I knew even while in school that, outside of their area of expertise, my teachers were just as idiotic as the rest of us. However, it took me awhile to discover that, even in their own specialized field, teachers rarely have a deep understanding of their subject. I have found this especially true for science. I’ve also realized that my math instructors along the way were not really operating above a detailed knowledge of a certain set of rules and algorithms they had memorized along the way. And even if they did go beyond this, the curriculum was designed merely to speed us through the lessons, hoping that we –at least the students with some mathematical aptitude– by tediously working through thousands of problems and memorizing rules and procedures by rote, would grow temporarily proficient at following the function-solving templates we were given.
It is not uncommon for me, once I begin studying a subject, to realize that ideas and discoveries in the history of that subject which I had taken as manifestations of genius, were actually arrived at in a very plodding sort of way… and that we don’t hear about all the silly notions held and wrong turns taken by history’s best and brightest. So many discoveries, I’ve found, are based upon accident or just poking things with a stick and seeing what happens. The intelligence, of course, comes in RECOGNIZING when some event has occurred which is jumping up and down waving its hands and screaming, “No, no– it works like this, stupid!”
We also don’t always hear about all the work, sometimes centuries’ worth, beneath the water-line marking the date of some specific person’s iceberg-capping contribution to the general ‘berg. Calculus is perhaps the best example of all concerning this. As far as I can tell, the fundamental ideas of Calculus were already out there before Newton and Leibniz came on the scene and gave us some shortcuts and argued that “approaching zero” could be treated mathematically as zero.
After studying Michael Starbird’s series of lectures, Change And Motion: Calculus Made Clear (2nd edition), I wasn’t sure how I’d do its representative post. After several false starts, I’ve decided that all I really want to write about (besides what I’ve stated above) is that any student is lucky who has a teacher like Professor Starbird, a man who can tell you much of the WHY behind the “here’s how you do it…”
Starbird took me through several areas which I’ve always been a little confused by, and made them clearer. He not only reminded me what a derivative IS, but really drove home that it can be used all over the place– anywhere in which we are examining how the change in one variable affects other variables. It was also very helpful to see him derive the derivative mathematically, and not just do the shortcut method that every calculus student knows.
He also mostly answered my confusion as to what a PARTIAL derivative is. The clearest case is when you have a function with three variables. Finding the partial derivative of a three-variable function just means keeping one of the variables steady so that you can see how the other two are moving relative to each other. The steady-held variable then behaves just like a CONSTANT, and as you know, when you take the deriative, the stand-alone constant-terms just drop out. Makes life easier.
Starbird additionally helped a little with another area I’ve been having trouble in, the Wave Function, but I have to tell you, the last third or so of the course, he was just zipping through, showing us examples of areas in which Calculus can be applied, without really delving very deeply and allowing us to understand it and have time to wrap our heads around the concepts and ideas.
He gives a few different ways to prove the Pythagorean theorem, which I think every math student should see– just as every mathie should see why the shortcut way of taking the derivative works (it ain’t magic, folks).
And I really liked seeing him derive some of the different area and volume formulas we just plug numbers into. I’d like a whole book just on that kind of stuff.
Starbird also explains why velocity is the derivative of the position function, and acceleration is the derivative of the velocity (and so, the second derivative of the position function). I was less clear why the derivative of the formula for circle area just happens to be the circumference.
He does some other cool things, too. Like showing us how, for the graph of a bank account growing over time, the slope IS the interest rate. He also demonstrated why the maximum profit to be made occurs where the marginal cost of production equals the marginal revenue of selling (the ol’ MR=MC thing we economist-types know so well). Basically, it’s because a max-profit point can occur where the derivative of the Profit Equation equals zero. And since the Profit Equation is P = Revenue – Cost, then setting its derivative equal to zero gives us R’ – C’ = 0, or… R’ = C’… with R’ and C’ being just another name for MR and MC.
By the way, the Professor also reminds us WHY maxes and mins occur where the derivative equals zero (remember?… it’s because: when the derivative equals zero, there the slope of the graph will be flat– going neither up nor down– thus, this is a place where the graph is forming either a peak or a trough).
This is probably one of those worthless posts that won’t get much traffic. Some posts start appearing in search results and take on an independent existence. Others, like this one, are just me talking to myself, saying.. way to go, kid… you’re still at it. Mark one more book off the list.
But I also have to remember, it’s not about the reading list, or the number of reads I’ve achieved. The whole reason I’m doing this is because I want the KNOWLEDGE in those books. It’s not enough for me just to “be”… I want to be MORE.