Teaching for the Twenty-Second Century: Whither (or Wither) Mathematics?Author(s): Jeff SuzukiSource: The Mathematics Teacher, Vol. 95, No. 4 (April 2002), pp. 244-245Published by: National Council of Teachers of MathematicsStable URL: http://www.jstor.org/stable/20871005 .

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HhJNDOFF!

Jeff Suzuki

Teaching for the Twenty-Second Century: Whither (or Wither) Mathematics?

Although technology frequently begins by

easing the servant's

workload, it ends by

eliminating the need for the servant

f we do not begin teaching mathematics, then in a

very few years, mathematics will cease to exist as an academic subject. Whether the mathematics is "reform" or "traditional," whether it uses the latest

technology or none at all, calculus and algebra will

join embroidery and woodworking as pursuits of

hobbyists, as topics not fit for serious study, and

certainly as subjects unworthy of a place in the halls of education. The rest of the existing mathe matics curriculum will soon follow.

This result will be an inevitable consequence of

technology. Cheap calculators perform all elemen

tary arithmetic. Slightly more expensive calculators can perform algebra and calculus?both numerical

ly and symbolically. Computer programs have been able to do the latter for a long time and have

recently begun to prove theorems. What remains to be taught?

Very few teachers refuse to allow students to use technology. They play the role of John Henry to the steam shovel, or of Canute to the tides; whatever the intellectual merit of their position, ultraconservative courses have little attraction and will wither through market forces alone.

Some reformers have embraced the new technol

ogy and have focused almost exclusively on making mathematics applicable. However, when mathe matics is turned into a practical subject, it also becomes a dependent subject that has no existence of its own. It changes from the queen of the sciences to the servant of the sciences?and although tech

nology frequently begins by easing the servant's

workload, it ends by eliminating the need for the servant.

The problem is that most mathematics courses, whether they are reform or traditional, do not teach

The views expressed in "Soundoff!" reflect the views of the author and not necessarily those of the Editorial Panel of the Mathematics Teacher or the National Council of Teachers of Mathematics. Readers are en

couraged to respond to this editorial by sending double spaced letters to the Mathematics Teacher for possible publication in "Reader Reflections." Editorials from readers are welcomed.

mathematics. They teach what was once known as

logistics, that is, the results and techniques of mathematics. The only difference between the stu dent who evaluates and solves a quadratic equation using a graphing calculator and the student who uses the quadratic formula is one of technique: nei ther student is actually engaged in mathematics.

One goal of the reform effort is to try to encour

age students to "think mathematically," but what does thinking mathematically entail? Is it solving problems? No; that is merely logistics and bears the same relation to mathematics that the ability to drive a car has to the ability to build one. Is it logi cal deduction and proof? No; they are merely the coda to the symphony, the final support of a conclu sion. In my view, the ability to think mathematical

ly can be summarized in three words: Observation

Conjecture-Test (OCT). I propose OCT as a model of the way that mathematics should be taught?how it must be taught if mathematics is to survive as an

academic subject. If, for example, we wish to talk about the process

of factoring the sum or difference of two cubes, we

might begin by asking what numbers can be writ ten as the sum of two cubes. This question, by itself, leads into an investigation of systematic ways to collect and organize data. Do we begin by trying to see whether 2, 3, 4, 5, and so on, can be written as the sum of two cubes, or does an easier

way exist? Do patterns occur in the results? The first few sums might suggest that the numbers are all squares (23 + l3 = 9, 23 + 23 = 16), but this obser vation is quickly demolished (l3 + l3 = 2).

But soon, we might make the Observation (0) that with the exception of 2, all the numbers that can be written as the sum of two cubes seem to be

composite. Is this result perhaps indicative of some

underlying rule? We make the Conjecture (C) that

Jeff'Suzuki, jeffs@bu.edu, teaches mathematics and physi cal science at the College of General Studies, Boston Uni

versity, Boston, MA 02155. He has recently published a book on the history of mathematics and is interested in the processes by which mathematics is created.

244 MATHEMATICS TEACHER

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no prime except 2 can be written as the sum of two cubes. Is our conjecture correct? We need to Test (T) it. Only fifteen numbers less than or equal to 250 can be written as the sum of two cubes; and except for 2, none are prime. Our conjecture seems to be well supported, and we can now spend time per

forming the ultimate test: finding a proof. Some people may object to this empirical

approach to mathematics. But the history of mathe matics includes many conjectures that were used

long before they were proved: the fundamental the orems of calculus and algebra, the binomial theo

rem, Fermat's lesser theorem, and many others.

Indeed, a good conjecture is often more valuable than a theorem that has been proved. Attempts to

prove conjectures made by Fermat, Goldbach, and Riemann have driven the development of mathe matics for centuries.

The simple conjecture that no prime but 2 can be written as the sum of two cubes can drive one's per sonal mathematical development. The sum of two cubes always seems to be composite; further, we

note that 9 = 23 + l3 has a factor of 3 = 2 + 1,16 = 23 + 23hasafactorof4 = 2 + 2,and35 = 33 + 23has a factor of 5 = 3 + 2, These results suggest another

conjecture: a3 + 63 has a factor of a + 6. This conjec ture can be used as a springboard for deriving the factorization. Once the factorization has been

found, it can then be used to prove the conjecture.

Thus, rather than learn yet another factorization, students are engaged in an activity that develops skills of organization, analysis, and reasoning. That

they might learn how to factor the sum of two cubes is incidental. The factorization has as much to do with mathematics as spelling has to do with the works of Shakespeare.

OCT shows the student that mathematical results do not fall out of the sky engraved on stone ^^^^^^HI^H tablets but are generated. Mathematics is no longer ^^^^^^^^^^H a daunting structure erected by distant giants but instead is a fundamental human activity. OCT ^^^^^^^^^^H transcends technology. The machine might make a ^^^^^^^^^^H pattern more obvious, but it cannot make the intu-

^^^^^^^^^^H itive leap from observation to conjecture. Indeed, ^^^^^^^^^^H OCT makes humans the masters of the machine. ^^^^^^^^^^H Instead of an all-knowing oracle that presents ^^^^^^^^^^H answers, the machine is a humble messenger that

^^^^^^^^^^H presents information. Jeff Suzukj

So the choice is clear. We can teach the essen tial mathematical process of OCT to ensure that mathematics remains a central part of the liberal arts despite all the changes that technology may bring. Or we can continue to teach logistics and find that technology has turned mathematics into the academic equivalent of hand weaving, a skill

pursued by a few specialists and hobbyists but entirely unnecessary in the days of machine looms and mass production. ?

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Vol. 95, No. 4 April 2002 245

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