In my earlier post I mentioned precalculus and pre-precalculus, but never really defined these terms, forgetting that what we mean at Polytechnic is different from what it means in the high schools and most universities.
Precalculus at most institutions is indeed more properly called Analytic Geometry or Advanced High School Algebra. It covers what I consider to be largely arcane and useless topics in algebra and co-ordinate geometry and is of no help to learning calculus or anything else.
We follow the lead of the Harvard Curriculum. To us Precalculus and Calculus is a unified set of courses devoted to the study of real-valued functions on the real line. We use the Harvard precalculus book for the pre-precalculus and precalculus courses.
Even though these courses are "remedial", we do not simply reteach the high school courses. I don't know about other schools, but this approach is a proven failure at ours. The students are bored to death by the lectures and just do not take the course seriously at all. So although we certainly review exponentials, logarithms, trigonometry, the focus of the courses is on really understanding what the concept of function is and how to use it effectively. Again, we are always focused on teaching mathematics as a USEFUL SKILL. The need for "understanding" is completely driven by this.
Over the years, I have always noticed that when students are weak in calculus, the root cause is not their lack of understanding of what a derivative or integral is but their lack of understanding what a function is. So now when I help students, I almost start by asking them to tell me what a function is. Usually, I just get a lot of inarticulate noises ending with "I know what it is, but I don't know how to say it". Others tell me it is a formula. Some try to recite from memory a precise mathematical definition involving ordered pairs. Some mutter something about the vertical line rule.
So here's my big beef. Mathematicians have to learn that a precise mathematical definition is NOT a useful working definition. At the same time students MUST be taught to articulate what they think they know.
And here is the definition we drill into our students at Polytechnic: A function is a box that eats and spits out numbers. Whenever you feed it a number, it spits one out. Different functions do different things. For example, one might be a very stubborn one; no matter what number you feed it, it always spits out a 3. That's commonly called the constant function 3. Another function simply spits out the same number you feed it; that's commonly called "y = x".
The key point we want students to understand is that a function is not an object in the usual sense; rather, it is a process. Once you establish this point, it can be constantly reinforced throughout the precalculus and calculus curriculum and helps motivate everything you do. On the other hand, if you do not constantly remind students of this, then the concepts of derivative and integral remain mysterious to students. They then simply think of derivatives and integrals as being symbol manipulation processes that are for some unclear reason useful to mathematicians.