Yes, Necessary Algebra

The latest MAA Horizons has a great opinion piece by Paul Zorn called “Necessary Algebra” (see http://horizonsaftermath.blogspot.com/ to read it).  In our age of change in the curriculum, we need to keep our eyes on the entire system and the important goals.  To the extent that we want people to be able to reason mathematically and apply their knowledge in powerful ways, algebra is not just necessary … it is essential.

In his article, Paul Zorn gives this informal definition of algebra:

You can manipulate unknowns and knowns to solve equations.

I had another of my discussions with a student about the problems of “PEMDAS”.  This student was having great difficulty keeping straight the algebra we are learning (fairly traditional at this point), partly because the rigid application of PEMDAS got her through the prior math course … and now she did not have a single pre-determined set of steps to get correct answers.  Algebra is all about the legitimate choices we have in working with quantities (with and without unknowns).  Reasoning is dependent upon both knowing that there are choices and understanding some of the implications of those choices.

One of the strong trends in our age is the ‘contextualization’ of learning, and the related method of ‘problem based learning’.  Algebra, and mathematics in general, is both practical now and cognitively useful in the future.  Paul Zorn points out that we typically don’t use much of the specifics from our education in any everyday job — whether we are talking about math, sciences, history, or almost any domain of knowledge.  To limit our education to the immediately practical is to take education out of our classrooms; education is about building capacity, not just about providing methods to solve specific problems that can be understood at the moment.

My own approach to algebra, and mathematics in general, is this:

I always want to include some useless and beautiful mathematics in all of my classes.

Education is the exciting work of strengthening human brains by exploring domains of knowledge.  Algebra has a role to play.  As we reform our curriculum we need to keep algebra as one of the core domains of knowledge.

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Are we serving STEM students well?

Here is a question for us to ponder deeply:  Does the traditional curriculum (starting with developmental mathematics and including a pre-calculus/college algebra course) serve STEM students well?  There seems to be some consensus that the traditional curriculum does not serve other students well, those not going in to a STEM field.  We seem to lack a cohesive view of what it even means to prepare students for calculus.

I do have a judgment on this issue, though I want to explain the context first.  Over my time in this profession, I have specialized in developmental mathematics and (more recently) general education mathematics (such as quantitative reasoning).  I have not taught calculus, nor pre-calculus.  This combination makes me quite dependent upon others for information and points of view.  Fortunately, many colleagues have been generous over the years in talking with a person like me.

No, absolutely not … that is my conclusion about our courses serving STEM students well.  If I had to boil down everything people have told me over the years about what students need to succeed in calculus, and in engineering, and in sciences, it would be this:

STEM-bound students need to develop perceptual abilities, flexible and adaptive reasoning, and a work ethic that allows them to acquire needed resources (such as procedural skills and technology usage).

Our current pre-calculus track (beginning with developmental courses but continuing throughout) is an anvil of symbolic procedures with occasional taps of reasoning.  Students will encounter roughly 400 discrete learning outcomes in approximately 20 containers in their experience, none of which prepares them for the cognitive challenges of calculus.  The occasional need to reason (we don’t want to make it ‘too hard’, apparently) are clearly less important to our students; they focus on what we focus on — procedures and correct answers.  As in Lockhart’s Lament, we submerge and disguise the beautiful … the exciting … and the real challenge.

Take a look at the Calculus Readiness (CR)  Test from the MAA (see http://www.maa.org/pubs/FOCUSFeb-Mar11_ccr.html) .  The items on this assessment are far more about perception and reasoning than the rational root theorem; the items are more about strong reasoning than they are about formulas for sum & products in trig functions.

Some people might be thinking that there may be some validity for my point of view if we are talking about a reform calculus curriculum (which is the framework that created the MAA CR test).  However, I see this is our fundamental flaw about STEM preparation:

STEM-bound students will eventually have to apply their mathematics within a content area.

I do not really see a reason to use a traditional calculus program as an excuse to avoid fixing the pre-calculus problem.  Both areas need work, so start somewhere.  Our good students consistently report that they did fine in our calculus courses but then really hit problems when they were required to apply those concepts within their program … whether this happens in the junior/senior courses in their major, or in their graduate courses.  One of my respected colleagues says:

We don’t know what a good pre-calculus course should be, but it is certainly not Pre-Calculus.

Developmental mathematics is in the current ‘hot seat’, in the target, on the radar, whatever your metaphor might be.  That’s fine, as the traditional courses are in severe need of renewal so that they actually help students.  However, the big difference maker will be when we extend the reform work into the pre-calculus and calculus courses.  For too long, we have meekly accepted the role handed to us … a role that places a high priority on ‘weeding out’ those not deemed good enough, by any means necessary.  This is our time to reclaim mathematics, to show all students the core ideas and provide experiences which expand their perceptual skills and reasoning abilities.  Math can be an a magnet, an attractor to STEM fields.

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Do we have a choice about “Intermediate Algebra”??

What would you do with a course that is packed with rules about problems that nobody really cares about?  What would you do with a course is based on a high school course that was last generally seen about 30 years ago?  What would you do with a course with little intrinsic value?

The typical intermediate algebra course is packed with rules about problems that nobody really cares about … based on a high school course not seen for 30 years in most areas … with little intrinsic value.  I’ve heard people say that the existing intermediate algebra course serves the “STEM” student well, but not the non-STEM student.  I am confused by a view that says STEM students do not need to reason mathematically, that performing procedures is enough.  If procedures are enough, we should certainly just give students a calculator with an algebraic package installed and a link to Wolfram Alpha.

Do we have a choice about intermediate algebra?  Yes, and it’s a better choice than banning the course … a course like “Transitions” (algebraic literacy) in the New Life model provides a different vision of what we can do.  If we look at what is required to succeed in a course like pre-calculus or college algebra, understanding of algebraic objects and behavior is more important than dozens of rules about procedures.  The Transitions course puts the focus on understanding and application, providing both numeric and symbolic skills for working with those objects.

For those coming to the AMATYC conference next month (Jacksonville, November 8 to 11), we are doing a workshop on the New Life courses (Friday afternoon, November 9 session W08).  The courses are “Mathematical Literacy for College Students” (MLCS) and “Transitions” (algebraic literacy), which are alternatives to beginning algebra and intermediate algebra.  During the workshop, we will look at the learning outcomes listed for each course … recognizing that there are more outcomes than a single course could provide.  Quite a few people are implementing MLCS this year, and those implementations are using most of the outcomes listed.  The Transitions course pilots will come a little later.

One of the advantages to the New Life model is that these two courses are flexible — they make sense as a set, and they make sense individually.  Both provide understanding about problems that people really care about (including mathematicians), based on modern course vision, with intrinsic value to our students.  The Transitions course emphasizes diverse models (linear, exponential, power, and even quadratic) with concepts such as rate of change, and includes a little bit of both geometry and statistics.

We have a choice about intermediate algebra … we can replace it with a better course, one that meets student needs.

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Do the Math! What does that mean?

I was at a conference this past week, when a keynote speaker used the phrase “do the math!”.  A redesign methodology states that one of the benefits is ‘students spend more time doing math’.  If we ever needed evidence that the mathematics curriculum is mis-directed, these comments would seem to be conclusive evidence of a problem — they are quotes from fellow mathematics faculty.

Perhaps we have lost track of what mathematics is.  According to a dictionary (Merriam-Webster in this case, though they are all similar), mathematics is

the science of numbers and their operations, interrelations, combinations, generalizations, and abstractions and of space configurations and their structure, measurement, transformations, and generalizations

Open any developmental ‘mathematics’ textbook, and you will see something that resembles mathematics … a bit like a scary Halloween costume, where the outside looks different than the true character underneath.  We have gone far from the path of mathematics, much to the detriment of our students.

In particular, we have lost all elements of science within mathematics (especially in developmental math, but also in gateway college courses).  Science is a ‘system of knowledge’.  If it were not for the undeserved special treatment of mathematics, our science colleagues would have long ago challenged our mathematics courses as being a ritualistic mis-education of the masses.  Two hundred types of problems with remembered procedures to manipulate values and symbols to acquire a ‘correct’ answer does not represent knowledge; the resemblance is stronger with uninformed rituals performed with no redeeming value (practical or intellectual).

The emerging models (New Life, Dana Center Mathways, Carnegie Pathways) are all movements towards mathematics.  We can, and must, reform our mathematics courses so that students learn mathematics more than rituals.  As mathematicians, we have knowledge systems that help people understand the world around them … and a knowledge domain that is enjoyable just for the learning.

The person who said “Do the math!” was simply saying “you need to agree with me, because my view of the data says you should”.  The person who said “students spend more time doing math” really meant that students spend more time in some activity that resembles mathematics … but was most likely engagement in the rituals that have taken the place of mathematics.  The fact that the majority of American students believe that they are bad at ‘mathematics’ says more about our curriculum than it does about them.

I still spend a large portion of my teaching time in courses where the content is still traditional; change is not instantaneous.  However, whatever the course, we can take a more mathematical approach by focusing on concepts and connections even as we get students to accurately perform the rituals.  We each need to start on this path towards teaching mathematics so that we are ready for larger changes; our curriculum in 10 years will have little resemblance to that of 5 years ago.  The good news is that we will be truly teaching mathematics when that change comes.

I hope that you will begin your personal journey towards teaching mathematics; perhaps you can even contribute to the professional work that will lead to the change that is coming.

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