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Monday, October 02, 2006

Topic: math teaching

Math and People Skills

It appears that the students consider me to be a bit more approachable this year. Hence, they ask more questions and therefore I can see what confuses them.

Of course, some of that startles me.

I am teaching differential equations and a current topic is linear second order differential equations.

For those who have had some calculus, a differential equation is an equation involving a function and its derivatives (possibly some of higher order).

Example: y' + y = 0 is an example of a differential equation; this reads: a function plus its derivative is always zero. A general solution is y = a*exp(-t) because
y' + y = -a*exp(-t) + a*exp(-t) = 0.

Now if we had instead: y' + y = cos(t) we would obtain:

y = a*exp(-t) + (1/2)(cos(t) + sin(t)) as a solution. The solution with the arbitrary constant is called the homogenous solution, and the one without is (the trig part, in this case) is called the particular solution.

There are many techniques for solving these; you know that you are done by the various existance and uniqueness theorems (e. g. differential equations meeting certain criteria have a unique solution, so if you get a solution by "any means necessary" (e. g. good guessing) then you have all of the solutions.)

So, we need to teach "good guessing" which is sometimes known as "the method of undetermined coefficients".

The basic principle is that

if yp(t) = exp(a*t) then all derivatives are of (something)*exp(a*t)
if yp(t) = (b*cos(r*t) + d* sin(r*t)) then all derivatives are sums of sin(r*t) and cos(r*t)
if yp(t) = polynomial, then all derivatives are polynomials.

That is, certain types of functions formed a closed class with respect to taking derivatives.

To see this, consider a function that does not fall into this category: y(t) = sec(t)
taking derivatives yields tan(t)sec(t), then sec^3(t) + tan^2(t)sec(t), etc., which are not of the form (some constant) * sec(t)

So the good guessing technique doesn't work with sec(t).

This is something I understood from day one as a student.

Most of my students (or at least many) don't have a natural understanding of this.

I never knew that.

Math Fun From An Old Friend

One of my old graduate student buddies is a professor at Pittsburg State University in Kansas.
He calls himself The Okie in Exile.

His reasearch area is geometric topology; he has published in the area of non-compact three manifolds. He is one of the world's experts in Whitehead manifolds.

He has put out a couple of Youtube intervals in which he teaches math topics via folksy, funny stories.

The first of these deals with the Chinese Remainder Theorem in a clever way; the second one talks about Egyptian fractions. Enjoy!


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