Matching Items (27)
Filtering by
- All Subjects: Mathematics Education
- Creators: Middleton, James
- Creators: Sloane, Finbarr
- Creators: Saldanha, Luis
Description
The research indicated effective mathematics teaching to be more complex than assuming the best predictor of student achievement in mathematics is the mathematical content knowledge of a teacher. This dissertation took a novel approach to addressing the idea of what it means to examine how a teacher's knowledge of mathematics impacts student achievement in elementary schools. Using a multiple case study design, the researcher investigated teacher knowledge as a function of the Mathematics Teaching Cycle (NCTM, 2007). Three cases (of two teachers each) were selected using a compilation of Learning Mathematics for Teaching (LMT) measures (LMT, 2006) and Developing Mathematical Ideas (DMI) measures (Higgins, Bell, Wilson, McCoach, & Oh, 2007; Bell, Wilson, Higgins, & McCoach, 2010) and student scores on the Arizona Assessment Collaborative (AzAC). The cases included teachers with: a) high knowledge & low student achievement v low knowledge & high student achievement, b) high knowledge & average achievement v low knowledge & average achievement, c) average knowledge & high achievement v average knowledge & low achievement, d) two teachers with average achievement & very high student achievement. In the end, my data suggested that MKT was only partially utilized across the contrasting teacher cases during the planning process, the delivery of mathematics instruction, and subsequent reflection. Mathematical Knowledge for Teaching was utilized differently by teachers with high student gains than those with low student gains. Because of this insight, I also found that MKT was not uniformly predictive of student gains across my cases, nor was it predictive of the quality of instruction provided to students in these classrooms.
ContributorsBurke, Margaret Kathleen (Author) / Middleton, James A. (Thesis advisor) / Sloane, Finbarr (Thesis advisor) / Battey, Daniel S (Committee member) / Arizona State University (Publisher)
Created2013
Description
In this mixed-methods study, I examined the relationship between professional development based on the Common Core State Standards for Mathematics and teacher knowledge, classroom practice, and student learning. Participants were randomly assigned to experimental and control groups. The 50-hour professional development treatment was administered to the treatment group during one semester, and then a follow-up replication treatment was administered to the control group during the subsequent semester. Results revealed significant differences in teacher knowledge as a result of the treatment using two instruments. The Learning Mathematics for Teaching scales were used to detect changes in mathematical knowledge for teaching, and an online sorting task was used to detect changes in teachers' knowledge of their standards. Results also indicated differences in classroom practice between pairs of matched teachers selected to participate in classroom observations and interviews. No statistical difference was detected between the groups' student assessment scores using the district's benchmark assessment system. This efficacy study contributes to the literature in two ways. First, it provides an evidence base for a professional development model designed to promote effective implementation of the Common Core State Standards for Mathematics. Second, it addresses ways to impact and measure teachers' knowledge of curriculum in addition to their mathematical content knowledge. The treatment was designed to focus on knowledge of curriculum, but it also successfully impacted teachers' specialized content knowledge, knowledge of content and students, and knowledge of content and teaching.
ContributorsRimbey, Kimberly A (Author) / Middleton, James A. (Thesis advisor) / Sloane, Finbarr (Committee member) / Atkinson, Robert K (Committee member) / Arizona State University (Publisher)
Created2013
Description
Research on combinatorics education is sparse when compared with other fields in mathematics education. This research attempted to contribute to the dearth of literature by examining students' reasoning about enumerative combinatorics problems and how students conceptualize the set of elements being counted in such problems, called the solution set. In particular, the focus was on the stable patterns of reasoning, known as ways of thinking, which students applied in a variety of combinatorial situations and tasks. This study catalogued students' ways of thinking about solution sets as they progressed through an instructional sequence. In addition, the relationships between the catalogued ways of thinking were explored. Further, the study investigated the challenges students experienced as they interacted with the tasks and instructional interventions, and how students' ways of thinking evolved as these challenges were overcome. Finally, it examined the role of instruction in guiding students to develop and extend their ways of thinking. Two pairs of undergraduate students with no formal experience with combinatorics participated in one of the two consecutive teaching experiments conducted in Spring 2012. Many ways of thinking emerged through the grounded theory analysis of the data, but only eight were identified as robust. These robust ways of thinking were classified into three categories: Subsets, Odometer, and Problem Posing. The Subsets category encompasses two ways of thinking, both of which ultimately involve envisioning the solution set as the union of subsets. The three ways of thinking in Odometer category involve holding an item or a set of items constant and systematically varying the other items involved in the counting process. The ways of thinking belonging to Problem Posing category involve spontaneously posing new, related combinatorics problems and finding relationships between the solution sets of the original and the new problem. The evolution of students' ways of thinking in the Problem Posing category was analyzed. This entailed examining the perturbation experienced by students and the resulting accommodation of their thinking. It was found that such perturbation and its resolution was often the result of an instructional intervention. Implications for teaching practice are discussed.
ContributorsHalani, Aviva (Author) / Roh, Kyeong Hah (Thesis advisor) / Fishel, Susanna (Committee member) / Saldanha, Luis (Committee member) / Thompson, Patrick (Committee member) / Zandieh, Michelle (Committee member) / Arizona State University (Publisher)
Created2013
Description
In 2007, Arizona voters passed House Bill (HB) 2064, a law that fundamentally restructured the Structured English Immersion (SEI) program, putting into place a 4-hour English language development (ELD) block for educating English language learners (ELLs). Under this new language policy, ELL students are segregated from their English-speaking peers to receive a minimum of four hours of instruction in discrete language skills with no contextual or native language support. Furthermore, ELD is separate from content-area instruction, meaning that language and mathematics are taught as two separate entities. While educators and researchers have begun to examine the organizational structure of the 4-hour block curriculum and implications for student learning, there is much to be understood about the extent to which this policy impacts ELLs opportunities to learn mathematics. Using ethnographic methods, this dissertation documents the beliefs and practices of four Arizona teachers in an effort to understand the relationship between language policy and teacher beliefs and practice and how together they coalesce to form learning environments for their ELL students, particularly in mathematics. The findings suggest that the 4-hour block created disparities in opportunities to learn mathematics for students in one Arizona district, depending on teachers' beliefs and the manner in which the policy was enacted, which was, in part, influenced by the State, district, and school. The contrast in cases exemplified the ways in which policy, which was enacted differently in the various classes, restricted teachers' practices, and in some cases resulted in inequitable opportunities to learn mathematics for ELLs.
ContributorsLlamas-Flores, Silvia (Author) / Middleton, James (Thesis advisor) / Battey, Daniel (Committee member) / Sloane, Finbarr (Committee member) / Macswan, Jeffrey (Committee member) / Arizona State University (Publisher)
Created2013
Description
Recently there has been an increase in the number of people calling for the incorporation of relevant mathematics in the mathematics classroom. Unfortunately, various researchers define the term relevant mathematics differently, establishing several ideas of how relevancy can be incorporated into the classroom. The differences between mathematics education researchers' definitions of relevant and the way they believe relevant math should be implemented in the classroom, leads one to conclude that a similarly varied set of perspectives probably exists between teachers and students as well. The purpose of this exploratory study focuses on how the student and teacher perspectives on relevant mathematics in the classroom converge or diverge. Specifically, do teachers and students see the same lessons, materials, content, and approach as relevant? A survey was conducted with mathematics teachers at a suburban high school and their algebra 1 and geometry students to provide a general idea of their views on relevant mathematics. An analysis of the findings revealed three major differences: the discrepancy between frequency ratings of teachers and students, the differences between how teachers and students defined the term relevance and how the students' highest rated definitions were the least accounted for among the teacher generated questions, and finally the impact of differing attitudes towards mathematics on students' feelings towards its relevance.
ContributorsRedman, Alexandra P (Author) / Middleton, James (Thesis advisor) / Sloane, Finbarr (Committee member) / Blumenfeld-Jones, Donald (Committee member) / Arizona State University (Publisher)
Created2012
Description
This study investigated the current state of the U.S. and Chinese urban middle school math teachers' pedagogical content knowledge (PCK) for the topic of functions. A comparative, descriptive case study was employed to capture the PCK of 23 teachers in Arizona and of 28 teachers in Beijing, regarding their instructional knowledge, understanding of student thinking and curricular knowledge--three key components based on Shulman's conceptualization of PCK--related to functions. Cross-case comparisons were used to analyze the PCK of teacher groups across countries and socio-economic statuses (SES), based on the questionnaire, lesson plan, and interview data.
This study finds that despite cultural differences, teachers are likely to share some commonalities with respect to their instructional decisions, understanding of student thinking and curricular knowledge. These similarities may reflect the convergence in teaching practice in the U.S. and China and the dedication the two countries make in improving math education. This study also finds the cross-country differences and cross-SES differences regarding teachers' PCK. On the one hand, the U.S. and Chinese math teachers of this study tend to diverge in valuing different forms of representations, explaining student misconceptions, and relating functions to other math topics. Teachers' own understanding of functions (and mathematics), standards, and high-stakes testing in each country significantly influence their PCK. On the other hand, teachers from the higher SES schools are more likely to show higher expectations for and stronger confidence in their students' mathematical skills compared to their counterparts from the lower SES schools. Teachers' differential beliefs in students' ability levels significantly contribute to their differences between socio-economic statuses.
This study finds that despite cultural differences, teachers are likely to share some commonalities with respect to their instructional decisions, understanding of student thinking and curricular knowledge. These similarities may reflect the convergence in teaching practice in the U.S. and China and the dedication the two countries make in improving math education. This study also finds the cross-country differences and cross-SES differences regarding teachers' PCK. On the one hand, the U.S. and Chinese math teachers of this study tend to diverge in valuing different forms of representations, explaining student misconceptions, and relating functions to other math topics. Teachers' own understanding of functions (and mathematics), standards, and high-stakes testing in each country significantly influence their PCK. On the other hand, teachers from the higher SES schools are more likely to show higher expectations for and stronger confidence in their students' mathematical skills compared to their counterparts from the lower SES schools. Teachers' differential beliefs in students' ability levels significantly contribute to their differences between socio-economic statuses.
ContributorsZou, Hui (Author) / Fischman, Gustavo (Thesis advisor) / Berliner, David (Committee member) / Sloane, Finbarr (Committee member) / Arizona State University (Publisher)
Created2014
Description
Industry, academia, and government have spent tremendous amounts of money over several decades trying to improve the mathematical abilities of students. They have hoped that improvements in students' abilities will have an impact on adults' mathematical abilities in an increasingly technology-based workplace. This study was conducted to begin checking for these impacts. It examined how nine adults in their workplace solved problems that purportedly entailed proportional reasoning and supporting rational number concepts (cognates).
The research focused on four questions: a) in what ways do workers encounter and utilize the cognates while on the job; b) do workers engage cognate problems they encounter at work differently from similar cognate problems found in a textbook; c) what mathematical difficulties involving the cognates do workers experience while on the job, and; d) what tools, techniques, and social supports do workers use to augment or supplant their own abilities when confronted with difficulties involving the cognates.
Noteworthy findings included: a) individual workers encountered cognate problems at a rate of nearly four times per hour; b) all of the workers engaged the cognates primarily via discourse with others and not by written or electronic means; c) generally, workers had difficulty with units and solving problems involving intensive ratios; d) many workers regularly used a novel form of guess & check to produce a loose estimate as an answer; and e) workers relied on the social structure of the store to mitigate the impact and defuse the responsibility for any errors they made.
Based on the totality of the evidence, three hypotheses were discussed: a) the binomial aspect of a conjecture that stated employees were hired either with sufficient mathematical skills or with deficient skills was rejected; b) heuristics, tables, and stand-ins were maximally effective only if workers individually developed them after a need was recognized; and c) distributed cognition was rejected as an explanatory framework by arguing that the studied workers and their environment formed a system that was itself a heuristic on a grand scale.
The research focused on four questions: a) in what ways do workers encounter and utilize the cognates while on the job; b) do workers engage cognate problems they encounter at work differently from similar cognate problems found in a textbook; c) what mathematical difficulties involving the cognates do workers experience while on the job, and; d) what tools, techniques, and social supports do workers use to augment or supplant their own abilities when confronted with difficulties involving the cognates.
Noteworthy findings included: a) individual workers encountered cognate problems at a rate of nearly four times per hour; b) all of the workers engaged the cognates primarily via discourse with others and not by written or electronic means; c) generally, workers had difficulty with units and solving problems involving intensive ratios; d) many workers regularly used a novel form of guess & check to produce a loose estimate as an answer; and e) workers relied on the social structure of the store to mitigate the impact and defuse the responsibility for any errors they made.
Based on the totality of the evidence, three hypotheses were discussed: a) the binomial aspect of a conjecture that stated employees were hired either with sufficient mathematical skills or with deficient skills was rejected; b) heuristics, tables, and stand-ins were maximally effective only if workers individually developed them after a need was recognized; and c) distributed cognition was rejected as an explanatory framework by arguing that the studied workers and their environment formed a system that was itself a heuristic on a grand scale.
ContributorsOrletsky, Darryl William (Author) / Middleton, James (Thesis advisor) / Greenes, Carole (Committee member) / Judson, Eugene (Committee member) / Arizona State University (Publisher)
Created2015
Description
This study explores teacher educators' personal theories about the instructional practices central to preparing future teachers, how they enact those personal theories in the classroom, how they represent the relationship between content, pedagogy, and technology, and the function of technology in teacher educators' personal theories about the teaching of mathematics and their practices as enacted in the classroom. The conceptual frameworks of knowledge as situated and technology as situated provide a theoretical and analytical lens for examining individual instructor's conceptions and classroom activity as situated in the context of experiences and relationships in the social world. The research design employs a mixed method design to examine data collected from a representative sample of three full-time faculty members teaching methods of teaching mathematics in elementary education at the undergraduate level. Three primary types of data were collected and analyzed:
a) structured interviews using the repertory grid technique to model the mathematics education instructors' schemata regarding the teaching of mathematics methods; b) content analysis of classroom observations to develop models that represent the relationship of pedagogy, content, and technology as enacted in the classrooms; and c) brief retrospective protocols after each observed class session to explore the reasoning and individual choices made by an instructor that underlie their teaching decisions in the classroom. Findings reveal that although digital technology may not appear to be an essential component of an instructor's toolkit, technology can still play an integral role in teaching. This study puts forward the idea of repurposing as technology -- the ability to repurpose items as models, tools, and visual representations and integrate them into the curriculum. The instructors themselves became the technology, or the mediational tool, and introduced students to new meanings for "old" cultural artifacts in the classroom. Knowledge about the relationships between pedagogy, content, and technology and the function of technology in the classroom can be used to inform professional development for teacher educators with the goal of improving teacher preparation in mathematics education.
a) structured interviews using the repertory grid technique to model the mathematics education instructors' schemata regarding the teaching of mathematics methods; b) content analysis of classroom observations to develop models that represent the relationship of pedagogy, content, and technology as enacted in the classrooms; and c) brief retrospective protocols after each observed class session to explore the reasoning and individual choices made by an instructor that underlie their teaching decisions in the classroom. Findings reveal that although digital technology may not appear to be an essential component of an instructor's toolkit, technology can still play an integral role in teaching. This study puts forward the idea of repurposing as technology -- the ability to repurpose items as models, tools, and visual representations and integrate them into the curriculum. The instructors themselves became the technology, or the mediational tool, and introduced students to new meanings for "old" cultural artifacts in the classroom. Knowledge about the relationships between pedagogy, content, and technology and the function of technology in the classroom can be used to inform professional development for teacher educators with the goal of improving teacher preparation in mathematics education.
ContributorsToth, Meredith Jean (Author) / Middleton, James (Thesis advisor) / Sloane, Finbarr (Committee member) / Buss, Ray (Committee member) / Atkinson, Robert (Committee member) / Arizona State University (Publisher)
Created2014
Description
This study contributes to the ongoing discussion of Mathematical Knowledge for Teaching (MKT). It investigates the case of Rico, a high school mathematics teacher who had become known to his colleagues and his students as a superbly effective mathematics teacher. His students not only developed excellent mathematical skills, they also developed deep understanding of the mathematics they learned. Moreover, Rico redesigned his curricula and instruction completely so that they provided a means of support for his students to learn mathematics the way he intended. The purpose of this study was to understand the sources of Rico's effectiveness. The data for this study was generated in three phases. Phase I included videos of Rico's lessons during one semester of an Algebra II course, post-lesson reflections, and Rico's self-constructed instructional materials. An analysis of Phase I data led to Phase II, which consisted of eight extensive stimulated-reflection interviews with Rico. Phase III consisted of a conceptual analysis of the prior phases with the aim of creating models of Rico's mathematical conceptions, his conceptions of his students' mathematical understandings, and his images of instruction and instructional design. Findings revealed that Rico had developed profound personal understandings, grounded in quantitative reasoning, of the mathematics that he taught, and profound pedagogical understandings that supported these very same ways of thinking in his students. Rico's redesign was driven by three factors: (1) the particular way in which Rico himself understood the mathematics he taught, (2) his reflective awareness of those ways of thinking, and (3) his ability to envision what students might learn from different instructional approaches. Rico always considered what someone might already need to understand in order to understand "this" in the way he was thinking of it, and how understanding "this" might help students understand related ideas or methods. Rico's continual reflection on the mathematics he knew so as to make it more coherent, and his continual orientation to imagining how these meanings might work for students' learning, made Rico's mathematics become a mathematics of students--impacting how he assessed his practice and engaging him in a continual process of developing MKT.
ContributorsLage Ramírez, Ana Elisa (Author) / Thompson, Patrick W. (Thesis advisor) / Carlson, Marilyn P. (Committee member) / Castillo-Chavez, Carlos (Committee member) / Saldanha, Luis (Committee member) / Middleton, James A. (Committee member) / Arizona State University (Publisher)
Created2011
Description
A fundamental motivation for this study was the underrepresentation of women in Science, Technology, Engineering and Mathematics careers. There is no doubt women and men can achieve at the same level in Mathematics, yet it is not clear why women are opting out. Adding race to the equation makes the underrepresentation more dramatic. Considering the important number of Latinos in the United States, especially in school age, it is relevant to find what reasons could be preventing them from participating in the careers mentioned. This study highlight the experiences young successful Latinas have in school Mathematics and how they shape their identities, to uncover potential conflicts that could later affect their participation in the field. In order to do so the author utilizes feminist approaches, Latino Critical Theory and Critical Race Theory to analyze the stories compiled. The participants were five successful Latinas in Mathematics, part of the honors track in a school in the Southwest of the United States. The theoretical lenses chosen allowed women of color to tell their story, highlighting the intersection of race, gender and socio-economical status as a factor shaping different schooling experiences. The author found that the participants distanced themselves from their home culture and from other girls at times to allow themselves to develop and maintain a successful identity as a Mathematics student. When talking about Latinos and their culture, the participants shared a view of themselves as proud Latinas who would prove others what Latinas can do. During other times while discussing the success of Latinos in Mathematics, they manifested Latinos were lazy and distance themselves from that stereotype. Similar examples about gender and Mathematics can be found in the study. The importance of the family as a motivator for their success was clear, despite the participants' concern that parents cannot offer certain types of help they feel they need. This was manifest in a tension regarding who owns the "right" Mathematics at home. Results showed that successful Latinas in the US may undergo a constant negotiation of conflicting discourses that force them to distance themselves from certain aspects of their culture, gender, and even their families, to maintain an identity of success in mathematics.
ContributorsGuerra Lombardi, Paula Patricia (Author) / Middleton, James (Thesis advisor) / Battey, Daniel (Committee member) / Koblitz, Ann (Committee member) / Flores, Alfinio (Committee member) / Arizona State University (Publisher)
Created2011