College Algebra is, to a large extent, recipe-driven. Consider:

• graphing rational expressions -- we lay out a sequential list of things to do
• Gauss-Jordan elimination -- an algorithm par excellence!
• polynomial long division
• finding the inverse of a function
• and the list goes on . . .

This is not to say that we should neglect mechanical skills. They constitute an important part of the College Algebra curriculum. College Algebra is not "Math Appreciation 101"; it is intended to provide students with the kinds of manipulative skills and math tools that they will need in order to attack "higher-level" problems as they travel down the road through "Trigonometry", "Functions and Graphs", and "Calculus".

On the other hand, I believe that mechanical skills alone are not enough. Our students also need to acquire the vocabulary, concepts and ideas that support the mechanics they are learning.  For some further insights into this issue, read the excellent article "Two dual assertions: the first on learning and the second on teaching (or vice versa)", by Guershon Harel, The American Mathematical Monthly, June-July, 1998.

How can we instill in our students both:

                                    (1) mathematical insight, and
                                    (2) the required mechanical skills?

I suggest that these are not antithetical objectives, but rather, synergistic. Many of the "recipes" that we concentrate on don't need to be learned by rote, but can be easily reconstructed whenever needed (test time, a year later in another math class), because they readily tumble out of a mathematical potpourri of fundamental ideas that the students have internalized.

This, then, is my thesis:

Detailed recipes do not always need to be memorized. Ofttimes, they can be (re)constructed as needed from a small set of internalized fundamentals.

AN EXAMPLE OF THIS THESIS AT WORK:

Consider the problem of graphing rational expressions of the form R(x) =P(x)/Q(x) . The key to this exercise is finding the asymptotes. Students don't need to "memorize" that you find the vertical asymptotes by solving the equation Q(x) = 0. This is M³. What they need is the fundamental insight that, of course, when Q(x) = 0, R(x) is, of necessity, undefined. Furthermore, if P(x) is not also 0, R(x) is going to go off-scale at such a singularity; ergo, these points naturally correspond to vertical asymptotes. So the picture a student should have in mind for those points for which Q(x)=0 is a vertical line, indicating "Here is a vertical asymptote".  So one is led immediately by fundamentals to the discovery each time one does such a graph, that finding the vertical asymptotes means you need to solve the equation Q(x) = 0.  You don't need a memorized recipe that says "find the vertical asymptotes by solving Q(x)=0."

ANOTHER EXAMPLE:

We may teach that for a quadratic function f(x) = ax² + bx + c, the vertex of its graph is given by (-b/2a , f(-b/2a)).  Students don't need to memorize the "formula" for that second coordinate of the point -- that's just more M³!  If they have the fundamental understanding that a graph is a depiction of function pairs, and that, given the first coordinate of a function pair, they can always compute the second coordinate simply by substitution in the function formula, they've got the recipe for free! Furthermore,  they don't even have to memorize the "-b/2a" part of the recipe, if they are already familiar with the quadratic formula!

Pursuing this a little further, a few more fundamentals will turn solving max and min problems for quadratic functions into an insightful, non-mechanical process, as well. If the student

        (1) knows that all quadratic functions are parabolas
        (2) can visualize a parabola and its vertex, and
        (3) understands that max and min occur at the coordinates of the vertex,

then all of the mechanics of max and min also come for free, with the quadratic formula as the only thing that will have been "memorized".  The rest will have been "internalized."

Reiterating:

The student, equipped only with the quadratic formula and a few fundamental concepts, can quickly and easily construct the answers to questions of the type "for what value of x does f(x) attain its maximum (minimum) value, and what is that maximum (minimum) value?". On the other hand, the M³ way lays out the whole recipe, formulas and all, with the student memorizing some of the same stuff in separate compartments of the brain, and consequently suffering from brain overload and its attendant amnesia.

We should, of course, do examples that illustrate the mechanics (unless we have lots of time to do "discovery pedagogy"). But the point is that we should strive for understanding, so that students are not forced to memorize recipes as the way to "do math".