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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Does College Mathematics Have a Future?

I have been wondering about something over the past few months. The concerns originated much earlier, as it seems that people are trying to avoid algebra within college math classes for non-STEM students.  More concerns were added as policy experts suggest that we align mathematics requirements with programs and, ideally, contextualize math for non-STEM students.  #CCA #STEM #MathPaths

There seem to be two premises at work:

1. STEM students need lots of algebra, like we’ve been doing.
2. Non-STEM students are harmed by algebra, and need something less ‘challenging’.

You can see by my phrasing that I am not objective about these premises.  Many people — mathematics educators, policy experts, and more — presume that STEM students, especially those headed towards calculus, are well-served by a college algebra experience.  The problem is that (1) the typical college algebra experience lacks development of covariational reasoning needed in calculus, and (2) our client disciplines have a more diverse need than we work with.  We continue to dig deep into symbolic calculus (which is one of our great achievements) but we downplay the usefulness of numeric methods that are heavily used in engineering, biology, physics, and more.

The STEM life is much more than putting calculus on top of algebra.

A brief story:  At a recent state MAA meeting, I attended a student session on mathematical modeling in biology.  The presenters where all about to get the BS in biology, and reported on fitting models using Matlab (Matrix Laboratory).  After the session, I asked one of the presenters where they learned the techniques … in a math class?  Nope — their biology professor taught them mathematical modeling because their math courses did not.

The non-STEM students are being tracked into statistics or quantitative reasoning, with statistics having the bigger push.  Policy experts push statistics because it is ‘practical’, and people will ‘use it’; these statements are true to some extent.  The problem is that almost all mathematical fields are practical.  In particular, algebra is practical.  Mathematics courses have failed to present algebra as a practical tool for living and for basic science & technology.

Even in a quantitative reasoning courses, we tend to de-emphasize great mathematical ideas.  Sure, we cover finances and statistics, voting and logic; however, the symbolic work combined with the concepts for transfer to new situations tends not to be there.  We use one of the best QR books on the market, and I supplement heavily on functions and related concepts; still, I do not think it is enough.  Some QR courses only apply a couple of concepts (such as proportional reasoning, or math in the news); great components of a QR course … terrible foundations for a QR course.

The risk I see is this: At some point, mathematics will be eliminated.  Non-STEM students get tracked into statistics and weak QR courses; mathematics is thereby eliminated for these students.  STEM students outside of mathematics are only required to show some basic background, and then all of their mathematics is taught by other departments (see biology story above).  The only mathematics students around will be mathematical science majors, and (in most institutions) this is far too small to support mathematics.

We need to do two difficult things:

• Get our heads out of the sand, in terms of modern mathematics (what we should be teaching)
• Effectively argue against the decay of mathematics requirements (especially in two-year colleges)

Fortunately, we have resources from people wiser than I … such as the Mathematical Sciences 2025 material (http://www.nap.edu/catalog/15269/the-mathematical-sciences-in-2025 ).  Please take a look at the diverse nature of mathematics needed in STEM fields, and think about how narrow of a focus we have.

The major threat to mathematics requirements comes from policy influencers (CCA, JFF, Lumina, etc).  Just because they say it, and have ‘data’, does not mean the idea is good or safe.  The degree requirements in institutions are the responsibility of faculty (including mathematics faculty).  It is our job to honor that responsibility, which does not belong to these external agencies.

Let’s keep mathematics as a valid component in a college education.

 
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Walking the STEM Path 2: One Course, or “APL Design”

In the early days of personal computing, it was clear that digital storage was very limited; initial on-board memory was often measured in kilobytes (great by those standards in the 1970s).  The computer speed was decent for that time; as a result, programming languages faced issues and constraints.

As a mathematician, the most beautiful programming language was “APL” … the acronym for the obvious name “A Programming Language”.  You say you’ve never seen this  language?  Well, take a look at the stuff over at http://en.wikipedia.org/wiki/APL_%28programming_language%29 .

APL used an applied mathematics approach to programming.  Need a matrix invert operation?  One symbol did that.  Need a row operation?  One symbol.  Each symbol in APL was a wonderful contraction of a big idea, just like mathematics.  Of course, you needed a special keyboard to use APL.  Small price to pay.

Here is the theme song for the person who ran the local training for APL back in the day:

If your program does not fit on one line, you have not thought about it enough!

In other words, if you have not analyzed the problem intelligently and with insight, your program becomes multi-line and shows that you have more work to do.  Of course, programming has gone in a totally different direction, where we worry about ‘time’ more than lines of code.

In the STEM path, we are talking about connecting developmental-level mathematics with Calculus I. Think about this path as a problem to solve.  If we can not write this program for one semester, we have not thought about it enough.

Over the years, we have developed several ‘solutions’ for this path. Some involve a two course sequence of ‘college algebra’ and trigonometry.  Others involve ‘college algebra’ then pre-calculus.  Some have 3 courses — college algebra, trig, and pre-calculus.  Some institutions have a one-semester option (often called ‘pre-calculus’ or ‘college algebra and trig’).  A few other combinations exist.

We often allow content inflation in these courses by focusing on procedures rather than capabilities.  A well-prepared student can either figure out a needed procedure, or look it up once.  On the other hand, a student who has experienced the “100 most important tricks before calculus I” will not be able to figure out much, and will lose most of these tricks quickly.

What are the capabilities needed for calculus I?  We have a very good starting point for that conversation.  Take a look at the MAA Calculus Concepts Readiness test (http://www.maa.org/publications/periodicals/maa-focus/maa-updates-its-test-for-calculus-readiness).  The first item on that web page shows this problem:

Suppose you have a ladder leaning against a wall. Now suppose that you adjust the slant of the ladder so that it reaches exactly twice as high on the wall.  The slope of the ladder [now] is:  a. Less than twice what it was   b. Exactly twice what it was …

A student knowing how to handle that problem is likely to be better prepared than a student who can correctly evaluate a difference quotient for some arbitrary function.

If your pre-calculus path has more than one course between developmental and calculus I, you have not thought about the problem enough.

This “one semester … if not, finish solving the problem so it is” approach has been a recent trend at the developmental level.  Many of us are replacing 3 (or 4) procedural courses with 2 courses which provide both skills and reasoning.

We need national leadership from MAA and AMATYC on these issues; those organizations are ready.  We need many of us involved with an effort to upgrade and reform the STEM path.  Are YOU ready?

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Walking the STEM Path I: Take Time to Smell the Functions

As we engage in a conversation and discussion about “pre-calculus” (or ‘college algebra’ to some), I am thinking of our curricular goals and how we emphasize what is apparently important.  When the two align, we have potential for success; when our goals differ from what we emphasize, non-success is guaranteed.

Our work in pre-calculus deals primarily with functions (of all kinds).  That makes sense.  However, take the case of ‘inverse function’; whether we are talking about a specific relationship (exponents and logarithms) or the general concept, the idea is important on the STEM path.  The emphasis for most of our courses is on the following:

1. Replace y with x (once), and x with y (all times).
2. Solve for y
3. This is the inverse function, called f^-1

We often feel good about this when combined with the identification of one-to-one functions.  Once we practice finding the inverse, we sometimes explore what the inverse does … sometimes, we present this in terms of composite functions.

This procedural emphasis on ‘finding the inverse’ hides the purpose:  All inverse functions are a matter of undoing.  Algebra starts with inverse operations to solve equations of limited types, where we almost always emphasize the WHY.  In pre-calculus, we take a remedial approach:

• The ‘why’ is too difficult, and we wait until calculus to deal with it.
• Correct answers are an accepted proxy for understanding mathematics.

The procedural approach submerges and prevents understanding; transfer of learning will not occur in most cases.  We can do better: Inverse functions can be approached from the ‘undoing’ perspective, in two senses:  We undo the operations in the function in the appropriate order, and the output for f, when substituted into f^-1 results in the original input.  [We should really create a more reasonable notation for inverse functions.]

Another example is ‘end behavior’ of rational functions.  Our typical approach is:

• If the leading term of the numerator is a higher degree than the leading term of the denominator, the function approaches positive or negative infinity as indicated by the coefficient of the numerator’s leading term.
• If the leading term of the numerator is a lower degree than the leading term of the denominator, the function approaches zero.
• If the leading terms have equal degrees, the function approaches the value of the quotient of the coefficients of those two terms.

Some textbooks do base this end-behavior topic on a discussion of limits (a good idea).  Seldom do we approach end-behavior with an understanding base, which might go something like this:

• End behavior analysis has nothing to do with reducing a fraction.
• Terms never ‘reduce’; factors do.
• End behavior is based on analyzing the terms with the greatest influence on the values of the numerator and denominator.

Our complaint in calculus is that students do not know algebra; however, many pre-algebra topics are approached in a way that avoids dealing with those algebraic struggles — like ‘when does a fraction reduce’.

The pre-calculus experience must involve deep work with functions, combined with a focus on fundamental algebraic ideas.  Procedures can help students become efficient; when presented without that deeper understanding of functions and basic algebra, we create our own potholes and ditches in calculus.

Unless your calculus students never struggle with function ideas, your pre-calculus course deserves a critical analysis — does the course provide a good sense (feeling, smell, vision, etc) for functions and covariation?  Unless your calculus students never make algebraic faux pas, your pre-calculus course deserves a critical analysis — does an emphasis on procedures avoid dealing with basic algebraic ideas?

 
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