Discourse practices in mathematics classes have been proven to lead to greater student achievement. Policy and standards require students are able to justify and critique mathematical reasoning. Literature on how high school mathematics teachers implement discourse practices and facilitate discussions is scarce. This research study examined how three high school mathematics teachers, who participated in a professional development course which focused on facilitating discussions in the classroom, used and described their use of discussions and specifically the teacher discourse moves (TDMs) in their classes. This study was situated in a high-achieving suburban upper-middle class district. Data sources included: journal reflections, responses to Use of Discourse Surveys, Beliefs Mappings, interviews (including post-observation Video Stimulated Recall (VSR) interviews) and classroom observations. Each participant was observed teaching four lessons. Qualitative analyses revealed that participants’ beliefs related to discourse and classroom expectations evolved. The results of this study confirmed that facilitating whole class discussions was challenging for high school mathematics teachers. In particular, some Teacher Discourse Moves (TDMs) were easier for participants to use over others and some changes were easier for participants to make such as utilizing different activity structures. Factors that contributed to participants’ use of discussion included: professional development, watching one’s own teaching, noticing changes in students’ behaviors, previous instruction on learning to teach, perceptions of student capabilities, perceptions of time constraints, and lack of reflective practice. Despite these challenges, participants were able to make positive changes in their instruction and notice an increase in student engagement as a result. Participating in VSR interviews had a dramatic impact on the participants’ beliefs, reflection and changes in practice.

abstract: Let $G=(V,E)$ be a graph. A \emph{list assignment} $L$ for $G$ is a function from $V$ to subsets of the natural numbers. An $L$-\emph{coloring} is a function $f$ with domain $V$ such that $f(v)\in L(v)$ for all vertices $v\in V$ and $f(x)\ne f(y)$ whenever $xy\in E$. If $|L(v)|=t$ for all $v\in V$ then $L$ is a $t$-\emph{list assignment}. The graph $G$ is $t$-choosable if for every $t$-list assignment $L$ there is an $L$-coloring. The least $t$ such that $G$ is $t$-choosable is called the list chromatic number of $G$, and is denoted by $\ch(G)$. The complete multipartite graph with $k$ parts, each of size $s$ is denoted by $K_{s*k}$. Erd\H{o}s et al. suggested the problem of determining $\ensuremath{\ch(K_{s*k})}$, and showed that $\ch(K_{2*k})=k$. Alon gave bounds of the form $\Theta(k\log s)$. Kierstead proved the exact bound $\ch(K_{3*k})=\lceil\frac{4k-1}{3}\rceil$. Here it is proved that $\ch(K_{4*k})=\lceil\frac{3k-1}{2}\rceil$. An online version of the list coloring problem was introduced independently by Schauz and Zhu. It can be formulated as a game between two players, Alice and Bob. Alice designs lists of colors for all vertices, but does not tell Bob, and is allowed to change her mind about unrevealed colors as the game progresses. On her $i$-th turn Alice reveals all vertices with $i$ in their list. On his $i$-th turn Bob decides, irrevocably, which (independent set) of these vertices to color with $i$. For a function $l$ from $V$ to the natural numbers, Bob wins the $l$-\emph{game} if eventually he colors every vertex $v$ before $v$ has had $l(v)+1$ colors of its list revealed by Alice; otherwise Alice wins. The graph $G$ is $l$-\emph{online choosable} or \emph{$l$-paintable} if Bob has a strategy to win the $l$-game. If $l(v)=t$ for all $v\in V$ and $G$ is $l$-paintable, then $G$ is t-paintable. The \emph{online list chromatic number }of $G$ is the least $t$ such that $G$ is $t$-paintable, and is denoted by $\ensuremath{\ch^{\mathrm{OL}}(G)}$. Evidently, $\ch^{\mathrm{OL}}(G)\geq\ch(G)$. Zhu conjectured that the gap $\ch^{\mathrm{OL}}(G)-\ch(G)$ can be arbitrarily large. However there are only a few known examples with this gap equal to one, and none with larger gap. This conjecture is explored in this thesis. One of the obstacles is that there are not many graphs whose exact list coloring number is known. This is one of the motivations for establishing new cases of Erd\H{o}s' problem. Here new examples of graphs with gap one are found, and related technical results are developed as tools for attacking Zhu's conjecture. The square $G^{2}$ of a graph $G$ is formed by adding edges between all vertices at distance $2$. It was conjectured that every graph $G$ satisfies $\chi(G^{2})=\ch(G^{2})$. This was recently disproved for specially constructed graphs. Here it is shown that a graph arising naturally in the theory of cellular networks is also a counterexample. / Dissertation/Thesis / Doctoral Dissertation Mathematics 2015

abstract: Research on combinatorics education is sparse when compared with other fields in mathematics education. This research attempted to contribute to the dearth of literature by examining students' reasoning about enumerative combinatorics problems and how students conceptualize the set of elements being counted in such problems, called the solution set. In particular, the focus was on the stable patterns of reasoning, known as ways of thinking, which students applied in a variety of combinatorial situations and tasks. This study catalogued students' ways of thinking about solution sets as they progressed through an instructional sequence. In addition, the relationships between the catalogued ways of thinking were explored. Further, the study investigated the challenges students experienced as they interacted with the tasks and instructional interventions, and how students' ways of thinking evolved as these challenges were overcome. Finally, it examined the role of instruction in guiding students to develop and extend their ways of thinking. Two pairs of undergraduate students with no formal experience with combinatorics participated in one of the two consecutive teaching experiments conducted in Spring 2012. Many ways of thinking emerged through the grounded theory analysis of the data, but only eight were identified as robust. These robust ways of thinking were classified into three categories: Subsets, Odometer, and Problem Posing. The Subsets category encompasses two ways of thinking, both of which ultimately involve envisioning the solution set as the union of subsets. The three ways of thinking in Odometer category involve holding an item or a set of items constant and systematically varying the other items involved in the counting process. The ways of thinking belonging to Problem Posing category involve spontaneously posing new, related combinatorics problems and finding relationships between the solution sets of the original and the new problem. The evolution of students' ways of thinking in the Problem Posing category was analyzed. This entailed examining the perturbation experienced by students and the resulting accommodation of their thinking. It was found that such perturbation and its resolution was often the result of an instructional intervention. Implications for teaching practice are discussed. / Dissertation/Thesis / Ph.D. Mathematics 2013

<p> What is mathematics? What does it mean to be a mathematician? What should students understand about the nature of mathematical knowledge and inquiry? Research in the field of mathematics education has found that students often have na&iuml;ve views about the nature of mathematics. Some believe that mathematics is a body of unchanging knowledge, a collection of arbitrary rules and procedures that must be memorized. Mathematics is seen as an impersonal and uncreative subject. To combat the na&iuml;ve view, we need a humanistic vision and explicit goals for what we hope students understand about the nature of mathematics. The goal of this dissertation was to begin a systematic inquiry into the nature of mathematics by identifying humanistic characteristics of mathematics that may serve as goals for student understanding, and to tell real-life stories to illuminate those characteristics. Using the methodological framework of heuristic inquiry, the researcher identified such characteristics by collaborating with a professional mathematician, by co-teaching an undergraduate transition-to-proof course, and being open to mathematics wherever it appeared in life. The results of this study are the IDEA Framework for the Nature of Pure Mathematics and ten corresponding stories that illuminate the characteristics of the framework. The IDEA framework consists of four foundational characteristics: Our mathematical ideas and practices are part of our <i><b>I</b></i>dentity; mathematical ideas and knowledge are <i><b>D</b></i>ynamic and forever refined; mathematical inquiry is an emotional <i><b>E</b></i>xploration of ideas; and mathematical ideas and knowledge are socially vetted through <i><b> A</b></i>rgumentation. The stories that are told to illustrate the IDEA framework capture various experiences of the researcher, from conversations with his son to emotional classroom discussions between undergraduates in a transition-to-proof course. The researcher draws several implications for teaching and research. He argues that the IDEA framework should be tested in future research for its effectiveness as an aid in designing instruction that fosters humanistic conceptions of the nature of mathematics in the minds of students. He calls for a cultural renewal of undergraduate mathematics instruction, and he questions the focus on logic and set theory within transition-to-proof courses. Some instructional alternatives are presented. The final recommendation is that nature of mathematics become a subject in its own right for both students and teachers. If students and teachers are to revise their beliefs about the nature of mathematics, then they must have the opportunities to reflect on what they believe about mathematics and be confronted with experiences that challenge those beliefs.</p><p>

Abstract not available

Diagram drawing is generally accepted as an important heuristic strategy for solving mathematical problems. However, novice problem solvers do not frequently choose to use this strategy. Further, when asked to draw a diagram, their attempts often do not result in a useful representation of the problem. The exploratory study, which used individual interviews with remedial mathematics students at the University of Massachusetts, identified five factors that influence whether a diagram is used and whether its use is successful: (1) Understanding of the mathematics involved in the problem and of basic arithmetic concepts (i.e. fractions, ratio); (2) Diagram drawing skills and experience; (3) Conceptions of mathematics; (4) Self-concept in mathematics; (5) Motivation to solve the problem correctly. The interviews also generated a set of diagram drawing subskills. The main study focused on factor two. It attempted to experimentally verify the importance of the subskills identified in the exploratory study. The list of subskills was translated into a series of external control suggestions for guiding the subjects' work during individual interviews. Subjects were precalculus students at the University of Massachusetts. These suggestions were provided by the experimenter as appropriate. Subjects who received these suggestions drew significantly higher quality diagrams than did subjects in the control group. The enhanced quality was particularly apparent in the area of completeness of the diagram. In addition, the study indicated several important metacognitive skills necessary for successful diagram drawing as well as a number of specific difficulties encountered by the subjects.

This study focuses on what comes to be constituted as mathematics when is presented in contextually embedded forms, how students constitute mathematical meaning from such texts, and what is constituted as mathematics when this particular form of mathematics is pedagogised.

Bibliography: leaves 97-101.

thesis is concerned with the production of an account of the relation between the reproduction of specialised knowledge and the moral discourse within pedagogic practice. The internal mechanism that knots together knowledge and moral discourse is elaborated by way of an analysis of texts produced by the originators of a pedagogic modality they refer to as the "problem-centred approach." The particular texts analysed are: (1) the Grade 1 to 4 textbooks and the corresponding teacher's guides, and (2) video records, supplied by the originators, of what they consider to be exemplary realisations of the pedagogy in practice of the "approach." The thesis opens with a discussion of a proposition, derived from Bernsteinian studies of curriculum and pedagogy, stating that everyday and academic know ledges are incommensurable, and from which it is claimed that the insistent contemporary attempts at incorporating the everyday into the academic in curricula and pedagogy, under the banner of "relevance," are educationally problematic. Against the Bernsteinian position, a central feature of the "problem-centred approach" is the extensive recruitment of extra-mathematical referents for the purposes of the reproduction of school mathematics. A more general examination of school mathematics texts that recruit the everyday reveal that such texts also associate the everyday with the pleasure of the student, so rendering "relevance," and hence moral discourse, as utilitarian. The manner in which the moral discourse operates within pedagogy was described in terms of Hegel's theory of judgement and Freudian-Lacanian accounts of imaginary and symbolic identification. Hegel enabled a description of pedagogic discourse at the level of the instructional content, and Freud-Lacan at the level of moral discourse. Hegel also enabled the location of the point at which the moral attaches to the instructional. What our analysis revealed is as follows: (1) the "problem-centred approach" is a competence-type pedagogy that employs strategies encouraging an initial imaginary identification with the everyday and pleasure, which is used to effect symbolic identification with school mathematics; (2) moral discourse drives pedagogic judgement by means of the imaginary-symbolic dialectic pertaining to identification; (3) evaluation drives pedagogic judgement aimed at the knowledge statements produced by students; and that (4) while the moral discourse is a pervasive and formally necessary component of pedagogy, it is ultimately embedded in the organisation and elaboration of the instructional contents, working in the service of the reproduction of instructional contents, but in accord with dominant ideological imperatives.

Mathematics courses should prepare students to use mathematics in a range of contexts, including science. However, students often struggle in applying mathematics in physics courses, even when they understand the mathematical content (Rebello et al., 2007). Often the root of this struggle is attributed to the student or the teaching; rarely is the form and function of the mathematics itself considered. Yet some have suggested that mathematics is used differently by physicists and mathematicians resulting in two different “languages” of mathematics (e.g., Redish & Kuo, 2015). This curriculum study identifies and describes differences in mathematics, specifically right triangle trigonometry, as manifested across high school physics and mathematics textbooks. Textbooks with different curricular approaches were selected for physics (i.e., Holt McDougal Physics, Active Physics) and mathematics (i.e., Holt McDougal Geometry and Algebra 2, Geometry Connections and Algebra 2 Connections). Since differences between these textbooks could come in a variety of ways, a broad definition of mathematics, including mathematical content, practices, and language, was used. Analysis included developing a coding scheme for understandings of trigonometry content using an iterative emergent process, coding tasks for opportunities for mathematical practices as described in the Common Core State Standards, and writing lesson memos to capture salient features of mathematical content, practice, and language. Findings expose significant differences in how right triangle trigonometry is used, the features of right triangle diagrams, and the opportunities provided for mathematical practices across physics and mathematics textbooks. In mathematics textbooks, sine, cosine and tangent ratios are used to find unknown measurements in a right triangle and tangent is also used to describe a slope. In physics textbooks, trigonometric ratios have more specialized uses, with tangent being used to find the direction of a vector and sine and cosine used to calculate the components of a vector. Right triangle diagrams in mathematics textbooks may be provided by tasks and are used to model static scenarios involving lengths in application tasks. In contrast, tasks in physics textbooks prompt a reader to generate their own right triangle diagram to model dynamic scenarios involving more abstract quantities (e.g., force, velocity) represented as vectors. Tasks in physics textbooks that involve the analysis of forces acting on an object provide different and specific opportunities for mathematical practices not found in mathematics textbooks. This study reveals that students may be experiencing mathematics differently between their physics and mathematics courses, which may explain why students struggle to use mathematics in physics class. Furthermore, this study demonstrates the value of comparing the mathematics situated in textbooks from different disciplines (i.e., not only mathematics textbooks) and opens new avenues for future research.