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This thesis is concerned with an unconventional understanding of the mathematics taught in schools and the role that students’ engagement plays for mathematics as a discipline. In this thesis, I explore mathematics through the lens of metaphor theory developed by George Lakoff and Mark Johnson. This area of scholarship is used to demonstrate that abstract concepts are conceived of metaphorically. From this perspective, the foundational metaphors of several major philosophies of mathematics are analyzed. This analysis concludes by asking if there might be another metaphor that allows for a different, more productive, understanding of school mathematics. The metaphor “Math is Alive” is offered as an alternative in the tradition of Humberto Maturana and Francisco Varela’s theory of autopoietic organization. The final chapter explores school mathematics as a part of mathematics as a discipline, and how “Math is Alive” might alter how educators view the role of children in mathematics. / May 2017

<p> This study investigates instructional moves by teachers in mathematics classrooms in which technology-based activities (i.e., student-oriented simulations) and features of those simulations influence classroom practices. Four teachers were studied over the course of a year as an exploratory study to build interpretive cases that described instructional practices in technology-based lessons. Teachers developed lessons using PhET simulations designed to support algebraic reasoning. Data sources included teachers&rsquo; process of selecting and designing lessons, observations of teachers&rsquo; non-technology and technology-based mathematical activities, and teacher interviews and reflections. </p><p> This work was based on a conceptual framework blending the ideas of Mathematical Tasks (Stein, Smith, Henningsen, &amp; Silver, 1998), Mathematical Pedagogical Content Knowledge (Ball, Thames, &amp; Phelps, 2008), and Technological Pedagogical Content Knowledge (Mishra &amp; Koehler, 2006), in which teachers&rsquo; instructional practices are determined by teachers&rsquo; mathematical pedagogical content knowledge, task selection and design, and use of technology. </p><p> Results indicated that teachers see simulations as having significant benefits in the classroom. Teachers leveraged these opportunities by increasing class discussions, engaging in higher levels of thinking and reasoning, and focusing on mathematical representations. When teachers used simulations, the teachers spent less time in direct instruction, focused more on the mathematics, and focused more on investigations rather than drill-oriented tasks. </p><p> Technology in the classroom, however, was problematic for some teachers. The very nature of students working independently with their own devices meant that student-student interactions decreased in some lessons. Furthermore, teachers&rsquo; discomfort in managing technology seems to limit ongoing use. </p><p> Specific features of the simulations that prompted instructional moves included the ability to support conceptual understanding and build student engagement. Simulations also provided a &lsquo;low floor, high ceiling,&rsquo; supporting differentiation, and a dynamic responsiveness, facilitating connections between representations. On the other hand, teachers raised concerns that some features of the simulation could do the math for the students. Furthermore, the perception of simulations as being a game may impact how and when simulations are used. </p><p> The emergent use of technology in math classrooms is under-supported. For simulations to be used in a more extensive fashion in mathematics classes, professional development and curricular materials are needed to support implementation. </p>

Thesis (M.M.E.)--Boston University

Thesis (Ed.M.)--Boston University / It was the purpose of this study to: 1. Determine the mathematical understandings possessed by a group of teachers in-service. 2. Compare the levels of mathematical understandings possessed by the different members of the group with the various grade levels of teaching experience within the group. 3. Compare the levels of mathematical understandings of the members of the group with their length of service. 4. Compare the levels of mathematical understandings of those members who have taken courses in the teaching of arithmetic with those who have taken no courses in this area. 5. Make a comparison between the levels of mathematical understandings of those members with Bachelor's degrees and those with Master's degrees.

<p> Graphing calculator programs have been used on high-stakes tests and teacher-created assessments at the secondary and college level for many years. These programs are even used on college-placement tests such as the ACT. Beginning with the 2011-2012 school year, the Mississippi Department of Education (MDE) made the decision to no longer allow the use of graphing calculator applications and/or programs on the Subject Area Testing Program (SATP2) for Algebra I.</p><p> Currently, limited research exists to address the influence of graphing calculator program use on high-stakes assessments. The programming capabilities of graphing calculators should not be ignored. Because of the 2011-2012 graphing calculator policy change, a unique opportunity exists in Mississippi to provide valuable information about this issue. Gaining insight about how the use of graphing calculator programs has affected assessment results in the past will give policy makers needed information for creating equitable assessment policies in the future. This research used a causal-comparative research design to determine the influence of the MDE&rsquo;s decision to no longer allow the use of graphing calculator programs on Mississippi&rsquo;s Algebra I SATP2. The research questions consider the influence of the policy change with regards to three groups: students, schools in general, and Title I schools in particular. </p><p> A chi-square test for association was used to examine the student-level data associated with research questions one and two. This analysis examined students&rsquo; ability to pass and their performance level on the Algebra I SATP2. The results for both of the chi-square analyses indicated significant results. For research question three, a two-way mixed ANOVA was used to examine the influence of the policy change on Title I schools. For this analysis, Title I schools represent disadvantaged populations. The results indicated no interaction between graphing calculator program use and type of school, but did reveal a main effect for the school type.</p>

Mathematics content courses for prospective elementary teachers have the potential to increase future teachers’ mathematical knowledge for teaching as well as model high quality instructional practices. This study investigated the instructional practices and curriculum usage of instructors of elementary mathematics-for-teaching courses. This mixed-methods study included a nationwide survey of instructors to identify the instructional practices and curriculum used in these courses. Additionally, this study compared the difference in reported use of instructional practices by survey participants’ academic and professional background characteristics. Two case studies of instructors who used instructional materials developed by the Elementary Pre-service Teachers Mathematics Project (EMP) were also conducted to more deeply describe instructional practices and use of curriculum materials in these courses. Results from the Instructional Practices and Curriculum Use (IPCU) survey (n = 458) indicate that college instructors of mathematics content courses for elementary teachers report using instructional practices supported by research and policy recommendations at higher levels than previous studies on general college STEM courses would suggest. In particular, survey participants reported using instructional practices such as engaging students in mathematical practices, attending to mathematical knowledge for teaching, pursuing students’ ideas, sharing mathematical authority with students, and supporting student-to-student interaction. Use of lecture, small groups, formative assessment, practices that lower cognitive demand, and efforts to achieve active participation varied substantially. The use of these instructional practices varied according to these characteristics, such as the subject and level of a participant’s terminal degree, their appointment to a mathematics department versus a school of education, their experience teaching in PreK–12 schools, at statistically significant levels. This study suggests that the common perception of mathematics content courses for pre-service elementary teachers as remedial and dominated by lecture is not the norm. Analysis of the case studies identified four ways that the participants used the EMP curriculum materials to create mathematically powerful experiences for their pre-service teachers. The case study instructors used the materials to (1) prompt pre-service teachers to examine and use mathematical relationships, (2) hold pre-service teachers responsible for engaging in rigorous mathematical work, (3) assess and make use of pre-service teachers’ thinking, and (4) support pre-service teachers to use mathematical language. The elements of the curriculum that supported the case study instructors were identified at the overall programmatic level, the unit and lesson level, and at the individual problem level. This study demonstrates that curriculum materials can support instructors in using research-based instructional practices, but the design of the materials impacts how instructors are able to use the materials to create mathematically powerful experiences for their students.

The purpose of the study is to examine students? understanding of the function concept by examining their concept images and concept definition when the students are introduced to function concept through a formal definition and informal approach. The participants were traditional college students enrolled in Intermediate Algebra at a large university in the southeast region of the United States. The students completed a questionnaire that asked them to identify functions and non-functions, mentally construct functions from verbal statements, and provide a definition for the function concept. The questionnaires were analyzed for correct answers, justifications related to the identification of a function or non-function, and the accuracy of the definitions provided. Often students do not possess concept definitions that match their concept images.

This study investigated the achievement of gifted students on mathematics problems that were designed to assess both conceptual and procedural knowledge of polynomial functions, and it attempted to determine the impact of the students? mathematical belief systems on this achievement. The students were enrolled in a three-week Algebra II course at a summer program for gifted mathematics students. Data sources were belief scales, in-class examinations, and in-depth interviews. Qualitative and quantitative analyses indicated that the students were able to make a variety of connections among concepts related to polynomials and functions, and they easily applied their mathematical knowledge to real world phenomena. The participants suffered, however, from several misconceptions relating to the understanding of the roles of the independent and dependent variables in functions. They also struggled with the concept of symmetry and how it relates to polynomial functions. Statistical analyses suggested that belief systems were correlated with achievement, but the conclusions from this study were ambiguous since the correlations were unexpectedly negative. Through its identification of potential conceptual difficulties that gifted students may encounter in their learning of polynomial functions, this study suggested specific topics that teachers of gifted students should consider when planning their instructional activities.

The purpose of this study is to investigate studentsâ uses and interpretations of mathematical symbols and the influence that symbols have on studentsâ goals and activities when solving tasks with and without a graphing calculator. The researcher conducted a multi-case study of pre-calculus college students with a focus on the goals and activities and they selected and the anticipations and reflections that they made as they worked on math problems in different interview settings Data was collected and analyzed under the conceptual lens of an activity-effect relationship framework and a symbol sense framework. Six different student cases were investigated, and both in-case and cross-case data analysis was conducted and reported. The researcher found that some symbols and symbolic structures had strong influences on studentsâ choices in problem solving. Graphing calculators were used as a way to abandon symbolic manipulation, although very few connections were made between symbolic and graphic or numeric forms. Students demonstrated a mixture of instances of symbol sense as they worked on symbolic mathematical problems.