The study in this paper examined how teachers engaged in their structured Professional Learning Community (PLC) time in a manner that focused their work on student mathematical thinking. The participants included two third-grade teachers and two fourth-grade teachers engaged in an alternative PLC process during their collaboration time. Interviews were conducted and focused on how the teachers thought about student thinking as the driving force of their collaboration. The teachers found their PLC time was more focused, student thinking led the discussions and lessons, they were more responsive to student needs, and their own mathematical understanding increased. The teachers also found they became facilitators of the mathematical discussions occurring in their classrooms and there was a stronger community present among the grade level team. This structure of PLC created an accelerated learning process for novice teachers, increased focus during PLC time, and supported teachers to feel valued in their meetings. Teachers also felt more excitement around student thinking and became more responsive to student needs, both in planning lessons and in assessing their students. Teachers also expressed how their PLC meetings supported accelerated learning of student ideas for novice teachers, while providing professional learning for all teachers that contributed to their generative growth. Further research could examine the alternative PLC process with a whole school and what this may look like with other content areas.

Practical concepts, such as area and perimeter, have an important part in today's school mathematics curricula. Research indicates that students and preservice teachers (PSTs) struggle with and harbor misconceptions regarding these topics. Researchers suggest that alternative instructional methods be investigated that enhance PSTs' conceptual understanding and encourage deeper student thinking. To address this need, this study examined and described what and how PSTs learn as they engage in anchored instruction involving web-based microworlds designed for exploring area and perimeter. Its focus was to examine the influences of a modified teacher development experiment (TDE) upon 12 elementary PSTs' content knowledge (CK) and knowledge of student thinking (KoST) regarding principles, relationships, and misconceptions involving area and perimeter as they develop simultaneously in a problem-solving environment. The learning of meaningful mathematics is a personal and independent activity, as one struggles to create and reason through their own mathematical realities and misconceptions. This study describes PSTs' reasonings, misconceptions, and difficulties as they grappled with new knowledge or reconciled new knowledge with prior understandings. Quantitative and qualitative research methods, including case-subject analysis, were used. Instructional sessions similar to Steffe's (1983) teaching episodes comprised this study's intervention. Results indicate that prior to intervention most of the PSTs possessed a procedural knowledge of area and perimeter and were bound by a dependency on formulas; their KoST pertaining area and perimeter was relatively underdeveloped. They seemed unaware of prevalent misconceptions students acquire while working with these concepts (specifically, units of measure and perceived relationships). The PSTs displayed an ineffective use of drawings to support their responses. Their preoccupation with finding what they judged as "the answer" to various problem-solving situations hindered their ability to properly diagnose and address student thinking and limited their meaningful interaction with the microworlds (MWs). A majority of PSTs felt the MWs were a valuable learning tool for themselves but not for their future students. The planned intervention played a role in the PSTs becoming more perceptive of the difficult mathematics involved with area and perimeter and better equipped to anticipate and address those difficulties with future students.

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The aim of this study is to investigate teachers&rsquo / perceptions of constraints on improving student thinking skills in schools, and to find out whether there are differences in teachers&rsquo / perceptions of constraints in terms of subject area, educational background, teaching experience, gender, geographical area, and school location. A survey design was used in this study. The questionnaire used in the study was developed by making use of the related literature, and it was administered to 522 teachers working in the public high schools in four different regions of Turkey during the fall semester of 2002-2003 academic year. The data gathered are analysed through descriptive and inferential statistics (one-way ANOVA and t-test). There were four major constraints on improving student thinking, namely, teacher-related, student-related, curriculum-related, and external factors to classroom. The results indicated that the most agreed constraints were the student-related ones. The results also showed that there were no statistically significant differences in teachers&rsquo / perceptions of the constraints on improving student thinking based on the background variables, i.e., subject area, educational background, teaching experience, gender, geographical region, and school location.

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/Thesis / Doctoral Dissertation Mathematics Education 2018

This study explored how a video club could be used to help develop teacher’s professional vision by investigating how teachers’ professional vision changed over time. In addition, the role of the facilitator was studied to determine how it contributed to the development of professional vision. The facilitation techniques appear to be the reason why the expected growth in professional vision did not occur. While video clubs are a valuable way of embedding professional development with artifacts from the classroom, care must be taken with the facilitation techniques employed.

This study explored how a video club could be used to help develop teacher’s professional vision by investigating how teachers’ professional vision changed over time. In addition, the role of the facilitator was studied to determine how it contributed to the development of professional vision. The facilitation techniques appear to be the reason why the expected growth in professional vision did not occur. While video clubs are a valuable way of embedding professional development with artifacts from the classroom, care must be taken with the facilitation techniques employed.

abstract: This dissertation describes an investigation of four students' ways of thinking about functions of two variables and rate of change of those two-variable functions. Most secondary, introductory algebra, pre-calculus, and first and second semester calculus courses do not require students to think about functions of more than one variable. Yet vector calculus, calculus on manifolds, linear algebra, and differential equations all rest upon the idea of functions of two (or more) variables. This dissertation contributes to understanding productive ways of thinking that can support students in thinking about functions of two or more variables as they describe complex systems with multiple variables interacting. This dissertation focuses on modeling the way of thinking of four students who participated in a specific instructional sequence designed to explore the limits of their ways of thinking and in turn, develop a robust model that could explain, describe, and predict students' actions relative to specific tasks. The data was collected using a teaching experiment methodology, and the tasks within the teaching experiment leveraged quantitative reasoning and covariation as foundations of students developing a coherent understanding of two-variable functions and their rates of change. The findings of this study indicated that I could characterize students' ways of thinking about two-variable functions by focusing on their use of novice and/or expert shape thinking, and the students' ways of thinking about rate of change by focusing on their quantitative reasoning. The findings suggested that quantitative and covariational reasoning were foundational to a student's ability to generalize their understanding of a single-variable function to two or more variables, and their conception of rate of change to rate of change at a point in space. These results created a need to better understand how experts in the field, such as mathematicians and mathematics educators, thinking about multivariable functions and their rates of change. / Dissertation/Thesis / Ph.D. Mathematics 2012

abstract: This action research addressed teacher effectiveness in supporting students’ critical thinking skills by implementing differentiated instructional strategies in eight 3rd- and 4th-grade, self-contained, inclusive classrooms. This study addressed how third- and fourth-grade teachers perceived their instructional effectiveness, how differentiated instructional strategies influence third- and fourth-grade teachers, and how third- and fourth-grade teachers make further use of differentiated instruction to support students’ critical thinking skills across cultures, linguistics, and achievement levels to increase student achievement. Out of the enrollment in a southwest Phoenix elementary school, there was a 35% mobility rate; 76%, free and reduced lunches; 35%, Spanish-speaking homes; 10%, ELL services; and 10%, special education. The school was comprised of 52 certified teachers, out of which there were five related arts teachers, and four teachers who served gifted and special education students. Participants included all eight third- and fourth-grade teachers, 75% female and 25% males; 75% identified as Caucasian and 25% Hispanic/Latina, middle-class citizens. Professional development training was provided to these eight individual teachers during four months on differentiated instructional strategies to support students’ critical thinking. At this study’s beginning, these teachers perceived an obstacle to supporting students’ critical thinking as they struggled to learn new curriculums. Persevering through this challenge, teachers discovered success by implementing design-thinking, developing students’ growth mindsets, and utilizing cultural responsive teaching. These teachers identified three differentiated instructional strategies which impacted students’ academic progress: instructional scaffolds, collaborative group work, and project-based learning. Building upon linguistic responsive teaching, cultural responsive teaching, and Vygotsky’s socio-cultural theory, teachers revealed how to support students’ critical thinking through the use of graphic organizers, sentence frames, explicit instructions, growth mindsets, cultural references, and grouping structures. In addition, the outcomes demonstrated teachers can make further use of differentiated instruction by focusing on instructional groups, teachers’ mindsets, and methods for teaching accelerated learners. This study’s results have implications on teachers’ perception toward using differentiated instructional strategies as a viable method to support the multiple ways all students learn. / Dissertation/Thesis / Doctoral Dissertation Leadership and Innovation 2018

abstract: This dissertation reports three studies about what it means for teachers and students to reason with frames of reference: to conceptualize a reference frame, to coordinate multiple frames of reference, and to combine multiple frames of reference. Each paper expands on the previous one to illustrate and utilize the construct of frame of reference. The first paper is a theory paper that introduces the mental actions involved in reasoning with frames of reference. The concept of frames of reference, though commonly used in mathematics and physics, is not described cognitively in any literature. The paper offers a theoretical model of mental actions involved in conceptualizing a frame of reference. Additionally, it posits mental actions that are necessary for a student to reason with multiple frames of reference. It also extends the theory of quantitative reasoning with the construct of a ‘framed quantity’. The second paper investigates how two introductory calculus students who participated in teaching experiments reasoned about changes (variations). The data was analyzed to see to what extent each student conceptualized the variations within a conceptualized frame of reference as described in the first paper. The study found that the extent to which each student conceptualized, coordinated, and combined reference frames significantly affected his ability to reason productively about variations and to make sense of his own answers. The paper ends by analyzing 123 calculus students’ written responses to one of the tasks to build hypotheses about how calculus students reason about variations within frames of reference. The third paper reports how U.S. and Korean secondary mathematics teachers reason with frame of reference on open-response items. An assessment with five frame of reference tasks was given to 539 teachers in the US and Korea, and the responses were coded with rubrics intended to categorize responses by the extent to which they demonstrated conceptualized and coordinated frames of reference. The results show that the theory in the first study is useful in analyzing teachers’ reasoning with frames of reference, and that the items and rubrics function as useful tools in investigating teachers’ meanings for quantities within a frame of reference. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2019