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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Saving Mathematics, part I

Because ‘developmental mathematics’ has been so much in the spotlight, we tend to treat the remainder of mathematics in the first two years as a stable curriculum with the presumption that it serves needs appropriately.  I suggest that the problems in ‘regular’ college mathematics are more significant than the problems in developmental mathematics.  #STEM_Path #MathProfess

We have indications that pre-calculus is not effective preparation for calculus (see David Bressoud’s note on “The Pitfalls of Precalculus” at http://launchings.blogspot.com/2014/10/the-pitfalls-of-precalculus.html).  The large data set used provides strong evidence for the fallacy of pre-calculus; the history of that course also suggests that it is ill-served for the purpose (see Jeff Suzuki’s talk “College Algebra in the Nineteenth Century” at https://sites.google.com/site/jeffsuzukiproject/presentations) .

The calculus sequence remains unchanged in any fundamental way over the past half-century, in spite of the changing needs of the client disciplines (engineering, biology in particular).  I believe that our calculus sequence is both inefficient and lacking.  In particular, our obsession with symbolic methods and the special tools that accompany them results in students who complete calculus but lack the abilities to do the work expected in their field (outside of mathematics or within).

So, just for fun, think about this unifying view of mathematics in the first two years.

Pre-college mathematics: 2 courses, at most

• Mathematical Literacy (prerequisite: basic numeracy)
• Algebraic Literacy (prerequisite: some basic algebra, or Math Literacy course)

College mathematics:  5 courses, at most

• Reformed Precalculus (one semester only)  (prerequisite: Algebraic Literacy, or intermediate algebra,, or ACT Math 19 or equivalent)
• Calculus and Modeling I (symbolic and numeric methods of derivatives, integration)
• Calculus and Modeling II (symbolic and numeric methods of multi-variable calculus)
• Linear Algebra and Modeling (symbolic and numeric methods, including high-level matrix procedures with technology)
• Intro to Differential Equations and Modeling (symbolic and numeric methods)

The current curriculum, over the same range, involves 3 to 5 pre-college courses and then from 6 to 9 college courses. The weight of this inefficiency will eventually be our undoing.

By itself, this inefficiency is not strong enough to be a strong risk to mathematics in the short term.  However, our client disciplines are not happy with our work … in many cases, they are teaching the ‘mathematics’ needed for their programs.  In general, those disciplines are focusing on modeling using numeric methods (MatLab, Mathematica, etc); symbolic methods are only used to a limited extent.

Our revised curriculum must be focused on good mathematics, central concepts, theory, and connections … implemented based on sound understanding of learning theory and diverse pedagogy.  The current pre-calculus course(s) offer a good example of what NOT to do — we focus on individual topics, procedures, limited connections, and artificially difficult problems. The capabilities needed for calculus are much more related to a sound conceptual basis along with procedural flexibility.  Take a look at the MAA Calculus Concepts Readiness material (http://www.maa.org/press/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness) .

We can continue offering the same college mathematics courses that the grandparents of our students took; OR, we can take steps to save mathematics.

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Generalities

I have to admit that I am a bit grumpy.  I have to even admit that I was grumpy yesterday in a meeting at my college about developmental education. #Pathways  #MathProfessors

As you may know, I have been in the profession of developmental mathematics for quite a while; in a way, I stumbled upon this work back in 1973, when I was not able to find a high school teaching job … I had spent a year working in a local car dealership doing odd jobs, and managed to get interviewed for an adjunct position at the local community college.

After some time in the work, I discovered that there were both rewards and significant challenges.  I stayed with the job, and eventually connected with AMATYC and our state affiliate; that connection was a key turning point in my life.  All of us involved with mathematics in the first two years of college have a responsibility to our profession and the professional group (AMATYC).

Our work is incredibly important; we make a difference in student lives every day.

So, the grumpiness … essentially, the profession that I have been committed to for over 40 years has been under attack for the past few years.  Reports, foundations, policy makers, and state lawmakers have stepped in to our work; many of declared that developmental mathematics is a failure, and many suggest that students would be better served by being placed directly in to college-level courses with ‘support’.  These attacks, filled with pseudo-data and articulated with propaganda features, seek to preempt the faculty responsibility to maintain the curriculum in colleges.

Generalities … the attacks take some valid criticisms of developmental mathematics, supported by external forces, to create the types of change that certain groups want to see.  Generalities … the challenge of speaking the truth while recognizing the extremes of variation in the work.

At one point in my meeting yesterday, I made some comments about the guided pathways work being started here.  My college has a long history of separation between academics, and between academics and service functions; my ‘generalities’ were meant to suggest that prolonged effort was needed to overcome our decades of certain work climates.

Generalities … when a person does not agree with generalities, the response is often “don’t speak in generalities” (which is what I was told yesterday).  Generalities are the only way to describe a system; this is comparable to having shared definitions in a mathematical system.  Generalities are not the end of the conversation, nor the only factor in decision making; a successful human system requires long-term effort among the community involved.

I am tired of the ‘generalities’ presented as attacks on developmental mathematics.  We know that much needs to be fixed, and I am confident that we (the professionals in the field) can create solutions which will serve our students better.  Some people lob generalities at us in the same way that people lob the “f bomb” in groups; there is an element of bullying involved when outsiders state generalities about how bad our work is.

Rare is the profession where non-professionals are able to implement specific procedures within the profession.

We need a “TEA Party” type movement; perhaps call it “Legislated Enough Already” (LEA) or “Bashed Enough, Dummies” (BED).

Thanks for reading!

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Algebraic Literacy: A Bridge to Somewhere (AMATYC 2015)

Here are the materials being used for the session at the AMATYC conference (November 21, 2015).

1. References, and the New Life Project curricular vision
References Bridge to Somewhere AMATYC 2015 final

2. Algebraic Literacy Goals and Learning Outcomes
Algebraic Literacy Goals and Outcomes 2015 cross referenced

3. Sample Lesson: Trigonometry
Algebraic Literacy Sample Lesson Trig Functions Basics 2015

4. Sample Lesson: Rational Exponents
Algebraic Literacy Sample Lesson Rational Exponents 2015

5. Sample Lesson: Rates of Change, Exponential
Algebraic Literacy Sample Lesson Rate of Change Exponential 2015

The sample lessons are licensed under a Creative Commons agreement … ‘by attribute’; the content can be used or modified, with attribution.

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