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The accounts that teachers and students write using UDED can be detailed, brief, or anywhere in between. At times, the "big picture" is precisely what students should absorb; at other times, a mini-term paper might be appropriate. In assigning the UDED account as a student project, the instructor can easily set the parameters for the UDED project.

One of my favorite abridged applications of the UDED model is using it to construct a brief chronicle of the acceptance of the principle of mathematical induction as a valid method of proof in mathematics. In the sixth century B.C.E., the Pythagoreans certainly used the ideas underlying this principle when, proceeding geometrically, they conjectured and accepted as "true" such number-theoretic patterns as theorem S, which states that the sum of the first n odd integers is equal to the nth square number (Burton 1999, pp. 91-93). Francesco Maurolico gave the first formal inductive proof in the history of mathematics when he proved theorem S by induction; his proof (discovery) can be found in his work Arithmeticorum Libri Duo, published in 1575, the year of his death (Burton 1999, p. 426). In the next century, Blaise Pascal explored and developed the technique of mathematical induction in connection with his work on the arithmetic triangle and its applications (Burton 1999, pp. 418-28). Although John Wallis and Augustus De Morgan helped name this procedure induction, only in the latter part of the nineteenth century did Richard Dedekind--and then Gottlob Frege and Giuseppe Peano--define it mathematically. When formulating their sets of categorical properties for the natural numbers, each included the principle of mathematical induction or one of its logical equivalents as an axiom (Katz 1998, pp. 735-37).

USING "UDED" TO DESCRIBE THE EVOLUTION OF COMPLEX NUMBERS

The UDED model can also be used to describe the evolution of the complex numbers, a more commonplace high school mathematical topic than induction. Girolamo Cardano and other sixteenth-century Italian algebraists reluctantly began to use complex numbers when they saw that negative values appearing under the radical sign in the Cardano-Tartaglia formulas for solving specific cubic equations sometimes corresponded to recognizable real roots and when Cardano attempted to solve the problem of dividing 10 into two parts such that the product is 40. In Ars Magna, his famous algebra text of 1545, Cardano showed by "completing the square" that the two parts must be 5 + Square root of -15 and 5 - Square root of-15. Although he checked that these answers formally satisfied the conditions of the problem, he still regarded them as being "fictitious" and useless; he was only halfheartedly using complex numbers.

A generation later, Raphael Bombelli discovered the complex numbers in analyzing the "irreducible case" of the cubic equation when all three roots are real and nonzero and yet negative values always appear under the radical when a Cardano-Tartaglia type formula is used. When he published his treatise Algebra in 1572, he became the first mathematician bold enough to accept the existence of "imaginary," or complex, numbers and to present an algebra for working with such numbers. He assumed that they behaved like other numbers in calculation and proceeded to manipulate them formally, with Square root of -a x Square root of -a = -a for a > 0 being his key observation.

During the next three centuries, many mathematicians explored and developed various aspects of the complex, that is, imaginary, numbers. For example, in conjunction with their formative work in analytic geometry, calculus, and algebra, such mathematicians as Rene Descartes, Isaac Newton, G. W. Leibniz, Leonhard Euler, Jean d'Alembert, Carl F. Gauss, and Bernhard Riemann all employed complex numbers in describing their theories of equations, formulating the general logarithmic and exponential functions, and devising analytic tools for modeling and solving real-world problems. Casper Wessel, Jean Argand, and Carl F. Gauss contributed a crucial development to accepting and understanding the nature of complex numbers when they began to represent them geometrically in the real plane, much as we do today.

Finally, William Rowan Hamilton established the theory of complex numbers on a firm mathematical footing when he defined them in terms of ordered pairs of real numbers in almost the same way that modern textbooks define them. This definition and his rules for performing arithmetical calculations with his ordered pairs can be found in his 1837 paper "The Theory of Conjugate Functions, or Algebraic Couples; with a Preliminary and Elementary Essay on Algebra as the Science of Pure Time." Additional details concerning this UDED account of the evolution of the complex numbers can be found in Burton (1999) and Katz (1998).

"UDED" AND THE EVOLUTION OF BRANCHES OF MATHEMATICS

The UDED paradigm can also be used to construct brief accounts of the evolution of such entire branches of mathematics as Euclidean geometry. Most ancient peoples used formulas to calculate the areas of simple rectilinear figures and to approximate the circumference and areas of circles. For example, the early Egyptians, Babylonians, and Chinese used algorithms to compute the volumes of rectangular blocks, cylinders, and pyramids. Furthermore, the latter two civilizations discovered the general Pythagorean theorem and used it in geometrical and astronomical applications. These civilizations had no real notion of an axiomatic system on which they could base "proofs" of their geometric formulas and theorems. As most students do today, they accepted their geometrical results on the basis of diagrams and intuition and often did not even distinguish between exact and approximate answers.

From the sixth century B.C.E. to the beginning of the third century S.C.E., Thales, Pythagoras, Eudoxus, Plato, Aristotle, and other Greek mathematicians and philosophers shaped mathematics into a deductive, axiomatic science and discovered Euclidean geometry. Around 300 B.C.E., Euclid compiled their accumulated discoveries in geometry and number theory and presented them axiomatically in his famous book, the Elements.

Over the next two millennia, Euclidean geometry was explored and developed by mathematicians from virtually every society that learned of the Elements. Such additional mathematical advances occurred as Archimedes' replacement of the Euclidean theorem "The areas of circles are to one another as the squares on their diameters" with a proof of the precise Babylonian formula "The area of any circle is equal to the area of a right triangle in which one of the legs is equal to the radius and the other to the circumference" (equivalent to the modem formula area = pi r[sup 2]). However, the principal explorations and developments did involve repeated attempts to prove that Euclid's fifth, or parallel, postulate followed as a theorem from his other four more self-evident postulates and his common notions. The celebrated attempts of Proclus, ibn al-Haytham, John Wallis, Girolamo Saccheri, Adrien-Marie Legendre, Johann Lambert, and untold others were doomed to failure because--as we now know from the work of Janos Bolyai, Carl F. Gauss, and Nikolai Lobachevsky in the early nineteenth century--Euclid was indeed on sound logical ground when he made his parallel postulate an axiom for his geometry. It is logically independent of his other four.

Finally, at the very end of the nineteenth century, David Hilbert completely and logically defined Euclidean geometry in his classic monograph Foundations of Geometry (1899). Hilbert began his treatment of Euclidean geometry by postulating three undefined terms (point, line, and plane) connected by three undefined relations--incidence (on), order (betweenness), and congruence. He then offered a set of twenty-one axioms on which a logically consistent and complete treatment of Euclidean geometry could be based. In axiomatic studies of Euclidean geometry today, authors often distill Hilbert's collection of twenty-one axioms down to a set of fifteen logically independent axioms by combining related ones and deleting those that are implied by the others.

The principal pedagogical message here is that anyone purporting to offer high school geometry students a complete, deductive study of Euclidean geometry will fail. NCTM's curricular standards and recommendations indicate that a school geometry course should emphasize discovery, applications, and a representative sample of truly accessible proofs of such theorems as the Pythagorean theorem. Additional details concerning this UDED account of the evolution of Euclidean geometry can be found in Burton (1999) and Katz (1998).

CONCLUSION

Topics in addition to those already noted to which the UDED paradigm can be applied without unduly forcing the issue include the evolution of the concept and theory of a function, limit, infinite series, the integral, the number zero, negative numbers, real numbers, the theory of equations, and numerical procedures. It can be applied to describing the evolution of such entire branches of mathematics as non-Euclidean geometry, analytical geometry, and algebra (both manipulative and structural); such subareas of modern algebra as group theory; and trigonometry.

I encourage classroom teachers of mathematics to use Grabiner's generic paradigm both as a tool for their own acquisition of authentic historical accounts of the evolution of mathematical topics and as a pedagogical stratagem for their students to do the same.

GENERALIZED FIBONACCI SEQUENCES

Source: Mathematics Teacher, Oct2000, Vol. 93 Issue 7, p604, 3p

Author(s): Bradley, Sean

Everyone loves the Fibonacci sequence. It is easy to describe, yet it gives rise to a vast amount of substantial mathematics. Physical applications and connections with various branches of mathematics abound. What could be better, unless someone told us that the Fibonacci sequence is but one member of an infinite family of sequences that we could be discussing? The generalization that follows has great potential for student and teacher exploration, as well as discovery, wonder, and amusement.

The Fibonacci sequence is defined by the recurrence relation F[sub 0] = 0, F[sub 1] = 1, and F[sub n + 1] - F[sub n] + F[sub n - 1], for all integral n is greater than or equal to 1. The Fibonacci numbers can be generalized in various ways. Horadam (1965) furnishes one example. He defines a collection of sequences that depend on the real numbers a and b, as well as arbitrary integers k and q, as follows: we let w[sub 0] = a, W[sub 1] = b, and W[sub n + 1] = k x w[sub n] - q x W[sub n - 1]. The Fibonacci sequence has a = 0, b = 1, and q = -1. For example, 8 = 1 x 5 - (-1) x 3.

A subset of these sequences is interesting enough to deserve wider recognition among teachers and students of mathematics. We consider Horadam's sequences with w[sub 0] = 0, w[sub 1] = 1, and W[sub n + 1] = k x W[sub n] + W[sub n - 1] for all n is greater than or equal to 1. That is, instead of adding two consecutive terms to find the next term, as in the Fibonacci sequence, we first multiply the current last term in the list by k, then add the result to the next-to-last term. When k = 1, the result is just the ordinary Fibonacci sequence. Table 1 gives the first few terms of several generalized Fibonacci sequences.

We begin by investigating a few properties of this infinite collection of sequences, bringing along only the quadratic formula.

FIBONACCI-LIKE PROPERTIES

For many, the attractions of the Fibonacci sequence are the many elegant identities that it satisfies and the curious properties that it possesses. This article offers eight to illustrate. For more, see Hoggatt (1969) or a variety of other sources. The first five properties follow:

(F1) The GCD of F[sub n] and F[sub n + 1] is I for all integral n is greater than or equal to 0.

(F2) F[sub n] divides F[sub n x m] for all positive integers m, for all integral n > 0.

(F3) F[sub n, sup 2] - F[sub n - 1] x F[sub n + 1] = (-1)[sup n + 1] for all integral n is greater than or equal to 1.

(F4) F[sub 2, sub 2] + F[sub n + 1, sup 2] = F[sub 2n + 1] for all integral n is greater than or equal to 0.

(F5) {F[sub 1] + F[sub 3] + F[sub 5] + ... + F[sub 2n - 1] = F[sub 2] {F[sub 2] + F[sub 4] + F[sub 6] + ... + F[sub 2n] = F[sub 2n + 1] - 1

The first four statements are still true if any of the generalized Fibonacci sequences W[sub n] replace the Fibonacci sequence F[sub n]. Property (5) needs only the minor modification