*paradigm*at the time ("normal science"), occasionally it is recognized that the current paradigm is incapable of solving important problems and what is needed is a new paradigm (a

*"paradigm shift"*)

*.*One example is the Copernican revolution in the 16-17th centuries where the paradigm of a geocentric universe was gradually replaced by that of a heliocentric universe. A second example is the chemical revolution caused by Lavoisier's discovery of oxygen that replaced the phlogiston theory that posited that all flammable substances contain a material ("phlogiston") that enables them to burn. A third example is the Einsteinian revolution that made Newtonian mechanics a limiting case of relativistic mechanics for small velocities.

What does this have to do with mathematics education. Well, read on...

*Calculators and the Elementary School Mathematics Curriculum*

Arithmetic has been at the heart of the elementary school mathematics curriculum for many centuries. Since counting has been important for as long as there has been human civilization and since, except for geometry, there was - and still is - just about no other mathematics accessible to children of elementary school age, arithmetic has been and still is a crucial, indeed a dominant part of the elementary school mathematics curriculum. And so, I believe, it should remain. The relevant issue here is: How should arithmetic be taught to students?

There is (almost?) no one who questions that an understanding of what addition, subtraction, multiplication and division

*are*and an ability to discern*when*each of these operations is the appropriate one to use should be in every child's mathematical arsenal at some point in his/her elementary school career. Neither is there any substantial disagreement among mathematicians or mathematics educators or anyone else that students need to achieve virtually instant recall of the addition and multiplication tables as early as possible in their educational careers.For at least the past two centuries the heart of the elementary school arithmetic curriculum has been instruction in the pencil-and-paper algorithms (hereafter PPA) of addition, subtraction, multiplication and division that all readers of this article surely recall from their own schooldays. Until the advent of the hand-held calculator these algorithms were of substantial practical value, being essentially the only way for almost everyone to add columns of figures, subtract one sizable number from another, multiply two multi-digit numbers or perform long division . But now the practical value of these algorithms has all but disappeared. The question is: Are these algorithms still the best way to impart the understanding of arithmetic necessary to prepare students for the further study of mathematics?

*The Math Wars*

Mathematics education, particularly in the United States , has been a subject of controversy for some 40 years now since the time of the New Math in the 1960s. Indeed, the New Math has been called the first of three "revolutions" in mathematics education [2], the other two being the back-to-basics reaction to the New Math and the changes wrought by the NCTM Standards (hereafter the

*Standards*) [10]. The first two of these, however, whatever you may think of either, were not revolutions in the Kuhnian sense and the last, whatever the long term impact of the Standards, proposed only a modernization of school mathematics education rather than anything revolutionary.

One thing the Standards did do, however, was to spark the so-called

*Math Wars*which for nearly 20 years now have pitted mainly research mathematicians but also some parents, business groups and teachers and others (Traditional Math Warriors - TMWs) against mainly mathematics educators but with significant support from some parents, teachers and others (Reform Math Warriors - RMWs) [5, 12]. At times these wars have led to acrimonious exchanges between the two sides; at other times the exchanges have been more genteel. There have even been recent attempts at truces [3] and fudges [9]. But an end to the Math Wars is not in sight nor, I believe, should it be because the essential issues are too important and the essential positions of the two sides are so far from each other that what is needed is victory for one side, not a pale compromise that, in the long run, would not be good for anyone.

Calculator technology has been the catalyst for almost all the disputation in the Math Wars. Since this technology almost wholly diminishes the practical value of PPA, it is essential to answer the question above about whether these algorithms are the best way, even an effective way to prepare students for further study of mathematics. In one sense, the answer to this question is clearly, yes, since there is no alternative to these algorithms that has ever been tried in schools anywhere.

Some places, notably California, have, after some experiments with calculators in elementary school mathematics, returned to a "back-to-basics" curriculum in which calculators are effectively banned in the classroom through the sixth grade. In many other places curricula are in use in which calculators are used in conjunction with traditional instruction in PPA. There are lots of variations in these curricula. Judging only by test results and anecdotal evidence, some of these curricula seem just as or more effective than traditional PPA curricula while others do not seem to perform very well. In some partly-calculator curricula, the traditional algorithms for addition, subtraction and multiplication are taught but long division is no longer taught. This is in line with a 25-year-old recommendation from Great Britain [4] that is, however, anathema to TMWs.

**The More Distant Future**

It is only reasonable to conclude this paper by asking the question: If the computer revolution presages a paradigm shift in elementary school mathematics education, will there also be paradigm shifts in other aspects of mathematics education?

It is true today that essentially all the manipulations of mathematics taught not just to elementary school students but also to secondary school students and to university undergraduates can now be performed by hand-held devices and, in virtually all cases, more rapidly and accurately than (almost) any human can perform them. What value then to teaching any of the old, almost entirely pencil-and-paper manipulations taught almost universally in secondary schools and universities? The answer to this question is not so simple as I believe it to be in the case of elementary school arithmetic. As with arithmetic, the

*practical value*of learning to do these manipulations is rapidly decreasing. But it cannot be said, at least it cannot be said*yet*, that learning these manipulations is not the best way to understand the underlying mathematics and, therefore, the best way to proceed to the study of further mathematics. Nevertheless, I would be surprised if more and more of the current secondary and university mathematics curricula were not replaced in coming decades by new curricula in which calculators and computers are fully integrated into them (mental algebra? mental calculus?).
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