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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Points, Lines, and Mental Maps

Do we consistently use different directions for problems involving discrete points as opposed to equations or functions?  Does “plot” mean something different than “graph”?  Most importantly, what mental maps do beginning students create for graphical representations of points and lines?

I am observing some patterns in my students’ behavior:

1. “Plotting” two distinct points is treated as if they represented a straight line.
2. Graphing a linear equation is done inflexibly, often by the first or last method seen or practiced.
3. Comparing and contrasting basic methods of graphing linear equations is seen as ‘confusing’.

We are doing our test today on linear equations in two variables, in our beginning algebra class.  The very first item on the test is plotting two distinct ordered pairs.  Some students have difficulty with the concept of ordered pairs, but most plot the two points accurately … and the majority of those connect the two points with a line.  This issue came up in our discussions, based on student questions … still, the programming seems too strong to resist: got two points?  Draw a line!!

Our course is quite traditional in coverage and outcomes at this time.  This chapter includes graphing equations by using specified input values, then by intercepts along with a 3rd point; we use a table of values via a calculator, and then graph using slope.  We’ve done quizzes with directions about a method, and worksheets with both method directions and ‘choose the most efficient method’.  However, each student tends to graph every problem by the same method.  Many students are using slope to graph every problem, even when intercepts would be easier … and even when the method is misapplied to a special case (missing x or missing y); graphing y=4 often results in a graph of y=4x.

Although a comparison of methods was part of our sequence of activities in class (as a secondary point), we tried to have a discussion of the methods as part of our review process in class.  This was not popular … most students did not want to think about how methods compared, only about how they can work problems correctly.  This is the same challenge in metacognition that students face when asked about what their improvement plan is … “I will do better” is confused with thinking about how to do better.

I’ve been working at this type of teaching and learning for quite a while now.  More students seem to have a simplistic view of learning in which having one method that gets mostly correct answers is seen as better than understanding how to choose the tool for the problem at hand; consistently getting one type of problem wrong is not seen as feedback about the learning process … it’s seen as an acceptable price to pay for getting other problems correct.

One way to look at these issues is to view them through a literacy framework.  Instead of listing behavioral outcomes for linear equations in two variables, we would ask the question: “How would we determine that students understand linear equations in two variables?”  We certainly include too many topics in the existing courses to assess understanding of most topics, and are left with simpler performance measures.  The mental map developed by each student responds to the conditions of learning, including assessment.

My own teaching and assessment is part of the problem I am seeing.  In some zones, I have limited flexibility … the basic content of the course is fixed.  However, I need to see how I can change the conditions of learning to encourage the more complex mental map that I think is important.

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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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TMI in the GCF and LCD of MATH: TTYL, PEMDAS!!

If we could tweet and text math, we would say things like “Need LCD, remove GCF, remember PEMDAS”.  Wait a minute, we already say those things.  Seems like math classes are ahead of the curve on not communicating well.  Let’s look a little deeper.

The human brain has some limitations that impact how well acronyms work in communication; as a teacher, I would say that communicating with acronyms tends to keep the information processing at the surface level — translating the acronym in to the words — rather than connecting ideas to important concepts.  We say “you need an LCD to add fractions” and have to remind students that an LCD is not needed for multiplying fractions.  Perhaps we would improve our instruction if we banned acronyms.

I’ve tried taking a compromise approach in my intermediate algebra class, where we are currently taking the test on rational expressions.  We use the label “LCD” after we’ve shown that terms need something in common before adding and subtracting.  I’d like to say that the approach improves student learning, but that is not likely to be measurable — partly because it is such a challenge to get students to reason mathematically instead of memorizing rules for getting answers.

Within five to ten years, we will have a different curriculum for most of our students (STEM and not-STEM) where we provide a better mathematical experience for all students.  In the meantime, many of us will struggle with better understanding in our students.  It might help to say “TTYL” (talk to you later) to all acronyms, as much as we can manage.

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