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The purpose of this research is to explore preservice teachers’ attitudes and beliefs towards mathematics, mental and written computations, and mental computation anxiety, to investigate their use of different mental computation strategies using different approaches (i.e., Direct Teaching (DT) and Open-Approach (OA)) among the three different groups, and to identify how the use of preservice teachers’ mental computation strategies affects their flexibility regarding mental computation. The participants were preservice teachers (PTS). Three classes were used for this study: two classes in a mathematics class (Course A) for experimental groups and one class for the control group. One class from professional education courses was selected. A mixed methods design was used, more specifically, the Mathematics Attitude Survey (MAS) was administrated before and after intervention to examine PTS’ attitudes towards mathematics, mental and written computation, and mental computation anxiety. In addition, to determine whether there is any statistically significant difference among the three groups, the one-way analysis of variance (ANOVA) was used. Then, the MAS was analyzed descriptively. Next, a pre-and post-Mental Computation Test (MCT) was given to investigate PTS’ mental computation knowledge in relation to whole numbers, integers, and rational numbers (i.e., fractions, decimals, and percentages). A one-way analysis of covariance (ANCOVA) was conducted to determine if there were significant differences in mental computation performance among the three groups (i.e., DT, OA, and Control) with different instructions. Further, before and after intervention, face-to-face interviews were given to both the experimental and control groups to identify how they arrived at their answers. During interviews, 38 interviewees in the pre-interviews and 36 in the post-interviews for all groups participated. The interview items were selected from the pre-and post-MCT problems. Three levels of problems (i.e., high, medium, and low difficulty) for each operation were selected. The results of the MAS showed that with respect to the attitudes towards mathematics, PTS were generally shown positive attitudes towards learning mathematics and were aware of the importance of learning mathematics; however, in reality, about half of them did not want to spend time on learning or studying mathematics. In terms of PTS’ attitudes towards mental and written computation, PTS were aware that learning mental computation is more useful in real life situations and provides benefits in their mathematics learning. However, they do not feel comfortable and safe when using mental computation because of their lack of confidence and teaching abilities. For the mental computation, PTS showed slightly higher anxiety levels from pre-to post-tests. The findings of Mental Computation Test (MCT) revealed that there was a statistically significant difference in post-MCT scores between the different instructional groups when adjusted for pre-MCT scores. In particular, PTS using Open-Approach (OA) performed better than the PTS in the group using Direct Teaching (DT). The PTS in the control group performed worst. Significant differences between pre-and post-MCT performance were found among the three groups in solving multiplication, fraction, and decimal operations. The results of interviews suggest that there was an association between each interviewee’s quintile level and their flexibility in the use of the mental computation strategies. Regarding the whole number operation strategies, the results revealed that the interviewees in the middle and upper quintiles in both DT and OA used more than two different strategies with higher accuracy and were more likely to use the strategies. Interviewees in the middle and upper quintiles for the DT and OA groups were more likely to use the strategies that reflect efficient number facts or number-sense (e.g., Adding by place, Decomposing, & Compensation). The mental image of the Traditional method was frequently observed in the OA group. In contrast, for the lower quintiles, alternative strategies were not provided for both groups. The interviewees in the control group offered the smallest range of strategies. For the integer and rational operations, the interviewees in the DT group showed strategies that focused more on conceptual understanding. Surprisingly, the interviewees in the OA group were more likely to apply teacher-taught methods, including the Traditional method. The control group was not able to provide any alternative strategies. Plans for future research are set forth to add to the body of knowledge that exists regarding mental computation.

The purpose of the study was to determine the validity and reliability of a revised observation and evaluation instrument of middle and secondary pre-service clinical teaching to be used as part of the clinical supervision cycle and for formative purposes. The North Texas Appraisal of Classroom Teaching (NTACT) serves as a performance assessment tool utilized by a south-central university-based educator preparation program for the evaluation and supervision of pre-service teachers during their last semester of their program. The researcher piloted and field-tested a redesigned observation and evaluation instrument (NTACT-V2) on observer participants with varying educational experiences in the south-central region. To accumulate evidence of validity and reliability, this study employed methods of factor analysis and generalizability study for developing a valid and reliable instrument to guide the refinement process of the NTACT observation and evaluation instrument. Some of the significant conclusions reached in this study were (a) the NTACT-V2 is a practical, user-friendly classroom observation and evaluation instrument; (b) the instrument refined and developed in this study exhibits appropriate content, face, and criterion validity as determined by a panel of experts and an extensive review of the literature; and, (c) a variety of observers can use the evaluation instrument with relative ease while achieving a high degree of reliability.

Lesson study is a form of professional development for teachers adopted in recent years from Japan. Introducing lesson study to U.S. teachers and researchers has been the focus of most of the literature on this subject. Much of the literature outlines how lesson study works and describes its essential features. One of the features of lesson study is anticipating student responses, also known as anticipating student thinking. Anticipating student responses is passingly described in lesson study literature. This research was conducted to understand what it means to anticipate student responses for preservice mathematics teachers in a lesson study group. Lesson study literature indicates that anticipating student responses is to anticipate conceptual development from the students' perspective, and the purpose is to be prepared to have meaningful discussions and questions to enable students to develop the understanding. Anticipating student responses is highly related to the hypothetical learning trajectory described by Simon (1995), the self directed anticipative learning model described by Christensen and Hooker (2000) and the expert blind spot discussed by Nathan and Petrosino (2003). While their work does not stem from lesson study, they add theoretical perspective to the idea of anticipating student responses. Their work indicates that anticipating student responses is difficult, valuable, that one gets better at it through experience, and that it is very useful in refining lessons. Participants were enrolled in the mathematics education methods class of a large private university in the U.S. A characterization of anticipating student responses was developed as the participants met in group meetings to create a lesson. They anticipated student responses in ways that facilitated lesson planning and task design. Participants did not anticipate student responses toward students' conceptual development. This research reports five particular ways that anticipating student responses was used as a tool to define and refine the lesson so that it ran smoothly toward lesson goals. These ways are related to: goals, tasks and materials, procedural mathematical reasoning, successful student efforts, and emotional responses. It is believed that anticipating student responses towards task design is a necessary precursor to anticipating student responses toward students' conceptual development.

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A classroom teaching experiment was conducted to document prospective teachers' development of whole number concepts and operations. The purpose of this mixed-methods study was to examine the collective understanding of prospective teachers in an elementary mathematics content course. Design research methodology, specifically a classroom teaching experiment was the methodology selected for this study since it allows learning to be documented in a classroom environment and is iterative in nature. A revised hypothetical learning trajectory and instructional tasks from a previous classroom teaching experiment were used in this study (Andreasen, 2006). Research about children's development of whole number concepts and operations was used in developing instructional learning goals. In addition, research regarding prospective teachers' development supported the instructional modification that all tasks would be presented and expected to be reasoned about in base-8. Both qualitative data and quantitative data were collected. Qualitative data included whole class dialogue that was videotaped and transcribed, as well as student work samples. Quantitative data included items from the Content Knowledge for Teaching Mathematics database that were administered prior to and subsequent to the instructional sequence in base-8 (Hill, Schilling, & Ball, 2005). It should be noted that the items selected from the database were in base-10. The emergent perspective served as the interpretive framework of the collected qualitative data. This perspective reflexively coordinates the social or group perspective simultaneously with psychological or individual perspective. As stated, this study sought to describe the communal mathematics understanding of prospective teachers in an elementary mathematics content course. Toulmin's (1969) model of argumentation and Rasmussen and Stephan's three-phase methodology served to document normative ways of group reasoning called classroom mathematical practices. The following classroom mathematical practices were identified as taken-as-shared by prospective teachers: (a) developing small number relationships using Double 10-Frames, (b) developing two-digit thinking strategies using the open number line, (c) flexibly representing equivalent quantities using pictures or Inventory Forms, and (d) developing addition and subtraction strategies using pictures or an Inventory Form. Quantitative results indicated that prospective teachers were able to apply mathematical understandings grounded in base-8 to whole number concepts in base-10. In the end, counting and calculating in base-8 provides a meaningful context for prospective teachers to reconstruct their knowledge of whole number concepts and operations.

Given current legislation to ensure education for students with disabilities and that institutions of higher education are required to use universal design for learning (UDL) principles, the purpose of this study was to explore the impact of video modeling on preservice teachers' knowledge, understanding and application of the three principles of UDL. Preservice teachers were randomly assigned to control or experimental groups to determine if video embedded with UDL principles impacted their thinking. Specifically, pre and posttest information of knowledge and understanding as well as self-perceived ability to teach students with disabilities using UDL was analyzed. In addition preservice teacher created lesson plans were analyzed for application of UDL principles after viewing the video intervention. Quantitative analyses were conducted to compare pre and posttest scores of the control group (n = 41) and experimental group (n =45). The quantitative analyses of knowledge, understanding

This research project aimed to extend the research literature by providing greater insight into the way individual prospective teachers develop their conceptual understanding of whole number concepts and operations in a social context. In this qualitative study, a case study analysis provided the opportunity for careful exploration of the manner in which prospective teachers' understanding changed and the ways two selected participants reorganized their mathematical thinking within a classroom teaching experiment. While previous research efforts insisted on creating a dichotomy of choosing the individual or the collective understanding, through the utilization of the emergent perspective both the individual and the social aspects were considered. Specifically, using the emergent perspective as a theoretical framework, this research endeavor has outlined the mathematical conceptions and activities of individual prospective teachers and thus has provided the psychological perspective correlate to the social perspective's classroom mathematical practices. As the research participants progressed through an instructional sequence taught entirely in base-8, a case study approach was used to select and analyze two individuals. In order to gain a more thorough understanding of the individual perspective, this research endeavor focused on whether teachers with varying initial content knowledge developed differently through this instructional sequence. The first participant initially demonstrated "Low-Content" knowledge according to the CKT-M instrument database questions which measure content knowledge for teaching mathematics. She developed a greater understanding of place value concepts and was able to apply this new knowledge to gain a deeper sense of the rationale behind counting strategies and addition and subtraction operations. She did not demonstrate the ability to consistently make sense of multiplication and division strategies. She participated in the classroom argumentation primarily by providing claims and data as she illustrated the way she would use different procedures to solve addition and subtraction problems. The second participant illustrated "High-Content" knowledge based on the CKT-M instrument. She already possessed a solid foundation in understanding place value concepts and throughout the instructional sequence developed various ways to connect and build on her initial understanding through the synthesis of multiple pedagogical content tools. She demonstrated conceptual understanding of counting strategies, and all four whole number operations. Furthermore, by exploring various ways that other prospective teachers solved the problems, she also presented a greater pedagogical perspective in how other prospective teachers think mathematically. This prospective teacher showed a shift in her participation in classroom argumentation as she began by providing claims and data at the outset of the instructional sequence. Later on, she predominantly provided the warrants and backings to integrate the mathematical concepts and pedagogical tools used to develop greater understanding of whole number operations. These results indicate the findings based on the individual case-study analysis of prospective elementary school teachers and the cross-case analysis that ensued. The researcher contends that through the synthesis of the findings of this project along with current relevant research efforts, teacher educators and educational policy makers can revisit and possibly revise instructional practices and sequences in order to develop teachers with greater conceptual understanding of concepts vital to elementary mathematics.

Cultivating preservice teachers' reflection on their practice has been a major objective in teacher education over the last twenty years. Teacher educators have designed a number of activities, usually related to coursework, to facilitate preservice teachers' reflectivity. These kinds of assignments might be described as asking preservice teachers to reflect on demand. Studies in the past have typically focused on the discussion of specific strategies used to elicit reflection or descriptions of reflective teacher education programs. This study explored how preservice and first-year teachers understand and engage in reflection in different contexts while learning to teach. A sample of seventeen respondents, education students enrolled in coursework, student teachers, and first-year teachers, was selected to participate in the study. The data consisted of interviews, classroom observations and the examination of related documents. Preservice teachers taking courses engaged in introspection or self-reflection. This time was spent exploring their emerging identities as teachers. As respondents got opportunities to work in school settings, they began to reflect more on content and students. First-year teachers applied some of the reflective strategies they learned during coursework in their own practice. Several factors appeared to enhance preservice and first-year teachers' ability to reflect as they moved from coursework into student teaching and their own classrooms: (1) building relationships with teachers and students in K--12 classrooms, (2) engaging in reflective dialogue with 'more knowledgeable others'---university faculty and practitioners, and (3) having opportunities to teach on a regular basis.