A Journey to a Mathematical Frontier with Multiple Computer Tools
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A Journey to a Mathematical Frontier with Multiple Computer Tools Sergei Abramovich • Gennady A. Leonov
Published online: 19 March 2011 Springer Science+Business Media B.V. 2011
1 Introduction Consider the difference equation fkþ1 ¼ afk þ bfk1 ;
k ¼ 1; 2; 3; . . .;
f0 ¼ f 1 ¼ 1
ð1Þ
where a and b are real parameters. It is well known, that in the case of a = b = 1 Eq. 1 generates the number sequence 1, 1, 2, 3, 5, 8, 13,… introduced into Western mathematics by a thirteenth century Italian mathematician The closed representation of Fibonacci. pffiffikþ1 pffiffikþ1 12 5 ; k ¼ 0; 1; 2; . . .; Fibonacci numbers Fk has the form Fk ¼ p1ffiffi5 1þ2 5 and the ratios Fk?1/Fk of two consecutive Fibonacci numbers oscillate about and eventually pffiffi converge to the number / ¼ 1þ2 5 called the Golden Ratio. The goal of this article is to demonstrate how the integration of multiple software tools into a familiar educational context of Fibonacci numbers allows for the discovery of new mathematical knowledge. It is a snapshot of the authors’ recent work (Abramovich and Leonov 2008, 2009a, b) on the development of technology-motivated methods of teaching topics in discrete mathematics at the university level, including programs for secondary mathematics teachers (referred to below as teachers). The snapshot highlights the potential of the joint use of a spreadsheet and computer algebra systems for the discovery of cycles *This column will publish short (from just a few paragraphs to ten or so pages), lively and intriguing computer-related mathematics vignettes. These vignettes or snapshots should illustrate ways in which computer environments have transformed the practice of mathematics or mathematics pedagogy. They could also include puzzles or brain teasers involving the use of computers or computational theory. Snapshots are subject to peer review. From the Column Editor Uri Wilensky, Northwestern University. Email: [email protected]. S. Abramovich (&) State University of New York at Potsdam, Potsdam, NY, USA e-mail: [email protected] G. A. Leonov St Petersburg State University, Saint Petersburg, Russia e-mail: [email protected]
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of different periods formed by the orbits of Eq. 1. Reflecting on work done with prospective secondary mathematics teachers in a capstone course, the snapshot will demonstrate the didactic significance of the joint use of experiment and theory in exploring mathematical ideas, thereby extending the notion of experimental mathematics (Borwein and Bailey 2004) to the domain of teacher education. The major assumption of the authors’ work was that Fibonacci numbers and associated concepts when explored in a computer environment provide an excellent context for promoting many ideas of mathematics education reform among the teachers allowing them to ‘‘take actions like representing, experimenting, modeling, classifying, and proving’’ (Conference Board of the Mathematical Sciences 2001, p. 8). In particular, by using technology, two distinct cases depending on t
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