Pre-Calc Post-Calc

Gary Davis (@republicofmath) wrote an article that caught my attention called What's up with pre-calculus?. In it, he presents a number of different perspectives on why Pre-Calc classes have low success rates and do not adequately prepare students for Calculus.

My perspective on pre-calculus is probably far from the typical student, but often times the study of "fringe cases" like myself can provide useful information on a problem. The reason why my experience with Pre-Calc was so atypical, is because I didn't take it. After taking Algebra I, I had started down a path towards game programming. By the end of the following year, where I had taken Geometry, this little hobby of mine hit a road block. I had come to the realization that in order to implement the kind of physics that I wanted in my game I would need to take Calculus. I petitioned my counselor to let me skip Algebra II and Pre-Calc to go straight into AP Calculus. They were skeptical at first, but eventually conceded to my determination and allowed me to follow the path I had chosen.

Skipping from Geometry to Calculus meant that there were a lot of things that I needed to learn that first month that many of my peers had already covered. I had never even heard the word "logarithm" before, had no idea what e was, and had only a cursory understanding of trigonometry. These were the topics I had missed by skipping Pre-Calc, and I was fully aware of that, so I "hit the books" and learned what I needed to know about them. By the end of that first month I had caught up to the rest of the class and by end of the semester I would be helping other students with those very same topics.

I think the most obvious difference between myself and the "typical Calculus student" was the level of motivation. Many of the students in Calculus were there because "it would look good on a college application". I was there because I wanted to be there. A common problem throughout math education is the "When am I ever going to use this?" attitude. I already knew where I was going to use the math I was learning. I had an unfinished game at home that needed a physics system, and every new piece of information I learned in Calculus made me one step closer to that goal. If you had ever wondered why a 4th order Runge-Kutta method is better than Euler's method, try writing a platformer.

The second difference was a little more subtle, but there were some conceptual differences in how I thought about exponential, logarithmic, and trigonometric functions. The constant "e" wasn't just some magic number that the textbook pulled out of thin air, it was the the unique number with the property that  \frac{de^x}{dx} = e^x and  \int e^x dx = e^x . When it came to sine and cosine, I would think of them like a circle while my other classmates would picture a right triangle. They would hear the word "tangent" and think "opposite over adjacent", but I thought of it more like a derivative. Sure, I had to learn the same "pre-calc" material as they did, but the context of this material was radically different.

A couple years ago I suggested that Pre-Calc should be abolished. The trouble with Pre-Calculus (at least in the U.S.) is that the course needs to cover a very diverse array of questions which includes exponential, logarithmic and trigonometric functions. I would argue that these concepts are not essential to understanding the basic foundations of Calculus. The math curriculum introduces the concept of "slope" in Algebra I, which is essentially the "derivative" of a line. There's no reason why we should be sheltering students from language of Calculus. The concepts of "rate of change" and "accumulation" can and should be connected with the words "derivative" and "integral", long before students set foot in the course we presently call Calculus. As students become more comfortable with these concepts as they relate to lines, parabolas and polynomials, then gradually step up the level of complexity. When students start to encounter things like surfaces of revolution, then they'll actually have a reason to learn trigonometry. Instead of trigonometry being the arbitrary set of identities and equations that it might appear to be in pre-calc, students might actually learn to appreciate it as a set of tools for solving problems.

I think this issue of Pre-Calc is really a symptom of a larger problem. The mathematics curriculum seems to be ordered historically rather than conceptually. I've heard Pre-Calc described as a bridge to Calculus. This makes sense when you consider the historical development of Calculus, but not when considering the best interest of students in today's society. Leibniz and Newton didn't have computers. Who needs bridges when you can fly?

4 Replies to “Pre-Calc Post-Calc”

  1. If trigonometry is "an arbitrary set of identities and equations", you're doing it wrong. There are legions of applied problems that don't require calculus.

    Having said that, I'm tempted to agree with your overall point. I believe a lot of what we teach is due to inertia (and people crying foul if we even broach changing things).

  2. I'd just like to add my personal view point that education should not be merely about content. In your article you highlighted the traditional view of students ("When am I ever going to use this?") and the view you advocate for, at least as I understood: "Motivation and Context should be relavent to the student".

    What I'd like to argue is that both those views are biased towards content, i.e. the information being communicated in the class. While there is a more important aspect which is the soft skill, that is not _directly_ teachable, which is called Mathematical Maturity. Apart from the name "mathematical", it do not really believe it is about "math" either. Basically, it is about the set of logical, reasoning, strategical planning, and problem solving skills that one gains during his education. Twenty years after graduation it is quite plausible that no one will remember what a cosine is, nor having actually needed to used one in real life. But what he will still have then, and for the rest of his life, is the logical skill of reasoning about problems he face everyday, which was _supposed_ to be the main goal of teaching.

    Why do I think this way? Well, look beyond school, beyond college, and even beyond grad school. A researcher who have had his PhD, doesn't really use cosines, nor does he even need to calculate them by hand. In fact, researchers, on daily basis, face new problems that no textbook out there describes how to solve, or even approach. All the "facts" they've learned in school, college, and grad school, come short to their rescue. The real asset those researchers have is the mindset they've gained from studying these "facts" and how whoever came up with them did, as a matter of fact, come up with them. More precisely, what was the problem those people had, and how they managed to think about it and produce the never-thought-of-before "fact" that we study nowadays in school. The "facts" themselves don't really matter, they are useless anyway in the face of new problems. What matters is the logical reasoning, and problem solving skills. As I hope is clear now, this is very far from the two points mentioned in the article, which stressed only "the use of what is taught". While I actually argue that "what is taught" and "its use" are useless to a student (except for learning the moral). "The use" would only be useful to a dumb worker who just repeats what he is being told to do, but not to a someone who actually faces new problems and needs to take decisions and adjustments. That is, educational settings is extremely far from industrial settings - it is not about "the use" or "the contents", but rather about "skill" and "inspiration".

    Have a nice day sir 🙂

  3. @Jason

    I agree that studying trigonometry has merits of its own, but my point is that it doesn't fit under the label of "pre-calculus". Just like there are many trig problems that don't require calculus, there are many calculus problems that don't require trig. The two are separate subjects, and both deserving of study, but the curriculum (perhaps erroneously) makes one dependent on the other. As you say, it's more a matter of social "inertia" than mathematics.

    @Mohammad

    I've made a similar argument before in "A Rebel Math Curriculum", which you might enjoy: http://www.suburbanlion.com/?p=134

    I think part of the problem is that students seldom have the opportunity to see what happens when the teacher encounters a novel problem. As a result, students don't have readily available models for problem solving behavior. In the U.S., a culture of over-testing has made this issue even worse. Teachers become the focus of blame when they don't "teach to the test" and as a result, many have become afraid of uncertainty in the classroom. If we can't even get teachers out of their "comfort zone" and into situations where the answer isn't known in advance, how do we expect their students to excel in such a situation?

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