Master of Science / Department of Mathematics / Andrew G. Bennett / The purpose of my research is to identify what features of a graph are important for college teachers with the intention of eventually developing a system by which a machine can recognize those features. In particular, I identify the features that college algebra teachers look at when grading graphs of lines and how much disagreement there is in the relative importance graders assign to each feature. In the process, eleven students from college algebra classes were interviewed and asked to graph six linear functions of varying difficulty. Eleven experienced college algebra graders were asked to grade the selected graphs, and interviewed to clarify what features of the graphs were important to them in grading. Altogether, a general grading rule appears to be: slope is worth 4 points, y-intercept is worth 4 points, labeling of intercepts, points and graph is worth 1 point. After that, add 1 point if everything is correct. All graders considered slope and y-intercept to be very important. Only some of them considered labeling to be important. Anything else was a matter of a single point adjustment. Furthermore, the graders judged slope and intercept from two points(the y-intercept and the first point to the right). Returning to the students’ work, I saw that the students also placed extra importance on points to the right of the y-axis. I conclude that this grading style may have a role in students’ learning to think only about two points in a line (but nothing else), and that replicating human grading may not be the best use of machine grading.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / "Just In Time Teaching" (JiTT) is a teaching practice that utilizes web based technology to collect information about students' background knowledge prior to attending lecture. Traditionally, students answer either multiple choice, short answer, or brief essay questions outside of class; based on student responses, instructors adjust their lectures "just-in-time." In this study, modified JiTT techniques in the form of online review modules were applied to a first semester calculus course at a large midwestern state university during the spring 2012 term. The review modules covered algebra concepts and skills relevant to the new material presented in calculus lecture (the "just-in-time" adjustment of the calculus lectures was not implemented in this teaching experiment). The reviews were part of the course grade. Instead of being administered purely "just-in-time," the reviews were assigned ahead of time as part of the online homework component of Calculus-I. While previous studies have investigated the use of traditional JiTT techniques in math courses and reported student satisfaction with such teaching tools, these studies have not addressed gains in student achievement with respect to specific calculus topics. The goal of this study was to investigate the latter, and to determine whether timing of the reviews plays a role in bettering student performance. Student progress on weekly Calculus-I online assignments was tracked in spring of 2012 and compared to student scores from weekly Calculus-I online assignments from spring 2011, when modified JiTT instruction was not available. For select Calculus-I online assignments during the spring 2012 term, we discovered that the review modules significantly increased the number of students receiving perfect scores, even when the reviews were not purely administered ``just-in-time." Analysis of performance, success of review assignments, and future implications are also discussed.

Doctor of Philosophy / Department of Physics / N. Sanjay Rebello / Calculus is used across many physics topics from introductory to upper-division level college courses. The concepts of differentiation and integration are important tools for solving real world problems. Using calculus or any mathematical tool in physics is much more complex than the straightforward application of the equations and algorithms that students often encounter in math classes. Research in physics education has reported students’ lack of ability to transfer their calculus knowledge to physics problem solving. In the past, studies often focused on what students fail to do with less focus on their underlying cognition. However, when solving physics problems requiring the use of integration, their reasoning about mathematics and physics concepts has not yet been carefully and systematically studied. Hence the main purpose of this qualitative study is to investigate student thinking in-depth and provide deeper insights into student reasoning in physics problem solving from multiple perspectives. I propose a conceptual framework by integrating aspects of several theoretical constructs from the literature to help us understand our observations of student work as they solve physics problems that require the use of integration. I combined elements of three important theoretical constructs: mathematical resources or symbolic forms, which are the small pieces of knowledge elements associated with students’ use of mathematical ideas; conceptual metaphors, which describe the systematic mapping of knowledge across multiple conceptual domains – typically from concrete source domain to abstract target domain; and conceptual blending, which describes the construction of new learning by integrating knowledge in different mental spaces. I collected data from group teaching/learning interviews as students solved physics problems requiring setting up integrals. Participants were recruited from a second-semester calculus-based physics course. I conducted qualitative analysis of the videotaped student conversations and their written work. The main contributions of this research include (1) providing evidence for the existence of symbolic forms in students’ reasoning about differentials and integrals, (2) identifying conceptual metaphors involved in student reasoning about differentials and integrals, (3) categorizing the different ways in which students integrate their mathematics and physics knowledge in the context of solving physics integration problems, (4)exploring the use of hypothetical debate problems in shifting students’ framing of physics problem solving requiring mathematics.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / The main objective of the study is to identify the general characteristics of groups within a typical Studio College Algebra class and then adapt aspects of the course to best suit their needs. In a College Algebra class of 1,200 students, like those at most state funded universities, the greatest obstacle to providing personalized, effective education is the anonymity of the students. Data mining provides a method for describing students by making sense of the large amounts of information they generate. Instructors may then take advantage of this expedient analysis to adjust instruction to meet their students’ needs. Using exam problem grades, attendance points, and homework scores from the first four weeks of a Studio College Algebra class, the researchers were able to identify five distinct clusters of students. Interviews of prototypical students from each group revealed their motivations, level of conceptual understanding, and attitudes about mathematics. The student groups where then given the following descriptive names: Overachievers, Underachievers, Employees, Rote Memorizers, and Sisyphean Strivers. In order to improve placement of incoming students, new student services and student advisors across campus have been given profiles of the student clusters and placement suggestions. Preliminary evidence shows that advisors have been able to effectively identify members of these groups during their consultations and suggest the most appropriate math course for those students. In addition to placement suggestions, several targeted interventions are currently being developed to benefit underperforming groups of students. Each student group reacts differently to various elements of the course and assistance strategies. By identifying students who are likely to struggle within the first month of classes, and the recovery strategy that would be most effective, instructors can intercede in time to improve performance.

Doctor of Philosophy / Department of Mathematics / Andrew Bennett / Engineering majors and Mathematics Education majors are two groups that take the basic, core Mathematics classes. Whereas Engineering majors go on to apply this mathematics to real world situations, Mathematics Education majors apply this mathematics to deeper, abstract mathematics. Senior students from each group were interviewed about “function” and “accumulation” to examine any differences in learning between the two groups that may be tied to the use of mathematics in these different contexts. Variation between individuals was found to be greater than variation between the two groups; however, several differences between the two groups were evident. Among these were higher levels of conceptual understanding in Engineering majors as well as higher levels of confidence and willingness to try problems even when they did not necessarily know how to work them.

Master of Science / Mathematics / Andrew G. Bennett / This study utilizes eye-tracking technology as a tool to measure college algebra students’ mathematical noticing as defined by Lobato and colleagues (2012). Research in many disciplines has used eye-tracking technology to investigate the differences in visual attention under the assumption that eye movements reflect a person’s moment-to-moment cognitive processes. Motivated by the work done by Madsen and colleagues (2012) who found visual differences between those who correctly and incorrectly solve introductory college physics problems, we used eye-tracking to observe the visual attention difference between correct and incorrect solvers of college algebra problems. More specifically, we consider students’ visual attention when presented tabular representations of linear functions. We found that in several of the problems analyzed, those who answered the problem correctly spend more time looking at relevant table values of the problem while those who answered the problem incorrectly spend more time looking at irrelevant table labels x, y, y = f(x) of the problem in comparison to the correct solvers. More significantly, we found a noteworthy group of students, who did not move beyond table labels, using these labels solely to solve the problem. Future analyses need to be done to expand on the differences between eye patterns rather than just focusing on dwell time in the relevant and irrelevant areas of a table.

Doctor of Philosophy / Department of Curriculum and Instruction / Michael F. Perl / This exploratory correlational study seeks to answer the question of whether a relationship exists between student average test score gains on state exams and teachers’ rating of values on the Schwartz Values Survey. Eighty-seven randomly selected Kansas teachers of math and/or reading, grades four through eight, participated. Student test score gains were paired with teachers and averaged. The results of these backward stepwise entries of multiple regressions using SPSS software are reported. Significant relationships with large effect sizes are reported for teacher values and student test score gains in reading and math. Models of teacher values are found that account for thirty-two percent of the average student test score gains in reading and for forty-three percent of the average student test score gains in mathematics. The significant model of values with the greatest adjusted relationship with reading test score gains is described as the Relational Teacher Value Type. The valuing of True Friendship (close supportive friends) and the valuing of Sense of Belonging (feeling that others care about me) proved to be the most powerful indicators of student reading score gains within this type. The significant model of values with the greatest adjusted relationship with mathematics test score gains is described as the Well-Being Teacher Value Type. The valuing of Healthy (not being sick physically or mentally), the valuing of Reciprocation of Favors (avoidance of indebtedness), and Self Respect (belief in one’s own worth) proved to be the most powerful indicators of student mathematics test score gains within this type. The significant value items within each of the above types’ models are discussed regarding possible reasons for their relationships to student test score gains. A value that is found significant for both reading and mathematics teachers in accounting for student test score gains is Moderate (avoiding extremes of feeling and action). Of the teachers in the study that taught mathematics and reading, their students’ mathematics score gains did not correlate in a statistically significant way with their students’ reading score gains, suggesting that a teacher’s ability to teach math has little to do with a teacher’s ability to teach reading.

Doctor of Philosophy / Department of Mathematics / Andrew G. Bennett / The goal of this work is to provide instructors with detailed information about their classes at each assignment during the term. The information is both on an individual level and at the aggregate level. We used the large number of grades, which are available online these days, along with data-mining techniques to build our models. This enabled us to profile each student so that we might individualize our approach. From these profiles, we began to investigate what can be done in order to get students to do better, or at least be less frustrated. Regardless, the interactions with our undergraduates will improve as our knowledge about them increases. We start with a categorization of Studio College Algebra students into groups, or clusters, at some point in time during the semester. In our case, we used the grouping just after the first exam, as described by Dr. Rachel Manspeaker in her PhD. dissertation. From this we built a naive Bayesian model which extends these student clusters from one point in the semester, to a classification at every assignment, attendance score, and exam in the course. A hidden Markov model was then constructed with the transition probabilities being derived from the Bayesian model. With this HMM, we were able to compute the most likely path that students take through the various categories over the semester. We observed that a majority of students settle into a group within the first two weeks of the term.

Master of Science / Department of Psychology / Gary L. Brase / In 2006, the Center for Disease Control and Prevention recommended the revision of state HIV testing laws. With these recommendations, more low-risk individuals are tested regardless of their risk group. However, there is a greater chance of a false positive test result for low-risk individuals than for high-risk individuals. Additionally, previous research found that doctors and HIV counselors in Germany did not accurately communicate the relationship between risk factors and false positive tests (Gigerenzer, Hoffrage, & Ebert, 1998). This study aimed to (1) compare the findings of the 1998 German sample to HIV hotline counselors in the United States in 2011; and (2) to investigate the ability of students to calculate the conditional probability of HIV for a low-risk individual after receiving a positive test, based on idealized transcripts of conversations with HIV hotline counselors. The first study found that HIV hotline counselors use both verbal expressions of risk and percentages to communicate HIV testing statistics. Additionally, 2011 American counselors were more aware of the chance of false positives and false negatives than compared to the 1998 German sample. However, no 2011 American counselors were able to provide an accurate positive predictive value for a low-risk woman. The second study found low performance among students in the calculation of the positive predictive value. Performance was facilitated by a natural frequency format for high numerate individuals. There were different patterns of results for the General Numeracy Scale and the Subjective Numeracy Scale. This would suggest that these two scales might be measuring different constructs. These findings are consistent with the two theories supporting the Frequency Effect, namely the Frequentist Hypothesis and the Nested Sets Hypothesis. Additionally, this research suggests computation of the conditional risk of HIV is facilitated by a natural frequency format. Teaching techniques have been developed and demonstrate long lasting improvement in health related computations. If a few hours of training is all that it takes to communicate these life and death statistics in a manner that is consistent with reasoning, health practitioners and students should be required to have more education in communicating and computing probabilities.

Doctor of Philosophy / Curriculum and Instruction / David S. Allen / This study addresses how the flipped method of classroom instruction differs from traditional classroom instruction when comparing student achievement measures in middle and high school mathematics classrooms. The flipped classroom is defined by the Flipped Learning Network (2014) as an instructional method that moves direct instruction outside of the classroom in order to make room in the classroom for a more interactive learning environment where students can actively engage in the content. The flipped classroom strategy theoretically allows teachers the time to develop mathematical ideas and the ability to facilitate that development. For the Common Core State Standards initiative to be effective, teachers need to engage students in new learning experiences that support college and career readiness. By implementing a technology based instructional approach, like the flipped classroom strategy, teachers are able to blend twenty-first century skills with the development of the essential habits of mind of mathematically proficient students (Brunsell & Horejsi, 2013). This study seeks to understand how the flipped method of classroom instruction can lead to improved student achievement in mathematics courses and improve student perceptions about math in order to encourage course consumption in the future (Zollman, 2011). A modified explanatory sequential mixed methods design was used, and it involved collecting quantitative data and then explaining the quantitative results with in-depth qualitative data. In the quantitative phases of the study, NWEA Mathematics MAP Assessment data were collected from middle school students and course common final assessment scores were collected from middle school and high school students in a large Midwestern suburban school district to determine how student math achievement was impacted for students in a flipped classroom as compared to a traditionally instructed classroom. The frequency of active learning incidents was also collected during classroom observations. The qualitative phase was conducted as a follow up to the quantitative results to help explain the quantitative results. In this exploratory follow-up, student and teacher perceptions of mathematics achievement as a result of the flipped classroom approach to instruction with middle and high school math students and how those perceptions might be different than those of students and teachers in traditionally taught classrooms along with descriptions of observable active learning incidents in the school district were explored.