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The
Brownstone Journal >>
Issues >> Vol.
IX Spring 2000
Plato Revisited: Towards a New Method
of Calculus Education
Sean Wright (CAS XX) is a freshman
in the University Professors program, studying cognitive science.
He would like to thank Professor James Schmidt and Mr. Joseph
Zimmerman for kindling his interest in Plato and in calculous
education, respectively. Sean is a recipient of the Trustee
Scholarship. His favorite mathematical constant is 47.
Calculus, the supreme development of the seventeenth
century in the realm of mathematics, has been described as the
gateway to higher mathematics. It has also frustrated countless
students. For all our modern ingenuity, the process by which
students learn math is fraught with difficulties. Thomas Judson
explains that the impetus for the calculus reform movement in
universities in the early 1980s was the perception “that students
were not learning necessary skills and important concepts. The
typical course was viewed as superficial, and the range of mathematical
problems and techniques that were taught were regarded as too
limited” (Judson l). In contrast to geometry, students have
no context for learning calculus. Students have an appreciable
knowledge of geometry (such as the area of a rectangle is the
product of its length and width) before they formally study
its more abstruse logical foundations. However, students first
learning calculus are taught how to use the Power rule (stating
the derivative of xn is nxn-l) and immediately proceed to a
proof of this rule. Perhaps the greatest obstacle is the lack
of synthesis that results from a lack of conceptual understanding.
Marc Bloch explains this fundamental problem of mathematical
instruction:
The grim esotericism, in which even the best of us sometimes
fall, the preponderance,
in our current writing, of those dreary textbooks which bad
teaching concepts have put
in place of true synthesis, the curious modesty, which, as soon
as we are outside the study,
seems to forbid us to expose the honest groping of our methods
before a profane public. (Davis and Hersh, 274)
A new pedagogical method is needed to correct the deficiencies
of the current system. To answer Bloch's criticism, I suggest
we look to the past for inspiration. Specifically, Plato’s theory
of Forms provides a metaphor for describing the process of learning
and reformulating the modern method of calculus instruction.
The new method consists of three components: rote memorization,
real-world exploration, and systematic study of the formal mathematical
properties.
Why use Forms at all? Plato wishes to distinguish between physical
entities and intellectual conceptions. The physical entities
are images of the “visible,” appetitive world, whereas the Forms
are the “sovereign of the intelligible kind” that inhabit the
unchanging world of reason (Plato 183). Before examining Plato's
illustrative examples, it is useful to note Alsdair MacIntyre’s
description:
[The Forms] refer to objects, and objects belonging not the
multifarious, changing
world of sense, but to another, unchanging world, apprehended
by the intellect
precisely through its dialectical ascent, whereby it grasps
the meaning of abstract
nouns, and of general terms. These objects are the Forms, through
the imitation of
which or participation in which the objects of sense perception
have the characters
that they have. (MacIntyre 42)
The ascension to the Forms represents a move towards truth
and philosophic thought. This combination allows one to apprehend
the Form and gain understanding. “Then it draws the soul towards
truth and produces philosophic thought by directing upwards
what we now wrongly direct downwards” (Plalo 199). Maclntyre
explains that one apprehends a Form through Plato's image of
the Line. How can this abstract theory concerning the acquisition
of knowledge be applied to modern calculus education? Consider,
as an example, simple multiplication. In elementary schools,
typically in second or third grade, students are taught rudimentary
multiplication. They are expected to memorize a certain number
of basic identities (perhaps all the elements in a l2 by 12
table). At first, the students’ entire knowledge of multiplication
is limited to rote memorization, which for Plato corresponds
to images of the visible world. With the image of the Line,
Plato distinguishes between the intelligible and the visible
world. In his own words:
It is like a line divided into two unequal sanctions. Then divide
each section–namely,
that of the visible and that of the intelligible–in the same
ratio as the line. In terms now
of relative clarity and opacity, one subsection of the visible
consists of images. And by
images I mean, first, shadows, then rejections in water... (Plato
l83)
The visible world, the domain of the non-philosophers, consists
of images and beliefs that may appear to have a certain quality,
such as beautiful. Plato asserts that this appetitive part may
be familiar with beautiful things without any understanding
of the concept of beauty itself. Each specific multiplicative
identity that the students learn by rote corresponds to Plato's
images. Students will learn specific calculus facts in the same
manner. That is, when students are learning by rote that four
times six equals twenty-four, they will also learn by rote that
the derivative of sine is cosine and the integral of x2 is x3/3.
The objection may be raised that students are learning nothing
but an arbitrary definition of the derivative, but this use
of incomplete definitions also occurs in the example of multiplying
four by six. Multiplication is frequently explained in terms
of apples or some other concrete object. This is an attenlpt
to form an imperfect association between the number of apples
and the abstract formulation of number. In the same way, the
derivative may be defined in intuitive terms such rate or qualilative
steepness. In contrast to these images of the visible world,
the intelligible world above the division of the Line encompasses
thoughts, “mathematical entities–which for Plato are closely
related to Forms,” and understanding associated with the Forms
(MacIntyre 44).
In the example of multiplication, this ascension includes the
algorithms learned in later grades to multiply numbers of arbitrary
length and thereby gain a deeper understanding of the mechanics
of multiplication. With the new method of calculus, students
in these intermediate grades explore real-world applications
that incorporate major ideas of the calculus. Through modeling,
students gain an intuitive understanding of the concepts of
limit, rate, and area under a curve. For instance, students
explore graphically and numerically sequence such as {2-n} which
has a limit of zero as n approaches infinity. To explore the
idea of rate, many concrete examples can be culled from the
realm of physics that students can link to real-world experience.
Students study the relationships between position, velocity,
and acceleration. Students show graphically and numerically
the basic relations that velocity = d(x)/dt and acceleration
= dv/dt. To calculate area under a curve, students simulate
a Riemann sum by fitting rectangles under the curve. Progressively
smaller sized rectangles give increasingly closer approximations
to the true amount of area under the curve. Using Riemann sums,
students use anti-derivatives (or equivalently, integrals) to
demonstrate that this procedure undoes the derivative: the anti-derivative
of acceleration is velocity and the identical operation of integration
on velocity yields the original position graph. Using these
intuitive examples, students demonstrate verbal understanding
of the physical situation. Thus, they will have mastered three
of the components of the “Rule of Four,” which comprises the
stated objectives (graphical, numerical, verbal, and analytic
understanding of functions) of the Advanced Placement committee
on calculus reform (Kennedy l).
The final stage consists in the ascension to the Form and concomitant
understanding associated with the Form. Understanding of multiplication
occurs in algebra. Algebraic theory demonstrates that any number
is represented by a function of increasing powers of x (where
x equals 10 in the standard representation). That is, a number
may be written as a polynomial, f(x) = a0x0+a1x1+a2x2+...+.
Multiplication can now be rigorously defined as an operation
applied to these polynomials. This algebraic definition of polynomial
multiplication provides the last component of the Rule of Four,
analytical understanding. The formal study of calculus acquaints
students with its fundamental mathematical objects: the epsilon-delta
form for limits, the definition of the derivative as the limit
of slope approximations, and the integral as a Riemann sum.
Analytic understanding of the underlying theory is expressed
in the Fundamental Theorem of Calculus, which demonstrates the
intimate relationship between integration and differentiation:
similar to the reciprocal connection between multiplication
and division, each operation reverses the other. Understanding
of the Fundamental Theorem of Calculus unlocks the gateway to
higher mathematics.
In conclusion, Plato's concept of the Forms provides a structure
for the learning process. The Forms present a milieu for understanding
the progressive refinements of understanding. The program of
calculus instruction is a practical analogue of the ascension
to the Forms. Beginning calculus education in earlier grades
allows more time to impress techniques to promote successive
sophistications that culminate in analytic understanding. Integrating
ancient wisdom into modern educational practices, the abstract
theory of Forms promotes a practical reformulation of calculus
education that affords comprehensive understanding, thus accomplishing
Bloch's “true synthesis.” TBJ
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