Graduating to three dimensions

Looking back at my academic career an interesting thing occurs to me: my teachers were always holding out on us with the third dimension.

Before I began learning the beautiful things calculus can represent and model, the only time I recall using the third dimension in any work I did was for measuring simple volumes (of spheres, cones, cubes, etc).  My entire world seemed to be made of algebraic curves and trigonometric triangles.  When someone traveled on an airplane for example, I made a line or curve across a flat map to track their progress.  Molecules had simple lewis dot structures to show their orientation and that was that.

As I learn about the true mechanics of waves, moving bodies beyond simple particles, and molecular arrangements due to certain bonding schemes I realize the main difference between college level and high school level learning seems to be paying attention to a third dimension.  Towards the end of my high school career I had some advanced classes in physics and some calculus that introduced concepts that consider the third dimension, namely the right hand rule for cross products (not that we were told that was what was going on) and to find where those volume formulas of yore were derived from.

I remember the right hand rule enraged me when my teacher explained that was how all the electromagnetic forces we would be considering were handled.  Why should I have to contort my hand and then align it with the forces explained in a problem?  Surely there must be an easier way?! 

There was were vectors.  I think our teachers teased us with our early introductions to vectors.  Planting the seeds of a third dimension then and there, possibly testing our potential for higher learning with those first rudimentary lessons that never seemed to go anywhere but was still "in the curriculum".  I must have passed those quiet tests, those subtle moments that lay my entire intellectual merit bare for inspection. 

I have had a lot of instructors put their faith in me as a competent student based on a prior teacher's recommendations.  I can't say with any confidence if I deserved that trust and opportunity, but here I am in college coming to terms with everything they didn't teach me.  However, I am also coming to realize how they still laid the groundwork for later education, whether I was going to make it here or not.  Now I understand why graduation felt like a hollow ceremony compared to my entry into calculus: I had been weighed and measured long before that decorated day.