Search

Background Students’ ability to construct and coordinate units has been found to have far-reaching implications for their ability to develop sophisticated understandings of key middle-grade mathematical topics such as fractions, ratios, proportions, and algebra, topics that form the base of understanding for most STEM-related fields. Most of the related research on unit coordination relies on time-intensive clinical interviews and teaching experiments. In this study, we investigate the work of 93 sixth-grade students on a written assessment containing whole number and fraction contexts using both continuous and discrete quantities, and how this work can be used to assess stages in students’ construction and coordination of units. Our investigation is guided by the following general research questions: (1) What forms of written work evidence the construction of and operation on composite units (units made up of other units)? (2) How does the categorization of students based on responses from a written assessment compare to written performance on a set of tasks conveying a continuous whole number multiplicative context? Results We documented the different ways students represented composite units in their written work. In particular, student written work on tasks that included figurative unit items provided the greatest variety of evidence regarding students’ construction of and operation on composite units. However, written evidence from partitioning tasks did not seem as promising for distinguishing student stages. Students’ performance on decontextualized bar tasks involving continuous quantities was found to be consistent with students’ level of unit coordination based on written work providing evidence for the validity of stage categorizations. Conclusions Our findings shed light on the affordances and constraints associated with particular stages in unit construction and coordination that a student brings to bear on tasks provided in a formal, written assessment. These findings provide promising evidence for scaling up the assessment of students construction and coordination of units through the use of written assessments instead of time-intensive clinical interviews. Published version

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.

This qualitative research seeks to identify and understand the beliefs of 10 high school geometry teachers that help resolve apparent inconsistencies between their espoused and enacted beliefs. Data was collected using an initial interview, classroom observations, and a follow-up interview to gather evidence of teacher beliefs based on what they say, do, and intend respectively. Open coding, analytical coding, cluster identification, coding memos, and analytical memos were used to analyze the data and write summaries of the teachers' explanatory beliefs with beliefs as the unit of analysis. It was identified that teachers consistently and inconsistently enact their espoused beliefs, but there are also instances when teachers both consistently and inconsistently enact particular espoused beliefs. This endeavor necessitates a shared understanding of terms, and it was found what it means to "understand" needs to be clarified with a definition and examples from teachers. When teachers appear to not enact their espoused beliefs, explanatory beliefs were pinpointed that resolve the conflict and found the explanatory beliefs exist in at least seven macro clusters. These explanatory beliefs interact with espoused beliefs by overriding, limiting, or preventing the espoused beliefs to resolve the apparent inconsistency in teachers espoused and enacted beliefs. The explanatory beliefs with limiting and overriding interactions were found to coexist for some teachers around a teaching practice as overriding interactions are connected to constraints on the classroom whereas limiting interactions are not. It was also found that belief clusters are nested within clusters of beliefs, and these clusters allow for beliefs to cluster in isolation in different ways. This work also shows empirically that some geometry teacher beliefs are socially constructed due to the presence of common cultural artifacts and influence from mathematics teacher educators. This work has implications and future research directions in the areas of using beliefs as the unit of analysis, mapping teacher's belief systems, considering the social construction of beliefs and role of community, connecting beliefs to specific teaching practices, and educating teachers. Doctor of Philosophy This research seeks to understand and interpret the beliefs of 10 high school geometry teachers that resolve apparent inconsistencies between what teachers say they believe and what they do in the classroom. Data was collected using an initial interview, classroom observations, and a follow-up interview to gather evidence of teacher beliefs based on what they say, do, and intend respectively. It was identified that teachers consistently and inconsistently enact their stated beliefs, but there are also instances when teachers both consistently and inconsistently enact their stated beliefs. When teachers appear to inconsistently enact their stated beliefs, it was found that teachers have logical reasons why they do so, and these reasons relate to specific teaching practices. It was also found that teacher beliefs interact with each other in different ways. Teachers' beliefs can limit or prevent the enaction of their other beliefs. In addition, school level constraints can override the enaction of some teacher beliefs. This research shows that some beliefs are held by different teachers from vastly different schools which suggests that some geometry teacher beliefs are held socially. The findings from this research have implications for teacher education.

This study examined the effects of computational labs in Business Calculus classes used at a single, private institution on student outcomes of conceptual understanding of calculus and attitudes towards mathematics. The first manuscript addresses the changes in conceptual understanding through multiple-method research design, a quantitative survey given pre and post study and qualitative student comments, found no significant gains in conceptual knowledge as measured by a concept inventory, however, student comments revealed valuable knowledge demonstrated through reflection on and articulation of how specific calculus concepts could be used in real world applications. The second manuscript presents results to the effects on attitudes toward mathematics, studied through multiple-method research design, using a quantitative survey given at two intervals, pre and post, and analysis of student comments, which showed that students that participated in the labs had a smaller decline in attitude, although not statistically significant, than students that did not complete the labs and the labs were most impactful on students that had previously taken calculus; student comments overwhelmingly demonstrate that students felt and appreciated that the labs allowed them to see how calculus could be applied outside the classroom. Overall students felt the labs were beneficial in the development of advantageous habits, taught some a skill they hope to further develop and study, and provided several recommendations for improvement in future implementation. Collectively, this research serves as a foundation for the effectiveness of computational tools employed in general education mathematics courses, which is not currently a widespread practice. Doctor of Philosophy Students from a variety of majors often leave their introductory calculus courses without seeing the connections and utility it may have to their discipline and may find it uninspiring and boring. To address these issues, there is a need for educators to continue to develop and research potentially positive approaches to impacting students' experience with calculus. This study discusses a method of doing so, by studying students' understanding of and attitude toward calculus in a one-semester Business Calculus course using computational labs to introduce students to calculus concepts often in context of a business scenario. No significant gains in conceptual knowledge were found as measured by a concept inventory; however, student comments revealed valuable knowledge demonstrated through articulation of how specific calculus concepts could be used in real world applications. Students that participated in the labs also had a smaller decline in attitude than students that did not complete the labs. Student comments overwhelmingly demonstrate that students felt and appreciated that the labs allowed them to see how calculus could be applied outside the classroom. The labs were most impactful on students that had previously taken calculus. Overall students felt the labs were beneficial in the development of advantageous habits such as persistence, utilizing resources, and precision, introduced them to coding, a skill they hope to further develop and study, and students provided several recommendations for improvement in future implementation. This research provides a foundation for the effectiveness of computational tools used in general education mathematics courses.

The purpose of this study was to investigate on a national scale the baseline level of p-value fluency of future researchers (i.e., doctoral students). To that end, two research questions were investigated. The first research question, Can a sufficiently reliable and valid measure of p-value misinterpretations (in a research context) be constructed?, was addressed via the development and validation of the Keller P-value Misinterpretation Inventory instrument (KPVMI). An iterative process of expert review, pilot testing, and field testing resulted in an adequately reliable measure (Alpha = .8030) of p-value fluency as assessed across 18 misinterpretations and 2 process levels as well as an independently validated sub-measure of p-value fluency in context as assessed across 18 misinterpretations (Alpha = .8298). The second research question, What do the results of the KPVMI administration tell us about the current level of p-value fluency among doctoral students nationally?, was addressed via analysis of a subset of the field test data (n = 147) with respect to performance on the subset of items considered sufficiently validated as developed in Phases I-III (KPVMI-1). The median score was 10/18 items answered correctly indicating that future researchers on the aggregate struggle to properly interpret and report p-values in context; furthermore, there was insufficient evidence to indicate training and experience are positively correlated with performance. These results aligned with the extant literature regarding the p-value misinterpretations of practicing researchers. Doctor of Philosophy The purpose of this study was to investigate on a national scale the baseline level of p-value fluency of future researchers (i.e., doctoral students). To that end, two research questions were investigated. The first research question, Can a sufficiently reliable and valid measure of p-value misinterpretations (in a research context) be constructed?, was addressed via the development and validation of the Keller P-value Misinterpretation Inventory instrument (KPVMI). The second research question, What do the results of the KPVMI administration tell us about the current level of p-value fluency among doctoral students nationally?, was addressed via analysis of a subset of the field test data (n = 147) with respect to performance on the subset of items considered sufficiently validated as developed in Phases I-III (KPVMI-1). Results indicate that future researchers on the aggregate struggle to properly interpret and report p-values in context; furthermore, there was insufficient evidence to indicate training and experience are positively correlated with performance. These results aligned with the extant literature regarding the p-value misinterpretations of practicing researchers.

Academic achievement involves complex processes that are not fully understood. That being said, the connection between working memory and academic achievement is well developed and emphasized in the literature. However considering the complex nature of academic achievement, other processes are likely involved. The current study examined the contributions of recollection, familiarity, working memory, and verbal IQ longitudinally in children at ages 6 and then 9. Recollection, but not familiarity, contributed to measures of both reading and math at age 6, but not 9. Path models suggested that the direct and indirect effects of working and episodic memory to academic achievement change from age 6 to 9. Furthermore, this study examined the contributions of the neural correlates of recollection and working memory to measures of academic achievement at ages 6 and 9. The neural correlates of working memory and recollection did not contribute to academic achievement, but additional research is needed to draw concrete conclusions. Overall, the results suggest that episodic memory should be considered in addition to working memory when examining academic achievement. Ph. D.

The purpose of this study was to determine the understandings and perceptions that teachers in one elementary school in Virginia had about the teacher evaluation system. Participants were recruited from one elementary school in Virginia. All of the teachers and specialists were asked to participate and 12 participants volunteered. General education teachers made up 25% of the participants while specialists who taught in areas such as Music, Art, Physical Education, or Special Education made up 75% of the participants. Participants were interviewed individually and were asked the same questions about the teacher evaluation process. Interviews were transcribed and primary codes were developed from the data. Interview questions asked participants to describe both their understandings of the evaluation system under which they worked and the perceptions they had about their experiences with evaluation. Even though there were no questions dealing with emotions and feelings, emotional language was prevalent throughout the data. The overall findings from this study indicated that teachers had a limited knowledge about a problematic evaluation system. The fact that elementary teachers have unique responsibilities means that they may need multiple ways to TEACHER UNDERSTANDINGS AND PERCEPTIONS OF EVALUATION iii show their effectiveness. Other findings suggested that inconsistent evaluation practices in this school caused teachers to see no benefit from the process. Finally, findings also suggested that there was a misalignment between the evaluation instrument and the jobs of teachers, particularly those of specialist teachers. Implications for future research include studies which focus on teacher learning over time in order to support teachers at any level of experience. Future research also needs to be conducted with elementary classroom teachers and specialist teachers to discover ways that may create a more aligned and fair process. Further research would also include studying the perceptions of evaluators and how they carry out the evaluation process and make decisions about its use. Ph. D.

In this research, I utilize the theoretical perspective Knowledge In Pieces to identify the knowledge resources students utilize while in the process of completing various differential equations tasks. In addition I explore how this utilization changes over the course of a semester, and how resources related to the concepts of function and rate of change supported the students in completing the tasks. I do so using data collected from a series of task-based individual interviews with two students enrolled in separate differential equations courses. The results provide a fine-grained description of the knowledge students consider to be productive with regard to completing various differential equations tasks. Further the analysis resulted in the identification of five ways students interpret differential equations tasks and how these interpretations are related to the knowledge resources students utilize while completing the various tasks. Lastly, this research makes a contribution to mathematics education by illuminating the knowledge concerning function and rate of change students utilize and how this knowledge comes together to support students in drawing connections between differential equations and their solutions, structuring those solutions, and reasoning with relationships present in the differential equations. Ph. D.

With the recent release of the Next Generation Science Standards (NGSS) (NGSS Lead States, 2014b) science educators were expected to teach engineering content and practices within their curricula. However, technology and engineering (T&E) educators have been expected to teach content and practices from engineering and other disciplines since the release of the Standards for Technological Literacy (ITEA/ITEEA, 2000/2002/2007). Requisite to the preparation of globally competitive STEM literate individuals is the intentional, concurrent teaching of science, technology, and engineering concepts. Many studies have examined the pedagogical content knowledge (PCK) (Shulman, 1987) of science and T&E educators, but none have examined the science PCK of T&E educators. The purpose of this study was to examine the extent of the relationship between T&E educator’s science and T&E preparation experiences, and their teaching of science content and practices. This study, which employed a fully integrated mixed methods design (Teddlie & Tashakkori, 2006), was conducted to inform the pre- and in-service preparation needs for T&E educators. A random sample of 55 Foundations of Technology (FoT) teachers across 12 school systems within one state participated in an online survey, leading to eight teachers being purposefully selected for classroom observations. Data collected from the surveys and classroom observations were analyzed through Spearman’s rho tests to examine relationships between preparation factors and teaching of science content and practices. These data were corroborated with curriculum content analyses, classroom observations, and interview responses to validate the results. Analyses of the data across all three methods revealed significant correlations between many preparation factors and the teaching of science content and practices. Specifically the amount of high school and undergraduate physics courses, and T&E and science in-service delivered were found to have statistically significant, strong positive correlations. These findings suggest T&E educators with increased amounts of these preparation experiences can be expected to teach science content and practices more proficiently. The findings and conclusions drawn from the data analyses provide implications for science and T&E educators, researchers, preservice programs, and in-service professional development efforts. The discussion and implications suggest the need to conduct replication studies in different contexts. Ph. D.

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.