Proof and Reasoning in Grades 9-12

NCTM Proof and Reasoning NCTM Proof and Reasoning StandardStandardInstructional 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

What is Mathematical Proof?

Different Kinds of Proof

Mathematical Language

Today’s Agenda

What is What is Mathematical Proof?Mathematical Proof?

Initial DiscussionInitial DiscussionWith your neighbors at your table, discuss the idea of “proof” or “proving” in each of the following fields:

•Biology•Philosophy•Mathematics•Psychology

DefinitionDefinition

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 quote from (Krantz, 2007)

A Mathematician’s View

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…

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.)

Checkerboard Problem

Two SolutionsTwo SolutionsTwo Solutions

A More Artistic SolutionA More Artistic SolutionA More Artistic Solution

Now remove two opposite corners of the checkerboard.

Following the same rules, can you still cover the remaining squares with dominos?

Try!

Mutilated Checkerboard Poblem

Failed AttemptFailed Attempt

I can’t fill in the bottom row with dominos, let alone the white square in the row above it.

Mounting EvidenceMounting EvidenceHow many different

arrangements have been tried in this room? 10? 50? 100?

Is that enough to conclude it is not possible?

The “Scientific Approach”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”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 PerfectNothing 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 Further Example: Atomic TheoryTheory

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 Back to the Mutilated CheckerboardCheckerboard

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?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 ApproachThe 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 Mathematical Mutilated CheckerboardCheckerboard

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 EternitySolved 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 Aside: Another Famous ExampleExample(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?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 If it Sounds too Good to be True…True…Alas,

Different Kinds of Different Kinds of ProofProof

Levels of ProofLevels 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 Carpenter’s Levels of JustificationJustificationAppeal to AuthorityAppeal to Authority

Justification by ExampleJustification by Example

Generalizable ArgumentGeneralizable Argument

The Chord-Chord-Power Theorem

Theorem: If PQ and RS are chords of a circle which intersect at A, then

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AP ⋅AQ = AR ⋅AS.

Random Examples

Extreme Examples

Carpenter’s Levels - Carpenter’s Levels - ExpandedExpandedAppeal to AuthorityAppeal to Authority

Justification by Naïve ExampleJustification by Naïve Example

Justification by Naïve and Extreme Justification by Naïve and Extreme ExamplesExamples

Generalizable ArgumentGeneralizable 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 IIProof 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 IIIProof 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 ProofQuestion 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:

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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

MathematicalMathematicalLanguageLanguage

Mathematicians are PickyMathematicians 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 ExampleIn your head, determine what sentence exactly expresses what it means for the sentence “All mathematicians wear glasses” to be false.

Futher Example: Student DataIn 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.

BaselineBaselineAssessmentAssessment

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 Assessment2. 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.