When
we first started discussing the values of proofs in class a few days
ago, my first thought was, "they're more trouble than they're
worth." Of course I was thinking about my high school
geometry class in which everyone complained about them and how this
reaction hasn't changed since I've come to college. But I
didn't truly understand the purpose and implications until a couple
days ago.

An
important implication that we discussed in class was that proofs
establish critical and logical thinking. They are essentially
arguments that, if written well enough, will never fail. That is a
very powerful skill, to write effectively enough that what you've
said can never be contradicted. So although proofs seem like they're
very specific to whatever math conjecture you're trying to prove and
require some foundation work to learn applicable theorems, they
really establish writing skills and logical thinking that can be
applied to many areas of life. Thus they're very important for
students to learn.

One
of the biggest problems I have with proofs is that the purpose of the
proof is lost in the details of memorizing theorems and definitions.
I find this particularly relevant in high school geometry where it's
expected that the student will memorize all of the relationships
between the angles and such. As teachers, I think it's important
that we practice the names of the relationships and theorems and make
sure students understand what they mean before they attempt to write
a proof. I believe that if we have them try to piece things together
too soon, the task will seem too daunting and students will become
too frustrated, which will lead to them writing proofs off as “too
hard.”

On
his blog, Daniel Schneider emphasizes the maze-like structure of a
proof. That is, given a beginning and an end, find the path
between those points. I think this is a very important connection to
establish because it takes the the bewildering nature of a proof and
simplifies it to a well understood maze. So with this simple concept
in mind, and with full understanding of applicable theorems, I think
students can easily tackle the daunting task of writing a proof and
be able to realize their importance.

Your line about proofs - "They are essentially arguments that, if written well enough, will not fail", I like it; I am keeping it to trot out anytime a student asks me why we are putting them through the torture of proof writing.

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