## Reasoning and Proof Standards

- Read through the text below on the difference between reasoning and
proof.
- See the directions that follow for the assignment
(If you are a math minor and want an assignment
that is less related to teaching, then come see me in office hours.)

### The Difference Between Reasoning and Proof

**Convincing Arguments**
There are many ways to convince someone that a statement is true, for example:
- Experimental evidence , e.g., physics experiments to validate theories
of General Relativity; medical
experiments to validate that rats that eat sugar get cancer.
- Statistical sampling e.g., Gallup polls to convince people that
someone will be elected President.
- Citing a reputable authority. "The professor/boss said the answer to the
problem was...".
- Acting confident, talking loudly. "I'm sure my program works now."
I don't see why not.
- Shifting the burden of proof to someone who disagrees with
you.
- Legal system. Uses "proof beyond a reasonable doubt"-convince a
judge and jury.

But these are not proofs, at least, not in the mathematical sense,
as these methods don't convince everyone and in fact can
lead to false conclusions. For example:
- Experiments can be contaminated or otherwise messed up.
- Statistics can be misleading. Consider the Florida election!
- Authorities can make mistakes, e.g., Intel said the original
Pentium chip worked fine.
- Juries can be fooled.

**Mathematical Proofs**
Mathematics uses a particularly convincing way to argue that something is
true:

A *mathematical proof* is a formal verification of a
proposition by a
chain of
logical deductions starting
from a base set of axioms.

*Formal language*:
Need a precisely defined language for expressing everything.

*Proposition*:
Statement written in the language, e.g., what you want to prove.

*Axioms*: Statements that you are assuming in your proof.

*Logical deductions*: Getting a new truth from old ones,
according to agreed-upon, formal rules of deduction.

The main idea is to specify everything so precisely that everyone is convinced.
If people accept the assumptions and deduction rules, they must also
accept the
conclusions.
Even if we can't understand the entire argument at once,
if we believe each deductive step, then we can be
comfortable accepting the conclusions.
This is a philosophical position, and thus debatable, but most of modern
civilization accepts it.
By learning to read and do proofs, you should be able to
- Convince others of your arguments.
- Find flaws in others' attempts to convince you.
- Find flaws in your own proofs.
- Develop the ability to reason
carefully about, attack, and defend ideas.

### Assignment

Read through the
National Council of Teachers of Mathematics (NCTM)
web page on
Reasoning and Proof Standard for Grades 9-12.

Username: 3610 Password:
- Reflect on your experiences as a student in mathematics classes
in grades Pre K-12. How do your experiences compare or measure up
to these standards?
- Choose one activity from 3610 and discuss its relationship to the
NCTM Reasoning and Proof Standards for Grades 9-12.

Respond to these questions with a typed paper.