id: 05777692
dt: j
an: 2010e.00422
au: Abramovich, Sergei; Leonov, Gennady A.
ti: Spreadsheets and the discovery of new knowledge.
so: Spreadsheets Educ. 3, No. 2, 42p (2009).
py: 2009
pu: Bond University, Faculty of Business, Gold Coast, Queensland
la: EN
cc: F65 H25 K25 R25 R75 U75
ut: spreadsheets; Fibonacci numbers; Fibonacci-like polynomials; difference
equations; cycles; generalized golden ratio; combinatorial identities
ci:
li:
ab: Summary: This paper shows how education-oriented spreadsheet-based
explorations with Fibonacci numbers can result in the discovery of
cycles of different periods formed by the orbits of a two-parametric
difference equation of the second order. This equation is motivated
through the introduction of the so-called Fibonacci sieve. The
occurrence of the cycles is interpreted in terms of Fibonacci-like
polynomials brought into being in the context of these explorations.
This new class of polynomials possesses a number of interesting
properties connected to the notion of a generalized golden ratio and
can be used as a background for a spreadsheet-enhanced teaching of
combinatorial identities and their formal demonstration. The paper
reflects on activities designed for a technology-rich mathematics
education course for prospective teachers of secondary mathematics. It
is argued that an appropriate experience with a mathematical frontier
can motivate the teachers to teach through a guided discovery mode.
rv: