Math Education Concepts

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Proofs and Geometry

Woman teaching geometry, from Euclid's Elements.

Woman teaching geometry, from Euclid’s Elements. (Photo credit: Wikipedia)

In all my years of tutoring, I’ve noticed that most students prefer algebra over geometry or the other way around.  Even when I was in high school, I preferred algebra.  Geometry required more effort on my part, especially when it came time to learn proofs.  I earned my first (and last) “F” on a geometry math test that had geometric proofs.  Of course I worked diligently to pull my grade up by the end of the year.  But I will always remember that test, the teacher (Mrs. Yarbrough), and the feeling I had when I got my test back.

This followed me for a long time.  Throughout the years, I was asked to tutor students in geometry.  I always had an internal conversation that went something like this:

“Why me, can they get someone else?  There has to be someone who loves geometry.  Now I have to tutor proofs.  Proofs made me fail my geometry test in high school.  Maybe they already learned proofs.  Okay, I can do this.”

Then I would learn that the students needed help with proofs.  Go figure!!!  Eventually, this happened a few times and through tutoring my students, I finally learned, understood, and appreciated proofs.

When I was in high school I only had access to the teacher, my textbook, and the books at the library.  With today’s technology it’s easier to get help with almost any subject.  I wish the internet were as accessible then as it is now.

Here are a few internet sites and videos to help you with proofs:

Sparknotes         Extensive notes about geometric proofs

For Dummies     Tips for “Mastering the Formal Geometry Proof”

WyzAnt                Steps for writing two column proofs

Your Teacher    A video of an example of a two column proof

Khan Academy  A video explaining how to solve geometric proofs

So, why do we teach proofs in geometry classes?  Who thought geometric proofs were important enough to confound even the best math students?  Joshua N. Cooper of the University of South Carolina made a compelling argument about the importance of proofs in math classes.  Read his article before you tackle your first proof!  It may help you accept proofs.

I hope this is helpful.  If you have other ideas or resources to help with geometric proofs, please share them in the comments section!

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One of My Favorite Quotes

A portrait of Pierre de Fermat, French lawyer ...

A portrait of Pierre de Fermat, French lawyer and mathematician. (Photo credit: Wikipedia)

“I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.”Pierre de Fermat

One of Fermat’s habits was to write proofs in the margins of the books he read.  He was most known for writing in his copy of Arithmetica.  One day he came across the book and worked through the many problems published therein.  The author of Arithmetica, Diophantus of Alexandria, shared his proofs and solutions in his text.  The story is told that at some point, about 1637, Fermat decided to expand the Pythagorean Theorem to similar equations with exponents greater than 2.  He concluded that there were no whole number solutions to the equation xn + yn = zn for values of n greater than 2.  However, in the margin of Book II of Arithmetica all he wrote was “I have a truly marvelous demonstration of this proposition which this margin is too narrow to contain.” (Singh, 62)  To his successors’ chagrin, Fermat did not write the proof to this, then, conjecture.  At least none that anyone have found and published.  Fermat died in 1665.  In 1670, Fermat’s son, Clément-Samuel, published Fermat’s discoveries, theorems, notes, and commentaries in Diophantus’ Arithmetica Containing Observations by P. de Fermat.  In the years to follow, mathematicians would unsuccessfully attempt to prove what was known as Fermat’s Last Theorem.  It was given this name because it was the only theorem that Fermat did not provide a proof for.

The quote rings out for me because it leaves an element of wonder, mystique, and brilliance, all at once.  Did Fermat have a proof?  Did he write out the proof?  Did he have it figured out in his mind?  Was this his last puzzle for the world?

WHAT IS YOUR FAVORITE QUOTE? WHY?

REFERENCE: Singh, Simon. Fermat’s Enigma. New York: Anchor Books, 1997