Characterizing Student Engagement in a Post-Secondary Developmental Mathematics Class and Exploring the Reflexivity between Social and Sociomathematical Norms

Traditionally, post-secondary developmental mathematics courses aspire to equip students with mathematical content knowledge needed to succeed in calculus and subsequent STEM courses. The literature shows that this goal alone is insufficient, as the emphasis on content acquisition often comes at the expense of developing higher-order skills such as argumentation, reasoning, and flexibility in mathematics problem solving (Chiaravalloti, 2009; Partanen & Kaasila, 2014; Star et al., 2015). Redesigning curricula with these additional objectives in mind requires providing students with opportunities to engage with mathematics in ways that may contrast with their past experiences or expectations. It requires changing patterns of classroom engagement and development of different classroom norms. This mixed methods research study incorporated a semester-long teaching experiment that aimed to support students\u27 development of higher-order skills by negotiating productive classroom norms. One of the primary interventions was a sequence of Multiple Solutions Activities that required groups of students to analyze and critique unfamiliar or erroneous mathematical solutions. The overarching goal of the research was to study students\u27 engagement during these activities across the semester by characterizing the nature of specific types of classroom norms. Social norms describe the classroom participation structure, while sociomathematical norms focus on aspects of student activity that are inherently mathematical, such as what constitutes an acceptable mathematical solution (Yackel & Cobb, 1996). Because of a reflexive relationship between norms and beliefs, students\u27 social and mathematical beliefs were also of interest to characterize the influence of the teaching experiment; these beliefs were assessed by a pre- and post-course questionnaire. The results paint a complex picture of student engagement and values. Despite quantitative analysis suggesting encouraging improvements in students’ mathematical engagement, qualitative analysis highlighted that this change was not homogenous. In particular, the analysis revealed variations in students’ perceptions of the value of multiple solutions and in the nature of the norms developed in student groups. Consequently, the study highlights the lasting impact of classroom norms on students\u27 beliefs, and vice versa, which may hinder the development of alternative norms in subsequent classes. The results of the project also expand upon Yackel and Cobb\u27s (1996) Interpretive Framework for characterizing classroom engagement by suggesting a reflexive relationship exists between social and sociomathematical norms. The data analysis describes concurrent development and mutual influence between the participation structure of a group and their taken-as-shared mathematical beliefs. In all, the project shows that deliberate attention towards negotiating productive classroom norms and students’ in-class engagement can positively affect students’ attitudes towards multiple solutions

Characterizing Student Engagement in a Post-Secondary Developmental Mathematics Class and Exploring the Reflexivity between Social and Sociomathematical Norms

Traditionally, post-secondary developmental mathematics courses aspire to equip students with mathematical content knowledge needed to succeed in calculus and subsequent STEM courses. The literature shows that this goal alone is insufficient, as the emphasis on content acquisition often comes at the expense of developing higher-order skills such as argumentation, reasoning, and flexibility in mathematics problem solving (Chiaravalloti, 2009; Partanen & Kaasila, 2014; Star et al., 2015). Redesigning curricula with these additional objectives in mind requires providing students with opportunities to engage with mathematics in ways that may contrast with their past experiences or expectations. It requires changing patterns of classroom engagement and development of different classroom norms. This mixed methods research study incorporated a semester-long teaching experiment that aimed to support students\u27 development of higher-order skills by negotiating productive classroom norms. One of the primary interventions was a sequence of Multiple Solutions Activities that required groups of students to analyze and critique unfamiliar or erroneous mathematical solutions. The overarching goal of the research was to study students\u27 engagement during these activities across the semester by characterizing the nature of specific types of classroom norms. Social norms describe the classroom participation structure, while sociomathematical norms focus on aspects of student activity that are inherently mathematical, such as what constitutes an acceptable mathematical solution (Yackel & Cobb, 1996). Because of a reflexive relationship between norms and beliefs, students\u27 social and mathematical beliefs were also of interest to characterize the influence of the teaching experiment; these beliefs were assessed by a pre- and post-course questionnaire. The results paint a complex picture of student engagement and values. Despite quantitative analysis suggesting encouraging improvements in students’ mathematical engagement, qualitative analysis highlighted that this change was not homogenous. In particular, the analysis revealed variations in students’ perceptions of the value of multiple solutions and in the nature of the norms developed in student groups. Consequently, the study highlights the lasting impact of classroom norms on students\u27 beliefs, and vice versa, which may hinder the development of alternative norms in subsequent classes. The results of the project also expand upon Yackel and Cobb\u27s (1996) Interpretive Framework for characterizing classroom engagement by suggesting a reflexive relationship exists between social and sociomathematical norms. The data analysis describes concurrent development and mutual influence between the participation structure of a group and their taken-as-shared mathematical beliefs. In all, the project shows that deliberate attention towards negotiating productive classroom norms and students’ in-class engagement can positively affect students’ attitudes towards multiple solutions

The Sociocultural Mediation of Metacognition During Problem Solving in Undergraduate Mathematics Classroom Communities of Practice

Metacognition has long been identified as an essential component of the problem-solving process. While the language of metacognition has been conveyed in teaching, research, and policy, much of the research on metacognition does not describe the explicit role metacognition plays during students’ real-time problem-solving process. Moreover, metacognitive interventions are typically disconnected from the natural mathematical activity and discourse within a classroom community. Research concerning metacognition and metacognitive interventions has historically adopted an acquisition metaphor for learning. This qualitative study takes a participationist lens to consider metacognition as a problem-solving habit of mind, a normative way of thinking to which students become attuned by participating in authentic problem-solving situations. This study explored one such situation, in which “portfolio” problem-solving sessions and write-ups were used to mediate metacognitive thinking in a first-year mathematics content course for pre-service elementary teachers. Six qualitative data sources were collected and analyzed: (1) recorded classroom sessions, (2) three individual interviews with 15 of the 24 students, (3) two interviews with the instructor of record, (4) students’ written artifacts, (5) recorded planning sessions with the instructor, and (6) journal reflections written by the instructor and myself, the researcher, after each class session. Two levels of analysis were employed to characterize sociocultural complexity surrounding students’ problem-solving activity. Results of micro-level analysis revealed a shift from product- to process-focused metacognitive norms. Through participation in authentic problem-solving situations, namely the portfolio problems, students problem-solving activity transformed in a way that afforded them opportunities to readily engage in process-focused metacognitive actions. Macro-level analysis utilized activity theory to operationalize the participation structure of the classroom and document the development of metacognitive norms, highlighting social mediators of activity and contradictions as catalysts for change. Results of macro-level analysis illustrated a correspondence between the shift in normative metacognitive actions identified in micro-level analysis, broader transformations of students’ problem-solving activity, and the teacher’s shifting goals and actions in response to students’ problem solving. This work extends previous research on metacognitive interventions, demonstrating that “embeddedness” of metacognitive activity during problem solving is beyond just the content, but also embedded in the collective classroom culture. Moreover, activity theory captured students’ agency in negotiating their problem-solving activity, suggesting its continued use by researchers wishing to adopt an anti-deficit framing. This research has additional implications for teaching content courses for pre-service teachers. Students’ metacognitive activity was very much situated in the sociocultural context of the classroom, especially their dual identities as current mathematics students and future teachers. For the pre-service teachers to value mathematical problem-solving habits of mind, legitimate participation meant as students, not just as future teachers, of mathematics. Finally, this study provides broader insight into how instructors can support undergraduate students’ process-focused metacognitive activity during problem solving through a combination of Inquiry-Based Learning (IBL) techniques and explicit reflection on real-time problem-solving processes

Exponential Growth and Online Learning Environments: Designing for and Studying the Development of Student Meanings in Online Courses

abstract: This dissertation report follows a three-paper format, with each paper having a different but related focus. In Paper 1 I discuss conceptual analysis of mathematical ideas relative to its place within cognitive learning theories and research studies. In particular, I highlight specific ways mathematics education research uses conceptual analysis and discuss the implications of these uses for interpreting and leveraging results to produce empirically tested learning trajectories. From my summary and analysis I develop two recommendations for the cognitive researchers developing empirically supported learning trajectories. (1) A researcher should frame his/her work, and analyze others’ work, within the researcher’s image of a broadly coherent trajectory for student learning and (2) that the field should work towards a common understanding for the meaning of a hypothetical learning trajectory. In Paper 2 I argue that prior research in online learning has tested the impact of online courses on measures such as student retention rates, satisfaction scores, and GPA but that research is needed to describe the meanings students construct for mathematical ideas researchers have identified as critical to their success in future math courses and other STEM fields. This paper discusses the need for a new focus in studying online mathematics learning and calls for cognitive researchers to begin developing a productive methodology for examining the meanings students construct while engaged in online lessons. Paper 3 describes the online Precalculus course intervention we designed around measurement imagery and quantitative reasoning as themes that unite topics across units. I report results relative to the meanings students developed for exponential functions and related ideas (such as percent change and growth factors) while working through lessons in the intervention. I provide a conceptual analysis guiding its design and discuss pre-test and pre-interview results, post-test and post-interview results, and observations from student behaviors while interacting with lessons. I demonstrate that the targeted meanings can be productive for students, show common unproductive meanings students possess as they enter Precalculus, highlight challenges and opportunities in teaching and learning in the online environment, and discuss needed adaptations to the intervention and future research opportunities informed by my results.Dissertation/ThesisDoctoral Dissertation Mathematics Education 201

Biology Faculty at Large Research Institutions: The Nature of their Pedagogical Content Knowledge

abstract: To address the need of scientists and engineers in the United States workforce and ensure that students in higher education become scientifically literate, research and policy has called for improvements in undergraduate education in the sciences. One particular pathway for improving undergraduate education in the science fields is to reform undergraduate teaching. Only a limited number of studies have explored the pedagogical content knowledge of postsecondary level teachers. This study was conducted to characterize the PCK of biology faculty and explore the factors influencing their PCK. Data included semi-structured interviews, classroom observations, documents, and instructional artifacts. A qualitative inquiry was designed to conduct an in-depth investigation focusing on the PCK of six biology instructors, particularly the types of knowledge they used for teaching biology, their perceptions of teaching, and the social interactions and experiences that influenced their PCK. The findings of this study reveal that the PCK of the biology faculty included eight domains of knowledge: (1) content, (2) context, (3) learners and learning, (4) curriculum, (5) instructional strategies, (6) representations of biology, (7) assessment, and (8) building rapport with students. Three categories of faculty PCK emerged: (1) PCK as an expert explainer, (2) PCK as an instructional architect, and (3) a transitional PCK, which fell between the two prior categories. Based on the interpretations of the data, four social interactions and experiences were found to influence biology faculty PCK: (1) teaching experience, (2) models and mentors, (3) collaborations about teaching, and (4) science education research. The varying teaching perspectives of the faculty also influenced their PCK. This study shows that the PCK of biology faculty for teaching large introductory courses at large research institutions is heavily influenced by factors beyond simply years of teaching experience and expert content knowledge. Social interactions and experiences created by the institution play a significant role in developing the PCK of biology faculty.Dissertation/ThesisPh.D. Curriculum and Instruction 201

Interaction and Mechanics: Understanding Course-work Engagement in Large Science Lectures

Post-secondary institutions have developed several interventions to address what Chamblis’ (2014) calls the arithmetic of classroom engagement. Large lecture courses limit the potential for student/instructor interaction. Instead, large lecture courses have historically relied on an industrialized model of information delivery. Very little is known about how students develop their strategies for completing their course-work in this context. The aim of this study was to outline a conceptual framework describing how undergraduates become engaged in their course-work in large science lecture courses. Course-work engagement refers to the set of practices that are part of students’ efforts to successfully complete a course. Course-work engagement is goal oriented behavior, shaped by the beliefs that individual holds about their self and the course. In the framework, I propose that students’ initial beliefs states catalyze their behavioral engagement in the course which is conditioned through feedback from working with peers, from performance assessments, and through interactions with the instructor. This study was conducted in a large (n=551) undergraduate introductory physics course. The course was composed of three lecture sections, each taught by a different instructor. Based on a review of the literature, I posed the following research questions: 1. What are the relationships among students’ peer interactions, their digital instructional technology use, and their performance on assessments in a physics lecture course? 2. How does the instructional system shape students’ engagement in peer interactions and their use of digital instructional technologies in a course? In this study, I employed three methods of data collection. First, I observed instruction in all three sections throughout the semester to characterize similarities and differences among the three lecture sections. Second, I administered two surveys to collect information about students’ goals for the course, their expectations for success, their beliefs about the social and academic community in the course, and the names of peers in the course who the student collaborated with in out-of-class study groups. Surveys were administered before the first and final exam in the course. Third, I used learning analytics data from a practice problem website to characterize students’ usage of the tool for study preparation before and after the first exam. Through the stochastic actor based modeling, I identified three salient factors on students’ likelihood of participating in out-of-class study groups. First, being underrepresented in the course may have shaped students’ opportunities to participate in out-of-class study groups. Women and international students both attempted to participate at higher rates than men and domestic students, respectively. However, women and international students were unlikely to have their relationships reciprocated over the semester. Second, when study tools are incorporated into out-of-class study groups, social influence appears to play a significant role in the formation of course-work engagement. For example, students who were non-users of the practice problem website tended to adopt the use behavior of their higher intensity peers. Third, changes in students’ beliefs about the course were significantly related to changes in their course grade. In terms of performance, students who experienced changes to their course beliefs, or what attempted to form new out of class study groups in the lead up to the third exam, were likely to experience academic difficulty. This study highlights the important role of time and the dynamic role of social interaction on the development of course-work engagement in large science lecture courses.PHDHigher EducationUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/138776/1/mbrowng_1.pd

Open, Online, Calculus Help Forums: Learning About and From a Public Conversation

This study is an exploration of participation, community, and mathematical understanding in an open, online, calculus help forum. These forums, populated by members from around the world, are locations where students post queries from their coursework and receive assistance from volunteer tutors. The site under investigation has a spontaneous participation structure, meaning that any forum member can respond to a query and contribute to an ongoing discussion. From earlier work, we know that such forums foster mathematical dialogue, contain exchanges with sophisticated pedagogical moves, and exhibit a strong sense of community. In this study, we delve deeper into the functional aspects of activity (such as student positioning and pedagogical moves), the benefits that accrue from participation in tutoring as a communal activity, and the mathematical understanding that is evident in the way problems on limit and related rates are framed and solutions constructed. Based on an observational methodology, we find that the forum provides tutoring for students and support for tutors that is unique from our expectations of other learning environments, such as one-on-one tutoring and computer-based tutoring systems. Students position themselves with authority in the exchanges by making assertions and proposals of action, questioning or challenging others' proposals, and indicating when resolution has been achieved. Tutors, who generally have more experience and expertise than students, provide mathematical guidance, and, in exemplary exchanges, draw the student into making a mathematical discovery. The dedication of tutors to the forum community was evident in the presence of authentic, honest mathematical practices, in the generous provision of alternative perspectives on problems, and in the sincere correction of errors. Some student participants picked up on these aspects of community and expressed excitement and appreciation for this taste of mathematical discourse. The primary contribution of the tutors was their assistance in supporting students as they constructed productive framings for the exercises, and this was the help that students were most in need of. As a result of eavesdropping on this public conversation, we conclude that the forums are a public conversation that should be listened to by educational researchers, teachers, and designers of tutoring systems

STUDYING EPISTEMIC COGNITION IN THE HISTORY CLASSROOM: CASES OF TEACHING AND LEARNING TO THINK HISTORICALLY

Building on the literature on epistemic cognition, epistemic beliefs, and historical thinking, three class-level case studies were conducted to investigate features of historical thinking and history-specific epistemic beliefs of high-school students and their teachers. These cases also considered teachers' pedagogical practices and the potential effects of those practices on students' historical thinking and epistemic beliefs. Two junior honors and one freshman US History classes were selected from a school system that fostered the preparation of students for AP History courses by encouraging the use of a variety of primary sources and analysis of documents in teaching history. Preliminary visits indicated that these classes' teachers used different pedagogical practices. Class observations spanned one semester of instruction. History-specific epistemic beliefs were explored using interviews structured around the items of the Beliefs about History Questionnaire (BHQ) and historical thinking was assessed through analysis of think-alouds collected while student informants (4 from each class) and their teachers read a set of 6 documents and responded to a constructed response task (CRT). Specifically, student data were collected at the middle and end of the semester, while teachers were interviewed only once, at the end of the semester. In one of the junior classes, 27 additional juniors responded in writing to the BHQ and to the CRTs. Additional questionnaires and interviews explored teachers' goals, rationales for their practice, and interest in history. In regard to history-specific epistemic beliefs, results indicated that students and teachers manifested ideas indicative of different developmental levels, suggesting that their epistemic beliefs are a complex system, not necessarily characterized by a high level of integration. Differences across students tended to be greater in regard to epistemic beliefs than to historical thinking. In addition, comparison of initial and follow-up data suggested different trajectories of change in regard to students' epistemic beliefs while changes in historical thinking were modest and not consistently suggesting progression in competence. These trends were confirmed by the analysis of students' written responses to the BHQ and the CRTs. The study identified a set of ideas and behaviors that tended to produce cognitive impasse and hindered the development of historical thinking and a series of pedagogical practices, mostly aligned with teachers' goals and beliefs, which might have fostered such outcomes

Adaptation of tertiary mathematics instruction to the virtual medium: approaches to assessment practice

Mathematics has been singled out as a challenging discipline to teach fully online (FO). Yet both the demand for and development of FO mathematics courses is increasing with little known about the quality of these courses and many calling for research. Whereas most research has investigated the nature of these courses by examining instructional outputs such as student grades this research seeks the same insight but by examining instructional inputs. Specifically, it seeks to investigate the nature of current assessment practice in FO mathematics courses. To conduct this investigation, deep learning (Marton & Säljö, 1976a, 1976b) is used as the principle theoretical framework. From the growing body of literature associated with deep learning, two studies are selected to investigate current FO mathematics instructors assessment practices. An additional framework based on empirical findings related to the use of different kinds of feedback is also used. In total, six study measures are used to conduct a mixed methods study in two parts. The target demographic and course context are tertiary instructors from Western nations that teach introductory level mathematics (particularly statistics and calculus). The first study explores current FO mathematics assessment practices using an online survey (n=70) where the majority of participants originate from US higher education institutions. In the second study six of the US survey participants are interviewed about how their assessment practices and approaches used in their FO mathematics courses differ from those used in their face-to-face (F2F) mathematics courses. This study represents the first known attempt to investigate the nature of tertiary FO mathematics instructors assessment practices using appropriate theoretical frameworks. In particular, it investigates mathematics instructors experiences of the affordances and constraints of the FO course context when adapting their F2F practice to this new environment. Findings suggest the FO course context is a challenging environment for instructors to orient their teaching and assessment practice in a way that helps develop students understanding of mathematics. Analysis of interview responses suggests the problem lies with the nature of interactivity provided in the FO course context