Mythical Course Holds Key for Reforming Mathematics

The concern is not that “those who ignore history are bound to repeat it”.  No, the concern is that those who ignore history create conditions that hurt students.  In the case of mathematics, the mythical course called intermediate algebra holds the key for reforming mathematics … and policy makers often fall in to self-defeating behavior because they ignore history.

Here is the core question:

Is it possible for a course to be ‘college level mathematics’ if it does not have an intermediate algebra prerequisite?

In an opinion (Sacramento Bee), Katie Hern writes of the challenges facing the dozens of California community colleges who have implemented an alternative statistics pathway (http://www.sacbee.com/2012/11/10/4974786/new-approach-to-remedial-math.html).  These colleges are either in the Statway network, or they are doing a stat path as part of the California Acceleration Project (http://cap.3csn.org/).  Policy makers in the state, along with math faculty unaware of history, may block this work.

Many problems exist within the current mathematics curriculum, and intermediate algebra is a core contributor to these problems.  With just a bit of cynicism, here are statements that define intermediate algebra in the 21st century landscape in this country:

1. Intermediate algebra is the course that protects faculty teaching ‘college math’ courses from students who might need extra help.
2. Intermediate algebra is a distorted version of a high school algebra II course from 1965.
3. Intermediate algebra is the perfect course to show students that mathematics can be totally without redeeming value.
4. Intermediate algebra is the last math course to employ technology in intelligent ways.
5. Intermediate algebra is the final course that you can assign to a high school math teacher with the directions “just do what you do in the day … the students are not likely to succeed anyway”.
6. Intermediate algebra is the course used to kill any dreams of being in a STEM field.

I do not know of any high school which offers a math course as mind-numbing as our intermediate algebra courses.  We have this belief that our intermediate algebra course is roughly equivalent to a second year algebra course in high school; even before Common Core … even before the NCTM standards … this was not true.  Back when community colleges were being born and growing rapidly in developmental math work (roughly 1965 to 1975), the curricular materials for our intermediate algebra courses were based on the general framework of an algebra II course that existed for a short time.  The high school curriculum changed — and we did not.

We might believe that our intermediate algebra course is still a good thing; after all, matching (or not matching) a high school course does not have anything to do with the merits of a course in college.  What good does an intermediate algebra course do our students?  Most readers will think something like “get students ready for college algebra or pre-calculus”; this would mean that the learning in intermediate algebra prepares students for the learning in those courses.  We confuse ‘covering the right topics’ with ‘preparing students’; college algebra and pre-calculus are more than finite sets of procedures to symbolically derive answers.  A college mathematics course is all about understanding mathematics as a science so that students both see the intellectual beauty and can apply their mathematics.  Does factoring the sum of cubes, or rationalizing a denominator, have anything to do with preparing students for that?

As a profession, we need to recognize the false nature of our beliefs about intermediate algebra.  Until we do, our students will continue to face artificially long sequences of math courses without any basic value.  If we can embrace a shared vision of college mathematics … ‘understanding mathematics as a science, can see the intellectual beauty and can apply it’ … we will open the doors to a better future.  Imagine a math curriculum where we emphasize good mathematics, the joy of learning mathematics, and developing reasoning abilities; perhaps we can build a curriculum which inspires students to consider STEM fields.

The mythical course (intermediate algebra) has been used as an artificial and false measure of ‘college mathematics’.  Our shared professional judgment, involving compromise as all shared work does, forms a reliable means to measure ‘college mathematics’.

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New Life math … at Parkland College (Illinois)

Change does not necessarily result in progress.  Change with a deliberate plan creates conditions for progress, and this idea is at the core of the New Life work in mathematics.  The math department at Parkland College has taken a professional approach to redesign and reform.

In their work, Parkland has designed a new path for non-STEM students, as have many other colleges.  However, they also examined their STEM-path to identify needs of those students which were not being currently met.

An overview of their work is in this document (which will also appear in the DMC newsletter … Developmental Mathematics Committee of AMATYC):Parkland College DMC article New Life 2012   [Thanks to Erin Wilding-Martin and Brian Mercer for sharing the info.]

Good job, Parkland College!!

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Algebraic Literacy: A Course to Help Students

At the AMATYC conference in Jacksonville (November 8 to 11), we explored the New Life model was a topic of much discussion and some sessions.  In these conversations, there is growing excitement about our Algebraic Literacy course as an alternative to ‘intermediate algebra’ … an alternative which also meets more students’ needs.

Conceptually, Algebraic Literacy was designed based on looking at the quantitative needs of students in several college courses … needs that exceed our MLCS course.  For example, the Algebraic Literacy learning outcomes deal with the concepts and capabilities needed to succeed in pre-calculus.  The outcomes also reflect much of what was learned by the MAA in its work with our client disciplines (biology, chemistry, and social sciences in particular); the needs identified are consistent with several fundamental concepts of mathematics … rate of change, symbolic and numeric representations, and basic types of relationships (linear, exponential, power, rational, etc).

Okay, so here is a document with the learning outcomes for Algebraic Literacy: Algebraic Literacy Course Goals & Outcomes Oct2012

One thing to keep in mind … Algebraic Literacy is not an intermediate algebra course like we have been used to.  You will see some familiar topics (equations and inequalities); you will also see some less familiar topics for this level (exponential and power equations).  And — you will see a deliberate coordination of symbolic and numeric methods, with some outcomes addressed only in numeric form (for solutions).

To give some idea of the nature of this course, I wrote a sample lesson for one of the function topics (rate of change).  Take a look: Algebraic Literacy Sample Lesson Rate of Change Exponential.

In designing this course to meet the needs of students, we discovered that the result is a course which is more attractive to mathematics faculty.  Rather than dealing primarily with procedures, the Algebraic Literacy course builds on key concepts (mostly from algebra) with an emphasis on both the symbolism and the application.  I hope that you will take a look at the Algebraic Literacy course.

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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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