College Algebra … an Archeological Study

As we make real progress on improving mathematics education in colleges, shown especially in developmental mathematics, our attention is going to focus on college algebra and the “STEM Path”.  Of course, the name “college algebra” is given to a variety of courses, some of which serve a pre-calculus purpose (and some do not).  For years, I have thought of the name (college algebra) as a statement of “not being remedial”.

Could be that I was wrong about that.  During some recent searching, I came across a paper that Jeff Suzuki gave a few years ago.  Most of that talk is available at https://docs.google.com/viewer?a=v&pid=sites&srcid=ZGVmYXVsdGRvbWFpbnxqZWZmc3V6dWtpcHJvamVjdHxneDo2MWI5YWE4YzU2MDM1MmY3; if you have trouble with that link, search for “Jeff Suzuki Project” to get a list of his presentations.

Assuming that the content of that history is essentially correct, here is a brief statement of what college algebra is today:

College algebra is a collection of mathematical topics for general education, taken in place of calculus.

Some of the information in the Suzuki paper is in the form of book references to the 19th century.  This led to a book, possibly the first, to use “College Algebra” in the title — George Wentworth’s “A College Algebra” (1888); a later edition (1902) is available at https://archive.org/stream/acollegealgebra07wentgoog#page/n12/mode/2up .  In the same period, Webster Wells authored “University Algebra” (1879) and “College Algebra” (1890); see the 1879 text at http://books.google.com/books?id=uKZXAAAAYAAJ&pg=PR7&source=gbs_selected_pages&cad=2#v=onepage&q&f=false

These courses were taught as universities (Harvard, Yale, Princeton, Bowdoin, etc) reduced their mathematics requirements.  The college algebra course was not designed to prepare students for calculus.

These early college algebra books did not contain some current topics (factoring and graphing, for example).  The addition of graphing (including properties of functions) is related to calculus preparation; factoring is generally not so related.  Overall, the current college algebra course is clearly a descendent of this earlier course.

One of my current projects is to study the math courses required before calculus in my state (Michigan); Michigan does not have a system for higher education, which results in diversity in mathematics — college algebra, precalculus, and other courses are used.  However, the overall approach (in Michigan and elsewhere) is to consider these as being an equivalence or subsets; either the college algebra course(s) equate to the pre-calculus course(s) OR the college algebra course is a prerequisite to pre-calculus (that is very rare in Michigan).

Therefore, I believe that this is our current method of preparing students for calculus:

After establishing that the student does not need further remediation on high school mathematics, the student enrolls in an antiquated general education math course with a few valid preparatory topics, with the unreasonable hope that this will prepare them for calculus.

Much of our apparent curricular dependency (stuff in college algebra that is needed for calculus) seems to be an artificially created dependency — we need this radical simplification because that technique is needed for a few problems in calculus, and those problems were created in calculus to show why we needed radical simplification; we need this multi-step factoring topic in college algebra because we have created a set of problems in calculus that require creative factoring.

I encourage us all to study the “Mathematical Sciences in 2025” (available at http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025 ).  Some parts of our curriculum are archeological artifacts from the 19th century, and some parts date from the mid-20th century.  Very little of our curriculum reflects either current needs of client disciplines; not much more of it reflects the needs of mathematical sciences.

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Just for Fun … Creative Factoring

I’m teaching a class focused on individualized learning and flexible pacing.  One student in that class took a test on factoring in our intermediate algebra course.  In the process, I experienced something very enjoyable — a creative way to factor a polynomial.

Here is the situation:

Problem:   Factor r^4 – 16

Student:   (r – 2)(r³ + 2r² + 4r + 8)

Initially, I found this a bit confusing; I was not expecting to see a proposed factor with 4 terms.  In the materials, we focus on patterns to factor binomials involving the difference of squares.  So, I asked the student why he did this; his answer was “it checks”.  [This is exactly what I tell students when they ask WHY we factor a polynomial in a specific manner.]

After a quick transition from confusion to mathematical thinking, I looked more closely at the cubic factor.  Sure enough, it factors to produce:

Correct answer:   (r – 2)(r +2)(r² + 4)

This particular student (planning to be an engineer of some sort) had a creativity I would like to see more of.  The only negative feedback I had to deliver was “Finish the factoring”.

I found this to be just a lot of fun (though I doubt this student enjoyed it as much as I did, though he did enjoy it).  Mathematical fun is meant to be shared.  In 40 years, I’ve not seen a student do this; it’s too good of a thing to keep to myself.

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Inequity in the Math Classroom

I had an experience last week which I just need to share; I’ll try to explain why.

One of my classes this semester is a “Math Lab” class in which we have no large group lectures; in fact, the class has students in 3 different courses.  Students can work faster, and take tests when they are ready.  The basic methodology involves students working problems in the online home systems.

The classroom used for this Math Lab class is clearly going to be different; we provide 10 computers (desk tops) in individual work stations around the wall.  Students can also bring in their laptops and notebooks to use, at tables.

On this particular day, 16 students were in this class.  Six of the students were minorities (african american in this case), and the other 10 were majority (white).  That is not unusual.

What was unusual is the classroom geography.  Every one of the minority students was at one of the computer work stations; every one of the majority students was at a table using a laptop or notebook.

This separation speaks to inequities — the minority students lacked the resources of their own, so were using the provided computers.  For the work being done, the computers were adequate … but the difference bothers me quite a bit.

Students with a portable device can move with their computer; they can socialize in different ways, and they can bring their computer to me with a question.  Students using our computers do not have those choices.

You might be thinking … “so, provide laptops instead of the desk top computers”.  Sure, we could do that (we’ve been trying).  However, it still bothers me: when a category of student tends to lack a resource, students in that category face additional challenges to completion.

I understand that the causes for this inequity are complex (employment, wages, and financial aid … as starters).  I understand that the situation is “not my job” as a math teacher.  Those facts do not change the moral dilemma: a category of students face additional barriers.

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Conversation II: Herb & Jack on the Why — Practicality of Theory

In response to a post about STEM students and the traditional developmental mathematics curriculum, Herb Gross began with this quote (from a prior talk he gave):

The music is not in the guitar.

I think Herb is saying that mathematics is not in the visible tools used, whether these tools are procedures written down or technology used to answer questions.  This is a great point, and it suggests that we question any suggestion that we limit the content of mathematics courses to just those things seen as ‘practical’.  Seeing mathematics as being bound by the practical (for STEM or non-STEM) is a self-defeating behavior; a health profession is based on continuing growth, and growth depends upon research both applied and theoretical (the two work together in surprising ways).  Our students are future policy makers — do we want them to only value mathematics that is practical NOW?  (Think University of Wisconsin budget cuts.)

The music, and the mathematics, is based on connections among concepts.  This speaks to the growth of mathematical reasoning and critical thinking.  Herb adds this comment:

So I am not overly impressed with the pass rate improving as much as I am in seeing what the effect is further down the road.  In fact one of the reasons I don’t like non-algebra/calculus based courses is that even the students who are most successful in these courses tend to know how to crunch numbers into the calculator but have little feel as to what to do when the distribution is anything other than normal.

I think Herb is speaking to a basic goal of education — the improvements retained over a longer period of time, meaning improved capabilities.  The comment Herb makes is important, and I think it applies to most algebra based courses; I also wonder about calculus based courses.  Look at this re-phrasing of a critical part of Herb’s comment:

Students tend to know how to manipulate symbols or numbers often with the use of tools but have little understanding as to what to do with mathematical concepts applied to a new situation. (JR)

Creating scalable change within an individual involves some of the same work as creating scalable change in a profession.  A more complete view of learning is required, with less focus on ‘passing’; passing is a great thing, but it can not be the core measure of our success.  We seek to create mathematical abilities, including the willingness to apply existing knowledge to new situations where this knowledge is not sufficient.

Students in STEM programs need a broad foundation in mathematics, combining procedural and conceptual fluency.  To some of us, we follow that statement with “Non-STEM students to not”; this is where we can make large mistakes.  The mathematical needs of citizens and the mathematical needs of our partner disciplines are not different in a basic way — they need procedural and conceptual fluency as well.  The difference, overall, is a matter of degree and extent.  STEM students need MORE, not so much ‘different’.

Our work in the AMATYC New Life project supports this single-source approach to mathematics — the Mathematical Literacy course serves the needs of all students.  The initial uses of the course have often been for non-STEM students; however, the outcomes of the course were designed back from the needs of all students.  I agree with the design of the New Mathways Project (Dana Center), which has a similar course serving all students.

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