This study explored the beliefs of pre-service secondary mathematics teachers about project-based learning (PBL), as they encountered a project-based learning high school where they implemented a project-based unit of instruction. A qualitative study was conducted with one undergraduate cohort in a higher education science, technology, engineering, and mathematics (STEM) national initiative that has 40 U.S. replication sites. Using teaching philosophy statements and focus group discussions, the beliefs of STEM pre-service secondary teachers are made visible. The findings from this study reveal a recurring theme: the process of how these pre-service teachers seemed to evolve and mature as teachers, from novice toward becoming an expert, as they asked themselves internal questions that are common to developing teachers. These pre-service STEM teachers experienced 1) internal questions about their own growth as a potential teacher, as evidenced through their verbal and written statements; 2) tension between PBL content and pedagogy; and 3) tension between practice and theory. The findings also infer that there are potential critical variables that may contribute to pre-service teachers’ beliefs. Those variables identified were the following: 1) the sequence of when the project-based instruction (PBI) course was taken; 2) time, as related to when the participant took the PBI course in relationship to the final semester when they were engaged in apprentice teaching; and 3) the field placement location during the apprentice teaching semester.

Concerns based adoption model (CBAM) instruments were used to examine instructional practices of six graduates from a highly stylized, inquiry-based secondary math and science preparation program. Teachers were in the first or second years of teaching mathematics in six different secondary settings, ranging from poverty to wealthy schools. CBAM assumptions were tested. The primary assumption about concerns was that new teachers’ highest concerns would be within the self and task dimensions. According to Hall and Hord, it was assumed that the levels of use are typically in the orientation and preparation stages as a new teacher begins to implement an innovation, in the case of this study, inquiry-based instruction. All three instruments of the CBAM model were used for data collection and included: the Survey of Concerns Questionnaire, Innovation Components Configuration Map, and Levels of Use matrix. Teachers were observed, interviewed, and surveyed, three times each, across a five-month period. The findings from this study showed that the teachers had similar concerns and levels of use, which supported the assumptions outlined by the CBAM principles. Across the six teachers, the self and task concerns were high, aligning with the assumptions. However, unrelated and impact dimensions were noted, in opposition to the assumption. Likewise, assumptions of the levels of use were upheld in the orientation and preparation levels of use noted in the observations. Some mechanical levels of use were observed for a few of the teachers, an anomaly to the assumption.

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Recently there has been a keen interest in the area of mathematics and finding the best methods of instruction. For instance, the No Child Left Behind Act (NCLB) has placed new levels of accountability on educators for the success of students with their education, especially in mathematics. Certain areas of mathematics, such as Algebra, have been known to challenge students to think abstractly. This has become a difficult task for educators to accomplish. The challenge of teaching algebra becomes apparent when students do not comprehend the abstract reasoning of algebra. Many students need help with the transition from numerical calculation to the abstract reasoning required for algebra. This dissertation focuses on the best approaches to helping students with this transition. This dissertation investigates a mathematics program called Hands-On Equations (HOE), which is designed to help students learn abstract concepts taught in algebra with less difficulty. The program concentrates on the transition from numerical calculation to abstract reasoning by utilizing manipulatives. The objective of the study is to investigate the effectiveness of HOE in 9 th and 10 th grade. The research uses three pretests, three posttests, a three week retention test, a six week retention test, and benchmark tests to evaluate the academic growth of students in two set groups. The collected data is then quantitatively analyzed by applying simple t-tests and an ANOVA. Analysis of the data endorses HOE as being effective with solving linear equations at Level 1, Level 2, Level 3, and with the three-week retention tests, which indicates HOE may be a positive factor in achieving success with linear equations. However, analysis of the data revealed HOE is not as effective with the six-week retention test or the benchmark test which indicates after six weeks and beyond, students do not retain the information to be successful on end of the year exams such as benchmarks. The findings from this study may be useful to educators who are contemplating implementing HOE or other such programs at their schools.

This study investigated the interaction of mathematics teachers with learners’ mathematical errors. The teachers’ verbal interaction with learners’ errors during learning periods and their written interaction in assessment tasks were explored. The study was contextualized in grade 9 secondary school classrooms in the Gauteng province of South Africa. The investigation was epistemologically underpinned by constructivism/socio-constructivism. The investigation was qualitatively approached through four case studies. Structured and semi-structured interviews, classroom observations and learners’ written assessment tasks were employed as sources of data. The participating teachers were described in terms of their beliefs about mathematics, their beliefs about learners’ mathematical errors, their observed prevalent teaching approach and their professed and enacted interaction with learners’ mathematical errors. Within-case and cross-case comparisons ensued. The findings proposed that when teachers believed that the value of learners’ errors was vested in the corrections thereof, rather than employing these opportunities for discussion, valuable opportunities for learners to develop and improve their meta-cognitive abilities might potentially be lost. The findings further indicated that a focus on the mere correction of learners’ errors probably denied learners opportunities to develop a mathematical discourse. The results of the investigation illuminated that an emphasis on achievement during assessment, together with a disapproving disposition towards errors among teachers and learners, were hindrances. They acted as barriers to engendering a socio-constructivist learning environment in which interactions with learners’ errors could enhance learning and establish a negotiating mathematical community. A concurrence between the teachers’ prevalent teaching approach and their mathematical beliefs was confirmed. However, in two of the four cases, a dissonance was revealed between their prevalent teaching approach and their interaction with learners’ errors. Interaction with learners’ mathematical errors was hence identified as a separate and discrete component of a teacher’s practice. The findings suggest the explicit inclusion of error-handling in reform-oriented teacher-training and professional development courses to utilize learners’ mathematical errors more constructively. / Dissertation (MEd)--University of Pretoria, 2010. / Science, Mathematics and Technology Education / unrestricted

Researchers have found that teachers' decisions affect students' opportunity to learn. Prior researchers have investigated teachers' decisions while planning, implementing, or reflecting on lessons, but few researchers have studied teachers' decisions and their reasoning throughout the teaching process. It is important to study teachers' reasoning for why they make the decisions they do throughout the teaching process. Furthermore, because inservice and preservice teachers differ in experience and available resources that they draw on while making decisions, it is helpful to consider the resources PSTs' draw on while reasoning. Curricular reasoning is a framework that describes teachers' thinking processes when making decisions during the teaching process. This study investigated secondary preservice teachers' decisions and curricular reasoning throughout the teaching process. Data were collected from two groups of secondary preservice teachers in a mathematics methods course focused on student thinking and mathematics. Results revealed that the preservice teachers used all seven curricular reasoning strands, especially drawing on mathematical meanings, mapping learning trajectories, and considering learners' perspectives. Specifically, this study demonstrates ways in which preservice teachers reason about their decisions and the intertwined nature of their curricular reasoning. The results from this study also imply that it may be helpful to consider the resources PSTs have access to, including their instructor, and that the order of their lesson planning may allow support for the mathematical learning trajectories within individual lessons. This study also provides validation for the curricular reasoning framework described by Dingman, Teuscher, Olson, and Kasmer (in press), provides subcategories of curricular reasoning strands, and has implications for teacher education.

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Thesis (M. Tech. (Education)) -- Central University of Technology, Free state, 2011 / This study examines the problems encountered by educators regarding the implementation of the National Curriculum Statement in mathematics in grades 10-12. The first aim of the study was to provide solutions to problems regarding training experienced by FET mathematics educators. The second aim was to identify problem areas in the NCS that frustrate mathematics educators teaching in the FET band and to identify areas that appeal to these educators. The third aim was to provide guidelines to assist educators with lesson planning in mathematics in the FET band. The fourth aim was to provide guidelines for appropriate assessment in mathematics in the FET band. The fifth aim was to provide guidelines for the effective integration of OBE in the teaching of mathematics in the FET band. The field work was executed by administering a questionnaire to a randomly selected sample of fifty two educators teaching in the FET band. Interviews were semi-structured, flexible and yielded additional information to that of the questionnaire. The questions of the interview were directly related to the objectives of the study and followed a given sequence that was adhered to in each interview process. The researcher arranged to interview one educator from each of the 15 randomly selected schools in the Motheo-district, but only 10 educators responded positively in the interview process, other educators could not avail themselves on that day. The researcher analysed the responses according to the respondent‟s personal particulars. Descriptive analysis of the sample data for section B of the questionnaire were then done, using respondent counting, percentages and the average for the responses of each statement. This study revealed that educators differ in terms of the problems that they encountered in implementing the NCS in mathematics. The findings from this study pointed out problems such as educators receiving inadequate training on implementing the NCS in mathematics. It was also revealed that educators had not been visited by the departmental officials in their schools for monitoring the implementation of the NCS in mathematics. The last finding showed that teaching and learning support material arrived late during 2008 and that there was a large shortage of such material. The result of the study provides invaluable baseline information with regard to the problems encountered by the educators in the implementation of the NCS in mathematics. On the basis of the findings of this study, a number of recommendations for the implementation of curriculum change in mathematics on FET level are given in Chapter 5.

Thesis (PhD (Curriculum Studies))--University of Stellenbosch, 2007. / This action research study was aimed at establishing the importance and role of communication, and determining to what extent it impacts on formative assessment in the mathematics classroom with particular reference to feedback. During the first cycle of research it was evident that conditions within the mathematics classroom were preventing this from being realised. If we as researchers were to assess the nature of communication patterns within the classroom situation, then those communication patterns should have existed. Our findings reflect that teachers were generally in control of all aspects of communication of their learners, and that communication was usually a type of monologue, with very limited response from learners to closed questions, (characterised by “yes” and “no” responses), which were frequently posed. The feedback from learners (perhaps inadvertently ignored), was not optimally utilised to enhance learning. Through observation it was determined that teachers’ ability to engage learners meaningfully for longer periods, or to consciously reflect upon their actions, needed to be developed. Praxis as research paradigm, which is based on reflection and appropriate response or action geared towards improving the circumstances or conditions of the people concerned (in this case teachers), underlies this study. This research is furthermore based on the following learning theories: enactivism, constructivism, facilitation theory, action learning, andragogy, reification, and situated learning...