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As part of their mathematical knowledge for teaching (Ball et al., 2008), teachers must have a well-connected understanding of the subject matter they teach and must know this content in deeper and different ways than other adults. This is essential for quality teaching and learning, as teachers’ knowledge and understanding impact the nature and effectiveness of instruction (e.g., Hill et al., 2005). Since decimal concepts are part of elementary curriculum (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010), and can be difficult for children and adults (e.g., Jacobson et al., 2020; Kastberg & Morton, 2014; Steinle & Stacey, 1998), mathematics teacher educators must consider how we can strengthen support for prospective teachers of elementary school (PTs), to deepen their mathematical knowledge for teaching decimals. This is of particular importance as existing research provides few rich characterizations of PTs’ decimal understanding and is limited in explorations into connections and mechanisms that may improve that understanding. In this dissertation, I attend to the research question, following engagement with rich conceptually focused decimal instruction, how may PTs’ conceptualizations of decimal place value and magnitude, and factors which have influenced this understanding, be characterized? I address some gaps in current literature by considering how robust decimal understanding for PTs may be connected to and grounded in their broader knowledge of the base-ten place value system, and to the decimal models which they use. Following an introduction to the problem in Chapter One, and a review of relevant literature in Chapter Two, Chapter Three reports on a study in which I examined how PTs characterized the base-ten place value system, distinguishing between responses crafted by PTs who had demonstrated different levels of decimal understanding. This allowed me to identify elements of base-ten place value understanding which likely supported PTs’ thinking about decimal place value and magnitude. In the study in Chapter Four, I explored the nature of PTs’ decimal understanding and its relationships with decimal square or number line models that they used, finding that certain model features facilitated PTs’ ability to think about decimal place value and magnitude in ways that are more likely to be productive and appropriate for teaching. These two empirical studies are both qualitative content analyses (Hsieh & Shannon, 2005) carried out in the context of the Elementary Mathematics Project (Chapin et al., 2021). Though implications for teachers and teacher educators are incorporated in Chapters Three and Four, Chapter Five is a practitioner article in which I focus more directly on these implications, making recommendations about important model features and areas of emphasis for decimal instruction. Chapter Six looks across the dissertation, discussing overarching themes and directions for future research. Results of this research may be used to support mathematics teacher educators in carrying out effective decimal instruction with their PT students, since better understanding of PTs’ thinking can help mathematics teacher educators to make informed curricular and pedagogical decisions to foster PT development. This is of high importance, since as PTs increase and enrich their decimal understanding, their students’ opportunities to learn will also expand. / 2027-04-30T00:00:00Z

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.

Doctor of Philosophy / Electrical and Computer Engineering / Steven Warren / Linear Systems (a.k.a., Signals and Systems) is an important class in an Electrical Engineering curriculum. A clear understanding of the topics in this course relies on a well-developed notion of lower-level mathematical constructs and procedures, including the roles these procedures play in system analysis. Students with an inadequate math foundation regularly struggle in this class, as they are typically able to perform sequences of the underlying calculations but cannot piece together the higher-level, conceptual relationships that drive these procedures. This dissertation describes an investigation to assess and improve students’ higher-level understanding of Linear Systems concepts. The focus is on the topics of (a) time-domain, linear time-invariant (LTI) system response visualization and (b) Fourier series conceptual understanding, including trigonometric Fourier series (TFS), compact trigonometric Fourier series (CTFS), and exponential Fourier series (EFS). Support data, including exam and online homework data, were collected since 2004 from students enrolled in ECE 512 - Linear Systems at Kansas State University. To assist with LTI response visualization, two online homework modules, Zero Input Response and Unit Impulse Response, were updated with enhanced plots of signal responses and placed in use starting with the Fall 2009 semester. To identify students’ conceptual weaknesses related to Fourier series and to help them achieve a better understanding of Fourier series concepts, teaching-learning interviews were applied between Spring 2010 and Fall 2012. A new concept-based online homework module was also introduced in Spring 2011. Selected final exam problems from 2007 to 2012 were analyzed, and these data were supplemented with detailed mid-term and final exam data from 77 students enrolled in the Spring 2010 and Spring 2011 semesters. In order to address these conceptual learning issues, two frameworks were applied: Bloom’s Taxonomy and APOS theory. The teaching-learning interviews and online module updates appeared to be effective treatments in terms of increasing students’ higher-level understanding. Scores on both conceptual exam questions and more traditional Fourier series exam questions were improved relative to scores received by students that did not receive those treatments.

The topic of students’ application of conceptual knowledge in physics is constantly being researched. It is a common occurrence that students are able to solve numerical problems without understanding the concepts involved. The primary focus of this dissertation is to investigate the extent to which a group of first year physics students are able to identify and use the correct physics concepts when solving problems set in different contexts. Furthermore, this study aims to identify underlying factors giving way to students not applying appropriate physics concepts. A questionnaire was designed in test-format in which all the problems dealt with two objects whose movement had to be compared to each other. The physical quantities describing or influencing the objects’ movement differed in each consecutive problem; whilst the nature of the concept under consideration remained the same. The problems were set in various contexts namely: i. Formal conceptual questions, some with numeric values; ii. Questions set in every day context with/without numeric values; iii. Questions on vertical upward, vertical downward and horizontal motion. The questionnaire was distributed to 481 students in the first-year physics course in 2014 at the Potchefstroom Campus of the North West University. It was expected that the percentage of correct answers would reveal discrepancies in the responses to contextual, numeric and formal conceptual questions. The outcome of the statistical analysis confirmed this expectation. In addition, it seemed that only a few students were able to correctly identify the appropriate variables when considering vertical and horizontal movement while the majority of the students did not apply the same physics principle in isomorphic vertical upward and vertical downward problems. It appears that the context in which the question was posed determined whether the problem was seen as an item that would require “physics reasoning” or as a setting where physics reasoning did not apply. The results revealed students inability to relate physics concepts to appropriate mathematical equations. Two important results from this work are: (1) the presentation of a questionnaire that can be implemented to investigate various aspects regarding the contexts of physics problems; and (2) expanding the concept of context to include the direction of movement as a separate context. / MSc (Natural Science Education), North-West University, Potchefstroom Campus, 2015

The topic of students’ application of conceptual knowledge in physics is constantly being researched. It is a common occurrence that students are able to solve numerical problems without understanding the concepts involved. The primary focus of this dissertation is to investigate the extent to which a group of first year physics students are able to identify and use the correct physics concepts when solving problems set in different contexts. Furthermore, this study aims to identify underlying factors giving way to students not applying appropriate physics concepts. A questionnaire was designed in test-format in which all the problems dealt with two objects whose movement had to be compared to each other. The physical quantities describing or influencing the objects’ movement differed in each consecutive problem; whilst the nature of the concept under consideration remained the same. The problems were set in various contexts namely: i. Formal conceptual questions, some with numeric values; ii. Questions set in every day context with/without numeric values; iii. Questions on vertical upward, vertical downward and horizontal motion. The questionnaire was distributed to 481 students in the first-year physics course in 2014 at the Potchefstroom Campus of the North West University. It was expected that the percentage of correct answers would reveal discrepancies in the responses to contextual, numeric and formal conceptual questions. The outcome of the statistical analysis confirmed this expectation. In addition, it seemed that only a few students were able to correctly identify the appropriate variables when considering vertical and horizontal movement while the majority of the students did not apply the same physics principle in isomorphic vertical upward and vertical downward problems. It appears that the context in which the question was posed determined whether the problem was seen as an item that would require “physics reasoning” or as a setting where physics reasoning did not apply. The results revealed students inability to relate physics concepts to appropriate mathematical equations. Two important results from this work are: (1) the presentation of a questionnaire that can be implemented to investigate various aspects regarding the contexts of physics problems; and (2) expanding the concept of context to include the direction of movement as a separate context. / MSc (Natural Science Education), North-West University, Potchefstroom Campus, 2015

The technology principle in the Principles and Standards for School Mathematics (NCTM, 2000) states that technology plays an important role in how teachers teach mathematics and in how students learn mathematics. The purpose of this sequential explanatory mixed-methods study was to examine the impact of interactive applets on students’ understanding of parameter changes to parent functions. Students in the treatment classes were found to have statistically significantly higher posttest scores than students in the control classes. Although the data analysis showed a statistically significant difference between classes on procedural understanding, no statistically significant difference was found with regard to conceptual understanding. Student and teacher interviews provided insight on how and why the use of applets helped or hindered students’ understanding of parameter changes to parent functions.

This mixed-method study investigated the impact of graphing calculator use on high school calculus students' reasoning skills through calculus problems when applying to concepts of the definite integral and its applications. The study provides an investigation of the effects on reasoning when graphing calculators are used, since it is proposed that, through reasoning, conceptual understanding can be achieved. Three research questions were used to guide the study: (1) Does the use of the graphing calculator improve high school calculus students' reasoning ability in calculus problems applying the definite integral? (2) In what specific areas of reasoning does use of the graphing calculator seem to be most and least effective? and (3) To what extent can students who have used the graphing calculator demonstrate ability to solve problems using pencil and paper methods? The study included a quantitative, quasi-experimental component and a qualitative component. Results of the quantitative and qualitative analysis indicate that (1) graphing calculators had a positive impact upon students' reasoning skills (2) graphing calculators were most effective in the areas of initiating a strategy and monitoring progress (3) students' reasoning skills were most improved when graphing calculators were used together with the analytic approach during both instruction and testing and (4) students who used the graphing calculator performed equally as well in all elements of reasoning as those who used pencil and paper to solve problems.

Selvan Naidoo, Student no: 0215998E. MSc Education, Faculty of Science, 2006. / This study used concept mapping to explore South African Grade 11 learners’ understanding of the function concept. Learners’ understanding of the function concept was investigated by examining the relationships learners made between the function concept and other mathematical concepts. The study falls within a social constructivist framework and is underpinned by the key educational notion of understanding. The research method employed was a case study. Data for the study was collected through a concept mapping task, a task on functions and individual learner interviews. In the analysis four key issues are identified and discussed. They are concerned with (a) learners who make most connections; (b) issues related to learners’ omission and addition of concepts; (c) learners’ use of examples in concept mapping and (d) the nature of connections learners made. The study concludes that concept mapping is an effective tool to explore learners’ understanding of the function concept. The report concludes with recommendations for classroom practice, teacher education and further research, particularly given the context of school mathematics practice in the South African curriculum where concept mapping (i.e. use of metacogs) has recently been incorporated as an assessment tool.