Aha! an illuminating perspective

Abstract

The 'Aha!' phenomenon is familiar to us in many domains including computer science and mathematics (e.g., [2,3,6]). It often stems from an unexpected point of view that illuminates an appealing solution path. The 'Aha' reaction is common to all. Its occurrence is related to the problem-solvers' common perspectives and solution repertoires. Whether more frequent or less frequent, 'Aha' occurrences enrich and strengthen perspectives and repertoires in a stimulating manner.Consider the following Ladder Problem: calculate the number of different ways to climb an N-stage ladder when each step is either one or two stages. One solution perspective may be 'forward reasoning', leading to a systematic accumulation of the possible climbing paths. Another perspective may be combinatorial, leading to the calculation of all the combinations of 1 and 2 that sum to N. A third perspective may be 'backward reasoning', yielding recursive decomposition of the Nth case into the N-1 and N-2 cases.Some problem-solvers may fairly quickly invoke the third perspective and elegantly obtain the Nth Fibonacci number. Others may first follow one of the other perspectives and later realize the illuminating third perspective. The 'Aha' reactions among the solvers may vary. However, both less experienced and more experienced solvers will gain from recognizing the relevance and elegance of the recursive decomposition and enhance their problem-solving repertoires.