Walking the STEM Path I: Take Time to Smell the Functions

As we engage in a conversation and discussion about “pre-calculus” (or ‘college algebra’ to some), I am thinking of our curricular goals and how we emphasize what is apparently important. When the two align, we have potential for success; when our goals differ from what we emphasize, non-success is guaranteed.

Our work in pre-calculus deals primarily with functions (of all kinds). That makes sense. However, take the case of ‘inverse function’; whether we are talking about a specific relationship (exponents and logarithms) or the general concept, the idea is important on the STEM path. The emphasis for most of our courses is on the following:

Replace y with x (once), and x with y (all times).
Solve for y
This is the inverse function, called f^-1

We often feel good about this when combined with the identification of one-to-one functions. Once we practice finding the inverse, we sometimes explore what the inverse does … sometimes, we present this in terms of composite functions.

This procedural emphasis on ‘finding the inverse’ hides the purpose: All inverse functions are a matter of undoing. Algebra starts with inverse operations to solve equations of limited types, where we almost always emphasize the WHY. In pre-calculus, we take a remedial approach:

The ‘why’ is too difficult, and we wait until calculus to deal with it.
Correct answers are an accepted proxy for understanding mathematics.

The procedural approach submerges and prevents understanding; transfer of learning will not occur in most cases. We can do better: Inverse functions can be approached from the ‘undoing’ perspective, in two senses: We undo the operations in the function in the appropriate order, and the output for f, when substituted into f^-1 results in the original input. [We should really create a more reasonable notation for inverse functions.]

Another example is ‘end behavior’ of rational functions. Our typical approach is:

If the leading term of the numerator is a higher degree than the leading term of the denominator, the function approaches positive or negative infinity as indicated by the coefficient of the numerator’s leading term.
If the leading term of the numerator is a lower degree than the leading term of the denominator, the function approaches zero.
If the leading terms have equal degrees, the function approaches the value of the quotient of the coefficients of those two terms.

Some textbooks do base this end-behavior topic on a discussion of limits (a good idea). Seldom do we approach end-behavior with an understanding base, which might go something like this:

End behavior analysis has nothing to do with reducing a fraction.
Terms never ‘reduce’; factors do.
End behavior is based on analyzing the terms with the greatest influence on the values of the numerator and denominator.

Our complaint in calculus is that students do not know algebra; however, many pre-algebra topics are approached in a way that avoids dealing with those algebraic struggles — like ‘when does a fraction reduce’.

The pre-calculus experience must involve deep work with functions, combined with a focus on fundamental algebraic ideas. Procedures can help students become efficient; when presented without that deeper understanding of functions and basic algebra, we create our own potholes and ditches in calculus.

Unless your calculus students never struggle with function ideas, your pre-calculus course deserves a critical analysis — does the course provide a good sense (feeling, smell, vision, etc) for functions and covariation? Unless your calculus students never make algebraic faux pas, your pre-calculus course deserves a critical analysis — does an emphasis on procedures avoid dealing with basic algebraic ideas?

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pre-calculus, Professional Development | Jack Rotman | May 2, 2015 4:12 pm

3 Comments

By schremmer, May 2, 2015 @ 8:49 pm

Re. Rational Functions. I do not understand what the problem is. Near \(\infty\) all you need is to divide the numerator by the denominator in descending powers of \(x\) and stop as soon as you get a term with concavity. Near a finite input, including possible poles and possible zero, let \(x = x_{0}+h\) and divide in ascending powers of \(h\) stopping as soon as you get a term with concavity.

Regards
–schremmer
By schremmer, May 2, 2015 @ 8:53 pm

I was hoping that you had mathjax installed. (It takes two lines of code.) So, sorry for the latex. (Mathjax automatically typesets it.)

Regards
–schremmer
By Jack Rotman, May 4, 2015 @ 8:30 am

Thanks for the tip … I had not seen mathjax before; it’s now installed, and I’m sure that you will find an occasion to test it on this site. [Initially, I thought ‘mathjax’ was a reference to me … as in “Math Jacks”. 🙂

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