# Math Education Concepts

## When Teaching Hurts

Standing in front of the class declaring all the interesting facts about mathematical concepts feels wonderful.  I enjoy math, I enjoy explaining mathematical concepts, and I enjoy watching students as they learn math.

The “hurt” is felt when it’s time to grade exams.  Some students are able to explain mathematical concepts, but have a hard time writing their explanations mathematically.  Some students can solve problems intuitively but cannot write the procedures the way they are taught.  Some students have anxiety attacks at the mere thought of taking a math exam, even when they know the material.

I understand the importance of tests, but I am a fan of assessments (not standardized, but individualized).  Most of my students understand the basic concepts that I teach and can explain them to me during class.  However, during quizzes and exams, those same students perform poorly.  This is when it hurts!  My heart just sinks when I know a student understands a concept, but cannot recall it during an exam.

The ultimate “hurt” happens when it’s time to submit final grades and students just don’t make the grade, so to speak.  My students are really “good” people who are trying to get through college so they can pursue their dreams.  Should one class get in the way?

Of course, the answer is obvious, but there are systems in place.  They are there for a reason, even when we disagree.

## It’s Just Substitution

The figure of a composite function. (Photo credit: Wikipedia)

I’ve always thought that teaching lessons on the composition of functions would be easy, but I was wrong.  Each semester poses a greater challenge.  This semester is the same.  The students look at me as if they have no idea what I’m talking about when I say “it’s just substitution.”  (My basic explanation for the process of completing the composition of functions). I show examples with numerals, then with single variables, then with binomials, and so on.  They follow along until I get to binomials, then they get “confused” again.  I encourage my students to complete their homework assignments (for practice and hands on learning) and to visit me during my office hours, so I can help them.  In the end some get it, some keep trying.

In their defense, the most challenging component of my composition of functions lesson is finding the domain of the composed functions.  That’s another blog entirely.

The one bit of satisfaction I received while teaching the early part of the lesson occurred when a student yelled out “it’s just like the difference quotient, right?”  I answered, “yes, exactly, if we let g(x) = x + h.”  I felt like someone finally got it and it felt great.  I only wish all the students made the same connection.  If only…

In any event, I found a nice explanation on Mathsisfun.com.  It explains the composition of functions really clearly, I think.

Enjoy and pass it along!

## Why Math Education

I am a math educator at heart.  I have tutored students from various backgrounds in mathematics for approximately 20 years (one-on-one and small groups).  During my high school years, I tutored my fellow classmates in math and have always had the reputation as the “Math Wiz.”  After high school, I earned a Bachelor of Arts degree with a major in Mathematics from the University of Pennsylvania.  After I graduated from Penn, I sub-matriculated into their Master of Arts in Education degree program, but later withdrew to focus on family issues.  Eventually, I became a Payroll Specialist for a corporation and worked there for approximately 7 years.  Although I held a full-time position in payroll, I did not ignore my passion for teaching math.  I tutored students privately to fulfill that passion.

In 2004 I enrolled in Arcadia University’s Master of Arts in Education degree program, but later withdrew.  In April 2006, the corporation I worked for outsourced my job.  I was liberated to pursue my true love and passion: math education.  I finally earned my Master of Arts in Education degree (Concentration: Secondary Mathematics) from Arcadia.

My immediate goal was to continue to teach students in a way that they could learn math more effectively (one-on-one instruction/tutoring).  I tend to direct my efforts toward students who have an interest in learning and excelling in mathematics, but I also work with students who simply need help getting through a math course.  Of course, the latter is more laborious.

I primarily work with high school students because the teachers who had the greatest impact on me were my high school teachers.  It takes a lot to reach high school students and I know because it took a lot to reach me when I was in high school (Martin Luther King, H.S.).  Today I tutor high school students and teach college students; an ideal combination!

I will always pursue higher learning.  The more I learn about math, the more I want to learn.  I want to devote as much of my time as possible to learn every aspect of math available to human kind.  Even if I don’t master every subject or topic, I want to be aware of its existence.  I believe that one must reach a certain level of mastery in their field in order to be the most effective in that field.  I believe in being the best and that means never ceasing to learn.  Personally, I am just fascinated with math for reasons I can not explain.  Math excites me!

## 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!