Two things spell certain doom for students in a math class, especially calculus, and they are on opposite ends of the spectrum: (1) not grasping big-picture elements of mathematics, such as the underlying concept of a construction or the overall strategy of a problem solution, and (2) not paying enough attention to (what seems like) the little details. Right now I am grading a set of tests that covered applications of the derivative, and the details are really getting some of the students. Such as:
- Paying attention to the units of measurement. Example: A problem asks for the rate of change in the surface area of a cube with respect to time. One student calculates two values of surface area, measured in centimeters, and divides them to get his/her answer. If s/he had followed the units, then whatever the correct answer might be, it’s not obtained by dividing two area values — centimeters over centimeters will not give the correct units, which would be centimeters per minute. How come calculus books never make a big deal out of dimensional analysis? Keeping track of units is a very easy way to avoid wrong moves, and I’ve never seen a calculus text that points out this fact, which every science major knows.
- Getting the independent variable right. Students have a natural tendency to call every independent variable “x” and every dependent variable “y”. (We teachers reinforce this by referring to the “x-axis” and “y-axis” in every graph we make…) This becomes deadly in problems that involve more than two variables, like the aforementioned related rates problem. The students are given that the volume of the cube is changing at a constant 10 cm3 per minute, and they are asked to find the rate at which the area is changing when the side length is 30 cm. The independent variable here is TIME. But when students (quite understandably) label the side length with an “x”, and then (correctly) come up with the volume at x3 and the area as 6x2, bad things will happen if they think that “x” is the independent variable here and start taking derivatives with respect to x, (because, you know, x is always the one you plug in) rather than time.
Mathematics is a chaotic system in which the omission or misuse of the smallest detail can lead to major errors. Calculus can often be overwhelming because students have to pay attention to so much — not only the grand, overarching concepts of the subject but also the little details like these. And given that many students have, er, issues with paying attention just in general, you can begin to see why calculus (and accounting and physics and….) can be hard and unsavory for students. (And why feel-good courses which have a minimum of content, or details, and focus instead on experiences always tend to be more popular than calculus.)