Slaying the Calculus Dragon

No doubt about it. Many students consider calculus scary, right up there with monsters under the bed. Calculus is the Minotaur or St George's dragon of math at school. Sadly, schools have done little to undermine its almost mythological reputation, what with “derivatives” and “integrals” and those frightening numberless equations recognizable by the initial elongated “∫.” There is like, what? 100 equations, that, according to most teachers, need to be memorized.

It is a pity, because calculus is really the Wizard of Oz, terrifying to behold, but quite tame behind the curtain. Did you know that most people do calculus in their heads all the time? In fact, because the numbers associated with calculus are ever-changing, moment by moment, doing calculus with numbers is a bit pointless. A mother filling the bathtub very often does not want to stand around watching the water. She knows that the water is coming out of the faucet at a certain rate. She knows the bathtub is filling at a certain rate. Every moment the volume of water is changing. Yet, she reliably comes back to check the tub before it overflows.

The high school quarterback and his wide receiver communicate an even more difficult calculus on the field, seemingly by telepathy. The quarterback never aims the ball at the place where the receiver is standing. That mental math is too easy, more like algebra or even arithmetic. No, he aims the ball toward the place he hopes the receiver will be. In his head, he calculates the trajectory of the ball, the amount of force necessary (oh my gosh, not physics, too!), the speed of the receiver, and every other factor. And most of the time he gets it right, and the pass is completed. The fun thing about calculus is that the numbers are ever-changing. It is like hitting a moving target, whereas algebraic numbers thoughtfully stand still.

At its core, calculus is nothing but slope. Remember humble slope, change in y over change in x. Slope is an expression of rate, such as, change in miles over change in time, most commonly called “miles per hour” or mph. The graph is a straight line, so you can pick any two points to find the slope. But what if the graph is curved? If you were to magnify each point and extend its line, every point has a different slope. That is because straight lines illustrate rates like speed (velocity), while curves illustrate rates like acceleration (getting faster and faster each moment), just like the football getting slower and slower until it reaches the top of its path, comes to dead stop (but only for a moment) and then gets faster and faster.

A graph has three basic pieces of information, the x data set, the y data set, and slope. “Derivatives” are used to find the slope of a curve at any point when the x and y data are known. When you know one of the data sets and the slope, you can use “integrals” to find the other data set. Techniques like finding the area under the curve are used to get as close as possible to the exact answer. First you divide the area under the curve into rectangles all having the same “x” length. Then you add up all the area. Obviously, some of the rectangles are a little too small for the curve and some are too big, so your answer is only an approximation. If you shorten the “x” length to make narrower rectangles, your approximation will be closer. Integration allows you to find an exact answer instead of an approximation, however close the approximation may be. But there are “limits” (you have no doubt heard of limits). Your “x” length may be very small indeed, but it can never be zero, because then the sum of the rectangles would illogically be zero.

If you can find a skilled dragon slayer, that is, a teacher who can demystify it, studying calculus can be great fun.

What do Employers Really Want from College Grads?

The summer 2015 issue of the Phi Kappa Phi Forum featured an article summarizing a recent survey of 300 employers entitled “The Value of Student Agency.” It was an interesting article in light of the fact that so many people are lobbying for universities to be job training centers. On line forums are replete with comments like, “No one forced students to accept substantial loans to finance their educations, all too often in fields for which there were no jobs following graduation.” Or, “That person making your latte probably has a masters degree in something no employer finds useful.” Or, “There is a lot of slop that colleges pass off as "education."”

A typical university mission state reads as follows:

ABCU aims to graduate lifelong learners with the courage to challenge boundaries, ask questions and ignite knowledge with creativity. ABCU students take charge of their own intellectual and artistic development and integrate an active, independent, critical and reflective perspective into their lives as a whole.

I suppose to many ears, that sounds like “slop.” So the question is what do employers want from College graduates.” The survey sought to answer that question. According to the results, employers want that “slop.” 90+% employers say they want employees who think critically, communicate clearly, solve complex problems, promote innovation and that these qualities are more important than any specific undergraduate degree. In other words, employers say they want employees who went to college to get educated, not to get job training.

Traditionally, college has been the place to get an education, and employers provided the job training. If college is for education, then, for example, whatever your major, one thing all students should learn at college is how to formulate and defend ideas without resorting to the logical fallacies and ad hominem so prevalent in online forums. However, if college is for job training, then choosing a marketable major/job is the important thing. Given that 65% of incoming freshman need remediation in English and/or math, apparently college is truly the new high school. Right now society needs to decide what it truly wants of college, education or job training.

The second question is whether the employers interview in the survey answered honestly, or told the interver what they thought he wanted to hear. Employees who actually have the desirable qualities say they often find themselves actively discouraged from displaying just those qualities. Thinking critically and communicating clearly implies a contradiction with being a team player. And “being a team player” is usually code for going along to get along.

The biggest problem I see with college students is their lack of seriousness. t would be great if students took their K-12 education way more seriously and qualified themselves for some of the millions of dollars in scholarship money that goes unawarded every year. Instead, what we have are students who think K-12 means "doing time," and then choose their college based on its party-school ranking. Think not? Ask any high school guidance counselor.

Data-Driven Fallacy

“Data- driven” sounds like a great way to make decisions. It even sounds scientific. What could possibly go wrong?

When data drives decisions, stakeholders, especially stake holders under pressure, will find a way to influence the data in their favor. Therefore Well Fargo employees, pressured to meet sales quotas, open dummy accounts. HRBlock tax preparers, pressured to demonstrate the success of the Second Look program, begin regular tax returns as Second Look returns and convert them midway to a standard tax return without the client’s knowledge, and teachers change answer sheets for standardized tests.

When statistics become the most important consideration, employees will create the statistics they need. It is only human nature. Performance evaluations MUST be holistic, not only to prevent fraud, but also, and most importantly, to form a comprehensive evaluation of the employee including factors not easily quantified, yet crucial to student success. Noth8ing is a panacea, not even and maybe not especially “data driven” decisions.