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Revisiting Fibonacci Numbers through a Computational Experiment (Paperback)
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Revisiting Fibonacci Numbers through a Computational Experiment (Paperback)
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The material of this book stems from the idea of integrating a
classic concept of Fibonacci numbers with commonly available
digital tools including a computer spreadsheet, Maple, Wolfram
Alpha, and the graphing calculator. This integration made it
possible to introduce a number of new concepts such as: Generalized
golden ratios in the form of cycles represented by the strings of
real numbers; Fibonacci-like polynomials the roots that define
those cycles dependence on a parameter; the directions of the
cycles described in combinatorial terms of permutations with rises,
as the parameter changes on the number line; Fibonacci sieves of
order k; (r, k)-sections of Fibonacci numbers; and polynomial
generalizations of Cassinis, Catalans, and other identities for
Fibonacci numbers.The development of these concepts was motivated
by considering the difference equation
f_(n+1)=af_n+bf_(n-1),f_0=f_1=1, and, by taking advantage of
capabilities of the modern-day digital tools, exploring the
behavior of the ratios f_(n+1)/f_n as n increases. The initial use
of a spreadsheet can demonstrate that, depending on the values of a
and b, the ratios can either be attracted by a number (known as the
Golden Ratio in the case a = b = 1) or by the strings of numbers
(cycles) of different lengths. In general, difference equations,
both linear and non-linear ones serve as mathematical models in
radio engineering, communication, and computer architecture
research. In mathematics education, commonly available digital
tools enable the introduction of mathematical complexity of the
behavior of these models to different groups of students through
the modern-day combination of argument and computation.The book
promotes experimental mathematics techniques which, in the digital
age, integrate intuition, insight, the development of mathematical
models, conjecturing, and various ways of justification of
conjectures. The notion of technology-immune/technology-enabled
problem solving is introduced as an educational analogue of the
notion of experimental mathematics. In the spirit of John Dewey,
the book provides many collateral learning opportunities enabled by
experimental mathematics techniques. Likewise, in the spirit of
George Polya, the book champions carrying out computer
experimentation with mathematical concepts before offering their
formal demonstration.The book can be used in secondary mathematics
teacher education programs, in undergraduate mathematics courses
for students majoring in mathematics, computer science, electrical
and mechanical engineering, as well as in other mathematical
programs that study difference equations in the broad context of
discrete mathematics.
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