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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