extension of algebra to transition into Calculus courses. Elk challenges the traditional notions that Calculus is new and different form of mathematics from the ones that the student has “a learned earlier,” the first course in analysis, and distinct from algebra and geometry (Elk, 1998, p. 351). He proposes introducing the underlying algebraic ideas imbedded in the concepts of the derivative and the integral (Elk, 1998). As an example of these underlying algebraic ideas, Elk notes the three situations that involve division by zero: 0/a, and 0/0, when is any number not equal to 0. He asserts that a/0, a differential Calculus gives meaning to the third case (0/0) under the appropriate conditions. From this, he introduces the idea of the change in two coordinates approaching zero and the y change in two coordinates approaching zero. As the two respective components of the points x decrease, the distance between the points decreases and becomes very small. Thus, in the case of the change of these y-coordinate over the change in these x-coordinates would cause both the numerator and the denominator to be zero, resulting in the 0/0 situation. The meaning of the case 0/0 becomes clear with an example. Consider the function f(x)= At the function reduces to 0/0, and at this point, the function is not defined. In (x2-4)/(x-2). x=2, order for the function to be defined at all points, an additional criteria needs to be added; for example, if were added, it would produce a continuous function for all values of x. f(2)=4 Furthermore, any where is any number not equal to 4 would produce a discontinuous f(2)=b, b function. Using this information, Elk presents an algorithm to compute the derivative of a given function (1998, p. 354-355). The algorithm goes as follows: From the given function form y=f(x), f(x+∆x). Subtract from a difference we call f(x) f(x+∆x) – ∆y. Divide by and simplify the algebra. ∆y ∆x
extension of algebra to transition into Calculus courses. Elk challenges the traditional notions that Calculus is “a new and different form of mathematics from the ones that the student has learned earlier,” the first course in analysis, and distinct from algebra and geometry (Elk, 1998, p. 351). He proposes introducing the underlying algebraic ideas imbedded in the concepts of the derivative and the integral (Elk, 1998). As an example of these underlying algebraic ideas, Elk notes the three situations that involve division by zero: 0/a, a/0, and 0/0, when a is any number not equal to 0. He asserts that differential Calculus gives meaning to the third case (0/0) under the appropriate conditions. From this, he introduces the idea of the change in two y coordinates approaching zero and the change in two x coordinates approaching zero. As the two respective components of the points decrease, the distance between the points decreases and becomes very small. Thus, in the case of the change of these y-coordinate over the change in these x-coordinates would cause both the numerator and the denominator to be zero, resulting in the 0/0 situation. The meaning of the case 0/0 becomes clear with an example. Consider the function f(x)= (x2-4)/(x-2). At x=2, the function reduces to 0/0, and at this point, the function is not defined. In order for the function to be defined at all points, an additional criteria needs to be added; for example, if f(2)=4 were added, it would produce a continuous function for all values of x. Furthermore, any f(2)=b, where b is any number not equal to 4 would produce a discontinuous function. Using this information, Elk presents an algorithm to compute the derivative of a given function (1998, p. 354-355). The algorithm goes as follows: From the given function y=f(x), form f(x+∆x). Subtract f(x) from f(x+∆x) – a difference we call ∆y. Divide ∆y by ∆x and simplify the algebra.
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