” or “proving”
in each of the following fields:
•Biology
•Philosophy
•Mathematics
•Psychology
Definition
A mathematical proof is a convincing
explanation that a given mathematical
statement is true.
THE BOOK
Paul Erdős, a famous Hungarian mathematician,
claimed God kept a book -- THE book -- full of the
most elegant and beautiful proofs.
When he saw a clever proof he would exclaim
“That’s it! That’s the one from THE BOOK!”
Is there a BOOK?
Erdős used to say that, whether or not a
mathematician believes in God, he or she ought
to believe in THE BOOK.
A Mathematician’s View
A quote from (Krantz, 2007)
The unique feature that sets mathematics apart
from other sciences, from philosophy, and
indeed from all other forms of intellectual
discourse, is the use of rigorous proof.
What’s the Big Deal?
Ok, so mathematicians highly value
the concept of proof. How is
“mathematical proof” different than
“proof” in other subjects?
We can explore the differences using
checkerboards…
Checkerboard Problem
A checkerboard has eight rows
and eight columns. A horizontal
or vertical domino can exactly
cover two squares on the
checkerboard.
Is it possible to cover all 64 squares on
the checkerboard with dominos?
(Dominos must be entirely on the
checkerboard and may not overlap.)
Two Solutions
A More Artistic Solution
Mutilated Checkerboard Poblem
Now remove two opposite
corners of the checkerboard.
Following the same rules,
can you still cover the
remaining squares with
dominos?
Try!
Failed Attempt
I can’t fill in the bottom row with
dominos, let alone the white square in
the row above it.
Mounting Evidence

How many different arrangements have
been tried in this room? 10? 50? 100?

Is that enough to conclude it is not
possible?
The “Scientific Approach”
Scientists make empirical observations about
the word and then attempt to devise an
explanation for these observations.
The strength of their argument is based on:
• How well does it explain the observations?
• Can it be used to explain other
phenomena?
Scientific “Theory”
The word “theory” is used for a scientific idea
if it has been tested over and over and
consistently predicts (or accounts for) real
world observations.
Think:
“Theory of Gravity”
“Theory of Evolution”
Nothing is Perfect
However, a scientific theory is still nothing
more than the best explanation currently
available.
Every scientific theory can (will?) be
contradicted by future evidence, requiring us
to revise or replace it.
Example: Newton’s Theory of Gravity has been
refined/replaced by Einstein’s Theory of
Relativity.
Further Example: Atomic Theory
Ancient Indians and Greeks suggested matter could be divided into
small, discrete pieces, but did not have the technology to investigate
it properly.
 ~1800: Dalton proposed a theory in which elements were
composed of small, indestructible atoms which could combine to
form molecules.
 ~1900: Thomson observed electrons, meaning the atom was made of
smaller pieces.
 ~1909: Rutherford discovered the nucleus. Bohr and others
continued to refine this model, discovering the nucleus could be
split into protons and neutrons.
Later we discovered even protons and neutrons can be split into
quarks!
At each point in this story, new experiments forced the scientists to
modify their explanation of how atoms work.

Back to the Mutilated Checkerboard
There are dozens of failed attempts by the
highly intelligent people in this room.
From the scientific viewpoint, we have nearly
irrefutable evidence that it is impossible.
Impossible… or Not?
As with any scientific “theory,” however, we can’t be
sure unless we check every single possible arrangement,
of which there are hundreds of thousands.
What if there is one very, very clever arrangement that
works? Then we’d have to modify our theory.
With the scientific approach this possibility, however
unlikely, is always lurking in the background. Doubt is
unavoidable!
The Mathematical Approach
In Math, “theory” has a very different meaning.
A “theorem” is something with an airtight
argument explaining why it is true, not a
“current best possible explanation” which
could be changed if new facts arise.
Once a theorem is demonstrated to be true, it
will always be true.
Mathematical Mutilated Checkerboard
Theorem: The mutilated checkerboard cannot be
covered by dominos.
Proof: Each domino covers exactly one black and one
white square. The mutilated checkerboard has 32
white squares and 30 black squares. After 30 dominos
have been laid down, only two white squares remain,
which can never be covered by one single domino.
Solved for All Eternity
After this proof, there is never any possibility of a “clever
arrangement” that everybody else missed. It simply can’t
be done!
Aside: Another Famous Example
(For teachers who want to caution their students about
jumping to conclusions, even when the evidence seems
insurmountable.)
Consider the quadratic polynomial:
On paper, compute f(1), f(2), and f(3). What do you
notice?
Prime Number Generator?
• f(1)= 41 -- prime!
• f(2)= 43 -- prime!
• f(3)= 47 -- prime!
• f(4)= 53 -- prime!
.
.
.
f(39)= 1523 -- prime!
f(40)= 1601 -- prime!
If it Sounds too Good to be True…
Alas,
Different Kinds of
Proof
Levels of Proof
Many researchers have proposed models
for different levels of proof and justification
by students, e.g. (Carpenter, 2003) or
(Balacheff, 1987).
Carpenter’s Levels of Justification

Appeal to Authority

Justification by Example

Generalizable Argument
The Chord-Chord-Power Theorem
Theorem: If PQ and RS are chords of a circle which intersect at
A, then AP  AQ  AR AS.

Random Examples
Extreme Examples
Carpenter’s Levels - Expanded

Appeal to Authority

Justification by Naïve Example

Justification by Naïve and Extreme Examples

Generalizable Argument
Other Methods of Proof…
From documents which have floated around online for 20+ years.
Have your students used any of these? Have you?
Proof by intimidation:
"Trivial."
Proof by vigorous handwaving:
Works well in a classroom or seminar setting.
Proof by omission:
"The reader may easily supply the details" or "The other 253 cases
are analogous"
Other Methods II
Proof by general agreement
"All in favor?..."
Proof by imagination
"Well, we'll pretend it's true..."
Proof by convenience
"It would be very nice if it were true, so..."
Proof by necessity
"It had better be true, or the entire structure of
mathematics would crumble to the ground."
Other Methods III
Proof by accident
"Hey, what have we here?!"
Proof by profanity
(example omitted)
Proof by lost reference
"I know I saw it somewhere..."
Proof by calculus
"This proof requires calculus, so we'll skip it."
Proof by lack of interest
"Does anyone really want to see this?"
Obstacles to Proof
Question for discussion: what do students
find intimidating about mathematical proofs?
A Proof that Proves
“A Magic Proof”
Prove: The sum of the first n positive integers is n(n+1)/2.
For n=1 it is true since 1=1(1+1)/2.
Assume it is true for some arbitrary k : S(k)=k(k+1)/2. Then:
S(k  1)  S(k)  (k  1)
 k(k  1) /2  (k  1)
 (k  1)(k  2) /2
Hence the statement is true for k+1 if is true for k. By
induction it is true for all n.

A Proof that Explains
Mathematical
Language
Mathematicians are Picky
Imagine a teacher tells her class “I promise that those who sit
quietly for the next ten minutes can go outside for recess,” but
then lets both the quiet and noisy children go outside. Did
she break a promise?
Language and Logic
Futher Example
In your head, determine what sentence exactly expresses what it
means for the sentence “All mathematicians wear glasses” to be
false.
Futher Example: Student Data
In your head, determine what sentence exactly expresses what it
means for the sentence “All mathematicians wear glasses” to be
false.
70% of college level calculus students selected
“No mathematician wears glasses.”
20% chose the correct statement
“Some mathematicians do not wear glasses.”
A Bad Joke…
An astronomer, a physicist and a mathematician are
on a train in Scotland. The astronomer looks out of
the window, sees a black sheep standing in a field,
and remarks, "How odd. Scottish sheep are black."
"No, no, no!" says the physicist. "Only some
Scottish sheep are black."
The mathematician rolls his eyes at his companions'
muddled thinking and says, "In Scotland, there is at
least one sheep, at least one side of which is black."
What to do?
1.
As time permits(!), continue to ask your students to show
their reasoning, and give them feedback.
2.
Logic Puzzles help students develop mathematical
reasoning skills (see Session 2) and work with precise
language.
3.
Activities such as the one with the shapes can help
students learn the difference between “all,” “every,” etc.
Baseline
Assessment
Baseline Assessment
1. Take an odd number and an even number and
multiply them together. Their product is always
an even number.
Provide a justification for this fact, explaining as
clearly as you can.
Baseline Assessment
2. Consider the statement that . Four students have provided
explanations below.
a. Which of the following students have proven this statement?
b. Whose explanation is best? Why?
Baseline Assessment
3. Write down exactly what you would have to do to
prove that the following sentences are false.
a. All High School students are lazy.
b. Some Major League Baseball Players have
taken steroids.
c. If the sun is shining, then it is at least 70
degrees outside.