Item 1. On the 1996 AB Advanced Placement Exam the first Free Response Question gave the graph of the derivative of a function and asked some questions about the function itself. The students were expected to glean information about the function from only the graph of its derivative. Fully 32.5 percent of the (~105,000) students writing the exam received no points for this question (none, zip, zero).
Item 2. I ran across an ad for DERIVE, a symbol manipulation software package, the other day. The left page extolled the virtues of the software and on the right was a listing of all the things DERIVE could do. The list was two columns running about 2/3 of the page's length. This list went from simple addition to multiple integration. I estimate that a typical student takes about six years to learn how to do what was listed on that 2/3 of a page.
There has been of late in these news groups a lot of discussion of the place of symbolic manipulation in the calculus curriculum and the mathematics curriculum in general. Some of it tied to criticism of the "reform calculus" movement. The gist was that all sorts of horrible things would result if much more time were not spent teaching students how to do algebra (by which is meant: how to manipulate symbols).
The question referred to in item 1 above is pretty close to a pure reform calculus question: interpreting a graph and writing the conclusions. I thought it was a great question and asked things very fundamental to some of the big ideas of calculus. A student who understood what he or she had been studying all year, should surely have been able to answer this question. There was no computation , no manipulation of symbols to do; in fact there was none possible. I suspect that students who had trouble with this question, could have answered the same questions had they been given the expression for the derivative to manipulate. And that's the point: they had in front of them everything the manipulation could have told them and were unable to interpret it.
On the other hand there was no way to use a calculator to do the problem either. However, if the students had spend some quality time with their calculators during the year investigating the relationships between the graphs of functions and their derivatives, I suspect they would have had no trouble with this question. The point is that the calculator in a high school or college classroom should be a tool for investigation, not just a shortcut to the answer.
Symbol manipulation is neither mindless nor unnecessary. It's what makes mathematics work. Students need to be able to use, interpret, read, write, decode and otherwise handle symbols. You really can't do mathematics without that ability. BUT there is a lot more to mathematics than that. Along with the symbols are the numbers, the graphs and the communication. Much more time needs to be spent with these rather than with the pure manipulation of symbols.
Machines can do the manipulation. Let them. People still need to understand what the machines are doing, what the symbols mean, how they apply to the problem under consideration and so on. Too many, far too many, students think that mathematics is only symbol manipulation. Why? Because that's what they spend so many years doing in math class. Because that's what shows up on the math tests. Because that's what shows up on the standardized tests.
Allow me to coin a word. The word is "algemetic" (al/ ge/ me/ tic rhymes with "arithmetic"). "Algemetic" means anything the TI-92, Derive, Mathematica, Maple or MathCad can do. "Algemetic" is not all there is to algebra. It is what students spend years learning.
Some instruction must include having students manipulate variables with pencil and paper. The goal is not to get them to be great "algemeticians," but rather to enable them to know what "algemetic" can and should be done to solve a particular problem or investigate a particular situation. To know when, where and what is appropriate is more important than the ability to do it by hand. In other words they have to know when to do it, but they don't have to be good at it.
Now before you all jump on that, consider:
(A) YOU balance your check book with a calculator, don't you? Admit it. Even though you probably have several degrees in mathematics and can do arithmetic more accurately and quicker than most, you still use a calculator when it really counts (it's your money).
(B) When scientific calculators came out I was still required for (too) many years to teach how to compute with logarithms. I can still see the students sitting there adding up logarithms with their scientific calculators.
We are still in the same continuum. People need more than ever to understand mathematics, particularly algebra and the calculus. But when it comes to "algemetics" what is the problem with letting the machines do it? They are better, faster and more accurate. If someone needs to do a computation or to solve an equation or to factor or to graph or to find a derivative or to find an antiderivative or even to add a few integers, what's so bad about pushing a few buttons?
The Reform Calculus ideas, the Rule of Three -- numerical, analytical, graphical and communication and technology (see they needed a calculator there) -- should be applied not just to calculus but to all mathematics starting in Kindergarten. Put the emphasis, spend the time, on teaching the mathematics not the "algemetics." Concentrate on the ideas, concentrate on the relationships (numerical, analytic, graphical) between the ideas and be able to communicate those relationships. Use technology not as a quick way to the answer, but as a tool for investigation. Use technology to go wider and deeper into the problem; don't use it to drive straight through to the answer. Use it as a way to keep focused on the mathematics by avoiding the "algemetics." Use the technology correctly and to its fullest extent -- not just to add logarithms.
The DERIVE ad: Six years of work reduced to less than a page! Think of all that time being used to really get into mathematics; all the time to investigate, solve problems, think, discuss and write about what's happening -- sure beats factoring x^2-4 for the hundredth time.