# Browsing by Subject "Mathematics--Study and teaching"

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Item Complex systems as lenses on learning and teaching(2007) Hurford, Andrew Charles; Stroup, Walter M.Show more From metaphors to mathematized models, the complexity sciences are changing the ways disciplines view their worlds, and ideas borrowed from complexity are increasingly being used to structure conversations and guide research on teaching and learning. The purpose of this corpus of research is to further those conversations and to extend complex systems ideas, theories, and modeling to curricula and to research on learning and teaching. A review of the literatures of learning and of complexity science and a discussion of the intersections between those disciplines are provided. The work reported represents an evolving model of learning qua complex system and that evolution is the result of iterative cycles of design research. One of the signatures of complex systems is the presence of scale invariance and this line of research furnishes empirical evidence of scale invariant behaviors in the activity of learners engaged in participatory simulations. The offered discussion of possible causes for these behaviors and chaotic phase transitions in human learning favors real-time optimization of decision-making as the means for producing such behaviors. Beyond theoretical development and modeling, this work includes the development of teaching activities intended to introduce pre-service mathematics and science teachers to complex systems. While some of the learning goals for this activity focused on the introduction of complex systems as a content area, we also used complex systems to frame perspectives on learning. Results of scoring rubrics and interview responses from students illustrate attributes of the proposed model of complex systems learning and also how these preservice teachers made sense of the ideas. Correlations between established theories of learning and a complex adaptive systems model of learning are established and made explicit, and a means for using complex systems ideas for designing instruction is offered. It is a fundamental assumption of this research and researcher that complex systems ideas and understandings can be appropriated from more complexity-developed disciplines and put to use modeling and building increasingly productive understandings of learning and teaching.Show more Item The discourse of mathematization: bilingual students reinventing mathematics and themselves as mathematical thinkers(2008-05) Dominguez, Higinio; Empson, Susan B.Show more In this paper, students' bilingualism and multicultural experiences are examined as cognitive resources for mathematization. Capitalizing on the view of language as action, and on students' familiarity with certain experiences through direct participation, the study includes a conceptual framework, never used with bilingual mathematics learners, to investigate how bilingual students organize and coordinate actions to solve mathematical problems about familiar and unfamiliar experiences in English and Spanish. The study used a research methodology to investigate two questions: (a) How do bilingual students' mathematize familiar experience problems and unfamiliar experience problems in Spanish and English? (b) What do differences and similarities in bilingual students' mathematization across problems and languages reveal about experience and bilingualism as cognitive resources? Findings show important differences. In problems about familiar experiences, students generated more productive actions, more reflective actions, and less unproductive actions than in problems about unfamiliar experience. As for the bilingualism, students used Spanish and English differently. When solving problems in Spanish, they framed actions more socially by including partners or sharing the action with partners, whereas in English they framed actions more individually, more depersonalized, excluding partners and instead relying on words in problems to justify their individual actions. This suggests that reinventing mathematics and themselves as mathematical thinkers is part of using their bilingualism and experiences as cognitive tools, and attention to how they use each language for each type of problem can reveal substantial knowledge about how bilinguals learn mathematics.Show more Item The effects of cognitive teaching techniques on ninth grade mathematics achievement : shifting the balance for special populations(2002-05) Breeding, Cynthia Ann; Wagstaff, Lonnie H.Show more The purpose of this study was to investigate the relationship between the within school and between-class effects of cognitive teaching techniques on the mathematics learning of special populations of urban high school students. Student mathematics learning was the dependent variable. Application of a teaching technique based on Reuven Feuerstein’s Instrumental Enrichment, a clinical questioning method used to increase cognitive functioning of adolescents with mental disabilities was the experimental intervention. In classrooms the technique was characterized by task analysis, definition of terms, and discourse designed to access students’ experiences and funds of knowledge. Questioning was designed to generate critical thinking, elicit interest, and assess student learning during classroom interaction between students and with the teacher. vi The independent variables were gender, ethnicity, income, ability, risk factors, special services, and teacher. A theoretical model of equity (Secada, 1992), equal opportunity (Coleman, 1967), and cognitive psychology (Feuerstein, 1983), provided the conceptual framework. Fifty-six ninth grade students formed the purposive sample and comparison group design for this quasi-experimental study. In addition, action research provided a model for qualitative elements of the study, including focus groups, interviews, journal-writing, and observations. Teachers who were in the second year of implementing the teaching technique provided the experimental treatment. Scores on the Texas Assessment of Academic Skills provided the measurement for mathematics achievement. Eighth grade TAAS scores provided the initial score for the participating students and the Texas Learning Index (TLI) was assigned to each of their scores. The TLI was a percentage probability that the student would pass the Exit Level TAAS in tenth grade with a score of at least 70. Subsequent interim scores were determined at five and ten months through test simulations, and the final score was recorded from each student’s Exit Level TAAS math score. Results showed significant relationships between student performance gains and the implementation of the teaching technique. Results also revealed a significant interaction between math ability and classroom behaviors associated with the teaching technique. Results of the study showed a significant narrowing of the historical gaps in performance of students grouped by background variables. The study also revealed that the in-class interactions between the teacher and students, and teachers’ attitudes toward the students had a strong effect on student performance. Finally, the results of the study showed that the effects of the teaching technique were not highly dependent on the level of implementation, but that the level of implementation and commitment of teachers to using the technique were dependent upon the support and facilitation skills of administrators.Show more Item Exploring an alignment focused coaching model of mathematics professional development: content of coach/teacher talk during planning and analyzing lessons(2007) Bradley, Janice; Empson, Susan B.Show more This exploratory case study examines an alignment-focused coaching model of mathematics professional development during a school district's second-year implementation of the coaching model. Specifically, the study describes the content of coach-teacher talk as five coach-teacher pairs, grades K-8, engage in planning and analyzing mathematics lessons. Using an alignment framework designed around the components of curriculum, instruction, and assessment to analyze talk, four patterns unfold. Issues of curriculum, instruction, and assessment were more often discussed in isolation than interconnected, mathematics was most often the content focus when teacher and/or coach were using the state standards document to plan, student thinking and learning were most often a focus when students were struggling, and teachers often talked about instruction as actions isolated from student thinking and learning. In addition, teachers reported changes to instruction as an outcome of participating in coaching. Self-reported benefits to teachers' practice included planning lessons that focused on student learning, that is, considering the mathematics in the standards and ways students would learn the content. Teachers also reported asking "better questions" more often and in different ways, using models such as manipulatives and representations for connecting mathematics ideas, thinking more about student learning, and analyzing and scrutinizing textbooks to align with the state standards.Show more Item The impact of Japanese Lesson Study on preservice teacher belief structures about teaching and learning science(2009-05) Fortney, Brian Scott, 1968-; Barufaldi, James P.Show more This study investigates how preservice teachers make sense of student-centered instruction with existing traditional beliefs about teaching. Teacher educators assume that university instruction translates directly into practice, yet, research is clear that beginning teachers revert to traditional teaching practice. For elementary teachers, one science methods course is assumed to be sufficient instruction in contemporary methods to successfully guide practice in their beginning years. Two main research questions are addressed: 1) Do preservice teacher belief structures change during the implementation of a Japanese Lesson Study cycle? 2) To what extent are preservice teachers teaching behaviors consistent with their belief structures? [...] To answer these questions, a case study methodology consisting of three preservice teachers, selected from a collective case study of 25 preservice teachers, was performed. The time periods of data collection were set with Lesson Study episodes. The time periods included pre-lesson study, during lesson study episodes, and post lesson study, with a conceptual framework synthesized from beliefs literature, Rokeach (1968), Fishbein and Ajzen (1975), and operationalized within the context of a Science Methods course using Richardson et al (1991) and Pajares (1992) as a guide. Findings indicate that even if preservice teachers have similar experiences with elementary science instruction, and have developed a traditional frame of reference (Kennedy, 1999) that guides their learning about teaching, each understands information idiosyncratically. When viewed in terms of Green's (1971) metaphor of belief structures, preservice teachers have widely differing frames of reference; thus, an individual's sensemaking about inquiry lessons within lesson study groups and the meaning conveyed within conversations are completely different. Ultimately, the participants in this study can be described, metaphorically, as having a Crisis of Belief (Green, 1971), an approach of Quiet Introspection, and a Crisis of Practice. For teacher educators, understanding preservice teacher understanding, and using that understanding in constructing lessons that facilitate evaluation of existing beliefs requires different lenses. The three lenses used are, Epistemological (Hewson [and] Hewson, 1984; Posner, Strike, Hewson, [and] Gertzog, 1982), Social/Affect (Pintrich, Marx, [and] Boyle, 1993; Tyson, Venville, Harrison, [and] Treagust, 1997), and an Expectational lens (Chi, Slotta, [and] de Leeuw, 1994). The selection of lenses is dependent upon the idiosyncratic nature of each preservice teacher's belief structure.Show more Item The lesson study professional development process : exploring the learning experiences of elementary and middle school teachers(2008-12) Harle, Carol Berg; Salinas, CinthiaShow more Lesson study has been introduced in the United States as a collaborative professional development process focusing on improving teachers’ content knowledge and instructional skills as teachers plan a research lesson, teach and observe students’ thinking and learning behaviors and then revise and re-teach the lesson. The origin of lesson study in the United States began shortly after, The Teaching Gap by James Stigler and James Hiebert was published in 1999. The researchers attributed Japanese students’ high achievement scores in mathematics and science to their teachers’ participation in the lessons study process. Lesson study professional development formally began with a grant from the U.S. Department of Education in 1999; however, research on effective professional development programs and practices have been studied and reported for decades. Results from these studies and findings on teacher and adult learning were examined and reported in this study. This case study examined the learning experiences of six elementary and middle school teachers as they participated in the lesson study process. The teachers first began with sessions on discussing lesson study research, the lesson study process and viewing other teachers participating in lesson study. The teachers later collaborated to create, teach, observe, revise and re-teach a 5th grade mathematics research lesson on elapsed time during summer school. The qualitative research incorporated teacher interviews, observations, written reflections and artifacts such as agendas, planning documents and lesson plans. The data collected helped provide insight in examining the three research questions: a) How do teachers understand lesson study as a professional development process?; b) How will engaging in the lesson study affect the teachers’ planning and practices?; and c) What are some challenges in learning and applying key lesson study concepts? The case study data affirmed existing research on quality professional development, effective teacher/adult learning and accurate lesson study practices and supported the emerging themes of the teachers’ valuing collaboration, teachers’ valuing the deepening of their mathematics understanding and enhancing of their instructional practices, and finally, the teachers’ valuing the incorporation of student thinking in their planning, teaching and re-teaching experiences.Show more Item Mathematical modeling and kinematics: a study of emerging themes and their implications for learning mathematics through an inquiry-based approach(2004) Carrejo, David John; Marshall, Jill Ann; Petrosino, Anthony J. (Anthony Joseph), 1961-Show more In recent years, emphasis on student learning of mathematics through “real world” problems has intensified. With both national and state standards calling for more conceptual learning and understanding of mathematics, teachers must be prepared to learn and implement more innovative approaches to teaching mathematical content. Mathematical modeling of physical phenomena is presented as a subject for new and developing research areas in both teacher and student learning. Using a grounded theory approach to qualitative research, this dissertation presents two related studies whose purpose was to examine the process by which in-service teachers and students enrolled in an undergraduate physics course constructed mathematical models to describe and predict the motion of an object in both uniform and non-uniform (constant acceleration) contexts. This process provided the framework for the learners’ study of kinematics. Study One involved twenty-three in-service physics and math teachers who participated in an intensive six-hour-a-day, five-day unit on kinematics as part of a professional development institute. Study Two involved fifteen students participating in the same unit while enrolled in a physics course designed for pre-service teachers and required in their undergraduate or graduate degree programs in math and science education. Qualitative data, including videotapes of classroom sessions, field notes, researcher reflections, and interviews are the focus of analysis. The dissertation presents and analyzes tensions between learner experience, learning standard concepts in mathematics and learning standard concepts in physics within a framework that outlines critical aspects of mathematical modeling (Pollak, 2003): 1) understanding a physical situation, 2) deciding what to keep and what not to keep when constructing a model related to the situation, and 3) determining whether or not the model is sufficient for acceptance and use. Emergent themes related to the construction of the learners’ models included several robust conceptions of average velocity and considerations of what constitutes a “good enough” model to use when describing and predicting motion. The emergence of these themes has implications for teaching and learning mathematics through an inquiry-based approach to kinematics.Show more Item The relationship between patterns of classroom discourse and mathematics learning(2008-08) Pierson, Jessica Lynn, 1976-; Martin, Taylor, 1970-Show more By creating opportunities for participation and intellectual engagement, standardized classroom routines are large determinants of the conceptual meaning students make. It is through repeated engagement in patterns of talk and intellectual practices that students are socialized into ways of thinking and habits of mind. The focus of this study is on moment-to-moment interactions between teachers and students in order to describe, identify and operationalize meaningful regularities in their discourse. Using classroom-level measures, I investigate the robustness of relationships between students’ mathematics achievement and discursive patterns across multiple classrooms with the statistical methods of Hierarchical Linear Modeling. Specifically, I investigated two theoretically significant constructs reflected in teacher’s follow-up moves -- responsiveness and intellectual work. Responsiveness is an attempt to understand what another is thinking displayed in how she builds, questions, clarifies, takes up or probes that which another says. Intellectual work reflects the cognitive work requested from students with a given turn of talk. After developing coding schemes to measure and quantify these discursive constructs, statistical analyses revealed positive relationships between the responsiveness and intellectual work of teachers’ follow-up and student learning of rate and proportionality (p=.01 and .08, respectively). Additionally, classroom communities with higher levels of responsiveness and intellectual work moderate the effect of prior knowledge on student learning by decreasing the degree to which pretest scores predict students’ post-test achievement (though neither are statistically significant). Based on these results, I conclude that classroom discourse and normative interaction patterns guide and influence student learning in ways that improve achievement. Recommendations are primarily concerned with ways the educational community can support and encourage teachers to develop responsive, intellectually demanding discursive patterns in their classrooms. In particular, we need to increase the awareness of the power of discourse, provide appropriate and sustained support for teachers to change current patterns, re-examine the design of teacher preparation programs, and develop ways to thoughtfully integrate responsiveness and intellectual work with core mathematics content. There is tremendous and often unrealized power in the ways teachers talk with their students; it is our obligation to help teachers learn how to recognize and leverage this power.Show more Item The role of mathematical aesthetic in network-supported generative design: a case study(2007-05) Mack, André Joseph, 1968-; Stroup, Walter M.Show more Use of a next-generation, classroom-based network technology for mathematics instruction illuminates possible connections between the aesthetic perceptions of mathematics and mathematics teaching practices. Generative activity design makes use of participatory classroom simulations with the technology to allow students to fully engage in the activities from various levels and trajectories of understanding. Moreover, the student engagement with these activities produces artifacts, the projections of which make mathematical aesthetic visible and a substantial topic in the classroom discourse. This investigation entails the study of one secondary mathematics teacher, examining her instructional practices in the context of a networked-supported environment. This case study, conducted within the framework of a design experiment, uncovers the ways in which the teacher's mathematical aesthetic perceptions acted to (1) constrain her process of generative activity design and (2) frame her role in the mathematical discourse during classroom implementation of the network. Findings suggest the need for augmentation of a generative activity design framework to include overt connections to aesthetic.Show more Item Student-to student discussions : the role of the instructor and students in discussions in an inquiry-oriented transition to proof course(2008-05) Nichols, Stephanie Ryan, 1979-; Smith, Jennifer ChristianShow more This study of student-to-student discussions focuses on a single inquiry-oriented transition to proof course. Mathematical proof is essential to a strong mathematics education but very often students complete their mathematics studies with limited abilities to construct and validate mathematical proofs (c.f. Harel & Sowder, 1998; Knuth, 2002; Almeida, 2000). The role of mathematical proof in education is to provide explanation and understanding. Both the research on mathematical discourse and the standards of the NCTM claim that participation in mathematical discourse provides opportunities for understanding. Although this link has been established, there is very little research on the role of students and the instructor during discussions on student generated proofs at the undergraduate level -- particularly in inquiry-oriented classes. This research analyzes the types of discussions that occurred in an inquiry-oriented undergraduate mathematics course in which proof was the main content. The discussions of interest involved at least two student participants and at least three separate utterances. These discussions fell along a continuum based on the level of student interaction. As a result of this research, the four main discussion types that were present in this course have been described in detail with a focus on the roles of the instructor and the students. The methodology for this research is qualitative in nature and is an exploratory case study. The data used for this research was video tapes of two to three class sessions per week of an Introduction to Number Theory course taught in the fall of 2005.Show more Item A study of pre-kindergarten teachers' mathematical knowledge for teaching(2011-05) Lee, Jae Eun; Brown, Christopher P., Ph. D.; Adair, Jennifer; Junk, Debra; Reifel, Stuart; Svinicki, MarillaShow more This dissertation investigates the ways in which pre-k teachers understood the math content that they were to teach and their math instruction. To investigate this, a qualitative case study examining five pre-k teachers was conducted. Data sources included observation field notes, teacher interviews, and documents such as state and district pre-k guidelines. The findings from this dissertation suggest that pre-k teachers’ knowledge entails both knowledge of subject matter and pedagogical content knowledge. In addition, this study identified what these pre-k teachers knew about math and teaching/learning math as well as what they still needed to know to provide high quality and effective math instruction. Chapter 1 introduces my research question and important terms, such as mathematical knowledge for teaching (MKT). Chapter 2 synthesizes relevant literature in the area of effective math instruction, theoretical framework of teachers’ mathematical knowledge for teaching and early mathematics education. The literature review seeks to highlight the importance of early childhood teachers’ deep understanding of mathematical content and of their math instruction. Chapter 3 forwards the specific conceptual framework for this study while detailing the methodology that guided this investigation including data gathering and analysis. Chapter 4 presents the findings from this research. It examines pre-k teachers’ understanding of mathematical content that they are to teach and their knowledge of how to teach mathematics. Chapter 5 addresses the significance of these two major findings. First, I discuss the four types of mathematical knowledge and skills that these pre-k teachers possess. I also compare and contrast them with the teacher knowledge examined in the literature. Then, by examining research literature on early math education, I suggest what mathematical knowledge and skills they still need to attain to offer high-quality and effective math instruction. This dissertation concludes with a discussion of implication for teachers, teacher educators, and suggestions for future research.Show more Item A study of the mathematical processes used in sixteen gainful occupations in Nashville, Tennessee(1931) Hale, William Richard; Not availableShow more Item Teaching mathematics and the problems of practice: understanding situations and teacher reasoning through teacher perspectives(2005) Junk, Debra Lynn; Empson, Susan B.Show more In this study, 4 teachers were asked to identify classroom-teaching situations that they “wondered” about. Each teacher was using an inquiry-based, National Science Foundation funded curriculum (Investigations in Data Number and Space or Connected Mathematics) to teach fractions. Results showed that teachers’ problems of practice centered on interactions in which they struggled to understand students’ strategies, both invented and school based. Though difficult, the teachers strove to find ways to support student thinking and instructional intentions of inquiry-based mathematics practices rather than resorting to more didactic approaches. Teachers recognized and valued children’s construction and use of representations for fractions that were often in the form of area models, and teachers wanted to find ways to interpret these strategies from the children’s point of view. Teachers in this study often perceived themselves as “stuck” rather then empowered because they did not have the strategies for teaching needed to support these novel uses of models and often unexpected strategies.Show more