Mathematics Education: Taking a Clue
From the Recent Technological Revolution
A Word of Warning:
Technology is not automatically useful.
That pedagogy is the essence of education is almost a tautology. Pedagogy was and remains the cornerstone of the instruction. Pedagogy has its dangers though. It always had. In a well publicized case, Socrates paid with his life when the Athenians rejected his pedagogical approach.
We are particularly concerned with the inclusion of technology into the instructional process. Let me state for the record the obvious: technology can't substitute for understanding. Developing a habit of using technology to achieve quick answers is counterproductive.
In the first chapter of his book Number: The Language of Science Tobias Dantzig mentions proposals to replace the decimal system with other bases. Wrote he: "In our own age, when calculating devices have largely supplanted mental arithmetic, nobody would take either proposal seriously." The book was written in 1930. The fourth, revised and augmented edition of the book appeared in 1954. Whether written in 1930 or 1954, I think the phrase about supplanting mental arithmetic was more a reflection of a popular sentiment than of a historical reality.
The sentiment is still with us while we are nearing the time when a fantasy may well become a reality. Below, I give a couple of examples that point to possible pitfalls.
Recently, I had the following exchange (italic is mine. A.B.):
- Visitor: How do I obtain the best equation for a curved line to describe a series of experimentally obtained points?
- A.B.: There is no best. Unless, of course, you specify your needs. I would go and have a look at the site Numerical Recipes
- Visitor: I found that Excel does this for me. To think that I was straining my brain for nothing! Thank you anyway.
Here's another example from the sci.math newsgroup:
Could anyone please solve this step by step? 3cos2x = sinx. x = ? Thank you
Actually, the problemas presented is insanely difficult as given. Essentially, it does, as you showed, boil down to 3 = y3y2, where y is in [0,1] (since y = sin x). But finding a solution to this, assuming a solution even exists, is pretty tricky.
When I in fact put this into Mathematica, I got the slightly bewildering y = (ProductLog(18log3)/(2log3))0.5 where ProductLog(k) is the principal solution to k = xex. Ick. And x is the Arcsin of that. Double ick.
Of course, when faced with the original function, Mathematica gave the illuminating error message:
The equations appear to involve transcendental functions of the
variables in an essentially non-algebraic way.
Umm, no. Here's a start:
3cos2x = sinx
31 - sin2x = sin x
Now we make life a little easier by denoting sin x by s:
31 - s2 = s
The graph of 31 - s2 looks vaguely like a bell curve and clearly has only one intersection with the graph of s. A bit of thought shows that s = 1 is the unique point of intersection, so you're left with finding the solutions to sin x = 1, an exercise left to the reader. (I assumed we were looking for real roots, BTW)
What mathematical tools are you allowed to use? (For what level course was this problem assigned?)
If you can sketch, even approximately, the left-hand side and the right-hand side as functions of x, where will the graphs meet?
After that, can you make a conjecture and prove it?
(And a suspicious question: This is the time of mathematical competitions, and after I solved the equation for myself, I suspect that it is a competition problem, and a clever one, if it is for high schools. If this is the case, are you sure you want to win by the work of others?)
Roughly sketch these functions?? You mean... actually _think_ about the question?? This is 2000 man!! We have Mathematica and Maple to think for us!!
Hello, maybe we should let these programs do the hard work, but try to think ourselves.
While it sometimes doesn't seem to make much sense to draw graphs of given functions, it might help in the long run to 'know' the shape of functions. Here students are sometimes unable to sketch easy formulas, because they never learned it.
I think its the same problem as with pocket calculators. It just helps to be able to perform simple calculations by hand.
I assume that x is required to be real.
What is the least possible value of cos2x?
So what is the least possible value of the left side?
What is the greatest possible value of the right side?
And one more example:
The area of the piece of property that measures 102ft x 290ft x 255ft x 230 ft is 43512.621769283 square feet. I don't know how accurate this value is. This value has been calculated by Visual Basic.