This study consisted of a comprehensive review of the consequential aspects of validity of a grade 10 mathematics assessment. This test is part of a larger state-mandated assessment system in which the studied test is one of two assessments that a student must pass in order to graduate from high school in the state of Massachusetts. Validity evidence was collected using three rigorous measurement methods. Qualitative and quantitative procedures were used to ensure a more complete collection and analyses of validity evidence. A survey was developed and administered to all participating high school mathematics teachers and key education personnel. Fifty-six percent of the surveys were completed and analyzed. In addition, focus group and one-on-one interviews were conducted within each participating school district. The results indicated that the Massachusetts' education reform initiative had created significant changes in high school mathematics curriculum and instruction. In addition, many positive and negative intended and unintended consequences connected to this assessment system were identified. The results were discussed based on a classification system in which a representative sample of school districts was selected from the state population. In this study, a comprehensive analysis of a few specific consequential validity questions was addressed using sound quantitative and qualitative research methods. This type of research, examining the consequential aspects of validity of a state mandated test as a component of a larger assessment system, represents a huge undertaking. The social, politic, and educational implications involved in any reform effort are complex and difficult to document. As education reform affects more and more students across this nation, answers to the outlined questions may assist key administrators in the state of Massachusetts, perhaps even other states in the middle of similar reform efforts, in making important mid-course corrections, and/or merely provide needed validity evidence regarding intended and unintended consequences of the program using solid, data-driven information.

This research investigation examined the effects of GO Solve Word Problems math intervention on problem-solving skills of struggling 5th grade students. In a randomized controlled study, 16 5 th grade students were given a 12-week intervention of GO Solve, a computer-based program designed to teach schema-based instruction strategies (SBI's) to solve math word problems and 16 control students continued with the standard school-based mathematics curriculum. A subset of items from the Massachusetts Comprehensive Assessment System (MCAS) as well as the Group Mathematics Assessment and Diagnostic Evaluation (GMADE) was used to measure student test performance. Examiner-made probes were given to both the treatment and control groups every other week to measure student progress. Results indicate that the mean difference scores of the experimental and control groups were statistically significant on a subtest of MCAS problems and a large effect size was reported. However, no statistically significant difference between the experimental and control groups was found on the on the Process and Application subtest of the GMADE. On examiner-made probes, there was a statistically significant difference between the experimental and control groups. Limitations of this study as well as implications for practice will be discussed.

This research study examined the importance of trust between instructor and students learning developmental mathematics in classrooms at 15 community colleges across Massachusetts. The study involved gathering new data using focus groups, field notes, surveys, and follow-up interviews with instructors and departmental chairs. Qualitative analysis was used to find emergent themes and connections to math anxiety and motivation. The study identified seven salient qualities of trust: respect, safe environment for students, belief in student's ability, care for student, competence of instructor, building over time, and destruction by betrayal. Instructors reported that increased classroom participation and student connection to instructor were factors that indicated trust was established. They also suggested teaching strategies to build trust: encouragement, mindful pedagogy, humanistic instruction, enabling student-to-student respect, situated applications, flexibility with assessments, and ensuring success. The causal impact of math anxiety was acknowledged: negative past experiences can block learning; lack of trust breeds hatred; fear and prejudice are barriers to learning; lack of trust can cause math anxiety; and trust lowers math anxiety and allows improved achievement. Instructors reported a "wall" as a metaphor that students use to describe their difficulties learning mathematics. Finally, the study recommends strategies for instructors to use to improve motivation for students learning mathematics: help establish a learning community, encourage self- efficacy, affirm students' questions, nudge students toward understanding, be a cheerleader, and utilize math support services.

Providing open access to higher education, community colleges extend the promise of a college education to all who can complete the degree requirements of a particular institution. One of the more universal requirements in community colleges is the completion of at least one course in college-level math. For female students, many of whom enter college without strong backgrounds in mathematics, the college-level math requirement often represents a particularly difficult challenge. The purpose of this study was to understand the ways in which personal, interpersonal, and cultural relationships influence achievement in math for women attending community college. By understanding these influences, community colleges can implement curricular and pedagogical changes in mathematics instruction that enhance opportunities for student success. Participants included six female community college students at various stages of completion of their math requirements. Qualitative methods were used to explore their stories with a particular focus on how interpersonal relationships, both inside and outside the classroom, affected achievement. A series of two interviews were employed to gather the data. Analysis consisted of coding interview texts, developing summaries of each participant's interviews, and comparing and contrasting participant responses between students with varying levels of mathematics achievement. The study found that all participants, even those with strong math skills, experienced some level of math anxiety, and that, in most cases, their math anxiety was strong enough to affect performance. Findings also confirmed that interpersonal relationships had a strong influence on student achievement. Those who experience the highest levels of achievement spoke of positive parental figures who took time to help them with math. In contrast, students who had the lowest levels of achievement spoke of negative experiences with teachers, and told of receiving little or no support from parents or others in their lives. Further research is indicated to help students reduce their levels of math anxiety in the community college classroom. Additionally, colleges must actively seek ways to counteract the negative experiences students bring with them to the mathematics classroom. Reducing student anxiety and promoting positive relationships with faculty and others should help to increase student achievement and success in mathematics.

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Spatial visualization--Mathematics and geometry achievement--Cognitive structure--Manipulatives--Lessons for triangles, quadrilaterals, polyhedra, polygons, Eulers Formula, and platonic solids.

This exploratory descriptive case study aimed to identify key issues in the transformation of mathematics teaching in elementary schools. This study told the story of what happened to sixteen elementary teachers who embarked on a quest to improve their mathematics teaching by participating in at least two in-depth professional development experiences over an interval of three years. It traced their ideas about how and why to improve instruction, identified their challenges with prevailing school organizational conditions, and reported perceived changes that were made in their teaching. The research data, comprised of quotations from teachers' writing while they were engaged in professional development experiences at SummerMath for Teachers of Mt. Holyoke College and the researcher's notes from interviews, formed the data for analysis to answer four interrelated research questions: (1) What changes in instruction do selected elementary teachers of mathematics report that they made as a result of participating in professional development for improving the teaching of mathematics? (2) What do selected elementary teachers of mathematics report are the reasons for the changes they made in instruction? (3) What changes in instruction do selected elementary teachers of mathematics report that they regard as most effective for improving student learning? (4) What organizational conditions in their local elementary schools do selected elementary teachers of mathematics report helped or hindered their changes in instruction? The major findings based on the collected data were summarized and presented according to the four research questions that guided this study. The major findings related to changes in instruction fell under five prominent categories: increased emphasis on student thinking and understanding, increase in student-centered activities, changes in classroom discourse, increase in conceptually-based mathematics content, and a shift in the teacher's role from an authoritarian model of instruction to one that is student-centered. Findings from this study suggested some recommendations for educational practice for institutions preparing elementary teachers of mathematics, for in-service teacher professional development programs, and for school policies and organizational structures. One primary recommendation involved engaging pre- and in-service teachers in revisiting the mathematics content that they currently teach or will teach within an inquiry-based teaching and learning environment so that they might draw their own conclusions as to the implications for teaching mathematics.

While the goal of mathematics education reform is to improve the mathematics achievement of all students (NCTM, 2000), at the core of these efforts is the teacher (Garet, Porter, Desimore, Birman, & Yoon, 2001). Educational change ultimately rests on the teachers who directly impact the students. Teacher leaders with expertise in all the dimensions of mathematics education can provide school-based professional development needed to support and maintain the teacher change process. The focus of this self-study is a critical examination of the influences on my development as an elementary mathematics teacher leader and on the strategies I develop as I coach teachers to improve, and change their practice. "Reciprocal Mathematics Coaching" was designed as a process to promote collaborative, job-embedded professional development. This model provided a means to meet each teacher's individual goals along a professional development continuum. Findings from the study indicate that teacher leader coaching interventions ranged along a continuum from the support of surface features of the curriculum, to co-teaching, and then to critical colleagueship in pursuit of a deeper pedagogy. Expected teacher outcomes occurred, but practice of reform-based instruction would require a longer time frame and the establishment of peer coaching support. My teacher leadership evolved from my approach to visualizing mathematics; elementary subject specialization; experience from teaching special education and regular education; beliefs in personalizing learning for students and teachers; self-reflective practice and practitioner research; and professional empowerment through collegial collaboration. Through "Reciprocal Mathematics Coaching", I came to learn how the pervasiveness of the affective domain impacts teachers even as they choose to improve their practice. Negative memories from their own mathematics education continue to influence their teaching. Formal teacher leadership can facilitate steps toward effective teacher growth and change. The strength of teacher leadership emanates from the nexus of teacher knowledge domains (Hill & Ball, 2004; Shulman, 1986;) situated within a learning community of reflective practice (Senge, 1990; Sergiovanni, 2000).

There is an inherent contradiction in a system of education in which a significant number of those who are to be taught, do not learn. The schools in the United States have contributed to a generations long pattern of non-achievement which contributes to many of society's ills. The author contends that many of these underachieving students have more ability than they are able to demonstrate easily. An experiment was conducted that empowered remedial mathematics students to pursue some mathematical problems in depth by removing the dual obstacles of poor computational skills and the lack of confidence that their computations produce correct answers.