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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Equity and Participation; our Response

This was one of those news reports that really got my attention as an educator.  There are many, of course, which should get my attention as a person and citizen; this report talked right to the ‘teacher’ in my brain.

The role of AP courses has been debated; in the mathematics community, we have some concern about how much benefit actually accrues to the majority of students in AP calculus.  However, as long as AP courses are offered … our goal needs to be equity: no group of students should be participating or not-participating at significantly different rates.

My college serves a blended district — a small-to-medium city (Lansing) and a surrounding area made up of suburbs and rural communities.  AP courses are offered in both the city and suburbs.  Here is a quick breakdown for two of the local districts for student population and AP population.

Lansing Public Schools (9-12) Holt Public Schools (9-12)
Category Student Pop AP Pop Student Pop AP Pop
Black 46% 34% 13% 10%
Hispanic 16% 13% 9% 3%
White 27% 40% 69% 76%  q

The school data came from a tool at marketplace.org; see http://www.marketplace.org/topics/education/learning-curve/spending-100-million-break-down-ap-class-barriers.  The city population data came from the 2010 Census.

One way to look at this data:  The Lansing high schools are about 73% ‘minority'; the AP classes in Lansing are about 60% minority.  Another view: the AP participation rate for white students is 50% higher than their proportion of the population would indicate, while the black student participation is 25% lower.

We might conclude that this discrepancy is a Lansing school problem; that is not the case.  The same pattern is present in Holt … just not quite as extreme, due to the smaller minority student population.

In case you are wondering, the national figures are:

Category Student Pop AP Pop
Black 15% 9%
Hispanic 21% 17%
White 54% 59%

We’ve known that minority students are over-represented in developmental math courses in college.  This recent data suggests that the equity problem extends through the whole range of abilities.  I respect the difficult work that our K-12 colleagues are doing, often without support or respect; this equity problem is not about the teachers … it’s about society and us.

We can, and must, do better.  How will we respond?  I think we recognize signs of a problem; what actions can be taken?

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What does ‘sin(2x)’ mean? Or, “PEMDAS kills intelligence, course 1″

My department is starting a conversation about pre-calculus and intermediate algebra.  I’m very pleased we are taking this step, and it is great that I do not know where our work will take us.

In our discussion yesterday, one of the concerns expressed was that students tend to not understand functions … including function notation.  People referred to the classic error:

sin(2x) = 2*sin(x)

The problem runs deeper than functions and function notation.

√(a²+b²) = a + b              and it’s corollary

(a + b)² = a² + b²

We cause these problems ourselves by allowing and even encouraging students to learn procedures by categorizing symbols, without needing to apply the meanings of those symbols.  In arithmetic or pre-algebra, this occurs in both order of operations and basic variables.

“Use PEMDAS to determine the correct order of operations”

“To add like terms, combine the numbers in front”

In a basic algebra course, the comparable statements for exponents and radicals might be:

“Negative exponents mean you write the reciprocal”

“Take out the perfect squares”

Much of our work in classes deals with getting students to correctly process the symbols we place before them.  When they generate mostly correct answers, we conclude ‘they understand’.  Seldom is that the case … because we seldom take the time to focus on the meanings of these objects along with the various correct choices we have in working with them.

When I say “PEMDAS kills intelligence”, I am using the pneumonic as a place-holder for the prescriptive procedures that we focus on.  As mathematicians, we are all about choices.  When we see:

3x(x – 2) + x(x – 2)

We think of two choices (combine ‘like terms’ first, or distribute first).  The PEMDAS-mentality boxes students in to the ‘one correct [sic] way’ approach to mathematics; the PEMDAS approach also encourages students to perform procedures on symbols without much regard for what that particular expression or statement meant.  We use these approaches in all kinds of math classes, from elementary classrooms to university classrooms, and it has got to stop.

In recent years, some of the reform efforts have de-emphasized symbolic work … partially as a response to this problem.  I applaud that work, and have contributed to the efforts.  However, sometimes we over-react and provide too little symbolic work.  We have course which emphasize ‘functions’ but never use basic function notation [f(x)], let alone variations such as ‘sin(x)’.  An irony is that most technologies that students use for our math courses (calculators, apps, web sites) generally use function notation.

Maintaining a strong focus on procedures and correct answers encourages a PEMDAS-mentality, causes problems for us later, and (I would suggest) limits student motivation to learn mathematics.  Think how much better it might be to have a balanced approach, where the key principles are:

  1. Meaning
  2. Properties
  3. Choices
  4. Application
  5. Extension and feedback on prior steps

Some of my colleagues have said that students should “do mathematics” in math classrooms, though they are mostly talking about step 4 (application).  I also believe that students should “do mathematics” in every math class by using all levels (1 to 5 in my list) with all topics.  If we are not willing, believe that student’s can’t, or think that we do not have time … well, then we should question whether we are really committed to teaching that mathematics.

Most of our collegiate math courses are overly ‘full’, not too full of topics but too full of wasted effort.  We focus so much on “simplify” and “solve” in the basic courses that students use the PEMDAS-mentality; of course they won’t remember most of it, and of course they can’t apply ideas to other contexts — we are training them to just process the symbols.

So, if you have been wondering what I would have us do to replace “PEMDAS” for basic expressions, we should focus on four items:

  • Meaning of each expression
  • Inherent priority of each operation (a generally predictable list, based on level of abstraction)
  • Properties for the type (meaning) of expression
  • Choices for this expression

It is almost useless to know that a student can correctly calculate “8 – 3(5)”.  Value comes from knowing that there are multiple procedures to correctly calculate “6(2x) + 8(4x)”.  It is also almost useless to know that a student can correctly solve “12 – 5x = 7″.  Value comes from knowing that we have choices for that equation and for “8(x – 2) + 4x = 48″.  [I also suggest that PEMDAS itself is both incorrect and incomplete.]

Mathematics did not become so valuable because we know how to correctly arrive at an ‘answer’.  Our work is indispensable because we can present alternatives, and in some cases one of those alternatives provides great benefit to people, companies or societies.  That is ‘doing mathematics’, and is the type of experience I want for our students … whether in pre-algebra, pre-calculus, or anything else.

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