Fibonacci numbers revisited: technology-motivated inquiry into a two-parametric difference equation

This article demonstrates how within an educational context, supported by the notion of hidden mathematics curriculum and enhanced by the use of technology, new mathematical knowledge can be discovered. More specifically, proceeding from the well-known representation of Fibonacci numbers through a second-order difference equation, this article explores its two-parametric generalization using computer algebra software and a spreadsheet. Combined with the use of calculus, matrix theory and continued fractions, this technology-motivated approach allows for the comprehensive investigation of the qualitative behaviour of the orbits produced by the so generalized difference equation. In particular, loci in the plane of parameters where different types of behaviour of the cycles of arbitrary integer period formed by generalized Golden Ratios realize have been constructed. Unexpected connections among the analytical properties of the loci, Fibonacci numbers and binomial coefficients have been revealed. Pedagogical, mathematical and epistemological issues associated with the proposed approach to the teaching of mathematics are discussed.