Today in seminar, Dr. Craig Cullen challenged our class of PhD students in mathematics education to consider the role of factoring. He posed the statement:
With the availability of tools today no students need to learn the skill of factoring.
Armed with CAS calculators that can perform a multitude of typical high school (and beyond) factoring problems, we were asked to consider if we agreed or disagreed with this statement. Then, we were to defend our stance. We had quite the range of responses. My favorite included:
What is factoring at a deep level? And, is factoring an essential component to algebra? What would algebra look like without factoring?
Factoring is needed for "more math" and is also a part of the gatekeeping process.
We were then given a series of interesting problems to consider. For example, how many quadratic polynomials are actually factorable over the integers? We considered different sets of integer coefficients for the quadratic and used an excel spreadsheet to examine small samples of polynomials. It turns out that the percentage of polynomials that actually factor over the integers is infinitely small. Not surprising, but this really got me thinking...and even led me a bit off topic...
What about calculus?
Everyone knows the two main elements of teaching calculus: derivatives and integrals.
The set of functions that one can take the derivative of is GREATER THAN the set of functions that one can take the integral of.
1. This is something I have thought about while teaching Calculus courses (particularly Calculus II).
As a teacher of Calculus, I am familiar with art of problem posing. Posing problems with derivatives that my students can solve hardly requires a second of thought. Whereas, problem posing for integrals that my students can ACTUALLY solve is a far greater challenge. As teachers we have to attend to the integration problems with an elegance, making sure to pose a problem that is "just right."
2. Just the existence of numerical analysis courses supports this conjecture. Students POUR over approximation techniques for integrating, with no sweat over derivatives.
(Mathematicians, please forgive me for my overgeneralization of numerical analysis).
3. Talk to an engineer, research scientist, or applied mathematician. They will often share that approximation techniques of integration rank NUMBER ONE in most important and useful mathematical concepts in the field (besides statistics).
4. Just think about it mathematically ... First, think about polynomials. Now, that is boring. Polynomials are beautiful; easy to differentiate and easy to integrate. Second, think about rational functions P(x)/Q(x), with polynomials P(x) and Q(x). Now this is interesting. We ALWAYS have a systematic approach to differentiating; but, we lack a systematic approach to taking the integral. In fact, there is no systematic approach to integrating. Sure, we have rules like u-subtitution and trig-subsitution. But, using those techniques is an ART FORM acquired by mathematicians and students of calculus over time. And these techniques work under certain conditions. Third, think about any other function... radicals, trig, etc. and there is similarities in the difficulty of integrating.
So, why is this conjecture important to talk about?
First, I think this conjecture is just interesting. I think it would be a fun experiment in the classroom. I think it would provide rich discussion for prospective secondary teachers, and particularly of interest to in-service secondary teachers of Calculus. Personally, I will be investigating/exploring this conjecture further, with a more math-tas-tic approach. :)
Second, if the set of functions we can differentiate is larger than the set of functions we can integrate, then what are the pedagogical implications of this for the teaching and learning of Calculus?
And finally, I think this conjecture points to the larger overarching theme of both of these discussions. What is the point of mathematics? What is the point of the things we teach? Now, this is not a novel thought or even a novel question. In the inauguration of Educational Studies in Mathematics in 1968, Hans Freudenthal posed a similar question in his editorial, "Why to teach mathematics as to be useful." This is something that as a field we need to reflect on. How can we change of the face of mathematics education or even improve mathematics without thinking about these ideas?