Solving proportion problems in schools is a very difficult task for most children. Too often children are taught to use techniques for solving fractions when working with ratio and proportion problems. Although these techniques may prove useful for obtaining a solution to a proportion problem, they do not provide rich learning opportunities for students to construct proportional relationships. Individual interviews, problem solving episodes, and personal journals were the primary tools used in collecting data for this study. The problem solving episodes became the key components in observing and interacting with the participants as they solved ratio and proportion problems. The researcher found that individuals must have many elaborated constructions to solve proportion tasks. Without these elaborated constructions the individual is unable to effectively coordinate the information needed to solve proportion tasks. That is, the individual is unable to reason proportionally. Another outcome of this study was the importance of finding meaningful and doable proportion tasks for the informants to complete. Unexpected outcomes of the study included the effect of engaging in problem solving on the identity of the child and the role of language in giving meaning to the tasks. / Typescript. / "Summer Semester, 1992." / "Submitted to the Department of Curriculum and Instruction in partial fulfillment of the requirements for the degree of Doctor of Philosophy." / Advisor: Grayson H. Wheatley, Professor Directing Dissertation. / Includes bibliographical references.

Thesis advisor: Ina V.S. Mullis / The eTIMSS mathematics PSIs were a new and pioneering effort to capitalize on the computer- and tablet-based mode of assessment delivery introduced in the eTIMSS 2019 assessments at the fourth and eighth grades. The PSIs were scenario-based mathematics problem solving tasks intended to enhance measurement of mathematics problem solving and reasoning skills and increase student engagement and motivation in the assessment. These unique tasks were designed to measure the same mathematics content as the rest of the mathematics items in the eTIMSS 2019 assessments, but because of their novelty, there was a question about whether the PSIs achieved this goal and could be reported together with the regular TIMSS mathematics items. Following a full-scale field test in 30 countries completed in May 2018, this dissertation conducted an in-depth investigation of the validity of the eTIMSS 2019 mathematics PSIs with the goals of informing analysis and reporting plans for TIMSS 2019 and providing insights for future assessments aspiring to capitalize on digital technology. This investigation involved three key tasks: 1) examining and documenting the methods and procedures used to develop the PSIs and promote validity by design, 2) investigating the characteristics of the PSIs in terms of the content coverage and fidelity of student responses, and 3) using the eTIMSS field test data to evaluate the internal structure of the PSIs. The results indicate that the PSIs are well-aligned with the TIMSS 2019 Mathematics Framework and elicited the intended interactions from students. The regular and PSI items were found to measure the same unidimensional construct, and therefore can be validly reported together on the TIMSS 2019 achievement scale. The lessons TIMSS learned in developing the PSIs for eTIMSS 2019 and suggestions for the future also are discussed. / Thesis (PhD) — Boston College, 2019. / Submitted to: Boston College. Lynch School of Education. / Discipline: Educational Research, Measurement and Evaluation.

by Lee Fong Lok. / Chinese title: / Bibliography: leaves 103-111 / Thesis (M.A.Ed.)--Chinese University of Hong Kong

by Wong Ka-Ming. / Thesis (M.A.Ed.)--Chinese University of Hong Kong, 1989. / Bibliography: leaves 164-173.

by Cheng Kar Wai. / Thesis (M.Phil.)--Chinese University of Hong Kong, 1990. / Bibliography: leaves [195-199] / ABSTRACT / Chapter CHAPTER 1 --- INTRODUCTION --- p.1 / Chapter 1.1 --- Problem --- p.2 / Chapter 1 .2 --- Approach --- p.3 / Chapter 1.2.1 --- Programming Approach --- p.4 / Chapter 1.2.2 --- Specification Approach --- p.5 / Chapter 1.3 --- Thesis Organization --- p.11 / Chapter CHAPTER 2 --- MODEL SPECIFICATION APPROACH --- p.12 / Chapter 2 .1 --- Overview --- p.14 / Chapter 2.2 --- Compilation Phases --- p.17 / Chapter 2.3 --- Array Graph --- p.22 / Chapter 2.4 --- Scheduling --- p.25 / Chapter CHAPTER 3 --- SURVEY --- p.31 / Chapter 3.1 --- Goodenough's Proposal --- p.31 / Chapter 3.2 --- Exception Handling Models --- p.34 / Chapter 3.3 --- Programming Languages --- p.40 / Chapter 3.4 --- Data-Oriented Exception Handling --- p.49 / Chapter 3.5 --- Specification Languages --- p.50 / Chapter CHAPTER 4 --- EXCEPTION HANDLING SPECIFICATION --- p.55 / Chapter 4.1 --- Data-Oriented Exceptions Specification --- p.55 / Chapter 4.2 --- Assertions for Exception Handling --- p.59 / Chapter 4.2.1 --- User-defined Exception Condition Assertion --- p.61 / Chapter 4.2.2 --- Fatal Condition Assertion --- p.62 / Chapter 4.2.3 --- Replacement Assertion --- p.64 / Chapter 4.2.3.1 --- Scenario 1: Immediate Replacement --- p.68 / Chapter 4.2.3.2 --- Scenario 2: Direct Dependency --- p.71 / Chapter 4.2.3.3 --- Scenario 3: Indirect Dependency --- p.72 / Chapter 4.2.3.4 --- Scenario 4: Lower Dimensionality --- p.74 / Chapter 4.2.4 --- Message Vector Assertion --- p.76 / Chapter CHAPTER 5 --- ARRAY GRAPH FOR EXCEPTION HANDLING --- p.78 / Chapter 5.1 --- Subgraph Embedding --- p.78 / Chapter 5.1.1 --- User-Defined Exception Conditions --- p.80 / Chapter 5.1.2 --- Fatal Conditions --- p.82 / Chapter 5.1.3 --- Pre-Defined Exception Conditions --- p.83 / Chapter 5.1.4 --- Replacement Assertions --- p.85 / Chapter 5.1.5 --- Message Vector Assertions --- p.89 / Chapter 5.2 --- Data Dependency Interpretation --- p.91 / Chapter 5.2.1 --- Immediate Replacement --- p.92 / Chapter 5.2.2 --- Direct Dependency --- p.92 / Chapter 5.2.3 --- Indirect Dependency --- p.93 / Chapter 5.2.4 --- Shared Data Variable --- p.99 / Chapter CHAPTER 6 --- SCHEDULING FOR EXCEPTION HANDLING --- p.104 / Chapter 6.1 --- Backward Path Tracing --- p.106 / Chapter 6.1.1 --- Forward Versus Backward Tracing --- p.106 / Chapter 6.1.2 --- Assertion-Marking Strategy --- p.113 / Chapter 6.2 --- Grain Scheduling --- p.116 / Chapter 6.2.1 --- New Constraints --- p.120 / Chapter 6.3 --- Delayed Exception Raise Event --- p.125 / Chapter 6.4 --- Enhancement of Scheduling Algorithm --- p.126 / Chapter 6.5 --- Control Flow Issues --- p.128 / Chapter 6.5.1 --- Immediate Replacement --- p.130 / Chapter 6.5.2 --- Direct Dependency --- p.131 / Chapter 6.5.3 --- Indirect Dependency --- p.132 / Chapter 6.5.4 --- Lower Dimensionality --- p.133 / Chapter CHAPTER 7 --- MORE COMPLICATED SCHEDULING --- p.135 / Chapter 7.1 --- Multiple Exception Handling Assertions --- p.137 / Chapter 7.1.1 --- Overlapped Scopes of Exception Grain --- p.138 / Chapter 7.1.2 --- Priorities in Scheduling --- p.156 / Chapter 7.2 --- Single Replacement Assertion --- p.160 / Chapter 7.2.1 --- Multiple Exception Conditions --- p.160 / Chapter 7.2.2 --- Conditional Replacement --- p.163 / Chapter 7 .3 --- Loop Optimization --- p.164 / Chapter 7.4 --- Modifications to the Scheduling Algorithm --- p.177 / Chapter 7.5 --- Implementation --- p.180 / Chapter 7.5.1 --- Syntax Checking --- p.180 / Chapter 7.5.2 --- Array Graph Construction --- p.182 / Chapter 7.5.3 --- Array Graph Analysis --- p.185 / Chapter 7.5.4 --- Generation of Schedule with Exception Handling Subgraph --- p.186 / Chapter CHAPTER 8 --- CONCLUSIONS --- p.187 / Chapter 8 .1 --- Future Work --- p.188 / APPENDIX / Chapter 1. --- Backward Tracing/Assertion Marking Strategy / REFERENCE

Yeung Kam-chuen Anthony. / Title also in Chinese. / Thesis (M.A.Ed.)--Chinese University of Hong Kong, 1990. / Bibliography: leaves 137-144. / ACKNOWLEDGEMENTS --- p.ii / ABSTRACT --- p.iv / LIST OF TABLES --- p.viii / LIST OF FIGURES --- p.xiii / CHAPTER / Chapter 1. --- Introduction --- p.1 / Context of the study problem --- p.1 / Statement of the Problem --- p.3 / Significance of the Study --- p.4 / Chapter 2. --- Review of Literature --- p.8 / From Concept Formation to Problem Solving --- p.8 / About Problem Solving --- p.12 / Information-processing Theory of Human Problem Solving --- p.16 / The Nature of Social Science Problems --- p.28 / Domain-specific Knowledge in Social Science Problem Solving --- p.31 / Social Science Problem Solving Strategies --- p.38 / Chapter 3. --- The Social Science Problem-solving Model --- p.40 / Early Development of the Social Science Problem-solving Model --- p.40 / The Problem-solving- reasoning Model --- p.41 / Chapter 4. --- Research Design --- p.48 / Statement of Hypotheses --- p.48 / Operational Definitions of Variables --- p.52 / Subjects --- p.57 / Instruments --- p.62 / Procedures --- p.71 / Chapter 5. --- Results and Discussion --- p.78 / Statistical Analysis of Data --- p.79 / Qualitative Analysis of Data --- p.108 / Discussion --- p.119 / Chapter 6. --- Conclusions and Recommendations --- p.130 / Conclusions --- p.130 / Implications --- p.132 / Limitations --- p.135 / Recommendations --- p.136 / BIBLIOGRAPHY --- p.137 / APPENDIX / Chapter 1. --- The Knowledge Test --- p.145 / Chapter 2. --- "The ""Locating a Ball Pen Factory"" Problem" --- p.150 / Chapter 3. --- "The ""Locating an Oil Refinery"" Problem" --- p.153

Wong Ngai Ying. / Chinese title in romanization: Zi wo jian cha you qiang hua zuo yong dui jie nan biao xian di ying xiang. / Thesis (M.A.Ed.)--Chinese University of Hong Kong, 1987. / Bibliography: leaves 89-99.

Thesis (Ed.D.)--Boston University / PLEASE NOTE: Boston University Libraries did not receive an Authorization To Manage form for this thesis or dissertation. It is therefore not openly accessible, though it may be available by request. If you are the author or principal advisor of this work and would like to request open access for it, please contact us at open-help@bu.edu. Thank you. / The purpose of the study was to investigate the algebraic reasoning of first through third grade subjects solving systems of two linear equations with two variables. The population consisted of 60 subjects, 20 from each of the grades, 1, 2, and 3, in an elementary school in a suburban city in Massachusetts. To assess algebraic reasoning abilities, the Assessment of Students' Equation Solving Abilities (ASESA) instrument was developed, one version for each grade level. In ASESA, three types of systems of equations were presented in three contexts: pictures of animals, geometric figures, and letters to represent the variables for a total of nine problems. Variations by grade level were due to the magnitude of the values of the variables. A predetermined sequence of hints was developed to be used in the event that subjects could not solve the problems independently. The hints were of three types: 1) Look, that focused subjects on key information; 2) Record, that requested subjects to record given problem information; and 3) Solve, that led subjects to the algorithm for solving the problems. Hints were scored for each subject and each problem. Strategies used to solve the problems were coded as algebraic or arithmetic. Statistical analyses were performed to determine the effect of grade level, problem context, problem type, and solution strategy on solution success when the subjects solved problems independently as well as with assistance in the form of hints. Differences in the numbers of hints of each hint type were also identified and analyzed. When solving problems both independently and when hints were provided, grade level was a significant factor for solution success. Subjects in Grades 2 and 3 performed significantly better than did subjects in Grade 1. Problem context was not a significant factor for solution success. The type of problem was a significant factor for solution success. Type 3 problems, involving four solutions steps, posed the greatest difficulty for all subjects. Subjects who used algebraic strategies were significantly more successful solving problems of all types and in all contexts than were subjects who used arithmetic strategies. / 2031-01-01

In this study, I examined the relation between various construct relevant and irrelevant variables and a math problem solving assessment. I used independent performance measures representing the variables of mathematics content knowledge, general ability, and reading fluency. Non-performance variables included gender, socioeconomic status, language proficiency and special education qualification. Using a sequential regression and commonality analysis, I determined the amount of variance explained by each performance measure on the Oregon state math assessment in third grade. All variables were independently predictive of math problem solving scores, and used together, they explained 58% score variance. The math content knowledge measure explained the most variance uniquely (12%), and the measures of math content and general ability explained the most variance commonly (16%). In the second analysis, I investigated whether additional variance was explained once student demographic characteristics were controlled and how this affected the unique variance explained by each independent performance measure. By controlling for demographics, the model explained slightly more than 1% additional variance in math scores. The unique variance explained by each independent measure decreased slightly. This study highlighted the influence of various construct relevant and irrelevant variables on math problem solving scores, including the extent to which a language-free measure of general ability might help to inform likely outcomes. The use of variance partitioning expanded understanding of the unique and common underlying constructs that affect math problem solving assessment. Finally, this study provided more information regarding the influence demographic information has on outcomes related to state math assessments.

Typescript (photocopy). / Digitized by Kansas Correctional Industries