Florida State University Libraries Electronic Theses, Treatises and Dissertations The Graduate School 2010 Informal Number-Related Mathematics Skills: An Examination of the Structure of and Relations Between These Skills in Preschool David J. Purpura Follow this and additional works at the FSU Digital Library. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS AND SCIENCES INFORMAL NUMBER-RELATED MATHEMATICS SKILLS: AN EXAMINATION OF THE STRUCTURE OF AND RELATIONS BETWEEN THESE SKILLS IN PRESCHOOL By DAVID J. PURPURA A Dissertation submitted to the Department of Psychology in partial fulfillment of the requirements for the degree of Doctor of Philosophy Degree Awarded: Summer Semester, 2010 Copyright © 2009 David J. Purpura All Rights Reserved The members of the committee approve the dissertation of David J. Purpura defended on October 30, 2009. __________________________________ Christopher J. Lonigan Professor Directing Dissertation __________________________________ Barbara Foorman University Representative __________________________________ Janet Kistner Committee Member __________________________________ Christopher Schatschneider Committee Member __________________________________ Jesse Cougle Committee Member The Graduate School has verified and approved the above-named committee members. ii I dedicate this dissertation to my amazing and beautiful wife Ashley. Without your love and support, none of this would have been possible. This degree is as much yours as it is mine. iii ACKNOWLEDGEMENTS I would like to thank Dr. Christopher Lonigan for his guidance and mentorship throughout my graduate career. You have pushed me to be a better student, writer, and researcher. For your continued efforts to challenge me, I am truly grateful. I would also like to thank the members of my committee, Drs. Foorman, Kistner, Schatschneider, and Cougle, for their valuable input and support throughout my work on this project. My thanks are also extended to the local preschool teachers and directors without whom this project would not have been possible. To the undergraduate testers who worked with me, your hard work has enabled me to complete this project successfully. My fellow graduate students, your friendship, and support have made graduate school an amazing experience. I could not imagine having gone through these years without any of you. To my parents who have pushed me to excel and not allowed me to settle for less than I could accomplish, I cannot thank you enough. The rest of my family, I thank you for the love and support you have shown to me over the years. Most importantly, I thank my beautiful wife Ashley. I could not have done this without you. The endless hours you have spent over the last few years reading drafts of papers, encouraging me when I was buried in work, and helping me to stop and smell the roses, has been a blessing to me. This work was supported by a Predoctoral Interdisciplinary Research Training Grant (R305B04074) from the Institute of Education Sciences. iv TABLE OF CONTENTS List of Tables ............................................................................................. vii List of Figures ........................................................................................... viii Abstract ............................................................................................... x 1. INTRODUCTION ................................................................................. 1 Formal and Informal Mathematics Skills ............................................. 2 Validity Concerns ................................................................................. 12 Overview of Current Study .................................................................. 19 2. PHASE 1: INITIAL MEASURE CONSTRUCTION .......................... 21 Method ............................................................................................... 21 Results ............................................................................................... 29 3. PHASE 2: FURTHER MEASURE REFINEMENT AND CONSTRUCT COMPARISON ........................................................................... 40 Method ............................................................................................... 40 Results ............................................................................................... 43 4. DISCUSSION ....................................................................................... 77 Measure Development.......................................................................... 77 Model Comparison ............................................................................... 79 Future Directions .................................................................................. 80 Early Numeracy and Third Variables ................................................... 82 Conclusion ............................................................................................ 83 APPENDICES ............................................................................................ 84 A Informed Consent Form .............................................................. 84 B Facility Consent Form ................................................................. 88 C Child Assent Script...................................................................... 91 D Phase 1 Counting Subtest ............................................................ 93 E Phase 1 Numerical Relations Subtest .......................................... 97 F Phase 1 Arithmetic Reasoning Subtest........................................ 101 v G Phase 1 Item Psychometric Information ..................................... 104 H Phase 2 Counting Subtest ............................................................ 129 I Phase 2 Numerical Relations Subtest .......................................... 133 J Phase 2 Arithmetic Reasoning Subtest........................................ 137 K Phase 2 Item Psychometric Information ..................................... 140 REFERENCES .......................................................................................... 166 BIOGRAPHICAL SKETCH ..................................................................... 172 vi LIST OF TABLES Table 1: Names, descriptions, and means of assessing counting skills.................. 5 Table 2: Names, descriptions, and means of assessing Numerical Relations skills ....................................................................................................... 9 Table 3: Names, descriptions, and means of assessing Arithmetic Reasoning skills...................................................................................................... 13 Table 4: Listing of important Counting, Numerical Relations, and Arithmetic Reasoning skills on five measures of informal mathematics skills and the National Council of Teachers of Mathematics standards ...................................... 16 Table 5: Demographic information of children participating in Phase 1 by group.................................................................................................................. 29 Table 6: Means, standard deviations, range, skewness, and kurtosis of the sum scores of the final early numeracy tasks in preschool children ................ 65 Table 7: Correlations between the sum scores of the final early numeracy skill tasks in preschool children ............................................................ 67 Table 8: Means, standard deviations, and F-values of the early numeracy skills for preschool children by sex ........................................................................ 69 Table 9: Model fit for the Counting, Numerical Relations, and Arithmetic Reasoning factors ................................................................................................... 70 Table 10: Fit indices for the models of the structure of early numeracy skills in preschool children ..................................................................................... 74 vii LIST OF FIGURES Figure 1: Test Information Functions for Counting tasks. High levels of information indicate high reliability of the task at a given point along the latent trait................................................................................................................ 45 Figure 2: Standard Error Functions for Counting tasks. Standard error values below .316 indicate high reliability of the task at a given point along the latent trait................................................................................................................ 46 Figure 3: Test Information Functions for Counting subtest (top) and Standard error (bottom) of the subtest. High levels of information and low standard error values indicate high reliability of the task at a given point along the latent trait. .. 51 Figure 4: Test Information Functions for Numerical Relations tasks. High levels of information indicate high reliability of the task at a given point along the latent trait. ........................................................................................................................ 52 Figure 5: Standard Error Functions for the Numerical Relations tasks. Standard error values below .316 indicate high reliability of the task at a given point along the latent trait. ......................................................................................................... 53 Figure 6: Test Information Functions