This study looked at what components are in student solutions to computational estimation problems involving fractions. Past computational estimation research has focused on strategies used for estimating with whole numbers and decimals while neglecting those used for fractions. An extensive literature review revealed one study specifically directed toward estimating with fractions (Hanson & Hogan, 2000) that researched adult estimation strategies and not children's strategies. Given the lack of research on estimation strategies that children use to estimate with fractions, this study used qualitative research methods to find which estimation components were in 10 eighth-grade students' solutions to estimation problems involving fractions. Analysis of this data differs from previous estimation studies in that it considers actions as the unit of analysis, providing a smaller grain size that reveals the components used in each estimation solution. The analysis revealed new estimation components as well as a new structure for categorizing the components. The new categories are whole number and decimal estimation components, fraction estimation components, and components used with either fractions or whole numbers and decimals. The results from this study contribute to the field of mathematics education by identifying new components to consider when conducting future studies in computational estimation. The findings also suggest that future research on estimation should use a smaller unit of analysis than a solution response to a task, the typical unit of analysis in previous research. Additionally, these results contribute to mathematics teaching by suggesting that all components of an estimation solution be considered when teaching computational estimation, not just the overarching strategy.

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This study uses qualitative methods to investigate the use of writing in a content course for elementary education majors in which writing was considered an important part of mathematical learning. The study differs from previous studies by investigating the role of writing in the everyday instructional activities, rather than investigating writing as a separate mathematical activity. An analysis of the instruction and class discussions that took place in this class reveals that components of writing that were addressed implicitly and explicitly in classroom instruction were developed simultaneously with conceptual understanding, suggesting a much stronger and more integral relationship between writing and learning than the relationship that has been hypothesized by previous research. Furthermore, specific ways in which the class was structured seemed to support the development of students' written explanations. Appropriate explanations of particular concepts were modeled by both teacher and students, and explanations of mathematical concepts were developed gradually in a relatively consistent progression that paralleled the development of the concepts themselves. The findings of this study contribute to the field of research by helping to describe the relationship between writing and learning and by illuminating some of the ways in which both student learning and student writing are affected by classroom instruction.

This study looks at conceptual explanations given in a reform-oriented mathematics class for preservice secondary mathematics teachers and extends Toulmin's argumentation framework to account for some of the complexities of the explanations given by these students. This study explains the complexities that arose in applying Toulmin's framework to explanations and extends the framework by accounting for the features of conceptual explanations. The complexities of these explanations are that they are made up of multiple arguments that build on each other to reach a final conclusion and that they are also dependant upon the social aspects of the class in which they are situated. Recognizing that some statements have dual purposes in the explanation and that there are varying levels of justification used in the explanations helped to account for the first complexity of explanations. The classification of class conventions helps to account for the social influences on explanations. This study differs from other studies that use Toulmin's framework to analyze formal proofs or to identify taken-as-shared understanding in a classroom. This study instead focuses on using the framework to analyze the components of explanations and to provide insight into the structure of conceptually oriented explanations. This study contributes to the existing research by extending Toulmin's argumentation framework to account for how social influences help determine the appropriate components of an explanation.

The primary concerns of mathematics educators are learning and teaching mathematics. It is, therefore, natural to ask "what implications and benefits might there be if learning were perceived as a risk-taking event?" (Atkinson, 1957, p. 266). The underlying motivation of this study is to analyze the risks students take in the mathematics classroom and how risk influences student creation of meaning and development of understanding. I define risk in the mathematics classroom to be any observable act that entails uncertain outcome. The research presented here focuses on a table of four students: Andrew, Carina, Kam, and Mark as they grapple with the mathematical uncertainties inherent in the Ticket Line Task. In analyzing student work and development of mathematical understanding, I identify risks that students take and the benefits they claim result from doing so. Contextualized Risk Theory (CRT) is introduced to improve our understanding of the risks students take in learning mathematics in a student-centered classroom where students exercise personal agency in mathematical problem solving. Findings include characterization of risks these students took, significant student mathematical activity, student enjoyment of their work, student development of personal understanding of purposes and meanings of specific mathematics, and students achieving mathematical success as defined by the researcher and the participants.

In the past, frameworks for what it means for students to understand elementary mathematical concepts like addition have been well-researched. These frameworks are useful for identifying what students must understand to have a good grasp of the concept. Few such research-based frameworks exist for secondary mathematical topics. The intent of this study was to create such a framework for what it means for students to understand logarithms, a topic that has been under-researched up to this point. Four task-based interviews were conducted with each of four different preservice secondary mathematics teachers in order to test a preliminary framework I had constructed to describe what it means for students to understand logarithms. The framework was adjusted according to the findings from the interviews to better reflect what it means for students to have a good understanding of logarithms. Also, a common practice taught to students learning logarithms, switching from logarithmic form to exponential form, was found to possibly have negative effects on student understanding of logarithms. The refined, research-based framework for what it means for students to understand logarithms is described in full in this document. The implications of the results of this study for mathematics teachers as well as for mathematics education researchers are also discussed.

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<p> This research was carried out within a qualitative research paradigm. The objective was to observe, analyze and enrich pedagogical practice through the use of pedagogical learning strategies. The learning strategy was a participative and individualized lab carried out during a research project in a non-Traditional Laboratory (LnT). The primary aim of this research was to observe if the LnT assist the students and in this way maximizes success and knowledge in the Introductory Math course (MATE3001) on the University of Puerto Rico campus. </p><p> The investigation questions were discussed in the light of each of the strategies of information collected, personal experience and revision of literature. The methodology used was of a qualitative nature in which the student reflected on the process experienced in the LnT. Seven participants of the math course (MATE3001) who formed part of the LnT in a voluntary manner were interviewed at the beginning and at the completion of the research. The purpose of the interviewed was to discover the participant opinion regarding the pedagogical impact of the LnT. Finally, the research professor made an observation in order to discover of the LnT strategy had the anticipated acceptance by the students.</p><p> The LnT contributed to: (1) students improved their study habits; (2) the students had greater participation in the solution of math problems, their practice and discussion; (3) they accepted that the research professor supervise their work as it was carried out and understood that the presence was for their benefit. Also, the findings of this research were contrasted with the Theory of reciprocal determinism, sources of self-efficacy and self-regulation of Bandura with the impact that these have on learning (Bandura, 1986, 1989a, 1989b). It was also found as the implicit theory (Yeager &amp; Dweck, 2012) resurges in the LnT the effects on interest, student&rsquo;s resilience and situational motivation (Nolen, Horn, &amp; Ward, 2015) which occurs during the living out of the lab experience. LnT stimulates the student, creates security and increases confidence in the solution of math problems.</p>