Math Education Concepts

Inspiring Motivating Empowering


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Teacher or Mathematician First?

Earlier this week a close friend, Lawrence, asked me a question.  And for the first time (in a long time) I had to think about the answer.

He asked me if I were a teacher or mathematician first, when I am in the classroom.  I paused for a moment to consider the question.  My first comment was that, since I have yet to earn my PhD, I am not necessarily considered a mathematician.  But I understood his question.  He wanted to know what drives me when in the classroom.  The example he used was that he is an architect first, then an engineer.  His education path is engineering, but his career path is architectural design (or something like that – sorry Lawrence).  But his real joy is designing blue prints for office buildings.  In fact, he is going to design my future institute (a post for another day).

After thinking about the question, I explained to Lawrence that it depended upon the class I taught.  This semester I am teaching a math course for students pursuing degrees in STEM related fields and a math education course for students pursuing education degrees in non-STEM related fields.  My initial answer was “both:  I am a mathematician first in the math class and a teacher first in the education class.”

This was my explanation:

In the math class the students really need to know and understand the concepts in order to proceed to the next math course.  I have to get the math concepts across to the students.  In the education class, the students need to pass a pre-service exam and satisfy this course requirement.  But they are future educators and I want to exemplify what that means to my students.  The goal for each course is different, so I teach each class differently.

My final answer, however, was that I am a mathematician first.  If you put me in a classroom and take away the math I would be less fulfilled.  I decided to teach to share my joy of math and to help others learn and appreciate math the way I do.  I know this will not happen for all of my students, but I want to reach as many as possible.  The classroom is the best place to do this!

So there you have it Lawrence:  I am a mathematician first, teaching is the vehicle I use to express and share my passion for mathematics!


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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.


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FERMAT’S ENIGMA

I recently read Fermat’s Enigma, written by Simon Singh.  It tells the story of the quest to prove Fermat’s Last Theorem, the last of Fermat’s conjectures to be proven.  Singh covers the story from the inspiration of the conjecture (Pythagoras’ time), through the time Fermat wrote the conjecture in the margins of Arithmetica, through the proof of the conjecture by Andrew Wiles.  I read this book because I wanted to know more about Fermat’s Last Theorem (FLT), but as I read along, I found myself critiquing the book.

My initial impression of the book, prior to reading it, was that it would be filled with a non-mathematicians’ attempt to explain mathematical concepts.  Then, I re-read the blurb about Singh in the beginning of the book and was reminded that he is a Physicist and, most likely, had a mathematical background.  I’m not sure how a non-mathematician would feel about reading Fermat’s Enigma, but I don’t think they’ll get the same enjoyment as a mathematician would.

One theme throughout Fermat’s Enigma is Singh’s peep into the world of mathematics and mathematicians.  The book tells the story of one man’s enjoyment of puzzles and riddles, hundreds of attempts to prove FLT, dozens of brilliant minds collaborating, and one man’s childhood dream come to fruition.  In telling the story, Singh covers a wide range of mathematics history.  This was an unexpected treat for me.  In addition to learning about the FLT story, I also got a brief account of a segment of mathematics history.

While covering the FLT story, Singh, along with John Lynch, a television editor, spent months talking with Andrew Wiles, conducting interviews with other mathematicians, and researching the history of FLT to give as accurate an account as they could about Fermat’s Last Theorem.  The intent of Lynch, with the help of Singh, was to create a television documentary of the story.  I haven’t seen the documentary, but I am satisfied with having read the book.

As I read the “Epilogue”, I felt sad.  Maybe I wasn’t ready to stop reading Fermat’s Enigma, maybe I was empathizing with Wiles on his feelings of finally proving Fermat’s Last Theorem.  He said “There’s no other problem that will mean the same to me… Having solved this problem there’s certainly a sense of loss.” (p. 285)  I felt that my brief, less intense, obsession with reading this book was finally concluding.  I was so engrossed in it that I had to set my alarm so I wouldn’t miss appointments.  When I’d finally finished the book, I felt that I needed something else to capture my attention the way this story did.  At the same time, I felt a sense of relief.  It was finished.

Fermat’s Enigma is fun and easy to read.  It explores some common mathematical concepts as well as a fair share of mathematics history.  Some other benefits of the book are the proofs and examples given in the “Appendixes,” as well as the list of dozens of references in the “Suggestions for Further Reading” section.  As a math educator, I think this book is a must read.  It is an informative and resourceful account of a momentous event in recent mathematics history.  However, if a reader wants a concise, less dramatic, overview of the story of Fermat’s Last Theorem, this may not be the best choice.