Some useful integer arrays for the teaching of generalization skills

The generalization of previously known results is both a key strategy and one of the delights of mathematical research. Casti comments that it is the ‘stock-in-trade of the mathematical enterprise’ [1]. Generalization often links previously unrelated results, gives a deeper insight into the original result and suggests alternative and more elegant methods of proof. Unfortunately, traditional mathematics syllabi pay scant explicit regard to its role. It seems to be assumed that better students will acquire the necessary mind-set (and rewards) naturally, while the rest are left to struggle with the more basic mathematical facts and methods. These latter are thus denied a part of the joy of mathematical discovery and suffer for the attendant lack of motivation. The purpose of this note is to provide resource material for use in guiding students to explore and consciously seek to generalize a range of mathematical results. In this case, the particular context is number sequences and arrays but it is hoped that th underlying mindset illustrated will itself be applied more generally.