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 curriculum in general, pre-calculus | Jack Rotman | November 18, 2019 9:47 am

2 Comments

By Eric Neumann, November 18, 2019 @ 1:19 pm

I used to take a very similar view with regards to credit-bearing vs. remedial. My view has changed not in the sense of softening towards College Algebra (I fully agree with you that it is an obsolete and also a remedial course), but in the sense of high school vs. college courses. What is the difference, in your mind and in your experience between U.S. History, the required high school course and U.S. History the 100-level college course? Why don’t history departments offer remedial history to prepare their students for college-level work? What about biology or chemistry? What about culinary arts – do they have remedial home ec coursework for those who never learned how to boil an egg or turn on an oven?

My understanding of the distinction between 11th grade U.S. History and 13th grade U.S. History is rigor in the Bloomian sense – higher expectations of synthesis and analysis. The same facts and dates and overview is presented, but students are expected to thinker more deeply about them. Students’ a priori knowledge base may help them somewhat, but the readings and lectures don’t presume a particular set of incoming knowledge.

Even for foreign languages, we start from scratch and expect students to now be able to master the same amount of vocabulary and grammar in four months for which younger students are given two years. We still give credit for Spanish 101, though.

My issue with College Algebra (really our whole sequence) is not its overlap with k-12 math content. It is with the identicalness of it. Students expect that they will be asked to think more deeply in a college history or science course than in the identically-titled high school course. And yet they expect their college math courses to remain at the purely surface-level that they are used to (for good reason, because we expect no more of them either). In fact, because of time constraints, my college courses (intermediate algebra and college algebra in particular) tend to be even MORE procedural than when I taught this content in high school.

We need to change the paradigm from granting credit based on some arbitrary starting point along the arithmetic to calculus trajectory to an (also arbitrary) consideration of analysis and thought. Your own work is thoroughly in this direction, and I applaud you for it. However, my observation that the most popular examples of work in this direction (Dana and Carnegie curricula) or of such low rigor (in my humble estimation) exposes the challenge before us. How have other disciplines come to consensus on what qualifies as college-level work in their intro courses? Let’s learn from them and try to start asking similar questions.
By Jack Rotman, November 19, 2019 @ 8:40 am

Re: History, chemistry, biology, foreign languages
The typical view of ‘remedial’ in college has been limited by assumption to ‘skill’ areas (math and ‘english’), especially since the typical HS graduation requirement involves n years of both subjects. Most other disciplines are a one-and-done (in terms of requirements).
I’d love to have conversations with other disciplines about defining college-level work. Based on reading a lot of syllabi, transfer lists, and some conversations, my conjecture is that the typical definition is one by existence — historically, course ‘y’ has been offered as a college-credit course (and, therefore, is still a college credit course).
I agree that all popular efforts in curriculum are of depressingly low rigor. In fact, my own classes are no different. A recent test included a slight variation of a non-standard problem solved in class, and almost nobody got it correct. My only success stories come from a QR course where I could focus on a semester-long process of building problem solving (and rigor). Teaching mathematics beyond procedures is incredibly difficult work; considerable content needs to be sacrificed to make room for that work. Small steps on rigor are equivalent to no steps on rigor; the ‘reasoning spring’ has to be stretched more than a trivial amount to prevent it from resuming prior rigidity.

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