College Algebra is Still Not Pre-Calculus :(

My colleagues at the college are having discussions about what the nature of a good precalculus course is; their discussions are interesting, though little consensus seems to be emerging.  I’ve posted on this before College Algebra is Not Pre-Calculus, and Neither is Pre-calc, so I am not going to repeat much of that.

I have also posted previously about some great data reported by David Bressoud, namely “The Pitfalls of Precalculus” (http://launchings.blogspot.com/2014/10/the-pitfalls-of-precalculus.html).  The basic message of that report is that precalculus does not help the students who really needed it, and actually caused harm to students who had little need.  This data, in fact, is less positive than the reviews of remedial mathematics.

The mess and dysfunctional curriculum in STEM mathematics seems to continue due to:

• The acceptance of a procedurally-bound preparation course or courses (college algebra, pre-calculus), without any design basis or data to support this arrangement.
• The rigidity of the calculus curriculum itself, which as resisted modernization for 3 decades (see http://launchings.blogspot.com/2017/05/re-imagining-calculus-curriculum-i.html)

Sure, we have made some progress in pedagogy; the recent MAA “Instructional Practices” guide outlines a bit of the progress (though without much consensus on ‘best’ practices); see https://www.maa.org/programs-and-communities/curriculum%20resources/instructional-practices-guide. The content — that which justifies the existence of a course — is horribly out of date, with a focus on symbolic manipulation and memorization that has caused our courses to not have respect within our partner disciplines.

My colleagues discussion does not generally include what to teach in calculus; that content is taken as a given, like a poorly written play for which our troupe must still put on a show.  No, the discussions generally turns on whether pre-calculus needs to serve the preparation needs for all calculus courses in the sequence or to focus on the first course in calculus.  I am hoping that the conversation will evolve to include the fundamental questions that determine whether calculus courses are STEM-encouraging or merely obstacles the ablest students need to overcome.

One of my colleagues refers to our pre-calculus course as “death by algebra”; I would put it a bit differently:

The precalculus content is a mixture of excessive algebraic procedure combined with a dysfunctional obsession with trigonometric formulae and identities; those who survive algebra are likely to meet their doom in the trig.

Our curriculum is all about serving the important goals.  We want more students to pursue STEM paths and dreams, and for most of them to be successful if they are willing to work.  Our current precalculus and calculus curriculum is an anthropological artifact worthy of study to help people understand how bad stuff can continue for decades; other than that, this curriculum is not worthy of any additional effort.  We need to create a better curriculum — SOON — so that we stop doing damage to the very students who could achieve success in STEM programs.

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Corequisiste Remediation as a STEM Recruiting Tool

Seems like much of the world (in higher education) has gone ‘crazy’ with reforms intended to remove mathematics as a barrier.  Are we happy with that vision of mathematics?  Are we content with a system which minimizes the learning of mathematics in college?

Perhaps we have not seen some potential opening doors which could support the vision for mathematics we would advocate.  Corequisite remediation has been implemented as a disruptive influence on an algebraic-based mathematics requirement … if a student does not qualify for “college algebra”, put them in a non-algebraic course (statistics, liberal arts math, QR, etc) with a support course which will cover a minimum of mathematics (just enough to learn that stat/Lib Arts/QR course).

Take a step back, and think about these questions.

• Do these non-algebraic courses typically have high needs for ‘remediation’? Or, did we have artificially high prerequisites for these courses … so now corequisite remediation allows us to save face while not providing any significant advantage to students?
• Do the initial STEM-enabling courses (such as college algebra and pre-calculus) have high needs for remediation and support?

To the extent that the answers are “no” and “yes” (respectively), the reform process has been mis-directed.

In addition, we have students who have the potential to be STEM majors — but are intimidated by the prospects of passing the STEM-enabling math course (college algebra, pre-calculus, calculus I).  The current reform work deliberately pushes these students into programs outside of STEM.

Let’s re-direct the reform work to meet student needs and enable many more students to achieve their STEM dream.  Instead of attaching co-requisite support classes to non-algebraic math, attach them intentionally to STEM-enabling math courses.  Whether a student barely places directly in to such a course, or minimally passes a prior math course, their prospects are not good currently.   Think about students within 1 standard deviation above the cutoff on a placement assessment, and those with 2.0, 2.5, C, and C+ grades.  Maybe something like this:

Where do we want students to succeed?  If we are okay with students succeeding if they avoid STEM-enabling mathematics courses, then continue doing the current reforms.  On the other hand, if you want students to choose STEM and succeed, it might be time to consider a better use for co-requisite support classes.

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Anti-Algebra College Mathematics: What are we DOING?

Much “cool-aid” has been distributed in recent years (as in “he/she has drunk the cool aid” … become a ‘convert’).  Our institutional leadership cadre sing the praises of ‘alignment’ and pathways, and celebrate the emphasis of non-algebraic courses in college curricula.

Of course, the word ‘algebra’ itself has multiple meanings. In this post, I am referring to polynomial algebra along with the reasonable connections to geometry, trigonometry, and modeling at the curricular level of first year of college.  The delivered curriculum in ‘algebra’ has degraded to the point that the primary student outcome is ‘survival’ that qualifies them to take another course.

This is not the same discussion as “Algebra II for All” in the K-12 world; we could debate the pros and cons of that issue, though in most ways that train has left the station.  Our interest is in college mathematics in the first two years.

At the  highest level, an observation is that the enrollments in STEM-enabling math courses is declining based on increased enrollments in courses aligned with programs (by which I mean statistics and quantitative reasoning [QR]).  As a general education course for students in non-scientific programs I think a rigorous QR course is the best option.  Such a rigorous QR course includes a significant focus on algebra and algebraic reasoning.  We probably don’t reach that goal very often in QR courses.  In any case, the STEM-enabling math courses are declining in enrollment.

Why?  Why does our leadership consider these non-algebra options to be superior?  Is it because they have conferred with us about the mathematical needs of students within the context of their programs and the issues of the 21st century?  Have some of us taken on the anti-algebra mantle to the extent that we encourage excessive emphasis on statistics and QR?

Sometimes, algebra has been used as a filter to weed out students who “can’t make it”.  Let’s be honest — that is not the nature of algebra, only the nature of algebra courses used to weed out students.  A positive … and accurate … conception of algebra is this:

• Algebra provides a set of tools for representing scientific and technical knowledge
• Algebra provides a framework for dealing with quantitative problems which are not primarily computational exercises
• Algebra encourages precise communication

If students do not need to deal with scientific or technical knowledge, AND will not need to deal with quantitative problems, then the emphasis of QR and statistics is not inappropriate.  As mathematicians, we value the precise communication aspect of algebra, and we might even make the case that this type of communication is just as foundational as the ‘regular’ communication areas (writing, speech, etc).  That rationale is probably insufficient to require students to take an algebraic STEM-enabling course.

Let’s just consider the first feature of algebra — representing knowledge.  Take a look at the occupations with the best employment prospects (above minimum wage), and I think you will find primarily scientific and technical fields (including health careers).  Some of the very best employment prospects are in highly quantitative professions.

We don’t need all of our students to declare a STEM major (though we can always dream of what this would be like).  However, I wonder if the rush to completion is putting a large portion of our students in programs for which they are either not prepared for the jobs available OR not prepared to handle the quantitative demands of those jobs.  That statement might not be clear; here’s an example of the latter condition: students in an associate degree nursing program take a statistics class to meet their math requirement, but they are not prepared to deal with problems requiring algebraic representations or algebraic reasoning.

The ‘elephant’ in the room is how poorly we have been delivering algebra-based courses in college.  In spite of fundamental changes in both the mathematics profession and in K-12 mathematics, we still emphasize courses which might be called “death by algebra” … which serve to weed out students rather than prepare students.  How could we, in good conscience, suggest to our leadership that these algebra courses should be used instead of the QR or statistics course?

The changes in college mathematics, so far, have been at the edges — developmental mathematics reform and co-requisites (usually for QR or statistics).  I believe that the external pressure will come to our algebra-based STEM-enabling courses:  either we make fundamental changes to those courses OR the leadership will make curricular changes that take our courses out of the normal set of student programs.  Within 10 years, we could be dealing with a situation in which the only students taking STEM-enabling math courses are those in ‘high’ STEM fields (physics, engineering, perhaps a few math majors).

What’s the future you want to see?  What’s the role of STEM-enabling math courses in your vision?

 
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Saving Mathematics, Part III: It’s Not Just Intermediate Algebra

It’s true (in my view) that intermediate algebra must die; that was discussed in a recent post.  We need to look for other places for basic change in the mathematical curriculum.  #STEM_Path #Pre-calculus

In response to that post, a long-term critic of our work with some good ideas (Schremmer) made this statement in part of a comment:

In fact, intermediate Algebra cannot be killed, as long as Precalculus, the reincarnation of College Algebra, has not been killed too. And Precalculus is not going to die either as long as it has not been reunited with the Differential Calculus. And, in spite of the few millions it spent in the late 80s, even the NSF was not able to reconstruct the Calculus

There are certainly challenges to changing these courses on the STEM-path (articulation being the paramount issue).  However, we have done little to work on the known problems.  Whether you think we can create a more efficient curriculum of 5 courses as I do (1 reformed precalculus course, 2 reformed calculus courses, 1 reformed differential equation course, 1 reformed linear algebra course) … or 3 courses as some others do (3 courses encompassing all of those topics) … nothing excuses our continuing past practice in the year 2016 or beyond.

The stakes are high.  If we do not fix this problem, our client disciplines will teach all of the mathematics they really need (much of which is already happening) — and they will stop using our courses in their programs whenever they have the option.  Most of our enrollment are from programs in these client disciplines.

If we do not fix this problem, we continue a curriculum that hides the modern nature of our work from students; who do we expect to become tomorrow’s mathematicians?  Using cool software to teach awful mathematics is a terrible trick to play on students; I compare that to putting a GPS on a 1975 Pinto … it looks, in a very small part of reality,  like we have modernized but the body of the work is mostly useless material.

This is our greatest challenge.  Will our legacy be that we had an opportunity to modernize the curriculum but wasted it … or will people see that the profession can work together to achieve something great?

We must step up; we must respond to the challenge with hard work and collaboration.  The rewards are too great, the risks too great, for us to take the easy path of ‘change nothing’.

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