Applying Kolmogorov’s Proofs to the Evaluation of Instruction: Generalized and Particular (PART ONE)

Successful sharing of information—positive (actual) knowledge about facts, skills that are imparted, abilities developed and expressed—is the implicit goal of instruction in all its varied forms. It is the goal of training athletes, dancers, and professionals in every walk of life from early childhood to the most advanced level of education. PART ONE introduces mathematical proofs showing that the interactional successes engineered by instructors, other things being equal, must trend toward 100% shared information—mastery of the course of study. Failed efforts trend toward a complete absence of shared information. All this holds independently for the subject-matter, methods of instruction, and the attributes conducive to instructional success. In Part One, the underlying proofs are united by a very simple proof from the theory of true narratives showing that every iota of knowledge that might be shared in any instructional context depends on the kind of representations found in true reports of actual experience. Empirical studies in Part One confirm the predicted agreement in diverse contexts on the elements of good teaching. In Part Two, Kolmogorov’s proofs from 1933 are generalized, amplified, and tested empirically showing successful instruction converging toward 100% agreement on 1) subject-matter, 2) which methods of presentation and assessment work, and even on 3) the abstract criteria for successful instruction. At the same time, as the proofs also show, the cumulative effects of failed communicative efforts must and do trend toward zero shared information.