Riemann sums

Sometimes technological advances mean things are less exciting.

The last half-century of technological advances have made mathematics, especially school mathematics, much easier to do.  What used to be done by slide rules with limited accuracy, or log tables taking much more time, now appears in a moment on the calculator.  And the laborious task of turning tables of numbers into hand-plotted curves on graph paper has now been replaced by an automatic feature of display software.  One can, of course, lament the loss of the skills the manual methods taught: how to estimate what your answer should be; choosing the right scale and size to show the important parts of your plot; how to extract the right number from a table rapidly but without error.  We’ll concentrate on teaching these separately, and be content with the ease that comes of modern machines.

There is also a difference in approach between the math textbooks of the Old Days and those our tutoring consultant deals with now.  There is more attempt to connect mathematics with applications in the real world, even if the problems sometimes seem contrived; more attempt to explain a concept in several ways; less of the deductive, almost scholastic formal exposition of yore.  Math remains a difficult subject for many, but is less tedious and painful than it was.

Most of the time.  Our tutor has noticed that students are now expected to calculate Riemann Sums when they get to Calculus.  This technique breaks up the area under a curve into several boxes, and adding up their area gives an approximation to that area.  When there are few boxes, the approximation isn’t very accurate; but adding up a lot of boxes is a lot of work.  In the Old Days we might have done one or two problems, just to show what it was like, but they took a long time to set up, and since one was trying to multiply and add many numbers accurately, a longer time to work through.  Very quickly, the time spent in such a tedious exercise did not justify the learning it produced, so we did few Riemann Sums.  There was some discussion of accuracy and how to do it most efficiently, but any serious work was left to those learning computer languages and performing advanced computations.

But now that any student can multiply ten-digit numbers instantly, and calculate the value of intricate functions to the same accuracy, doing Riemann Sums is less difficult.  The average student is now expected to be able to deal with details that only computer geeks had to worry about before.  Each of the relevant lessons have several such problems in their homework sets.  We haven’t actually measured the time it takes to do them, but strongly suspect it’s longer than in the couple of example problems we once dealt with.  Certainly the unexciting but exacting work of setting up the calculations is greater than it used to be.

So in this case, the technological advances that made calculation so much quicker and easier have increased the tedium of learning mathematics.