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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

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
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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

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
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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

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
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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

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.
ContributorsOrletsky, Darryl William (Author) / Middleton, James (Thesis advisor) / Greenes, Carole (Committee member) / Judson, Eugene (Committee member) / Arizona State University (Publisher)
Created2015
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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

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.
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
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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

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
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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

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
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This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet

This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet vector calculus, calculus on manifolds, linear algebra, and differential equations all rest upon the idea of functions of two (or more) variables. This dissertation contributes to understanding productive ways of thinking that can support students in thinking about functions of two or more variables as they describe complex systems with multiple variables interacting. This dissertation focuses on modeling the way of thinking of four students who participated in a specific instructional sequence designed to explore the limits of their ways of thinking and in turn, develop a robust model that could explain, describe, and predict students' actions relative to specific tasks. The data was collected using a teaching experiment methodology, and the tasks within the teaching experiment leveraged quantitative reasoning and covariation as foundations of students developing a coherent understanding of two-variable functions and their rates of change. The findings of this study indicated that I could characterize students' ways of thinking about two-variable functions by focusing on their use of novice and/or expert shape thinking, and the students' ways of thinking about rate of change by focusing on their quantitative reasoning. The findings suggested that quantitative and covariational reasoning were foundational to a student's ability to generalize their understanding of a single-variable function to two or more variables, and their conception of rate of change to rate of change at a point in space. These results created a need to better understand how experts in the field, such as mathematicians and mathematics educators, thinking about multivariable functions and their rates of change.
ContributorsWeber, Eric David (Author) / Thompson, Patrick (Thesis advisor) / Middleton, James (Committee member) / Carlson, Marilyn (Committee member) / Saldanha, Luis (Committee member) / Milner, Fabio (Committee member) / Van de Sande, Carla (Committee member) / Arizona State University (Publisher)
Created2012
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This study purposed to determine the effect of an endogenously designed instructional game on conceptual understanding of the associative and distributive properties of multiplication. Additional this study sought to investigate if performance on measures of conceptual understanding taken prior to and after game play could serve as predictors of game

This study purposed to determine the effect of an endogenously designed instructional game on conceptual understanding of the associative and distributive properties of multiplication. Additional this study sought to investigate if performance on measures of conceptual understanding taken prior to and after game play could serve as predictors of game performance. Three versions of an instructional game, Shipping Express, were designed for the purposes of this study. The endogenous version of Shipping Express integrated the associative and distributive properties of multiplication within the mechanics, while the exogenous version had the instructional content separate from game play. A total of 111 fourth and fifth graders were randomly assigned to one of three conditions (endogenous, exogenous, and control) and completed pre and posttest measures of conceptual understanding of the associative and distributive properties of multiplication, along with a questionnaire. The results revealed several significant results: 1) there was a significant difference between participants' change in scores on the measure of conceptual understanding of the associative property of multiplication, based on the version of Shipping Express they played. Participants who played the endogenous version of Shipping Express had on average higher gains in scores on the measure of conceptual understanding of the associative property of multiplication than those who played the other versions of Shipping Express; 2) performance on the measures of conceptual understanding of the distributive property collected prior to game play were related to performance within the endogenous game environment; and 3) participants who played the control version of Shipping Express were on average more likely to have a negative attitude towards continuing game play on their own compared to the other versions of the game. No significant differences were found in regards to changes in scores on the measure of conceptual understanding of the distributive property based on the version of Shipping Express played, post hoc pairwise comparisons, and changes on scores on question types within the conceptual understanding of the associative and distributive property of multiplication measures. The findings from this study provide some support for a move towards the design and development of endogenous instructional games. Additional implications for the learning through digital game play and future research directions are discussed.
ContributorsDenham, Andrew (Author) / Nelson, Brian C. (Thesis advisor) / Atkinson, Robert K. (Committee member) / Middleton, James (Committee member) / VanLehn, Kurt (Committee member) / Arizona State University (Publisher)
Created2012
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The primary purpose of this study is to examine the effect of knowledge for teaching mathematics and teaching practice on student mathematics achievement growth. Thirty two teachers and 299 fourth grade students in three elementary schools from one school district in urban area participated in the study. Most of them

The primary purpose of this study is to examine the effect of knowledge for teaching mathematics and teaching practice on student mathematics achievement growth. Thirty two teachers and 299 fourth grade students in three elementary schools from one school district in urban area participated in the study. Most of them are Hispanic in origin and about forty percent is English Language Learners (ELLs). The two level Hierarchical Linear Model (HLM) was used to investigate repeated measures of teaching practice measured by Classroom Assessment Scoring System (CLASS) instrument. Also, linear regression and a multiple regression to examine the relationship between teacher knowledge measured by Learning for Mathematics Teaching (LMT) and Developing Mathematical Ideas (DMI) items and teaching practice were employed. In addition, a three level HLM was employed to analyze repeated measures of student mathematics achievement measured by Arizona Assessment Consortium (AzAC) instruments. Results showed that overall teaching practice did not change weekly although teachers' emotional support for their students improved by week. Furthermore, a statistically significant relationship between teacher knowledge and teaching practice was not found. In terms of student learning, ELLs have significantly lower initial status in mathematics achievement than non-ELLs, as were growth rates for these two groups. Lastly, teaching practice significantly predicted students' monthly mathematics achievement growth but teacher knowledge did not. The findings suggest that school systems and education policy makers need to provide teachers with the chance to reflect on their teaching and change it within themselves in order to better support student mathematics learning.
ContributorsKim, Seong Hee (Author) / Sloane, Finbarr (Thesis advisor) / Middleton, James (Committee member) / Flores, Alfinio (Committee member) / Arizona State University (Publisher)
Created2012