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Background Fractions continue to pose a critical challenge for students and their teachers alike. Mathematics education research indicates that the challenge with fractions may stem from the limitations of part-whole concepts of fractions, which is the central focus of fractions curriculum and instruction in the USA. Students’ development of more sophisticated concepts of fractions, beyond the part-whole concept, lays the groundwork for the later study of important mathematical topics, such as algebra, ratios, and proportions, which are foundational understandings for most STEM-related fields. In particular, the Common Core State Standards for Mathematics call for students to develop measurement concepts of fractions. In order to support such concepts, it is important to understand the underlying mental actions that undergird them so that teachers can design appropriate instructional opportunities. In this study, we propose a learning progression for the measurement concept of fractions—one that focuses on students’ mental actions and informs instructional design. Results A hierarchy of fraction schemes is charted outlining a progression from part-whole concepts to measurement concepts of fractions: (a) part-whole scheme (PWS), (b) measurement scheme for unit fractions (MSUF), (c) measurement scheme for proper fractions (MSPF), and (d) generalized measurement scheme for fractions (GMSF). These schemes describe concepts with explicit attention to the mental actions that undergird them. A synthesis of previous studies provides empirical evidence to support this learning progression. Conclusions Evidence from the synthesis of a series of research studies suggests that children’s measurement concept of fractions develops through several distinct developmental stages characterized by the construction of distinct schemes. The mental actions associated with these schemes provide a guide for teachers to design instructional opportunities for children to advance their construction of a measurement concept of fractions. Specifically, the collection of quantitative studies suggest that students need opportunities to engage in activities that support two kinds of coordinations—the coordination of partitioning and iterating, and the coordination of three levels of units inherent in fractions. Instructional implications are discussed with example tasks and activities designed to provoke these coordinations. Published version

Students often ask "when is this ever going to be useful?", "why are we doing this?", etc. when speaking about mathematics. If we take this as a question about 'meaningfulness', how can instructors respond and how do they even understand the terms 'meaningful' and 'meaning'? My dissertation looked at how college instructors see their instruction as meaningful or not. Drawing on social and cognitive perspectives of learning, I define four ways to think of what's 'meaningful' about mathematics. From a cognitive perspective, instructors can understand 'meaningful' as mathematical understanding versus understanding the significance of mathematics. From a social perspective where meaning is taken as the experiences of everyday life within communities, teachers can understand 'meaningful' as anything that engages students in practices the mathematics community engage in versus practices non-mathematics communities engage in (e.g. pushing computation or critical thinking as a means for maintaining social hierarchies). Using these four conceptions to categorize instructors' goals, this work focuses on how four undergraduate mathematics instructors thought of their instruction as meaningful and contextual and background factors that influenced those views. Doctor of Philosophy Students often ask "when is this ever going to be useful?" when speaking about mathematics. If we interpret this as seeking the meaning or purpose of their education, how can teachers respond and how do they even understand the terms 'meaningful' and 'meaning'? I wanted to look at how college instructors thought of this and how they addressed such a question in their classrooms. Drawing on different theories of learning, I outlined four ways to think of what's 'meaningful' about mathematics and then used these four ways to categorize how instructors think of their instruction as meaningful. To meet this end, I looked at some accounts of instructors' goals. My data came from college instructors of different mathematics classes: math for elementary education, math for liberal arts, statistics, and calculus. One important thing I found was that experiences with underserved communities or of not being 'a math person' corresponded with instructors' ability to attend to different kinds of 'meaningful' goals. What this might suggest is that educators may not feel prepared to respond to students' pursuit of meaning in diverse ways unless they have also personally struggled with it growing up or have personally experienced the consequences of disenfranchisement.

Improving the learning and success of students in undergraduate science, technology, engineering, and mathematics (STEM) courses has become an increased focus of education researchers within the past decade. As part of these efforts, discipline-based education research (DBER) has emerged within STEM education as a way to address discipline-specific challenges for teaching and learning, by combining expert knowledge of the various STEM disciplines with knowledge about teaching and learning (Dolan et al., 2018; National Research Council, 2012). Particularly important to furthering DBER and improving STEM education are interdisciplinary studies that examine how the teaching and learning of specific concepts develop among and across various STEM disciplines... Ph. D. Dissertation proposal

In the basic understanding of measurement, students are expected to be able to subdivide a given whole into a unit and then change the position of that unit along the entire length of the whole. These basic operations of subdivision and change of position are related to the more formal operations of partitioning and iterating. In the context of fractions, partitioning and iterating play a fundamental role in understanding fractions as measures, where students are expected to partition a whole into an iterable unit. In the context of angle measurement, students are expected to measure angles as a fractional amount of a full rotation or a circle, by partitioning the circle into a unit angle and then iterating that unit angle to find the measure of the given angle. Despite this link between measurement, fractions, and angles, research suggests that there is a disconnect between students' concepts of measurement and geometry concepts, including angle and angle measurement. Therefore, one area of study that might help us understand this disconnection would be to investigate the relationship between students' concepts of measurement and their concepts of angle measurement. This current study documents sixth, seventh, and eighth grade students' measurement schemes for fractions and their quantifications of angularity, and then investigates the relationship between them. This research is guided by the following question: What is the relationship between middle school students' measurement schemes for fractions and their quantifications of angularity? Results indicate that the majority of students involved in this study do not possess a measurement concept of fractions nor a measurement concept of angularity. However, these results demonstrate that there is a relationship between students' measurement schemes for fractions and their quantifications of angularity. It is concluded that students who construct more sophisticated fraction schemes tend to construct more sophisticated quantifications of angularity. Doctor of Philosophy Although the concepts of measurement, fractions, and angle measurement are related, research suggests that there is a disconnect between students' concepts of measurement and geometry concepts, including angle and angle measurement. Therefore, one area of study that might help us understand this disconnection would be to investigate the relationship between students' concepts of measurement and their concepts of angle measurement. This current study documents sixth, seventh, and eighth grade students' understandings of measurement, as indicated by their fraction schemes, and angle measurement, as indicated by how they quantify angularity or the openness of an angle. This study then investigates the relationship between them. This research is guided by the following question: What is the relationship between middle school students' measurement schemes for fractions and their quantifications of angularity? Results indicate that the majority of students involved in this study do not possess a measurement concept of fractions nor a measurement concept of angularity. However, these results demonstrate that there is a relationship between students' measurement schemes for fractions and their quantifications of angularity. It is concluded that students who construct more sophisticated fraction schemes tend to construct more sophisticated quantifications of angularity.

In this study, I will examine the beliefs and conceptual understanding of instructors and students from two abstract algebra classes. This research takes the form of a case study in which I answer four research questions, each addressing a relationship between instruction and beliefs or conceptual understanding. Specifically, these research questions are: 1. What beliefs do the instructors have about math, teaching, and learning and what relationship exists between these beliefs and instructional practice? 2. What is the relationship between instructional practice and students' beliefs about math, teaching, and learning? 3. What conceptual metaphors do the professors use to describe isomorphisms and homomorphisms and what relationship exists between these metaphors and the mathematical content in instruction? 4. What is the relationship between the mathematical content in instruction and conceptual metaphors the students use to describe isomorphisms and homomorphisms? In terms of beliefs, the instructors articulated considered positions on the nature of math, math learning, and math teaching. These beliefs were clearly reflected in their overall approaches to teaching. However, their instruction shifted in practice over the course of the semester. Students' beliefs seemed to shift slightly as a result of the ways their instructors taught. However, their core beliefs about math seemed unchanged and some lessons students took away were similar in the two classes. In terms of conceptual understanding, the instructors provided many conceptual metaphors that related to how they understood isomorphism. They struggled more to provide an image for homomorphism, which requires thinking about a more complicated mathematical object. Their understandings of isomorphism and homomorphism were largely reflected in their instruction with some notable differences. Students took away similar understandings of isomorphism to the instructors, but did not all take away the same level of structural understanding of homomorphism. In short, relationships between instructors' beliefs and instruction and between instructors' conceptual understanding and instruction were evident. However, certain elements were not made as clear as they perhaps intended. Relationships between instruction and students' beliefs and between instruction and students' conceptual understanding were also evident. However, relationships between instruction and beliefs were subtler than between instruction and conceptual understanding. Doctor of Philosophy In this study, I will examine the beliefs and conceptual understanding of instructors and students from two abstract algebra classes. I address four relationships: between instructors’ beliefs and instruction, between instruction and students’ beliefs, between instructors’ conceptual understanding and instruction, and between instruction and students’ conceptual understanding. Relationships between instructors’ beliefs and instruction and between instructors’ conceptual understanding and instruction were evident. However, certain elements were not made as clear as they perhaps intended. Relationships between instruction and students’ beliefs and between instruction and students’ conceptual understanding were also evident. However, relationships between instruction and beliefs were subtler than between instruction and conceptual understanding.

Teaching and learning mathematical functions and graphs pose significant challenges for teachers and students. Students often have difficulty in understanding a functional relationship between two quantities such as distance and time, temperature and precipitation, and gas price and number of gallons. Teaching students to have quantitative thinking about functions can help them understand the rate of change for complicated functions and later succeed in learning Calculus. Traditional educational methods such as static graph images and some learning tools usually have some limitations. Teaching students the dynamic changes of quantities within the static picture has serious difficulties. Compared to the learning tools, the game-based learning increases interest when students are learning complicated functions. This thesis presents a game-based learning application called Functions Fun, which runs on iPad tablet computers. The game is created to teach / learn the following functions: Linear, Quadratic, Exponential, Logarithmic, Trigonometric, and Polynomial with degrees over four. Each function is covered under a game level. The game setting is a jungle environment. Each game level has its own scene, challenging the player to take an action while teaching a function and its graphical representation. Functions Fun enables students to play and learn functions and graphs in a more effective and entertaining manner. Master of Science Teaching functions and graphs plays an important role in mathematics education. Teachers and researchers emphasize the need for students to form the quantitative thinking habits when they are learning different function graphs. This thesis presents Functions Fun, an iPad educational game that aims to help secondary students to understand function graphs by varying two quantities x and y such as distance and time, temperature and precipitation, and gas price and number of gallons. Adventure game is chosen as the game genre and six real-life models related to the following six functions are created: Linear, Quadratic, Exponential, Logarithmic, Trigonometric, and Polynomial with degrees over four. Each function is covered under a game level. The game setting is a jungle environment. Players are rewarded for their accomplishments. Through game effects and animations, students can have a better understanding of how the value changes of variables x and y influence the shape of graphs.

Nowadays cross platform software development is more expensive than ever before in terms of time and effort. Meantime with increasing number of personal devices, it is harder for local applications to synchronize and connect to the Internet. In terms of educational games, they can be divided into "local educational game" and "web educational game." "Local game" indicates the ones either on tablets, mobile devices or PC, which is an application on the corresponding platform. This kind of game mostly does not have backend support nor cross platform features such as the iPad version of CandyFactory. For one specific game, if the developer wants it to run on iPad and Android tablets, they need to develop two applications based on corresponding development framework, which is time and effort consuming. "Web game" indicates the ones on websites, which support cross platforms, but do not have backend support. Usually they are pure JavaScript or flash games with no backend recording the performances and the achievements. Software development for each individual platform is time and effort consuming. In order to achieve cross platform development, many programming languages and platforms like Java, Python, and JVM appear. Among all the cross platform approaches, cloud-based software development is the most universal solution to this problem. With web browsers built into every operating system, cloud software can be compatible with almost any device. Moreover, "Software-as-a-Service" (SaaS) is becoming a new software engineering paradigm and cloud-based software development is more popular because of its flexible scalability and cross platform features. In this thesis, we create a cloud-based educational game, CandyFactory, based on an iPad version of CandyFactory, and add backend to it to record user performance as well as achievements. Firstly, we re-develop the whole game from the iOS platform to the cloud-based Java EE platform. Secondly, we add new features to improve the game play such as ruler functionality and achievements animation. Thirdly, we add backend support to CandyFactory, including user account creation, course creation and performance report generation. With this functionality, teachers can monitor their students' performances and generate course reports. Moreover, teachers can view a specific student's report in order to provide more specific and effective help to their students. Lastly, with the advantages of cloud-based software development, we can update the whole application at any time without forcing the user to reinstall the update or re-download the game. With the hot update, the cloud-based CandyFactory is highly maintainable. The cloud-based CandyFactory runs on any computer that supports minimum 1024x768 screen resolution. The computer could be iPads, Android or Microsoft tablets, Windows or Mac laptops and desktops, and any other computer with a web browser. The advantages of cloud-based educational games over local educational games and web educational games are: firstly, they have cross platform features; secondly, they have backend data collection support; thirdly, they are consistent even if users log in with different computers, their game record and history will always be the same; lastly, the teacher can always keep track of his/her students' performance and provide more specific help and feedback. Master of Science Providing services on the cloud has become universal. The term “Cloud-Based” indicates that the software application runs on a server computer and users access the application by using a web browser anywhere and anytime. This thesis presents a cloud-based educational game called CandyFactory to teach fractions. The users can use CandyFactory under a web browser on an Internet-connected tablet, laptop, or desktop computer with minimum 1024x768 screen resolution. User’s game performance data is recorded on the server computer regardless of which tablet, laptop, or desktop computer the user uses to play the game. Cloud-based CandyFactory has four kinds of users: Individual, Teacher, Student, Administrator. Individual users can play the game to learn fractions as well as generate performance reports. Teachers can create a course, automatically generate student accounts under a course, and generate performance reports for individual students or for the whole class. Students can play the game under the account provided by the teacher and view their performance reports. Administrator is a built-in account user for maintaining the cloud-based software application. By developing the cloud-based CandyFactory educational game, we provide the users a crossplatform and cross-computers solution which helps the teachers and students learn fractions more efficiently and effectively.

Development of a game in the form of software for game-based learning poses significant technical challenges for educators, researchers, game designers, and software engineers. The game development consists of a set of complex processes requiring multi-faceted knowledge in multiple disciplines such as digital graphic design, education, gaming, instructional design, modeling and simulation, psychology, software engineering, visual arts, and the learning subject area. Planning and managing such a complex multidisciplinary development project require unifying methodologies for development and software quality evaluation and should not be performed in an ad hoc manner. This dissertation presents such methodologies named: GAMED (diGital educAtional gaMe dEvelopment methoDology) and IDEALLY (dIgital eDucational gamE softwAre quaLity evaLuation methodologY). GAMED consists of a body of methods, rules, and postulates and is embedded within a digital educational game life cycle. The life cycle describes a framework for organization of the phases, processes, work products, quality assurance activities, and project management activities required to develop, use, maintain, and evolve a digital educational game from birth to retirement. GAMED provides a modular structured approach for overcoming the development complexity and guides the developers throughout the entire life cycle. IDEALLY provides a hierarchy of 111 indicators consisting of 21 branch and 90 leaf indicators in the form of an acyclic graph for the measurement and evaluation of digital educational game software quality. We developed the GAMED and IDEALLY methodologies based on the experiences and knowledge we have gained in creating and publishing four digital educational games that run on the iOS (iPad, iPhone, and iPod touch) mobile devices: CandyFactory, CandySpan, CandyDepot, and CandyBot. The two methodologies provide a quality-centered structured approach for development of digital educational games and are essential for accomplishing demanding goals of game-based learning. Moreover, classifications provided in the literature are inadequate for the game designers, engineers and practitioners. To that end, we present a taxonomy of games that focuses on the characterization of games. Ph. D.

With the advancement in information technology, online assessments are getting more attention and online examinations are regarded as important parts of online learning. Online examinations can be easily taken by remote students, help the students get exam results quickly and save their time; online examinations also aid instructors in collecting students' exam answers and generating the exam reports effectively. In addition, online examinations can help reduce cost and save trees for our world. Multimedia elements like images, graphics, video and audio have been widely integrated into online learning environments. They not only help instructors design more engaging online learning content, but also help provide more interactive and pleasant learning experience for learners. However, integrating multimedia elements into online examination systems is rarely reported. Multimedia elements generally consume amounts of computing resources in a separated software system running on a single computer. "Software-as-a-Service (SaaS)" has become a new software paradigm and cloud-based software systems are becoming more attractive due to their dynamic scalability and effective usage of computing resources. Yet, how to effectively integrate multimedia elements into a cloud-based software system for online examinations is not significantly investigated. Although a variety of online-assessment tools have been developed, few of them adopt the "Software-as-a-Service (SaaS)" paradigm and most of them focus on the assessment in a specific domain or an application area with short of multimedia elements. There is a lack of a comprehensive software solution for online multimedia examinations. This thesis tries to utilize the "Software-as-a-Service (SaaS)" paradigm, design and develop a cloud-bAsed softwaRe systEm for oNline multimediA examinationS (ARENAS), and explore a comprehensive software solution for the online assessment field. ARENAS employs a multi-tiered client-server architecture and includes five subsystem modules: user module, question repository module, exam module, exam report module and configuration module. The developed cloud-based software system can present online questions with multimedia elements, and also support a myriad of question types, flexible accounts to the exam-takers, randomized question order in an online exam, flexible grading mechanisms, and analytical exam reports. For instructors, the developed system can help design more engaging online questions; for exam-takers, the developed system can help provide more user-friendly experience; for other educators and researchers, the design and development processes of ARENAS can be taken as a reference to designing and developing other large-scale cloud-based educational software systems. Master of Science

In this research, I set out to elucidate the relationships that might exist between students' conceptual understanding upon which they draw in their proof activity. I explore these relationships using data from individual interviews with three students from a junior-level Modern Algebra course. Each phase of analysis was iterative, consisting of iterative coding drawing on grounded theory methodology (Charmaz, 2000, 2006; Glaser and Strauss, 1967). In the first phase, I analyzed the participants' interview responses to model their conceptual understanding by drawing on the form/function framework (Saxe, et al., 1998). I then analyzed the participants proof activity using Aberdein's (2006a, 2006b) extension of Toulmin's (1969) model of argumentation. Finally, I analyzed across participants' proofs to analyze emerging patterns of relationships between the models of participants' understanding of identity and inverse and the participants' proof activity. These analyses contributed to the development of three emerging constructs: form shifts in service of sense-making, re-claiming, and lemma generation. These three constructs provide insight into how conceptual understanding relates to proof activity. Ph. D.

Advances in information technology are driving massive improvement to the education industry. The ubiquity of mobile devices has triggered a shift in the delivery of educational content. More lessons in a wide range of subjects are being disseminated by allowing students to access digital materials through mobile devices. One of the key materials is digital-based educational games. These games merge education with digital games to maximize engagement while somewhat obfuscating the learning process. The effectiveness is generally measured by assessments, either after or during gameplay, in the form of quizzes, data dumps, and/or manual analyses. Valuable gameplay information lost during the student's play sessions. This gameplay data provides educators and researchers with specific gameplay actions students perform in order to arrive at a solution, not just the correctness of the solution. This problem illustrates a need for a tool, enabling educators and players to quickly analyze gameplay data. in conjunction with correctness in an unobtrusive manner while the student is playing the game. This thesis describes a client-server software architecture that enables the collection of game-based data during gameplay. We created a collection of web services that enables games to transmit game-data for analysis. Additionally, the web application provides players with a portal to login and view various visualization of the captured data. Lastly, we created a game called "Taffy Town", a mathematics-based game that requires the player to manipulate taffy pieces in order to solve various fractions. Taffy Town transmits students' taffy transformations along with correctness to the web application. Students are able to view several dynamically created visualizations from the data sent by Taffy Town. Researchers are able to log in to the web application and see the same visualizations, however, aggregated across all Taffy Town players. This end-to-end mapping of problems, actions, and results will enable researchers, pedagogists, and teachers to improve the effectiveness of educational games. Master of Science

Numerous researchers have argued early adolescent behavior patterns are among the best predictors of later achievement and social-psychological adjustment outcomes. In the current study, a secondary data analysis was conducted to determine the extent to which four prosocial behaviors (cooperation, assertiveness, self-control and prosocial behaviors toward peers) and two forms of aggression (overt and relational) influence academic performance (as indexed by GPA and standardized achievement scores). Additionally, the potential moderating role of two school environment variables (perception of school climate and teacher bonding) were also considered in order to further examine the social-emotional environment of middle schools. Examining concurrent relations between grade 5 social behaviors and academic performance revealed all four forms of prosocial behavior were positively related to higher academic performance. Predictive relations between grade 6 social behaviors and academic outcomes at grade 9 indicated that of the four prosocial behaviors and two forms of aggression, cooperation alone predicted the likelihood of later academic achievement. Finally, it was hypothesized that prosocial behaviors, perception of school climate and teacher bonding would serve as moderators between aggression and academic performance; however, this prediction was unsupported. By drawing attention to the limited typology employed in studying prosocial behavior, as well as the methodological challenges that arise when examining these behaviors during adolescence, the hope is to spur research geared towards a more comprehensive understanding of prosocial development. Master of Science

Primarily via constructivist teaching experiment methodology, units coordination (Steffe, 1992) has emerged as a useful construct for modeling students' psychological constructions pertaining to several mathematical domains, including counting sequences, whole number multiplicative conceptions, and fractions schemes. I describe how consideration of units coordination as a Piagetian (1970b) structure is useful for modeling units coordination across contexts. In this study, I extend teaching experiment methodology (Steffe and Thompson, 2000) to model the dynamics of students' units coordinating activity across contexts within a teaching experiment, using the construct of propensity to coordinate units. Two video-recorded teaching experiments involving pairs of sixth-grade students were analyzed to form a model of the dynamics of students' units coordinating activity. The modeling involved separation of transcriptions into chunks that were coded dichotomously for the units coordinating activity of a single student in each dyad. The two teaching experiments were used to form 5 conjectures about the output of the model that were then tested with a third teaching experiment. The results suggest that modeling units coordination activity via the construct of propensity to coordinate units was useful for describing patterns in the students' perturbations during the teaching sessions. The model was moderately useful for identifying sequences of interactions that support growth in units coordination. Extensions, modifications, and implications of the modeling approach are discussed. Ph. D.

This descriptive qualitative study examines six eighth-grade students' thinking while solving probability problems. This study aimed to gather direct information on students' problem solving processes informed by the heuristics and biases framework. This study used purposive sampling (Patton, 1990) to identify eighth-grade students who were knowledgeable about probability and had reached the formal operational stage of cognitive development. These criterion were necessary to reduce the likelihood of students' merely guessing answers and important so that the researcher could distinguish between reasoning and intuition. The theoretical framework for this study was informed by Kahneman and Fredrick's (2002) recent revision to the heuristics and biases framework grounded in the research of Amos Tversky and Daniel Kahneman. Kahneman and Fredrick (2002) drew on dual process theory to explain systematic and predictable heuristic ways of thinking. Dual process theory hypothesizes that human thinking is divided into two different modes of processing. One mode, called System 1, is fast and linked to intuition, and the other, called System 2, is slow and linked to reasoning (Evans, 2008; Stanovich and West, 2000). Within dual process theory, System 1 thinking provides a credible system for explaining why people use heuristic thinking (Kahneman and Frederick, 2002). The recent revision to the heuristics and biases framework is focused on three heuristics, representativeness, conjunction fallacy, and availability. These three heuristics are believed to share the same mental process identified by Kahneman and Fredrick (2002), as the attribute substitution process. The clinical task based interview method was used in this study. This technique allowed the researcher to better observe and interact with the participants while exploring the students' probability thinking. The researcher also used think-aloud protocols to better reveal the organic thinking patterns of the students in real time (Ericsson and Simon, 1980; Fox, Ericsson, and Bets, 2010; Van Someren, Barnard, and Sandberg, 1994). The data from the interviews were analyzed using the constant comparison method (Glaser, 1965). This analysis revealed three categories that were combined with other analyses to create profiles for various thinking patterns observed by the researcher. The researcher identified patterns of thinking by students that were consistent with System 1 thinking and associated with the attribute substitution process (Kahneman and Fredrick, 2002). There were also situations in which students demonstrated ways of thinking consistent with System 2 thinking. However, unexpected ways of thinking were also identified by the researcher. For example, there were occasions when students substituted their fraction knowledge when solving probability problems and even seemed to equate probability with fractions. This type of thinking was referred to as the content substitution process in this study. This process occurred when students were using System 1 thinking as well as other types of thinking. In addition, the researcher observed students with thinking patterns that contained characteristics of both System 1 and System 2, which is referred to as slow intuition in this study. Slow intuition seemed to affect students' problem solving strategies as they wavered between multiple problem solving strategies that included either of the two substitution processes: attribute substitution and content substitution. This study contributes to the body of knowledge related to probabilistic thinking. In particular, this study informs our understanding of heuristic thinking used by eighth-grade students when solving probability problems. Further, teaching practices that draw on Fischbein's (1975, 1987) general notion of intuition might be developed and used to improve probability reasoning skills. These teaching practices target students that depend on the attribute substitution process and/or the content substitution process. Each of these heuristic ways of thinking may require different instructional techniques to help students develop more sound ways of thinking about probability. Regardless, teachers need to be informed of the extent that some students rely on their fraction knowledge when solving probability problems. Ph. D.

Subitizing, a quick apprehension of the numerosity of a small set of items, is inconsistently utilized by preschool educators to support early number understandings (Sarama & Clements, 2009). The purpose of this qualitative study is to investigate the relationship between children’s number understanding and subitizing activity. Sarama and Clements (2009) consider students’ subitizing activity as shifting from reliance upon perceptual processes to conceptual processes. Hypothesized mental actions carried into subitizing activity by children have not yet been empirically investigated (Sarama & Clements, 2009). Drawing upon Piaget’s (1968/1970) three mother structures of mathematical thinking, the theoretical implications of this study consider expanding the scope of Piaget’s (1968/1970) definition of topological thinking structures to include patterned orientations. Increasing the scope of this definition would allow for the investigation of the development of topological thinking structures and subitizing activity. An 11-week teaching experiment was conducted with six preschool aged children in order to analyze student engagement with subitizing tasks (Steffe & Ulrich, in press). To infer what perceptual and conceptual processes students relied upon when subitizing, tasks were designed to either assess or provoke cognitive changes. Analysis of interactions between students and the teacher-researcher informed this teacher-researcher of cognitive changes relative to each student’s thinking structure. Results indicated that students rely upon the space between items, symmetrical aspects of items, and color of items when perceptually subitizing. Seven different types of subitizing activity were documented and used to more explicitly describe student reliance upon perceptual or conceptual processes. Conceptual subitizing activity was redefined in this study, as depending upon mental reversibility and sophisticated number schemes. Students capable of conceptual subitizing were also able to conserve number. Students capable of conserving number were not always capable of conceptual subitizing. The symmetrical aspects of an item’s arrangement elicited students’ attention towards subgroups and transitioning students’ perceptual subitizing to conceptual subitizing. Combinations of counting and subitizing activity explained students’ reliance upon serial and classification thinking structures when transitioning from perceptual subitizing to conceptual subitizing. Implications of this study suggest effectively designed subitizing activity can both assess students’ number understandings, and appropriately differentiate preschool curriculum. Ph. D.

Over the past twenty years, journal writing has become more common in mathematics classes at all age levels. However, there has been very little empirical research about journal writing in college mathematics (Speer, Smith, & Horvath, 2010), particularly concerning the relationship between journal writing in college mathematics and college students' motivation towards learning mathematics. The purpose of this dissertation study is to fill that gap by implementing homework journals, which are a journal writing assignment based on Powell and Ramnauth's (1992) "multiple-entry log," in a college mathematics course and studying the relationship between homework journals and students' motivation towards learning mathematics as grounded in self-determination theory (Ryan & Deci, 2000). Self-determination theory predicts intrinsic motivation by focusing on the fundamental needs of competence, autonomy, and relatedness (Ryan & Deci, 2000). In addition, the purpose of this dissertation study is to explore and describe the relationship between homework journals and students' attitudes towards writing in mathematics. A pre-course and post-course survey was distributed to students enrolled in two sections of a college mathematics course and then analyzed using a 2Ã 2 repeated measures ANOVA with time (pre-course and post-course) and treatment (one section engaged with homework journals while the other did not) as the two factors, in order to test whether the change over time was different between the two sections. In addition, student and instructor interviews were conducted and then analyzed using a constant comparative method (Anfara, Brown, & Mangione, 2002) in order to add richness to the description of the relationship between homework journals and students' motivation towards learning mathematics as well as students' attitudes towards writing in mathematics. Based on the quantitative analysis of survey data, no differences in rate of change of competence, autonomy, relatedness, or attitudes towards writing were found. However, based on the qualitative analysis of interview data, homework journals were found to influence students' sense of competence, autonomy, and relatedness under certain conditions. In addition, students' attitudes towards writing in mathematics were strongly influenced by their likes and dislikes of homework journals and the perceived benefits of homework journals. Ph. D.

This design research study explored the potential of Curriculum Support Materials for promoting teaching as design. Conducted over a four-month period, the study traced the design, development, and pilot testing of a web site intended to serve as a professional development resource for teachers. The purpose of this exploratory study was to evaluate the web site's potential for promoting elementary mathematics teachers' understanding of teaching as a design activity and for supporting teachers in engaging in teaching as design. A team of four second grade teachers tested the web site during a two-week pilot unit on introducing the concepts of multiplication and division. Qualitative data were collected from these teachers through a planning and teaching log, a post-unit questionnaire, and a post-unit focus group interview. The findings indicated that the web site had the potential to promote teachers' understanding of teaching as design, but that the web site's potential as a stand-alone resource for supporting teachers in engaging in teaching as design was limited. Two specific features of the web site, the Unit Checklist and the videos addressing the related mathematics content, were identified as potentially valuable resources that could be incorporated into an ongoing professional development experience. Suggestions for revisions to the web site are discussed along with recommendations for further study. Ed. D.

This research investigates relationships between undergraduate students' understanding of proof and how this understanding relates to their conceptions of mathematical definitions. Three students in an introductory proofs course were each interviewed three times in order to assess their proof schemes, understand how they think of specific mathematical concepts, and investigate how the students prove relationships between these concepts. This research used theoretical frameworks from both proof and definition literature. Findings show that students' ability or inability to adapt their concept images of the mathematical concepts enhanced and impeded their proof schemes, respectively. Master of Science

The distance to uncontrollability of a controllable linear system is a measure of the degree of perturbation a system can undergo and remain controllable. The definition of the distance to uncontrollability leads to a non-convex optimization problem in two variables. In 2000 Gu proposed the first polynomial time algorithm to compute this distance. This algorithm relies heavily on efficient eigenvalue solvers. In this work we examine two alternative algorithms that result in linear matrix inequalities. For the first algorithm, proposed by Ebihara et. al., a semidefinite programming problem is derived via the Kalman-Yakubovich-Popov (KYP) lemma. The dual formulation is also considered and leads to rank conditions for exactness verification of the approximation. For the second algorithm, by Dumitrescu, Şicleru and Ştefan, a semidefinite programming problem is derived using a sum-of-squares relaxation of an associated matrix-polynomial and the associated Gram matrix parameterization. In both cases the optimization problems are solved using primal-dual-interior point methods that retain positive semidefiniteness at each iteration. Numerical results are presented to compare the three algorithms for a number of benchmark examples. In addition, we also consider a system that results from a finite element discretization of the one-dimensional advection-diffusion equation. Here our objective is to test these algorithms for larger problems that originate in PDE-control. Master of Science

This dissertation focuses on the development and testing of a conceptual framework for student motivation in mathematical discussion. Specifically, this document integrates Yackel and Cobb's (1996) framework with aspects of Self-Determination Theory (SDT), as described by Ryan and Deci (2000). Yackel and Cobb articulated the development of students' mathematical dispositions through discussion by facilitating student autonomy, incorporating appropriate social norms and co-constructing sociomathematical norms. SDT mirrors these factors and describes a similar process of self-regulation through fulfillment of the individual needs of autonomy, social relatedness, and competence. Given the conceptual overlap, this dissertation examines the connection of SDT with mathematical discussion with two studies. The first study examined the effect of student frequency of explaining mathematics on their perceived autonomy, competence and relatedness. Results of HLM analyses found that more frequent explanation of mathematics had a positive effect on students' perceived mathematics autonomy, mathematics competence, and relatedness. The second study used a triangulation mixed methods approach to examine high school geometry students' classroom discourse actions in combination with their perceived autonomy, competence, and relatedness. Results of the second study suggest a higher perceived sense of autonomy is indicative of more engagement in mathematical talk, but a measure of competence and relatedness are needed for such engagement to be fully indicative of mathematical discourse. Rather, students who lacked a measure of perceived competence or relatedness would cease participation in mathematical discussion when challenged by peers. While these results need further investigation, the results of the second study provide evidence that indicates the necessity of fulfilling all three SDT needs for engagement in mathematical discussion. Evidence from both the first and second studies presented in this dissertation provides support for the conceptual framework presented. Ph. D.

The goal of this paper is to evaluate commonly held criticisms of the practice of ability tracking in high school mathematics. To do so, I employ data from the National Education Longitudinal Study of 1988 and follow-ups to model classroom selection and education production. This paper will focus only on the causes and effects of tracking on students who were tracked as low-ability in eighth grade. From this, we can see how many students, if any, switched out of the low-ability track by tenth grade and how various switches have affected their test scores in mathematics. I find that students exercise mobility between ability-tracks as late as tenth grade and that ability-track placement is largely determined by test scores. In addition, I find evidence that there would be minimal, if any, test score improvement among low-ability students if they were all moved to a class of heterogeneous ability. Master of Science