Saving Mathematics, Part II … Diversions in our Curriculum

Among the threats to mathematics is the ‘diversion’ strategy, wherein colleges look for the least-mathematical option when choosing program requirements.  The original diversion course was “Liberal Arts Mathematics”, or early relatives.  Since then, “Statistics” has gained ground and “Quantitative Reasoning” is sometimes used as a title to make a Liberal Arts course sound more mathematical.  #MathProfess #MathVsStat

I am using the word ‘diversion’ in reference to courses that are used for a ‘math’ requirement, often offered as a ‘math’ course (academic department), while not being a ‘good math course’ (see below).  Quite a few of these Liberal Arts Math or Statistics courses are diversions from mathematics, just like basic math and pre-algebra are diversions (and dev math in general).  The developmental courses have, at least, the excuse that they are not claiming to meet a mathematics requirement … though that is not a always-true statement.

Think about this way of approaching the question of ‘what mathematics’ is required for a college degree … the student is taking course(s), which are samples from the population ‘mathematics’.  Like all good samples, this sample needs to be representative of the population in the important ways.  The question becomes: what are the important characteristics of ‘mathematics’?

Here is one possible list of characteristics:

• use of standard mathematical language and symbolism
• almost all content follows from use of properties in the mathematical system, applied in consistent manners
• the content represents multiple (2 or more) domains of mathematics
• the mathematical reasoning would transfer to other samples of mathematics
• learning can be demonstrated in both contextual and generalized ways

The purpose of this approach is to assess whether a student’s general education math requirement provided them with a valid ‘mathematical’ experience.  If that sample was not representative, then the student experienced a biased sample and is not likely to know what mathematics is (making the reasonable assumption that most students do not have an accurate view of mathematics, prior to the course in question).

In the classic Liberal Arts Math (LAM) tradition, the content is either ‘appreciation’ or specialized with little generalized knowledge; in some cases, the majority of the course derives from proportional reasoning with applications across non-mathematical disciplines.  The tradition of LAM is based in both liberal arts colleges or in ‘math for non-math-able students’.  In the former case (liberal arts colleges), the LAM course would make sense as one of the capstone courses, with an earlier math course that is more of a representative sample.  The latter (‘non-math-able students’) speaks more to our problems in teaching than it does to student problems learning mathematics.

The Quantitative Reasoning (QR) tradition is fairly new, and the QR name is sometimes used as a re-branding of a LAM course.  A strong QR course meets the requirements for a representative sample.  The QR course at my college is our best math course, combining both contextual and generalized results.  However, some QR courses are arithmetic-based applications courses; learning can not be generalized because the symbolic language (algebra in this case) is not required nor utilized.

The comments about statistics being a diversion from mathematics might be the least-well received due to the current popularity of ‘introductory statistics’ as a math course for general education.  The intro statistics course has a lot to offer … in particular, the fact that it is a fresh start in mathematics for most students.  However, the content is mostly from one domain (stat) with just enough probability to support that work.  The primary ‘non-representative sample’ issue, however, is the one about properties — where the vast majority of the intro stat content deals with concepts (good), and reasoning (good) but without a unifying structure (properties).  There is, of course, the irony in suggesting that a statistics course is a non-representative sample.

When a math course is a non-representative sample, students are being diverted from mathematics for that course and the students reach invalid conclusions about mathematics.  Such diversions tend to reinforce negative attitudes about mathematics OR suggest that the student is now good at ‘mathematics’.

All of this is written from a general education perspective.  Some programs clearly need knowledge of statistics, and I suppose a few of these needs can actually be met by an introductory statistics course.  The most common use of statistics in general education is the same as the original “LAM” (liberal arts math): a course that looks like mathematics for students who we do not believe can handle a representative sample of mathematics.

A good QR course is a representative sample of mathematics; although most students in a QR course do not take another math course, the QR course itself is not a diversion from mathematics.

The primary drawback to “QR” is that we lack consensus about that is ‘covered’ in a QR course.  In general, I am likely to be happy with any QR course that meets the standards above for being representative. Sometimes we worry far too much about the ‘topics’ in a course, and attend way too little to the important criteria related to what makes a ‘good math course’.

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