A KNOWLEDGE STRUCTURE FOR THE ARITHMETIC MEAN: RELATIONSHIPS BETWEEN STATISTICAL CONCEPTUALIZATIONS AND MATHEMATICAL CONCEPTS by Mark A. Marnich B.S., Mathematics, Carnegie Mellon University, 1994 M.A., Mathematics, University of Pittsburgh, 2000 Submitted to the Graduate Faculty of the School of Education in partial fulfillment of the requirements for the degree of Doctor of Education University of Pittsburgh 2008 UNIVERSITY OF PITTSBURGH School of Education This dissertation was presented by Mark A. Marnich It was defended on August 11, 2008 and approved by Dr. James G. Greeno, Professor, Department of Instruction and Learning Dr. Nancy Pfenning, Senior Lecturer, Department of Statistics Dr. Margaret S. Smith, Professor, Department of Instruction and Learning Dr. Gabriel Stylianides, Assistant Professor, Department of Instruction and Learning Dissertation Advisor: Dr. Ellen Ansell, Associate Professor, Department of Instruction and Learning ii Copyright © by Mark A. Marnich 2008 iii A KNOWLEDGE STRUCTURE FOR THE ARITHMETIC MEAN: RELATIONSHIPS BETWEEN STATISTICAL CONCEPTUALIZATIONS AND MATHEMATICAL CONCEPTS Mark A. Marnich, Ed.D. University of Pittsburgh, 2008 This study examined cognitive relationships between the fair-share and center-of-balance conceptualizations of the arithmetic mean. It also hypothesized the use of these conceptualizations as blending spaces for the mathematical and statistical domains within a proposed knowledge structure for the arithmetic mean. Twenty-nine undergraduate liberal arts students completed pre/post verbal protocols with written solutions to arithmetic mean problems. The problems emphasized either the fair-share or center-of-balance conceptualization, or mathematical concepts related to the arithmetic mean. The participants were divided into three groups: those that received fair-share instruction, those that received center-of-balance instruction, and a control group. The data was analyzed using statistical methods, including contingency tables and ANCOVA, to investigate the effects fair-share and center-of-balance instruction had on knowledge of fair-share, center-of-balance, and mathematical concepts regarding the arithmetic mean. A qualitative analysis of the verbal protocols helped explain any statistically significant connection between the fair-share and center-of-balance conceptualizations, or between either conceptualization and mathematical concepts related to the arithmetic mean. Analysis of the data indicated participants increased their knowledge of the fair-share conceptualization after receiving instruction that was focused on center-of-balance. Similarly, iv participants increased their knowledge of the center-of-balance conceptualization after receiving instruction that was focused on fair-share. In either case, the concept, ‘the sum of the deviations from the mean is zero,’ was used to transfer knowledge between the conceptualizations. In addition, instruction in either the fair-share or center-of-balance conceptualization increased knowledge of the mathematical concepts related to the arithmetic mean. However, only specific mathematical concepts were impacted by each of the conceptualizations. The results suggest that both the fair-share and center-of-balance conceptualizations are pertinent to pedagogical decisions regarding the arithmetic mean. Furthermore, the concept, ‘the sum of the deviations from the mean is zero,’ is a viable cognitive connection between the fair- share and center-of-balance conceptualizations. v TABLE OF CONTENTS ACKNOWLEDGMENTS ........................................................................................................ XII 1.0 INTRODUCTION........................................................................................................ 1 1.1 BACKGROUND.................................................................................................. 3 1.1.1 Differences between Mathematics and Statistics ....................................... 3 1.1.2 Relationship between Mathematical and Statistical Knowledge.............. 4 1.1.3 Arithmetic Mean as a Subject of Study ...................................................... 6 1.2 PURPOSE OF STUDY........................................................................................ 8 1.2.1 Research Questions....................................................................................... 8 1.3 CONTRIBUTION TO THE DISCIPLINES..................................................... 9 1.4 LIMITATIONS OF STUDY............................................................................. 10 1.5 ORGANIZATION OF THE DISSERTATION.............................................. 11 2.0 REVIEW OF LITERATURE ................................................................................... 13 2.1 DEVELOPING UNDERSTANDING OF THE ARITHMETIC MEAN ..... 14 2.1.1 Averages and Their Development ............................................................. 14 2.1.2 Arithmetic Mean and Its Development..................................................... 16 2.2 MATHEMATICAL AND STATISTICAL KNOWLEDGE DOMAINS..... 19 2.2.1 Mathematical Knowledge of the Arithmetic Mean.................................. 20 2.2.1.1 Mathematical Procedural Knowledge of Arithmetic Mean............ 21 2.2.1.2 Mathematical Conceptual Knowledge of Arithmetic Mean ........... 21 2.2.2 Statistical Knowledge of the Arithmetic Mean ........................................ 23 vi 2.2.3 Connection between the Mathematical and Statistical Domains ........... 24 2.3 FAIR-SHARE AND CENTER-OF-BALANCE ............................................. 28 2.3.1 Fair-Share.................................................................................................... 28 2.3.1.1 Fair-Share as a Mathematical Concept ............................................ 29 2.3.1.2 Fair-Share as a Statistical Concept................................................... 30 2.3.2 Center-of-Balance ....................................................................................... 31 2.3.2.1 Center-of-Balance as a Mathematical Concept ............................... 32 2.3.2.2 Center-of-Balance as a Statistical Concept ...................................... 33 2.3.3 Relationship between Fair-Share and Center-of-Balance....................... 35 2.4 A KNOWLEDGE STRUCTURE FOR THE ARITHMETIC MEAN......... 37 2.4.1 Relating Mathematical Concepts to Concepts in Other Disciplines ...... 39 2.4.1.1 Cognitive Blending.............................................................................. 42 2.4.2 Using Fair-Share and Center-of-Balance for Connecting Domains....... 44 2.5 INSTRUCTIONAL INVESTIGATIONS........................................................ 46 2.6 SUMMARY AND IMPLICATIONS OF LITERATURE ............................. 49 3.0 METHODOLOGY..................................................................................................... 51 3.1 RESEARCH DESIGN....................................................................................... 52 3.2 PARTICIPANTS ............................................................................................... 55 3.3 DATA COLLECTION...................................................................................... 57 3.3.1 Instrument ................................................................................................... 59 3.3.1.1 Fair-Share Problems .......................................................................... 60 3.3.1.2 Center-of Balance Problems .............................................................. 64 3.3.1.3 Mathematical Concept Problems ...................................................... 66 vii 3.3.1.4 Assignment of Problems..................................................................... 69 3.3.2 Procedure..................................................................................................... 70 3.3.2.1 Verbal Protocol Pretest and Posttest ................................................ 71 3.3.2.2 Knowledge Instruction Modules ....................................................... 72 3.3.3 Data Collection Summary .......................................................................... 74 3.4 DATA CODING................................................................................................. 75 3.4.1 Inter-rater Reliability ................................................................................. 83 3.5 DATA ANALYSIS............................................................................................. 84 3.5.1 Quantitative Statistical Analysis................................................................ 84 3.5.1.1 Association Study within Groups...................................................... 85 3.5.1.2 Comparison Study between Groups ................................................. 86 3.5.2 Qualitative Analysis.................................................................................... 90 3.5.3 Data Analysis Summary............................................................................. 94 3.6 SUMMARY OF METHODS............................................................................ 94 4.0 RESULTS ................................................................................................................... 96 4.1 GENERAL CHARACTERISTICS OF PRETEST DATA............................ 96 4.2 FAIR-SHARE AND CENTER-OF-BALANCE RELATIONSHIP............ 104 4.2.1 Hypothesis Testing for Research Question #1........................................ 104 4.2.2 Tabular Depiction of