Saving Mathematics, Part III: It’s Not Just Intermediate Algebra

It’s true (in my view) that intermediate algebra must die; that was discussed in a recent post.  We need to look for other places for basic change in the mathematical curriculum.  #STEM_Path #Pre-calculus

In response to that post, a long-term critic of our work with some good ideas (Schremmer) made this statement in part of a comment:

In fact, intermediate Algebra cannot be killed, as long as Precalculus, the reincarnation of College Algebra, has not been killed too. And Precalculus is not going to die either as long as it has not been reunited with the Differential Calculus. And, in spite of the few millions it spent in the late 80s, even the NSF was not able to reconstruct the Calculus

There are certainly challenges to changing these courses on the STEM-path (articulation being the paramount issue).  However, we have done little to work on the known problems.  Whether you think we can create a more efficient curriculum of 5 courses as I do (1 reformed precalculus course, 2 reformed calculus courses, 1 reformed differential equation course, 1 reformed linear algebra course) … or 3 courses as some others do (3 courses encompassing all of those topics) … nothing excuses our continuing past practice in the year 2016 or beyond.

The stakes are high.  If we do not fix this problem, our client disciplines will teach all of the mathematics they really need (much of which is already happening) — and they will stop using our courses in their programs whenever they have the option.  Most of our enrollment are from programs in these client disciplines.

If we do not fix this problem, we continue a curriculum that hides the modern nature of our work from students; who do we expect to become tomorrow’s mathematicians?  Using cool software to teach awful mathematics is a terrible trick to play on students; I compare that to putting a GPS on a 1975 Pinto … it looks, in a very small part of reality,  like we have modernized but the body of the work is mostly useless material.

This is our greatest challenge.  Will our legacy be that we had an opportunity to modernize the curriculum but wasted it … or will people see that the profession can work together to achieve something great?

We must step up; we must respond to the challenge with hard work and collaboration.  The rewards are too great, the risks too great, for us to take the easy path of ‘change nothing’.

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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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Clarifying the Curricular Vision of the New Life Project

The ‘map’ showing how the New Life Project courses (Math Literacy, Algebraic Literacy) fit into the collegiate mathematics curriculum has been updated.

Here is the version intended for mathematics professionals:

We also have a ‘simplified’ version, intended for those outside of mathematics departments:

These new versions continue the same concepts.  The clarifications involve (A) the eventual use as replacements for the traditional developmental mathematics courses (from 3 or more, down to 2)  and (B) placement into algebraic literacy (more than can go into intermediate algebra).

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