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 cm^{3}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 x^{3}and the area as 6x^{2}, 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.)

Ah yes, we have the same problem teaching stats, particularly multiple regression analysis. Students have a hard time determining which are independent variables, and which is the dependent variable, not to mention forget that there is only one dependent variable. If they’ve done problems like the one you mention, that could actually create problems in statistics — because there are different types of variables (data), and time is technically not a numeric variable (usually). One of our problems asked students to do a simple, one variable regression to see if the amount of money spent on ads affected the total sales. Two variables: ad cost (independent) and total sales (dependent). Several students, however, used the time series variable (quarters, 1, 2, etc.) as if it were a numeric variable, and I got quite a few bizarre and meaningless regressions trying to incorporate it. Trying to explain the errors in class was a nightmare. They just didn’t understand why the arbitrarily assigned number to differentiate one quarter from another wasn’t a numeric variable, and therefore, why it could not be used in a meaningful regression analysis.

I remember encountering the idea you’re referring to when I was a psychology major — that numbers do not always represent numerical data. To me, that made sense immediately — ZIP codes for instance are just labels and have no inherent mathematical meaning — but that was a hurdle that drove some of my fellow psych majors to become PE majors.

That’s one of those things some people just don’t seem to ever understand. Quarter two isn’t twice quarter one in any sense, I tried that. You could just as easily call them quarter A, B, and so forth, I tried that. And these students should have known this, since they had already had half a semester of database design (and of course you do the whole data type thing when you’re assigning data types to fields).

I mean, what are you saying if you run a regression to see if the ad costs affect the quarter number?

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Why don’t they mention dimensional analysis in calculus books? You have the answer right there – every science major knows it already. In addition, there’s that attitude that says fact-checking and error-catching isn’t

reallypart of math. It’s something I encounter in a huge number of introductory calculus (and physics) students – dimensional analysis won’t be on the test, it’s not an equation, and math – math’s notsupposedto make sense, it’s all just abstractions without any connection to the real world. If the equation says to divide centimeters by centimeters, well that’s what they’re going to do, regardless of whether the answer is supposed to be a rate, or a duration, or a volume.It’s a bad way of thinking about math, and a worse way of thinking about science.