Learning

I'm sitting in the EWS [engineering workstation] computer lab on the fourth floor of Engineering Hall. The guy next to me is pretty busy doing Physics 212 homework online (electricity and magnetism), and he has so much going on that his papers are spilling over onto my desk. One of these interlopers is an old Differential Equations exam. This guy could have been me, way back in second semester freshman year: same classes, same problems, same computer lab, same paper overload.

Ever curious, I take a closer look at the diff. eq. exam. The visible problem asks the student to find the general solution for the homogeneous problem given an inhomogeneous linear second order differential equation, y'' + (w^2)y = x cos(x). The exam goes on to ask the student formulate the first step toward solving the equation using the method of undetermined coefficients.

I took the class Differential Equations, and I like to think I learned something about the subject of differential equations from the course: It was an important pre-req that has been built upon in many of my subsequent classes.

There's a problem, though. I can't remember what it is that distinguishes the homogeneous differential equation from the inhomogeneous one. I certainly can't remember the method of undetermined coefficients. I can't solve this exam problem. I know that I learned these concepts, in part because I was tested on them while I took the course, in part because I've learned higher level concepts in later courses that use these concepts as building blocks, and in part because I have the hard-to-define feeling of simply knowing that I had learned them. But if I learned it, why can't I solve it?

Which brings me to the topic at hand: How do I know if I've learned something? I know only if I'm tested--in an academic setting or in any other experience in general. I know I've learned something if I approach a situation I've encountered before in a new way. I know I've learned something if I act differently than I would have before I learned. I know I've learned something if I can apply what I've learned.

This makes it sound like I must be cognizant that I'm applying what I've learned, and, in the vast majority of cases, I think I am. Whether it's something small, like the fact that the hot and cold faucet handles are switched in my grandparents' bathroom, or something large, like a cultural awareness fostered by Introduction to American Indian Studies, I think that I am usually aware that I am acting in a new way based on what I've learned.

[A 'small' thing would be something that only impacts a few of my actions/interactions/aspects of my life--like washing my hands when I visit my grandparents. A 'large' thing would be something that impacts me daily--like my perceptions of culture, my perceptions of my culture, and my attitudes towards interpersonal interaction.]

This sentiment of learning manifested as changing behavior is certainly not new. Virtually all of my classmates wrote variations on the same thing in this past week's reflections. [I was going to link to all of them, but that would have been ridiculous, so you get this obnoxious aside instead.] All had slightly different takes on the subject, but the essence of most of what I read was that learning is proven by demonstration of that learning, or as I would say it, that learning involves the changing of thought processes, and that the outward manifestation of these changes is behavioral.

If you are so inclined, see my previous post for my views on the workings of the mind. From this lens, learning is like trying to communicate with yourself: you can only learn if you are able to overcome your current thought processes and forge new connections in your brain. People have different learning methods because different things help them to form these new connections--for some (e.g. me) the auditory stimuli of a lecturer combined with diligent note-taking and a receptive attitude is the preferred combination.

A quick dip into the internet (<10 min.) has reacquainted me with homogeneous differential equations and the method of undetermined coefficients. I was able to 're-learn' this material much faster than I learned it the first time. I think that this is because the connections to deal with differential equations already exist in my brain; they had just become re-discovered. Someone exposed to differential equations for the first time would obviously need much longer than 10 minutes to decipher the problem given on this exam, much less begin to solve it. The upshot seems to be that learning isn't permanent: just as connections can be made, so too can they be lost.


Incidentally, for anyone who cares, the solutions are:
y=A cos(w*x) + B sin(w*x)
yp [which is the particular solution] =(Ax+B)cos(x) + (Cx+D)sin(x)