Quantifying depth and complexity of thinking and knowledge

Qualitative approaches to cognitive rigor and depth and complexity are broadly represented by Webb's Depth of Knowledge and Bloom's Taxonomy. Quantitative approaches have been relatively scant, and some have been based on ancillary measures such as the thinking time expended to answer test items. In competitive chess and other games amenable to incremental search and expert evaluation of options, we show how depth and complexity can be quantified naturally. We synthesize our depth and complexity metrics for chess into measures of difficulty and discrimination, and analyze thousands of games played by humans and computers by these metrics. We show the extent to which human players of various skill levels evince shallow versus deep thinking, and how they cope with 'difficult' versus 'easy' move decisions. The goal is to transfer these measures and results to application areas such as multiple-choice testing that enjoy a close correspondence in form and item values to the problem of finding good moves in chess positions.