A Computational Theory of Early Mathematical Cognition by Albert Goldfain June 1, 2008 Dissertation Committee: William J. Rapaport (Major Professor) Stuart C. Shapiro Douglas H. Clements A dissertation submitted to the Faculty of the Graduate School of State University of New York at Buffalo in partial fulfillment of the requirements for the degree of Doctor of Philosophy Department of Computer Science and Engineering Copyright by Albert Goldfain 2008 ii Contents 1 Introduction 1 1.1 Mathematical Cognition . 1 1.2 Motivation . 2 1.3 Two “Foundations” Programs . 2 1.4 The “Burning” Questions . 4 1.5 Computational Math Cognition . 4 1.5.1 Understanding . 5 1.5.2 SNePS for Mathematical Cognition . 5 1.5.3 Focus . 6 1.6 Significance . 6 1.7 Methodology . 7 1.8 Outline . 8 2 Previous Work on Mathematical Cognition 9 2.1 Research from the Branches of Cognitive Science . 9 2.1.1 Psychology . 9 2.1.1.1 Piaget . 10 2.1.1.2 Gelman and Gallistel . 11 2.1.1.3 Wynn . 13 2.1.1.4 Butterworth . 14 2.1.2 Neuroscience . 15 2.1.2.1 Dehaene . 16 2.1.3 Philosophy . 17 2.1.3.1 Shapiro . 17 2.1.3.2 Wittgenstein . 18 2.1.3.3 Glasersfeld . 19 2.1.4 Linguistics . 20 2.1.4.1 Lakoff and Nu´nez˜ . 20 2.1.4.2 Wiese . 23 2.1.5 Education . 24 2.1.5.1 Schoenfeld . 24 2.1.5.2 Sfard . 25 2.1.6 Anthropology . 26 iii 2.1.6.1 Sapir and Whorf . 27 2.1.6.2 Hutchins . 27 2.2 Research in Mathematics . 28 2.2.0.3 Polya . 28 2.2.0.4 Mac Lane . 29 2.3 Research in Artificial Intelligence . 30 2.3.1 Automated Theorem Proving . 30 2.3.2 Production-System Simulations . 31 2.3.2.1 Klahr . 31 2.3.2.2 Fletcher and Dellarosa . 31 2.3.3 Heuristic Driven Models . 32 2.3.3.1 Lenat . 32 2.3.3.2 Ohlsson and Rees . 33 2.3.4 Connectionist Models . 34 2.3.5 Tutoring Systems and Error Models . 35 2.3.5.1 Nwana . 36 2.3.5.2 Brown and VanLehn . 36 2.3.6 Word-Problem Solving Systems . 37 2.3.6.1 Bobrow . 37 3 Understanding and Explanation: Towards a Theory of Mathematical Cognition 39 3.1 Claims . 39 3.1.1 Multiple Realizability . 39 3.1.2 Representability . 40 3.1.3 Performability . 42 3.1.4 Empirical Testability . 42 3.2 SNePS . 43 3.2.1 Overview of SNePS . 43 3.2.1.1 GLAIR . 43 3.2.1.2 SNIP . 44 3.2.1.3 SNeRE . 45 3.2.1.4 SNeBR . 46 3.2.2 SNePS for Mathematical Cognition . 46 3.3 Understanding . 47 3.3.1 Modes of Understanding . 47 3.3.2 Understanding in SNePS . 49 3.3.3 Characterizations of Understanding . 50 3.3.3.1 Logical Characterizations . 50 3.3.3.2 Educational Characterizations . 52 3.3.3.3 Computational Characterizations . 55 3.4 From Knowledge to Understanding . 55 3.4.1 Formalizations of JTB . 56 3.4.1.1 Belief . 56 iv 3.4.1.2 Truth . 57 3.4.1.3 Justification . 57 3.4.2 JTB in SNePS . 58 3.4.3 First-person knowledge . 59 3.4.4 Understanding as a Gradient of Features . 60 3.5 Exhaustive Explanation . 60 3.5.1 A Turing Test for Mathematical Understanding . 61 3.5.2 The Endpoint of an Explanation . 62 3.5.3 Multiple Justifications . 64 3.5.4 Behavior and Procedural Understanding . 65 4 Abstract Internal Arithmetic 68 4.1 Representations for Abstract Internal Arithmetic . 68 4.1.1 A Case-Frame Dictionary for Mathematical Cognition . 69 4.1.1.1 Counting Case-Frames . 70 4.1.1.2 Arithmetic Acts . 72 4.1.1.3 Evaluation and Result Case-Frames . 73 4.1.1.4 Results as Procepts . 75 4.2 A Taxonomy of Arithmetic Routines . 76 4.3 Semantic Arithmetic . 77 4.3.1 Counting . 77 4.3.1.1 The Greater-Than Relation . 79 4.3.2 Count-Addition . 81 4.3.3 Count-Subtraction . 82 4.3.4 Iterated-Addition Multiplication . 83 4.3.5 Iterated-Subtraction Division . 85 4.3.6 Inversion Routines . 87 4.4 Cognitive Arithmetic Shortcuts . 88 4.5 UVBR and Cognitive Plausibility . 90 4.6 Syntactic Arithmetic . 91 4.6.1 Multidigit Representations . 92 4.6.2 Extended Arithmetic . 92 4.7 Questions in SNePS . 93 4.8 Why Questions and Procedural Decomposition . 95 4.9 Conceptual Definitions . 97 4.10 Questions in Natural Language . 98 4.11 A Case Study: Greatest Common Divisor . 99 4.11.1 Human Protocols . 100 4.11.2 A Commonsense Natural-Language GCD Algorithm . 101 4.11.3 An Idealized Dialogue.