The use and application of geometric transformations is a fundamental standard for the Common Core State Standards. This study was developed to determine current high school teachers’ prior mathematical content knowledge and develop their content knowledge of transformations and their applications. The design of this study was guided by the questions: “Why is there a level of reluctance amongst secondary teachers when it comes to teaching geometric transformations?” and “How can their content knowledge become deepened to apply geometric transformations to prove that two figures are congruent?” The study provided teachers a chance to gain experience with transformations and use transformations to develop an understanding of congruence and similarity. The teachers’ work with transformations also enhanced their understanding of how transformations are the foundation for Euclidean geometry and begin to lay a foundation for the basics of rigid motion in the plane, with or without the use of coordinates. The results supported the claim that teachers’ transformation content knowledge needs to be deepened overall and in particular with respect to the application of transformations to prove that two figures are congruent. The results also showed that, with an increase of understanding of the mathematical properties of transformations, teachers are better prepared to teach them in their classrooms.

The beauty of mathematics can arguably be found in the way in which all concepts are interrelated and interwoven to create a massive web of knowledge and in the ways this can be applied to all aspects of life and technology. Given this inextricable interrelationship amongst several mathematical topics, many students encounter issues in learning mathematics due to gaps in their understanding of previously taught material. As a result, mathematics education in the K-12 setting has emphasized the need for interventions in order to help students grasp the progressively complex concepts that are required by our current society and education system as they advance throughout their academic career. This literature review researches effective and non-effective indicators of future mathematics proficiency as an initial step towards identifying the most beneficial cognitive and non-cognitive areas of focus, and consequently early interventions, in order to support student learning especially for underperforming students. Specifically, this research synthesizes research about three essential questions: (1) What skills, conceptual understandings, or student traits can serve as possible predictors of future mathematics proficiency? (2) Which of these identified skills, conceptual understandings, or student traits are stronger predictors of future mathematics proficiency? and (3) What is the degree of accuracy of these predictors? The research was conducted through the review of articles retrieved from diverse research studies. The literature revealed that the single most effective indicator of future mathematical proficiency is knowledge of fractions, specifically, conceptual understanding of and operations with fractions as well as fluidity with rational operations. Other less effective indicators included early knowledge of whole number division, functional numeracy, students’ attitudes and dispositions towards mathematics, gender, early mathematics achievement/ability, and literacy/linguistic ability. Other skills, conceptual understandings and student traits investigated in the relevant research included whole number arithmetic knowledge, number system knowledge, verbal & non-verbal IQ, working memory, and family education & income. These indicators did not exhibit a significant correlation to future mathematics performance and thus were classified as non-effective.

Inconsistent math assessment practices do not accurately represent and communicate student mathematics achievement. Because of inconsistencies in assessment practices, local middle school mathematics teachers in an urban school district in the northwestern United States piloted the use of multiple formative assessments. The purpose of this study was to compare mathematics achievement, growth, and course percentage grades for students who have multiple formative assessment attempts compared to students who are not provided multiple assessment attempts. Theoretical foundations originated from Black and Wiliam, supporting the use of formative assessment for a positive impact on student learning. A quantitative, ex post facto quasi-experimental design was used. The research question focused on the statistical differences in course percentage grade, state standardized testing score, and growth score on state standardized math tests between groups of students who were allowed multiple formative assessment options and those who were not. Data were analyzed using an independent samples t test and a one-way MANOVA, which showed a statistically significant difference for student course percentage grade. Findings were used to produce a 3-day professional development program supporting teachers' use of formative assessment in mathematics classes. The findings will inform educational stakeholders' decisions regarding the use of multiple assessment attempts and differences between this specific formative assessment strategy and student mathematics assessment performance to promote positive social change. Positive social changes may include increased awareness of how multiple assessments may affect student growth, course percentage grades, and state testing scores.

This study is an investigation into the science literacy of college genetics students who were given a modified curriculum to address specific teaching and learning problems from a previous class. This study arose out of an interest by the professor and researcher to determine how well students in the class Human Genetics in the 21st Century responded to a reorganized curriculum to address misconceptions that were prevalent after direct instruction in the previous year's class. One of the components to the revised curriculum was the addition of a multimodal representation requirement as part of their normal writing assignments. How well students performed in these writing assignments and the relationship they had to student learning the rest of the class formed the principle research interest of this study. Improving science literacy has been a consistent goal of science educators and policy makers for over 50 years (DeBoer, 2000). This study uses the conceptualization of Norris and Phillips (2003) in which science literacy can be organized into both the fundamental sense (reading and writing) and the derived sense (experience and knowledge) of science literacy. The fundamental sense of science literacy was investigated in the students' ability to understand and use multimodal representations as part of their homework writing assignments. The derived sense of science literacy was investigated in how well students were able to apply their previous learning to class assessments found in quizzes and exams. This study uses a mixed-methods correlational design to investigate the relationship that existed between students' writing assignment experiences connected to multimodal representations and their academic performance in classroom assessments. Multimodal representations are pervasive in science literature and communication. These are the figures, diagrams, tables, pictures, mathematical equations, and any other form of content in which scientists and science educators are communicating ideas and concepts to their audience with more than simple text. A focused holistic rubric was designed in this study to score how well students in this class were able to incorporate aspects of multimodality into their writing assignments. Using these scores and factors within the rubric (ex. Number of original modes created) they were correlated with classroom performance scores to determine the strength and direction of the relationship. Classroom observations of lectures and discussion sections along with personal interviews with students and teaching assistants aided the interpretation of the results. The results from the study were surprisingly complex to interpret given the background of literature which suggested a strong relationship between multimodal representations and science learning (Lemke, 2000). There were significant positive correlations between student multimodal representations and quiz scores but not exam scores. This study was also confounded by significant differences between sections at the beginning of the study which may have led to learning effects later. The dissimilarity between the tasks of writing during their homework and working on exams may be the reason for no significant correlations with exams. The power to interpret these results was limited by the number of the participants, the number of modal experiences by the students, and the operationalization of multimodal knowledge through the holistic rubric. These results do show that a relationship does exist between the similar tasks within science writing and quizzes. Students may also gain derived science literacy benefits from modal experiences on distal tasks in exams as well. This study shows that there is still much more research to be known about the interconnectedness of multimodal representational knowledge and use to the development of science literacy.

This study focused on two themes which have recurred in education since the 1980's: equity of educational outcomes for all students and reform in mathematics education. The problem addressed in this study concerned the apparent inability of large- scale reforms to meet equity goals. The purpose of the study was to increase understanding of this problem from both theoretical and practical perspectives. The study was influenced by feminist perspectives in the choice of theoretical framework and methodology. Focusing specifically on gender equity, the study was set in the context of a large-scale reform in the USA, Ohio's Statewide Systemic Initiative, Project Discovery.There were three major objectives in this study. First was to synthesise the literature concerning gender equity in mathematics education to produce a definition of the ideal Connected Equitable Mathematics Classroom (CEMC). There were two parts to the literature review: one concerning explanations for observed gender differences in mathematics education, and another concerning initiatives implemented to try to bring about gender equity in mathematics education.The second objective was to use the definition of the ideal CEMC, derived from the literature, to determine the extent to which reform had occurred in mathematics classrooms in Ohio. This was accomplished through the analysis of quantitative data collected from a random sample of teachers and principals across the state, and qualitative data collected from seven case study sites. The third objective was to determine the barriers to and facilitators of the realisation of equity goals in middle-school mathematics classrooms involved in Project Discovery. This was accomplished through a cross-site analysis of data collected at the seven case- study sites, with the analysis framed around the characteristics of the CEMC.The outcomes of the study are ++ / set out in terms of these three objectives, culminating in a discussion of the implications and challenges which the findings of this study pose for researchers, reformers, equity advocates and practitioners.

This thesis addresses the internationally recognised problem of transforming classroom teaching that is underpinned by transmissionist epistemologies, especially in the field of mathematics education. A constructivist-related theory of pedagogical reform was derived from the research literature in the fields of teacher cognition and conceptual change theory for the purpose of facilitating the radical reconstruction of teachers' centralist classroom roles and the development of pedagogies aimed at enhancing students' conceptual development. The thesis reports a collaborative action research study which was conducted with a teacher of high school mathematics who attempted to develop a constructivist- oriented teaching approach in his Grade 12 mathematics class.An interpretive research approach (Erickson, 1986) was employed to generate an understanding of the complex network of teacher beliefs and their constraining influence on the radical reform of the teacher's centralist pedagogy. An emergent research focus was the efficacy of the theory of pedagogical reform, especially its underpinning constructivist theory which shaped my discourse with the teacher during the collaborative phase of the study. Data were obtained from classroom participant- observations, teacher and student interviews, the teacher's report on the study, and an application of the repertory grid technique that was designed to investigate the teacher's implicit pedagogical beliefs.The major pedagogical reforms that resulted from the collaborative phase of the study included (1) the teacher's adoption of the role of teacher as learner; and (2) the refinement, rather than radical reform, of his centralist classroom role of teacher as informer. Although the reforms provided enriched opportunities for the development of students' algorithmic abilities, they largely failed to enhance students' mathematical ++ / conceptual development.Interpretive analyses were conducted from multiple constructivist-related perspectives (radical constructivism, social constructivism, critical constructivism) in order to generate a better understanding of the narrow scope of the pedagogical reforms. These analyses revealed (1) that rationalist preconceptions of mathematics and mathematical cognition underpinned important aspects of both the theory of pedagogical reform and the teacher's refined centralist classroom role of teacher as interactive informer, and (2) that the teacher's technical rationality, which was buttressed by personally constraining beliefs, maintained the ascendency of his technical curriculum interests and sustained his centalist classroom role of teacher as controller.The results of this thesis suggest strongly, therefore: (1) that pedagogical reforms which are based largely on a cognitivist theory of constructivism are susceptible to being subsumed by a powerful technical-rationalist ideology, and (2) that the hegemonic nature of this ideology is capable of disempowering reform-minded teachers from realising the practical viability of their constructivist-related ideals.The thesis recommends that future constructivist-related pedagogical reform in high school mathematics be based on a critical constructivist perspective which focuses attention on the curriculum interests that govern teaching and learning activities. In particular, it is recommended that reform-minded teachers establish a Critical classroom discourse which aims to attain a balanced rationality by making visible and subject to critical examination the hidden frames of reference that constitute the prevailing rationality of the traditional mathematics classroom.

This thesis records a study of major change. The study was designed to reveal and address the implications for teachers of primary mathematics, of moving from test-based assessment to a base built upon a balanced blend of norm-referenced and criteria-based assessments. In developing embedded authentic assessment through a process portfolio model, the teachers looked to change from the assessment of learning to assessment for learning. Consequently, through the efforts of the teachers involved, their students and those students’ parents, the study traced a substantial pedagogical restructure. Based on an interpretative methodology, this study of significant assessment restructure used mainly qualitative approaches to data collection and analysis, supplemented by limited quantitative data. Interviews, participant observer interactions, surveys and joint teacher discussion and planning sessions were effective in mapping the change. Through frequent interaction, participating teachers shared their emerging understandings, along with difficulties and successes in the evolution and implementation of an effective, flexible process portfolio. From the beginning of the evolution, teachers working together to bring about improvements that would lead to students perceiving mathematics as meaningful, engendered a strong feeling of excitement, curiosity and ‘team’. As the change progressed the team identified and met a range of challenges, not the least of which was gaining an understanding of the nature and function of a process portfolio strategy as against the product portfolio which was in use at that time in the study school. The resultant change was not implemented without barriers. / Of prime concern across the group of teachers involved was the perennial problem of finding development time in what were already busy teaching days. However, for the change to be meaningful and lasting, it was imperative that the teachers invested considerable time in assuming ownership through genuine engagement in the evolution of the new concept. The engagement saw teachers experience first-hand the application of constructivist learning theory. It was an approach to learning that was largely unfamiliar to them and one they needed to understand in developing a successful process portfolio model. The study of that learning and the resultant change illustrated that a well-designed process portfolio structure offers widely diverse opportunities for teachers and students to work meaningfully with authentic mathematics. The enthusiastic prolonged engagement on the part of the students, with notable parental support, was deemed by the participant teachers to be suitable reward for the time and effort that they invested over the two years of the study. Following the teachers’ prolonged commitment, the emergent portfolio was shared through an in-house booklet written to encourage other teachers to adopt authentic assessment, Process Portfolios in Primary Mathematics: A Guide. Within the booklet, explanation and illustration of the rationale, form and function of the unique process portfolio model offers starting points for others, should they embark on a similar course of assessment change in search of real student engagement in understanding mathematics. Subsequent sharing of the results of the study with the wider profession through journal articles and conference workshops is to be based on the contents of the guide booklet.

Mathematics plays a key role in many of today’s most secure and financially rewarding careers. In almost every sector of the economy, a substantial core of mathematics is needed to prepare students both for work and for higher education. The impact of computers and information technology in areas as diverse as manufacturing and advertising means that understanding mathematics becomes more important because it provides students with basic prerequisites in other useful areas such as problem-solving. Technical students in Brunei are trained with the skills needed in the world of industry and commerce to become competent workers and many of them continue to pursue higher education. They need the right balance of mathematics that can prepare them for both purposes. Considered to be academically weak, and coming from the system (high schools) whose teaching approaches benefit abstract learners, these students need to be motivated and have their interest in mathematics nurtured. This study is an attempt to improve the mathematical skills of technical students in Brunei by developing a teaching and learning package that can be used by mathematics instructors with their students. The package was designed to provide student-centred instruction and focuses on the learning environment aspects of “Teacher Support”, “Innovation”, “Cooperation”, “Task Orientation” and “Relevance”. These learning environment aspects were incorporated into each category of the ARCS motivational model (Keller, 1983b) for the purpose of enhancing motivation. It was anticipated that students’ mathematical understanding and attitude would be improved when their learning environment and thus their motivation was enhanced. / When the package was implemented among a group of technical students, they experienced an approach to the teaching of mathematics that shifted from instruction fostering the procedures of practice and memorisation toward instruction that emphasised mathematical inquiry and conceptual understanding. Integrated curricula and cooperative learning techniques were used to link both the mathematics understanding of materials and their composition to the application of materials in the world of work. The use of technology to pursue mathematical investigations by way of learning aids was encouraged because the impact of technology on education today cannot be ignored. A group of students from two classes were involved in the implementation of the package to determine its effectiveness, for a duration of eight weeks. By applying the pre-experimental design methodology to the study, pre-test and post-test were used to measure students’ cognitive and affective changes. Mathematics proficiency in the categories of procedural skills, conceptual understanding and problem solving abilities were measured and examined by comparing students’ pre- and post-test results. Other forms of assessment such as projects and graded class-work (and homework) and also the communication that took place between the students during discussions were analysed to further validate their mathematical understanding. The learning environment and attitude factors mentioned were identified and validated through surveys, observations and interviews. A learning environment instrument called the College Classroom Environment Inventory (CCEI) was adapted for the purpose of measuring students’ perception of the learning environment. / Another instrument named the Attitude Towards Mathematics survey was designed to measure students’ attitude towards mathematics. Both instruments were created, validated and then used to measure students’ affective changes (before and after package implementation) and thus evaluate the efficacy of the package. Besides the quantitative data obtained, the qualitative data from observations and interviews was used to confirm, explain and verify results. The results obtained from this study demonstrated students’ improved cognitive outcome in all areas of mathematical proficiency measured. As for the affective outcome, there were improvements in students’ perception of the classroom environment and also in the attitude category of “Importance” where more post-test than pre-test students agreed on the importance of mathematics in everyday life. The result also indicated associations between cognitive outcomes and a number of the learning environment scales. Students who experienced the package also demonstrated better mathematical understanding compared to those who did not. Students, instructors, curriculum developers and administrators should benefit from the results of this study. The study also provides a starting point for more research of this kind to be carried out for the benefit of technical students in Brunei in particular, and for mathematics students generally.

Research suggests that many high school students are not learning mathematics of value from a personal or an employment perspective. School mathematics often consists of applying memorised algorithms to exercises that do not meaningfully connect with the student's experience, and hence do not lead to the construction of meaningful mathematics concepts by the student. Moreover, most high school mathematics curricula give students a false idea of the essence of mathematics: Instead of understanding mathematics as another powerful lens through which to view the world, and a creative, enjoyable endeavour, it is seen as mere calculation or esoteric gobbledegook. Authentic learning experiences involve a different perspective on both what passes as mathematics and how students learn to mathematise. The study examined high school mathematics knowledge from several perspectives, and sought, through an empirical study, to enhance student learning and attitude towards mathematics through authentic learning. A class of Year 8 students learnt several units of mathematics primarily by authentic methods, using problems or interesting phenomena in the students' own experience, or otherwise meaningful to the student. Qualitative data was collected by multiple methods, including video recordings. Surveys were administered to five classes of Year 8 students and their parents at the beginning and at the end of the semester in which most of the empirical research took place. This allowed a comparison of attitudes towards mathematics between the experimental class and the other classes. A comparison of achievement was also made. / The results indicate that employing authentic learning experiences may enhance learning and attitude towards mathematics. However, prior transmission teaching methods presented a significant barrier to student acceptance of authentic learning. Furthermore, there remain grave problems with other aspects of current high school mathematics curricula, specifically the mathematics content and the assessment style, which act against the full implementation of authentic learning. These problems are investigated and possible future paths considered.

This action research study was conducted to investigate the efficacy of a constructivist approach to the training of first-year Chinese student mathematics teachers in the Hong Kong Institute of Education where I am employed. A four-stage teaching model was designed, based on the learning theory of constructivism and taking into particular consideration the characteristics of Chinese learners: the maintenance of hierarchical and group harmony and high achievement motivation. In order to determine whether the application of this model in a methodology could alter the teaching beliefs of newly enrolled students, a two-phase procedure was employed. First-year students in each phase of the study were involved in solving a teaching problem. Through self-articulation, group- and class-discussions and self-reflection, the students were examined to determine any change in their beliefs about teaching mathematics. Prior beliefs about mathematics teaching, and beliefs held at the end of the methodology module were determined and compared in order to determine if new learning was in evidence. The creation of an authentic interactive learning environment to foster the kind of learning desired - a potentially safe, trusting and non-judgemental environment for free disclosure of students' opinions and feelings about mathematics teaching - was investigated. Data was generated by different quantitative and qualitative methods. Findings were cross-checked by a critical colleague and through my observation and reflections, and these were recorded as clearly, orderly, and accurately as possible. The first phase results were employed to inform and to improve the teaching of the same methodology module in the second phase. / Findings in the two phases were indicative of the creation of a genuine social constructivist learning environment in which student teachers enjoyed their learning. Student teachers in the second phase implementation of my study indicated an understanding of their role in a constructivist classroom - to construct their own theories of teaching mathematics, to assist their peers in knowledge construction and to learn to learn. Student teachers in the two cohorts were found to hold entrenched constructivist beliefs about teaching mathematics. They agreed that the teacher's role was a facilitator of learning and that persistent questioning could alter knowledge about mathematics. However, at the conclusion of the module, the Phase I students seemed to re-adopt traditional approaches to teaching, whereas the Phase II students exhibited two different perspectives - an indication of the instability of their teaching beliefs. The Phase II student teachers, nevertheless, showed that they became more aware of sequencing the various interactive activities for their pupils in secondary schools. In the actual teaching, they professed their inability to realise their teaching ideals because of their inexperience in teaching and of the unexpected situations in the school settings. The present research study adds to the paucity of literature in two areas. First, the employment of a constructivist approach in the preparation of teachers of junior secondary mathematics (for pupils of age between twelve and fifteen), especially in the training of Chinese student mathematics teachers. / Second, the study of a higher education lecturer conducting research to improve his/her own practice. Undeniably, further research on models to change student teachers' prior knowledge (about mathematics, about the nature of mathematics, and about the teaching and learning of mathematics), on factors affecting the instability of beliefs, and on models to facilitate continuous development of the teaching professionals are necessary if not exigent.