June 20, 2010
June 20, 2010
June 23, 2010
15.1191.1 - 15.1191.15
Because points of view and techniques differed so sharply before and after the year 1807, from this point forward in this paper, the mathematics preceding 1807, will be referred to as Euler’s calculus and the subsequent developments as Cauchy’s analysis. The new analysis of Cauchy and the new definition of functions called for clear descriptions of the domain and range of the functions. But for the algebraic functions of calculus the domains are relatively simple, excluding the isolated zeroes that occur in the denominator and the “excluded intervals”, which occur when taking an even root of a negative value.
Euler, being unencumbered by the “monstrosities” developed during the 19th century, could rely on his intuition to develop into “the master of us all.” However, the ingenious examples and techniques invented during the 19th century led mathematicians to distrust Euler’s intuition and visualizations. By the mid 20th century, visual interpretations were considered misleading and were disparaged in math classes.
The success of Cauchy’s analysis led to authoritative abuse. A student who is not considering Fourier series has no reason for these definitions. Why should a student want to study ordered pairs. Why should a student, who may not understand what a transcendental number is, want to follow a delta-epsilon argument? On what basis could a student object to or question the authority of both the teacher and the text? Any student who questioned or objected to the concept of “ordered pairs” was considered as either unintelligent or mathematically immature. If students dropped out, it did not matter; other students, more obedient, would take their place.
By the late 20th century, mathematicians noticed that too many students were dropping out and that changes were needed. The reform movements were initiated. The admission that graphing was needed was a helpful step. Additionally helpful was that tables of functions remained as useful in applications. Digital calculators and computers were now widely available so they could contribute to the reform whether their use constituted conceptual reform or not. These instruments changed mathematics by eliminating much of the drudgery of calculating, allowing us to address problems of conception. However, effective pedagogy requires something more that the mathematicians missed.
Mathematicians believe that they are disseminating the “truth.” But nothing is disseminated when the “truth” is not understood. Answers are provided to questions which are beyond the students’ horizon. Issues of convergence of Fourier series do not appear in K-12 math or even in the first calculus course. In addition, issues of functions defined only on the rationals, or say the Cantor ternary set, are not yet in view either. A student unfamiliar with the place of irrationals on the real number line is not in a strong position to approach delta-epsilon arguments. Providing a student with definitions capable of treating all the “monstrosities” of Cauchy’s analysis denies him the ability to understand the intuitive problems at hand.
Every mathematical concept has a story that provides its meaning. If the narrative is not related correctly, the student will be left confused. In too many texts, concepts are given abstract definitions, which appear seemingly out of nowhere, that are designed to be the foundation of the proofs that the mathematicians admire and some students may need later. But when the introductory story is needed, it is missing. It is possible to make ‘true’ statements about a new
Grossfield, A. (2010, June), Technical Enrollments And Mathematical Pedagogy Paper presented at 2010 Annual Conference & Exposition, Louisville, Kentucky. https://peer.asee.org/15887
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