**Garakai Campbell on mathematics**:

Recently, a group of friends commended me for having earned a Ph.D. They ceremoniously read aloud the title of my thesis: "...Elliptic Curves Defined over Q...". It did not take long for the inevitable to occur; someone asked: ``what the heck does that mean" More than being accustomed to such a response, I in fact enjoy hearing it. I am given the opportunity to do what I deeply enjoy - lead a mathematics discussion, show off the beauty of the subject and possibly even excite someone to learn more - in short, teach mathematics. Moreover, this particular question invokes a response that exemplifies my philosophy on teaching. Let me explain as I present a condensed response to this person's question.

First let me draw a circle. Suppose I say the radius of this circle is 1; you may remember from high school that the equation for such a circle is x^{2}+y^{2}=1.

My first goal is always to find a familiar place to begin the discussion. Some students are intimidated by mathematics and before the real work can begin, I must liberate them from that fear. Beginning every discussion with material that the student knows or feels comfortable with does just that.

Try to find some points on the circle where the x and the y are rational.

Mathematics is learned by actively doing it, not by passively watching someone else do it. In many classes, I try to spend half of the class time lecturing and half of the time letting students work on problems in groups. Walking around to each group, I can give students more personal and directed attention. Working in groups not only facilitates the flow of ideas, but it also reinforces the necessity of being able to articulate your ideas well. Ever watchful of students who may not be full participants, I make sure that everyone is contributing in their group's discussion by frequently choosing someone to present a problem to the class.

What if I draw a line with rational slope that goes through (-1,0) Notice that the line hits the circle in one other place. What can you tell me about this other point

Students are fascinated by this particular discussion because it illustrates so nicely the interplay between different areas of mathematics -- in this case between algebra and geometry. Students can see what's going on and then use algebra to solve for the other point of intersection. They eventually conclude that it too has to be rational. This interplay is not unique to this problem of course, but is in fact the norm for most of mathematics. I try to emphasize this in my classes by asking students to draw pictures, do computer experiments, make hypotheses and finally, to choose appropriate methods of attack. Students often surprise themselves by how excited they are by this scientific method of discovery.

Given one rational point on any conic, this idea (of drawing lines with rational slope) can be used to find all rational points. So what next

Mathematics could not have solutions without questions. Whenever possible, I get my students to think about the next step, to formulate a slightly more difficult question and to understand that this inquiry is as much a part of mathematics as finding solutions.

So what if you want to find the rational points on something like x^{3}+y^{3}=1

The joy I find in teaching is that the question above is generally not my own; it belongs to that same person who asked ``what the heck does that mean" And so, in the end,

they have found their own path to understanding (at least the meaning of) Elliptic Curves Defined over **Q**. Affirmed by student responses to my approach, I am currently working on ways to use technology to increase each student's active participation in mathematics while at the same time reducing the amount of class time needed for lecturing.