Algebra as Calculus’ Farm System

I was struck by a comment left on a recent post of mine by a person called “Rich”:

Algebra leads to Calculus, ask any professor and Calculus is the gold standard that leads to the fun stuff with mathematics.  Unfortunately, there’s no short-cut to go from arithmetic to Calculus.  Many people talk about “real world” problems, but most of them are calc based or need a deeper understanding of higher mathematics.  I see algebra as the foundations of calculus, kinda like what high school football is to college football, and what college football is to the NFL. (emphasis mine)

His reference to football makes for an interesting analogy. Let’s push this one. If Algebra I is high school football and calculus is the NFL, then I agree with the following points:

1. There are way more [high school football players/Algebra I] students than [NFL players/calculus students].

2. It takes more skill to successfully compete in the [NFL/calculus] than it does to compete in [high school football/Algebra I].

3. Some [high school football players/Algebra I students] will go on to the [NFL/calculus]. Many won’t.

Here are some ways that I would extend this analogy that seems to disagree with the commenter.

1. High school football exists for far more reasons than to produce NFL talent. In fact, I would go so far as to say that developing NFL talent is pretty much the last thing that most high school football coaches are considering when they coach their team.

2. High school football has players, pads, refs, marching bands, and all of the things that make it REAL football. It isn’t pretend football. It has the same rules. It is played on the same-sized field. Similarly,  algebra should stand on its own as a class that contains all of the elements of an effective math class.

Here is a big question in my mind right now: Are we concerned that algebra isn’t good enough? Does a rich understanding of algebra have any meaning of its own? Armed with an airtight understanding of basic high school algebra, is a person able to be a proficient, patient solver of life’s everyday problems?

Well, it seems to me that attempting to motivate and inspire high school Algebra I and Algebra II students by treating them like minor leaguers is probably not going to work. In order to draw these students out, we have to treat them to an experience that shows them the stand-alone value that being able to reason algebraically has.

“Rich” seems to be suggesting that algebra finds its value in calculus. Perhaps in that mantra lies the key to the disengagement problem for secondary algebra students. How many classes are taught as if “the fun stuff” was all somewhere else because engaging in “the real world” requires a “deeper understanding of higher mathematics”?

2 thoughts on “Algebra as Calculus’ Farm System

  1. I wonder if seeing the virtues of algebra as a path to calculus isn’t a reflection of the truism that one really only understands any class when one goes onto something that has it as a prerequisite. Or whether the ability to do stuff the calculus way makes so many problems easier to do that you forget to even try algebraic approaches (for example, to find roots of an equation, if all I require is a numerical approximation, I’d probably think to try a Newton-Raphson iteration, as so comfortable and easy, before I’d try manipulating the equation even if that offered an exact solution).

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