Math Education Concepts

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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

 

 


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COMPLEX NUMBERS …in brief

I don’t remember exactly when I learned about imaginary/complex numbers, but I do remember feeling excited about this “new” math; well, it was new to me.  I easily grasped the idea of complex numbers, but my classmates struggled with their meaning and purpose.  For me, it was easy; I treated i like a variable when performing operations on the complex numbers.  However, I understood very well the value of i as the square root of negative one.  I didn’t understand why it was so difficult for others to comprehend complex numbers.  Even now, some of my students do not “get it.” 

When approaching the topic, I often ask my students to calculate the square root of negative one.  They look at me puzzled and say “one” or “negative one.”  I then ask “what is one squared” or “negative one squared” and they say “one.”  I repeat this cycle of questioning until they finally ask “what is the square root of negative one?”  Ah-ha!!  Now we’re getting somewhere.  This is usually how I introduce my students to complex numbers.  It never fails; my students almost always say “that’s stupid” or “what’s the point?”  After I allow them to vent their frustrations, I explain that i is short for “imaginary” (the name given by René Descartes in 1637) and represents the value of the square root of negative one.  I tell them the letter was assigned by a mathematician since the exact value is unknown; that i simply represents the value of the square root of negative one.  They buy it (well some of them), open their minds to it, and learn enough to get through their lessons.  In the meantime, I sigh with relief that they don’t ask additional questions. 

For years, I’d wanted  research i and complex numbers so I could give a more accurate account of their beginnings, history, and existence.  Finally, the day came that I actually learned a brief history of these obscure numbers known as “complex numbers,” a term given by Carl Friedrich Gauss in 1831.  I’ll spare you the details.

As with many topics in mathematics, complex numbers became useful, and necessary.    Therefore, it was important for mathematicians to carefully analyze, define, introduce, and explain complex numbers.  This did not happen without opposition.  Some mathematicians argued against the usefulness of complex numbers.  Even Leonhard Euler did not care for the idea of these “impossible” numbers, but realized their necessity.  In spite of the arguments stacked against complex numbers, the quest to validate them did not end.  Many mathematicians have worked with complex numbers and found uses for them. 

Delving any deeper into the world of complex numbers would require extensive research into other fields, which time does not allow.  Maybe one day I will have the resources to discover all the truly amazing characteristics of this number system.  As with many mathematical concepts, it took many centuries and multiple mathematicians to finally come up with yet another set of numbers.  I can’t help but wonder if there are any other number systems that have not yet surfaced.

NOTE:  Some of the information shared here was inspired by “Math through the Ages: A Gentle History for Teachers and Others,” written by William P. Berlinghoff and Fernando Q. Gouvêa.  A must read for Math Educators!


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Chichen Itza… A Mathematical Enlightenment

I recently visited Cancun, Mexico and decided to take a tour of Chichen Itza in Yucatan Peninsula (Pictures posted are courtesy of me).  My friend of 20+ years visited Chichen Itza and made the recommendation (thanks Margie).  I did my research and booked the tour.  I was in for a wonderful surprise! 

Although Chichen Itza is breath-taking, and is one of the New 7 Wonders of the World, what piqued my interest the most is the Mayan contributions to math and their use of math in building El Castillo – The Kukulcan Pyramid.

Here is one really basic fact: there are 4 faces on the pyramid with 91 steps on each face (364 steps).  The top platform counted as a step makes 365 steps in all, the number of days in our calendar year (not including leap year, of course).

We often hear, very loosely, about the contributions ancient civilizations made to math, but we rarely have the opportunity to experience the reality of those contributions.  It was a fantastic vacation and an enriching mathematical experience for me.  Hopefully, I will have the opportunity to visit similar ruins throughout my lifetime.

Have you visited Chichen Itza or another ancient ruin that inspired you?  Please share you experience.