The purpose of this study is to analyze the patterns of classroom discourse when high school students move from performing prescribed algorithms in order to solve problems for which the process and solution are well-defined to spoken proof, in which ideas are discussed and arguments are formulated and formalized. The study uses a modified version of discourse analysis developed by Arno Bellack and refined for usage in a mathematics classroom by James T. Fey. The analysis framework is supplemented by codes borrowed from Maria Blanton, Despina Stylianou, and M. Manuela David (2009), which is in turn a modified version of a coding system developed by Kruger (1993) and Goos, Galbraith and Renshaw (2002). Twelve mathematics lessons involving two mathematics teachers were recorded, transcribed and coded. Eight of the lessons were classified as “proof-related” and four were designated “non-proof-related.” A lesson designated “proof-related” contained more than half activity that was actively concerned with the construction of proof; whereas a lesson in which no proofs were formulated was designated “non-proof.” Using the codes described above and a variety of qualitative and quantitative measures, the transcripts were examined for constructivist behavior on the part of the teachers and modes of participation on the students’ part. The findings suggest a relationship between a teacher’s beliefs in constructivist principles and the way in which that teacher instructs proof vs. non-proof. More specifically, a teacher who views her/himself as informed by constructivist pedagogical principles may not evince a sharp distinction between her/his teaching of proof vs. non-proof; but a teacher who does not attempt to incorporate constructivist principles on a daily basis may exhibit more constructivist tendencies when teaching proof.

本研究以"教学性解释"这一课堂会话形式为研究对象，调查小学数学课堂教学中教学性解释的数学丰富性特征和结构特征，以及它们与学生学习的关系。其中，数学丰富性是指课堂教学活动与数学内容的相关程度,以教学性解释中的学术词汇比例和教学性解释的概念水平为指标。 / 本研究数据源于项目"课程改革的成效──教师课堂教学实践变化"（Ni, Li, Cai, & Hau, 2009），选取使用新课程教材的17名小学数学教师及其1013名学生作为研究样本。教学性解释来源于这17个班级的51节课录像（每个班级3节课录像，教学内容为"分数运算"）。同时，也对学生的认知学业表现（计算、简单问题解决和复杂问题解决）和数学学习的情感表现（学习兴趣、课堂参与、数学观、交流素养）进行了两次测查。 / 研究一考察了教学性解释的结构特征。研究结果表明：小学数学课堂中教学性解释主要是教师进行引导，由学生来提供解释，教师的引导方式包括提问引导和回应性引导。就引导水平而言，教师将选择性引导、产品性引导、过程性引导和元过程引导这四种不同水平的引导方式相结合，其中高水平的引导方式（过程性引导和元过程引导）所占比例近四成，教师倾向于让学生表达观点和看法，提供解释，并且对自己的观点进行论述。 / 研究二考察了教学性解释的数学丰富性特征。研究结果表明：教学性解释的学术词汇比例和概念水平可以作为数学丰富性的两个有效指标。小学数学课堂中教学性解释的丰富性水平较高：绝大部分教学性解释是数学解释，并且原理性解释是比例最高的数学解释。 / 研究三考察了教学性解释的结构特征与丰富性特征与学生学习结果的关系。研究结果表明：教学性解释的丰富性对学生简单问题解决能力表现具有正向预测作用，但与学生计算能力表现呈负相关，同时，与学生复杂问题解决能力表现的关系更为复杂，受到学生原有的知识和技能水平的调节作用。教学性解释的丰富性与学生的数学兴趣、数学观和交流素养呈现负性相关。 / The main purpose of the present study was to investigate the mathematical richness and structural features of instructional explanation in 17 primary mathematics classrooms, and their relations to students’ cognitive and affective performance in learning mathematics. Mathematical richness in the present study refers to the extent to which classroom instruction is related to mathematics, or the extent of doing or talking mathematics in classroom instruction. The indicators of mathematical richness of instructional explanation included the ratio of academic words and conceptual level of instructional explanations. / The data source of the study was from the project "Has curriculum reform made a difference? Looking for change in classroom practice" (Ni, Li, Cai, & Hau, 2009). The current study selected 17 primary mathematics classrooms and the 1013 students from the database. 477 episodes of instructional explanation were identified from the 51 videotaped lessons of the classrooms (3 lessons for each class). The content of all 51 lessons was about division with fractions. The identified episodes of instructional explanation were analyzed in terms of the indicators of mathematical richness. The students’ achievement data included two times of assessment on three aspects of cognitive performance (calculation, simple problem solving, complex problem solving) and four aspects of their indicated attitude towards mathematics and learning mathematics (interest in learning mathematics, classroom participation, views of mathematics, literacy about mathematical communication). / Study One analyzed the structural features of instruction explanations in the 17 primary classrooms. The results indicated that teachers were used to guiding the students to provide explanations when constructing instructional explanation in the mathematics classrooms. Teachers’ elicitations consisted of questions and responsive elicitations. Four levels of elicitation were identified. They were choice elicitation, product elicitation, process elicitation and metaprocess elicitation. The higher levels of elicitations (process and metaprocess elicitation) accounted for a significant amount in the classrooms. The teachers tended to let students express their views, provide explanations and arguments of reasoning. / Study Two investigated the mathematical richness of instructional explanations. The results showed that the ratio of academic words and conceptual level of instructional explanations could be valid and useful indicators of mathematical richness. The mathematical richness of instructional explanation was high for the observed classrooms in terms of the two indicators. Majority of the instructional explanations were mathematical and involved mathematical concepts and principles. / Study Three examined whether the richness and structural features of instructional explanation were able to predict student learning outcomes in the cognitive and affective domain. The results indicated that the mathematical richness positively predicted students’ simple problem solving performance, but was negatively related to students’ computation performance. Furthermore, its relation to students’ complex problem solving performance was complicated, which was moderated by the students’ prior status in the knowledge and skills. Lastly, mathematical richness was negatively associated with students’ indicated interest in learning mathematics, view of mathematics and literacy about mathematical communication. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / Detailed summary in vernacular field only. / 鄒君. / Thesis (Ph.D.) Chinese University of Hong Kong, 2014. / Includes bibliographical references (leaves 148-172). / Abstracts also in Chinese. / Zou Jun.

Thesis (Ph.D.)--University of the Witwatersrand, Faculty of Science, 2004 / This research is in the field of ethnomathematics. The study was carried out in three phases with the Tchadji-game being its principal focus. Tchadji is a traditional game played at Ilha de MOyambique, a small Island situated on the northeast coast of Mozambique. Tchadji belongs to a class of board games known as Mancala, which are thousands of years old and are played throughout the African continent as well as in other parts of the world. Mancala games have only recently received due attention as a topic for research. The first phase of the research was carried out in the school mathematics classroom. The outcomes of this phase indicated that games like Tchadji and the three stones (a variant of the Muravarava game) as a part of Mozambican culture are also rich in providing opportunities for activities in the mathematics classroom related to the development of key probabilistic concepts. Quasi-experimental research involving the researcher, 4 mathematics teachers and 162 students showed statistically significant positive effects on attitude towards the learning of, and performance in probability. These outcomes were corroborated by qualitative research. The second phase of this study consisted of ethnographic research amongst master Tchadji players, which explored the mathematical ideas embedded in the Tchadji-game and in the procedures for playing the game. The results of this phase of the research revealed that, the Tchadji-players had mathematical knowledge, skills and ways of thinking such as counting, logical thinking, calculation, visualisation, recognition of different numerical patterns and infinity. These results are described in terms of examples taken from different critical moments of the recorded matches of Tchadji. This way of iii Abstract (PhD thesis) presenting the results gives insight into the complexity of the methodology used. It also uncovers different aspects of playing Tchadji, like rules, strategies and tactics. A need for further investigations of this nature for uncovering mathematical ideas in traditional culture is also indicated by this research. As advocated in ethnomathematical research an intervention with (pre and in-service) student teachers formed the third phase of the research. The results of this phase indicated that the 24 teachers, who participated in the research, showed enthusiasm, satisfaction and excitement in experiencing the mathematical richness of Tchadji and in appreciating possibilities for the use of Tchadji in the mathematics classroom. They were able to analyse the game independently and to identify embedded mathematical ideas in the game, like logical thinking, counting and empirical and mental calculation. The research makes contributions to the field of ethnomathematics itself, to ethnographic research methodology and to the pedagogy of mathematics.

As a young and dynamically evolving discipline, statistics evokes many conceptions about its purpose, the nature of its development, and the tools and mindset needed to engage in statistical work. While much research documents the perceptions of statisticians and experts on these matters, little is known about how the disciplinary perspectives of statistics instructors may interact with the work of teaching. Such connections are likely relevant since research has shown that teachers’ and instructors’ views about the discipline they teach inform their instructional approaches. This work specifically focuses on the disciplinary views of graduate teaching assistants (GTAs), who continue to serve a critical role in undergraduate instruction. Using multiple case study design, I document the views, experiences, and teaching practices of four statistics GTAs over the course of a full year—from their induction into the department in the fall, until their first solo-teaching opportunity the following summer. From the literature, I organized important disciplinary themes in statistics, including disciplinary purpose, epistemology, and disciplinary engagement. Targeting issues and questions stemming from these areas, I documented the various perspectives, models, and tensions that characterized the disciplinary views of the participants. I also documented the relevant experiences and influences that motivated these views. Additionally, I explored the GTAs’ pedagogical views and vision for teaching introductory statistics while looking for possible connections (and glaring disconnects) between these views and their disciplinary views. Finally, I observed their instruction and considered the participants’ teaching reflections as I looked for alignment between their expressed views and actual instructional decisions. From the data, I found that several of the GTAs expressed sophisticated views and expert notions about the discipline. There was a clear disconnect, however, between their perceptions of disciplinary work and the work of students in an introductory statistics course. Despite recognition that statistical questions typically do not have right answers, that statistical methods are often quite flexible and contextually-driven, or that many disciplinary elements developed through community negotiation rather than discovery, the GTAs struggled to bridge these considerations to the tasks being posed and the practices being emphasized in introductory courses. The participants also expressed a basic desire to engage students in practice problems and activities, yet their instructional visions were not specific and well-grounded in rich classroom experiences that modeled student-centered pedagogy. As a result, all four GTAs converged on a singular vision for introductory statistics. This vision involved focusing on “the basics,” acquainting students with a wide array of procedures, honing students’ computational abilities, and emphasizing statistical problem-solving as a pursuit for right answers. This dissertation study provides insights into disciplinary tensions that may be of value in developing an instrument for assessing the disciplinary views of instructors and students alike. GTAs without well-developed views may need opportunity to engage in rich, open-ended tasks that serve to develop their disciplinary perspectives. Additionally, this work reveals how GTAs may struggle to bridge their perceptions of advanced disciplinary work to the work of their own students. Acquaintance and experience engaging in tasks that promote informal inferential reasoning or exploratory data analysis, coupled with connections to situated and constructivist learning theories, may enrich GTAs’ instructional visions as they see how disciplinary and instructional spaces may interact and inform one another. / A Dissertation submitted to the School of Teacher Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2019. / March 28, 2019. / Disciplinary Views, Epistemology, Statistics Education, Teaching Assistants / Includes bibliographical references. / Ian Whitacre, Professor Co-Directing Dissertation; Elizabeth M. Jakubowski, Professor Co-Directing Dissertation; Eric Chicken, University Representative; Lama Jaber, Committee Member; Jennifer J. Kaplan, Committee Member.

Inquiry-oriented teaching is a specific form of active learning gaining popularity in teaching communities. The goal of inquiry-oriented classes is to help students in gaining a conceptual understanding of the material. My research focus is to gauge students’ performance and conceptual understanding in inquiry-oriented linear algebra classes. This work is part of a broader NSF funded project; Teaching Inquiry-Oriented Mathematics: Establishing Support (TIMES) (Grant # 1431393), and TIMES project was designed to support instructors to shift towards inquiry-oriented instruction/teaching. Being part of the TIMES team, a broader goal of my dissertation is pragmatic to the project that is to measure the effectiveness of inquiry-oriented teaching on students learning of linear algebra concepts. Through my research, my contribution to math education field is the development of a valid and reliable assessment instrument for instructors teaching linear algebra concepts in their classes. My dissertation is a mixed method research and follows a three-paper format, and in these papers I discuss (1) the development and validation of a reliable linear algebra assessment tool, (2) comparison of performance of students in inquiry-oriented classes with the students in non-inquiry-oriented classes by using the tool developed in the first paper, and (3) development of research-based choices and distractors to convert the current open-ended assessment into a multiple-choice test by looking into students’ ways of reasoning and problem-solving approaches. The first paper is a quantitative study in which I establish the validity of the linear algebra assessment, and I also measure the reliability of the assessment. In the second paper, I use the linear algebra assessment to measure students’ conceptual and procedural understanding of linear algebra concepts and to compare the performance of students in inquiry-oriented classes with the students in non-inquiry-oriented classes. In the final paper, I focus on the analysis of patterns in student responses, particularly to open-ended response items, to inform the multiple-choices and distractors for the open-ended questions on the linear algebra assessment. This analysis will help me to convert the existing linear algebra assessment into a multiple-choice format research tool that linear algebra researchers can use for various comparisons to gauge the effectiveness of interventions. Additionally, the multiple-choice format of the assessment will be easy to administer and grade, so instructors can also use the assessment to measure their students’ conceptual and procedural understanding of linear algebra concepts. / A Dissertation submitted to the School of Teacher Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Fall Semester 2018. / October 18, 2018. / Assessment Validation, Inquiry-Oriented Teaching, Linear Algebra / Includes bibliographical references. / Christine Andrews-Larson, Professor Directing Dissertation; Giray Ökten, University Representative; Ian Whitacre, Committee Member; Russell Almond, Committee Member.

The purpose of this study was to examine the effects of a vocabulary tutoring intervention with defining key vocabulary terms and algebraic problem-solving skills of students who struggle with mathematics. Literature shows that there is a need to further explore how students with mathematical learning difficulties learn mathematics vocabulary at the post-secondary level. The participants for this study included five college-aged students, 18 years or older, who self-identified as struggling with mathematics. Each participant completed two vocabulary tutoring sessions each week and complete layered-look books during each session. The layered-look books included the vocabulary word, definition, an example, and non-example. The dependent variable was the percentage of correct answers on a six-question test. Each test will contain three vocabulary short answer questions and three multiple-choice algebraic exercises. The researcher used a multiple probe across behaviors, replicated across participants design to determine what effect mathematics vocabulary tutoring has on a student’s ability to define vocabulary terms and what effect mathematics vocabulary tutoring has on a student’s algebraic problem solving. The study included three phases: baseline, vocabulary tutoring (intervention), and maintenance. The researcher followed a modeling and guided practice teaching strategy to tutor the student. Based on the results of this study, it was concluded that the vocabulary tutoring intervention did help students learn the vocabulary. Three of the five participants showed a functional relationship between the vocabulary intervention and defining key vocabulary words. However, the vocabulary tutoring intervention did not help participants with the algebraic problem-solving examples. None of the five participants had three demonstrations of effect. / A Dissertation submitted to the School of Teacher Education in partial fulfillment of the requirements for the degree of Doctor of Philosophy. / Spring Semester 2019. / March 7, 2019. / Mathematics, Tutoring, Vocabulary / Includes bibliographical references. / Elizabeth M. Jakubowski, Professor Directing Dissertation; Robert A. Schwartz, University Representative; Kelly J. Whalon, Committee Member; Jenny Rose Root, Committee Member.

The purposes of the study were threefold: (1) to assess whether the math program of a small, midwestern elementary school was meeting the district's established cognitive mathematics objectives, (2) to make recommendations for improvement of the existing mathematics program and (3) to provide a basis for the cognitive components of elementary mathematics program evaluation suitable for adoption by school corporations of a similar size.The study was designed to determine the following: Does the existing mathematics program currently conducted by a small, midwestern elementary school meet the stated program learning objectives as measured by the Metropolitan Achievement Test and the program's criterion-referenced tests?The review of literature considered pertinent for the study was reviewed and categorized as. follows: (1) history and overview of achievement assessments, (2) evaluation of mathematics achievement, (3) mathematics teaching today, and (4) mathematics program recommendations. The population for the study was defined as those students in kindergarten through grade six in a small, midwestern elementary school enrolling 506 students who had been administered the Metropolitan Achievement Test during the week of April 28, 1985.The assessment of the achievement of the cognitive mathematics objectives was dependent upon results of the mathematics subtest scores of the Metropolitan Achievement Test, Form JS, Survey Battery and the criterion-referenced tests of the U-SAIL Mathematics Program. Data obtained from the tests were analyzed, summarized and presented in a narrative report.Based upon the results of the study using the MAT the following conclusions have been drawn:1. At all grade levels, the mathematics program learning objectives as measured by the MAT are generally met, however, the proportion of the curriculum measured is not adequate for assessing achievement of the district's established mathematics cognitive objectives.2. The Metropolitan Achievement Test does not measure enough objectives to adequately assess achievement of the program's mathematics cognitive objectives.Based upon the results of the study using the U-SAIL criterion-referenced tests, the following conclusions have been drawn:1. At kindergarten, first and second grade levels, the objectives are adequately met. The program is effective.2. At third, fourth, fifth and sixth grade levels, the objectives are being inadequately met. The program is ineffective.68

published_or_final_version / Education / Master / Master of Education

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