Monday, July 25, 2011

Hot Topics: But what about factoring?

One of the common questions I get about the MLCS course has to do with the big guns of algebra:  factoring, quadratics, and rational expressions.  Instructors want to know if those topics are included.  Here's the short answer:  yes and no. 

First, the yes.

Each of these topics is seen in the course and materials I'm working on. 

Now, the no.

They are not developed in the depth comparable to a traditional algebra text.

Here's why:

These topics are lightening rods in the debate over curricular reform at the developmental level.  Scrapping them entirely alienates some faculty which is never the goal.  However, we are limited on the time we have in the classroom.  There simply isn't enough time to develop all the finesse with these topics normally seen in a traditional algebra course and accomplish the goal of this course.  That goal is specific:  in one semester give students the mathematical maturity necessary to be successful in a general education math class or an elementary statistics class.  The reality is that procedural fluency in each of these three topics is not necessary to succeed in either of the college level courses listed.  So each topic is seen but not developed deeply.  That is intentional.

These are good topics and do naturally appear.  Our goal is to expose students to them on a conceptual and applied level.  Let them see what factoring does first and why it could be used along with its inherent limitations (that is, most things do not factor).  Let them see that not all functions are visualized by a straight line.  We spend more time on exponential functions than quadratic but quadratic functions do make their presence known.  Likewise, there are some functions that appear that naturally include rational expressions.  It has been quite interesting and exciting to see functions arise through another problem or question and see the form they take on. 

For example...

Recently I was writing a problem where students need to make a conversion using pi and average several numbers.  NOTE:  This was not an exercise in calculation but the calculation that arose from a realistic problem being solved.  As I was writing the solution, I thought about how students would approach the calculation.  They could convert all the numbers and then average them but that was cumbersome.  It would be faster to average, then convert.  But are the results the same and why?  When I started proving this to myself, I saw rational expressions and their arithmetic including division and addition.  It was fascinating to see this occur naturally and look absolutely nothing like any application I've seen in the rational expressions chapter of an intermediate algebra book. 

Because this question is real, involves proof, and includes algebra, we will ask students to work through this question.  It won't be simplistic but it's worthy of time and attention.  It also does not take 3 weeks on rational expressions to answer it.  But it does expose students to something they haven't seen that is definitely worth the time.  This is not the only place rational expressions appear but I've listed it for illustration purposes. 

The key thing to remember about this course is that it takes students on a journey where skills are developed but are not the end goal.  So some skills won't be seen and that's ok.  That's why we have intermediate algebra for the students who will eventually need those skills in a precalculus or college algebra course.  We are not trying to be all to all but instead give to a certain population what is more in line with their needs and goals.

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