School Mathematics can NOT be Aligned with College Mathematics

How do we help students become ready for college mathematics?  How do avoid students earning credit for learning that should have occurred before college?  Perhaps our conceptualization of these problems is flawed in fundamental ways.

As I write one of my final posts for this blog, I am pondering history and future … and the intersection called the present.  Some of this pondering has been pleasant reflection, while much of the pondering has been either professional regret or stimulating conjecture.  I hope to put some of each ‘pile’ in this post.

[Here, “common core” is a place-holder for school mathematics.]

As usual, a problem and its solutions are based on definitions.

• School mathematics is defined operationally by the curricular materials and accepted pedagogical practices.
• School mathematics is usually characterized by a closed system focused on experiencing a constrained subset of mathematics at constrained levels of learning.
• College mathematics is ill-defined with conflicting goals of historical course content and service to the discipline.
• College mathematics is characterized by a closed system serving history competing with components seeking to build mastery of modern mathematics.

The fact that one system is reasonably well-defined while the other is ill-defined suggests that any goal of alignment is unreasonable.  In other words, the reasonable-sounding effort to create a smooth transition from one level to the next is foolish.

Just as groups sought to deliberately disrupt the work of developmental education, groups using ‘alignment’ are also seeking to disrupt the world of college mathematics.  In their view, college mathematics should be more like school mathematics where the system is well-defined operationally by a limited collection of curricular objects (‘courses’).  The presumption is that the core of the college mathematics system is valid and that we can apply the school mathematics process to standardize the alignment.

All of this ignores two related and critical flaws:

1. School mathematics was (nominally) designed to prepare students for college mathematics.
2. College mathematics (as known today) is a collection of obsolete tools along with a bit of valuable mathematics.

At a CBMS meeting a few years ago, I raised the question “When are we going to question the college mathematics courses consisting of excursions into issues that we don’t care about?”  Some in attendance thanked me for saying that we should change the applications in our courses; sadly, that is not at all what I was saying.  I was suggesting that much of college mathematics presented mathematics that we no longer care about as mathematicians.

Advocating for alignment does not mean such alignment is possible; it’s not.  Advocating for alignment does not mean that people support our curricular goals; they have their own agenda (not ours).

Before we worry overmuch about ‘alignment’, we had better make basic corrections to our own system.  College mathematics could be an exciting world for our students to explore with colorful vistas combining symbolic and computational methods supported by conceptual knowledge.  Do not look to MAA and AMATYC to ‘tell us’ how and when … our organizations are too fearful of offending part of ‘us’.

Build local alliances to support experimentation in modernizing mathematics in college.  Do not let ‘alignment’ lock you in to an obsolete and harmful set of mathematics courses.

Remedial College Algebra

We are all familiar with ‘predictions’ based on societal trends which are seldom validated by reality — whether it is flying cars, Facebook’s “population”, or economic stability.  Predictions are often based on a presumption of continuity within the determining forces; people attempt to apply modeling concepts to an open (or semi-open) system.  As mathematicians and mathematics educators, however, we often fail to notice the interaction between forces impacting our curriculum.

At the collegiate level, the most dramatic example of such a disconnect is the course called “college algebra”.  I’ve written before about how ill-designed this course is, considering the role it plays; see College Algebra is Not Pre-Calculus, and Neither is Pre-calc, Cooked Carrots and College Algebra, College Algebra Must Die! and also about it’s history (see College Algebra … an Archeological Study. This post, however, deals with the conditions we are operating within in approximately the year 2019.

For reference, I will be using information about the Common Core Math Outcomes.  (See http://www.corestandards.org/Math/).  I recognize that the Common Core has many detractors, as well as structural problems within (such as insufficient guidance about which outcomes have a higher priority).  However, there is no dispute with this statement:

In spite of ‘problems’ with the Common Core, the Math Outcomes listed are the only usable reference for national conversations about K-14 mathematics.

So, here is the bottom line statement: if one compares the set of Common Core Math outcomes for K-12, they exceed the outcomes normally listed for a college algebra course required prior to pre-calculus.  Even the standard pre-calculus course is repeating content described in the Common Core.  [ACT conducts regular research on ‘national curriculum; the surveys are at http://www.act.org/content/act/en/research/reports/act-publications/national-curriculum-survey.html ]

Complex numbers?  Vectors? Matrices?  Connect zeros to factors? Binomial Theorem? Polynomial functions?  Rational functions?  Those, and more, are listed as Common Core outcomes for high school mathematics for ‘all students’.  I am not trying to equate the high school courses to a college algebra course; that is not a required element for the conclusion about college algebra as a course preceding pre-calculus:

College algebra is a remedial course.

The traditional remedial mathematics courses received that designation primarily because people saw that the content was what students SHOULD HAVE HAD in K-12 mathematics.  We maintained developmental mathematics courses which taught 9th to 11th grade mathematics, and denied college credit for them because students ‘should have already learned this stuff’.  [I am not suggesting that we allow remedial math to get credit towards a degree; in particular, I don’t think intermediate algebra should meet a math requirement.]

My claim is that the college algebra course preceding pre-calculus materially meets the same conditions which resulted in the determination that our traditional ‘dev math’ courses were remedial.  Substantially, every topic in the college algebra course should have already been learned in the K-12 experience.  Certainly, not every student had that opportunity (just as before).  Certainly, not every K-12 school does a quality job in mathematics (just as before) … though this statement also applies to “us” as college math professionals.

At the college level, we often function in isolation from K-12 mathematics; in general, we also continue to work as if the client disciplines exist now as they did 50 years ago.  We have not been sensitive to the dramatic changes in intent within the K-12 curriculum, and sometimes we seem to take pride in our ignorance of school mathematics.  We presume continuity as it relates to our curriculum, in contrast to our intense efforts to improve pedagogy.   I continue to believe something I have been saying for years:

Improving our pedagogy without modernizing our curriculum is like putting a GPS on a 1973 Ford Pinto — sure, we can see a map to help us drive, but it is still a 1973 Pinto.

We teach the importance of continuity within our courses.  I find it ironic (and tragic) that we tend to make basic assumptions concerning continuity within the world around us.  College algebra is a remedial math course.

Math Education in the Face of Climate Change

Our profession is denying climate change.  We continue to create greenhouse gasses with little regard for the planet nor for the vulnerable organisms who become collateral damage for our ignorance.  Our organizations celebrate the isolated experiment in ‘smaller carbon footprint’ while the vast majority of our companies are focused more on tradition than on science.  We are the enemy.

This metaphor is meant to convey the tragic condition of mathematics education in the year 2019.  Some of us have had major changes imposed on us relative to developmental mathematics, which generally leave all other college mathematics unscathed.  Even in some of those places, we still offer just as many traditional developmental math courses.  In almost all cases, the important processes in our industry remain as they were 40 and 50 years ago.

One of the attack lines against developmental mathematics has been “remedial math is where dreams go to die”, and it is true that our traditional developmental math courses did not serve our students well.   The response data to this criticism is trimodal — some of us replaced the traditional courses with fewer and more modern courses, some of us eliminated dev math with ‘corequisite’ strategies, and the rest of us continue business as usual.

If you really want to see dream death in mathematics, study our ‘pre-calculus’ content and courses.  Students enter into the prep for calculus with dreams of being a scientist or engineer or computer scientist; they almost always experience a brain-deadening mix of algebraic procedures and memorization which seems to have the goal of eliminating the ‘unfit’ before calculus I.

In mainstream college mathematics, we hold to tradition … all of this ‘stuff’ is needed for calculus; the rationale:  we have always taught this ‘stuff’ in pre-calculus.  We sometimes can justify the greenhouse gasses of pre-calculus by citing a contrived calculus problem which happens to require this contrived pre-calculus topic.  Our current books — including “OER” materials — for pre-calculus are still descendants of a gen-ed college algebra course never intended to be in a calculus path (see College Algebra … an Archeological Study).

Evidence of this bizarre mix of dream killing curriculum is our habit of having “college algebra for pre-calculus” as the prerequisite for pre-calculus.  College algebra has nothing to do with pre-calculus, just as pre-calculus has nothing to do with calculus (see College Algebra is Not Pre-Calculus, and Neither is Pre-calc and College Algebra is Still Not Pre-Calculus 🙁 ).

Pre-calculus is where STEM dreams go to die.

The most egregious contribution to ‘climate change’ in mathematics?  The fact that numerical methods and modeling are not integrated into the curriculum (at all levels).  All of our client disciplines are heavily dependent on a collection of matrix and modeling methods and technology.  Nobody ever needed all of the manual calculus methods we taught, but many of them were critical before computational mathematics.  With computational mathematics, fewer manual methods are needed — more conceptual rigor is required, along with content to support appropriate numerical methods.

This image comes from a page on this Envisioning Our Future, and is an attempt to envision a solution to our climate change problem.  Eliminating wasted energy and keeping dreams alive are essential criteria for judging the validity of such solutions, and I have no doubt that my ‘vision’ will not be our shared solution.  We need to work together to create viable solutions locally, share these solutions regionally, and eventually develop a national pattern of ‘good college mathematics’ courses.

Are you so attached to tradition that you are willing to contribute to this ‘climate change’ in higher education?  Or, do you want STEM dreams to live and thrive?  Perhaps you are willing to consider fundamental change to our curriculum just based on the criteria “teach good mathematics”. We might have pride in our individual teaching practices, but none of us can have pride in delivering bad or awful mathematics to tomorrow’s scientists.

Our traditional courses create dangerous levels of greenhouse gasses (bad mathematics) and contribute directly to climate change (the death of dreams).  We need to reduce our carbon footprint (more ‘good mathematics’) and actively improve the climate (student dreams).

What are you doing to ‘fight climate change’ in college mathematics?

College Algebra Must Die!

Sadly, many people look at situations with such a strong bias, born of history, that obvious problems are hidden.  Such is the case with the American “college algebra” course.  College Algebra is a glacier which has trapped college mathematics faculty for decades, causing harm to students and society.

College Algebra must die!

In support of this assertion, let’s begin with the origins of “college algebra”.  Based on the research of Jeff Suzuki (see https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxqZWZmc3V6dWtpcHJvamVjdHxneDo2MWI5YWE4YzU2MDM1MmY3) our contemporary college algebra course is a descendant of the original ‘math for liberal arts’ course in the 19th century.  Certainly, the content has changed since the 19th century — we now cover factoring and graphing, which were not so much in the original.  However, the topics in a college algebra course indicate a ‘survey’ course orientation — not a focused preparation for success in mathematics.

This college algebra course was created before any standardized high school mathematics existed.  If we created such a course today, we would classify it is remedial — the content is primarily a subset of Common Core objectives.  Even if we live in a state or region not ‘implementing Common Core’, the local K-12 districts have an intended curriculum with similar objectives and goals.

Any effective preparation for calculus, coming from college algebra, is a coincidence of epic proportions.

Many of us combine a college algebra course with a trigonometry course as the prerequisite to calculus I.  Since this trig work — identities and memorization emphasized — is the most recent mathematics students experience before calculus, it is no wonder that our pre-calculus courses often harm the students they are supposed to help.  See the fascinating study by Sonnert & Sadler (https://www2.calstate.edu/csu-system/why-the-csu-matters/graduation-initiative-2025/academic-preparation/Documents/IJMEST-Sonnert-Sadler-Precalculus.pdf)

The other central reason for the assertion that “College Algebra Must Die” is based on the current best thoughts of both preparation for calculus as well as the improvements in calculus content.  Both of these sets of perspectives are based on long-term work of MAA and AMATYC, and have been articulated fairly consistently over the past 20 years.

Take a look at the MAA “Calculus Readiness” instrument (https://www.maa.org/press/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness) and some research on the CR (https://math.la.asu.edu/~carlson/…/CCR-Carlson,%20Madison%20&%20West.pdf).  Our college algebra courses have little to do with these outcomes and goals, even we use the politically correct title of “pre-calculus”.

You might also want to look at the current MAA CUPM report on calculus (https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/cupm/2015-cupm-curriculum-guide).  College Algebra is an antiquated non-preparation course for an out-of-date calculus sequence.

Many people and institutions (my own included) have worked hard to make a college algebra course a more reasonable general education course, which is ironic given the history of this course.  The result is usually a ‘modeling and functions’ course, clearly not rigorous.  Although these modeling and functions courses are very valuable, they are no longer college algebra courses.

We need to develop potential precalculus courses which have the content validity to justify an expectation that they will be effective preparation for calculus.  Much is known and understood of the basic design issues for such courses; before we can do this work, we need to escape from the College Algebra Glacier that we have been trapped within.

College Algebra Must Die!

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