Developmental Mathematics Revival!

Good Algebra

A recent comment on this blog basically asked the blunt question: Basic math seems fine, but WHY did I have to learn algebra? Mathematicians know that the word ‘algebra’ has multiple meanings. In developmental math courses, the ‘algebra’ is usually various procedures relating to polynomials with integer exponents, with a collection of procedures for rational exponents. The traditional algebra course packages material that is either (A) thought to be important for pre-calculus and/or calculus, or (B) what students should have had in high school.

Given this, my honest answer to the question is “There is no good reason for you to learn that algebra.” If you need calculus, we probably are not building your understanding deeply enough; we certainly are not developing your reasoning in the way you will need in calculus. If you do not need calculus, what you experience in ‘algebra’ is unrelated to any mathematical need you might have (such as science classes, technical careers, or life in general).

A reasonable follow-up question would be: “If this algebra is sort-of okay for calculus bound students (and could be improved), and this algebra is not helpful to most students in the course, WHY does the profession maintain these courses built around an amazingly consistent content package?”

I believe that we, as a profession, are committed to helping students … that we want to provide the mathematics they need. We seem to be ignoring a logical analysis of the situation; there must be a strong reason for us continuing the traditional ‘dev math’ package. I believe that there are two processes which combine to create this reason (an illusion of a valid reason):

Myth 1: Algebraic manipulation is evidence of either understanding or mathematical reasoning; quick and correct execution are evidence of better understanding and/or mathematical reasoning.

Myth 2: Developmental students can not be expected to deal directly with abstractions (core mathematical ideas); the best we can do is provide basic skills.

For my college, we use a common departmental final exam for these courses … a practice which I support. However, the final exam for our intermediate algebra course is a set of 40 problems to be completed in 2 hours; the 40 problems represent 40 learning ‘objectives’ in the course … no item on the final involves applying synthesis or learning based on multiple objectives. Good algebra seems to be seen as quick algebra … good algebra seems to be seen as repetitive algebra.

Every day, people make mathematical claims. Whether it is economics, environmental, or political … somebody says “this is growing exponentially”. Do our algebra courses help students understand this phrase? Would students have any idea what conditions allow truly exponential growth … could students tell when the phrase is being used as a rhetorical tactic? Does the phrase “we expect 150000 jobs per month to be added to the economy” imply an equation for our students … could they estimate when we will have replaced the number of jobs lost in the recession? Given a graphical representation of either an equation or data, can our students determine if the representation is accurate or if it is distorted (by inappropriate scales, for example)?

Yes, we have good algebra we can and should provide to our students. Good algebra is not quick algebra (except for experts like us); good algebra involves abstractions and reasoning, and can be messy. We need to have faith that our students are capable of doing good algebra; if we do not have this faith and act on that, we are enabling students to be ‘bad at math’ as a way of life.

It’s time for us to step out of our constraints created by history and myths … step out of that cage, and build a new experience centered on good algebra for our students.

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Content of developmental math courses, Math curriculum in general | Jack Rotman | May 5, 2012 4:17 pm

2 Comments

By Peter Brown, May 5, 2012 @ 5:49 pm

I would conclude that our algebra classes, intermediate algebra being the terminal class for much of the student body is a disjointed collection of procedures that provide no context, no real applications (except for trains), and no relevancy for students. It is basically stemming from a belief that they must somehow be fixed. I have even been told the purpose of one class is to prepare you for the next one in the sequence.
By schremmer, May 7, 2012 @ 9:18 am

Re. “WHY did I have to learn algebra?” Answer: At this stage, for no reason whatsoever.

The problem with developmental mathematics is that, regardless of our intentions, it turns out to be a very effective—and yet completely unnecessary—barrier if only because of the length it adds to any approach to college level course.

For instance, at worse, one semester is quite enough for students to be able to access precalculus—assuming the precalculus is not the usual cookbook. In my experience, with a properly designed precalculus, students quickly realize that the only thing that separates their correct understanding and their flawed implementation thereof is some relatively trivial issue in algebra and they are then quite motivated to resolve said issue which, at this stage, is itself trivial.

I would be interested in alleged counter-examples.

Regards
–schremmer