Newton's Method

Readers familiar with calculus may know another way of arriving at the iterative scheme, ay ex squared minus cee, all over two ay ex plus bee. In particular, "Newton's method" enables us to use the concepts of tangent lineand derivativeto find the zeros of f(x) = ax² + bx + c = 0 by implementing the same iterative scheme

that we have derived from the Babylonian method for finding square roots.

Setting aside for the moment the question of why these two approaches are equivalent, it seems appropriate to explain why IADM starts by developing a special case of Newton's method at the pre-calculus level. That is, "What's wrong with just teaching the quadratic formula at high school and deferring iterative methods until calculus-based tools are at hand?"

The answer lies in the remarkable computer technology that has so profoundly affected our civilization. Had computers existed 500 years ago, the subject that Arab scholars called al-jabr would have evolved in very different ways. Now that computers are part of our everyday lives, it seems appropriate to re-examine such 500 year old curricula with an eye to reconciling them with the modern world.

To make this argument more specific, one need only look to the next logical step in the algebra curriculum, namely the solution of cubic equations of the form

ax³ + bx² + cx + d = 0.

In the context of the middle ages, this problem generated an ingenious technique called Cardano's method. While the mathematical questions surrounding Cardano's method are both interesting and profound, no mathematician faced with actually solving an equation such as

x³ -2x² + 2 = 0

Another reason for making iterative techniques part of the algebra curriculum is their close connection to "models for change." Given an ability to implement iterative schemes of the form x_i+1 = F(x_i), the student is only a step away from being able to solve "difference equations" of the form

x_i+1 - x_i = f(x_i).

The ability to program such difference equations on a graphing calculator or spreadsheet opens up a world of new ideas. In particular, students can now engage in profound forms of mathematical modeling at a pre-calculus level, thereby casting light on topics of importance to their lives and future welfare.

Finally, we note the great interest in the role technology can and should play in schools and in mathematics instruction. "Iterative algebra" brings technology to bear on questions central to established school curricula in a mathematically meaningful way. It also establishes direct connections between these curricula and emerging fields of study such as fractals and chaos.

Indeed, establishing the litany ay ex squared minus cee, all over two ay ex plus bee is not primarily a matter of solving quadratic equations. Rather, it is one of developing mathematical tools and computer skills that are part of bringing school mathematics into the third millennium.

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