Level of visual geometry skill towards learning style Kolb in junior high school

This study aims to conduct an in-depth analysis of the visual thinking level of junior high school students with the learning style of assimilators, converges, accommodators, and divergers in solving geometry problems. The type of research used is qualitative research with a grounded theory and case study design. The subjects studied were junior high school students consisting of 6 of 56 students. Data were collected through a learning style inventory (LSI) test given to 56 students to group participants based on the learning style of the Kolb model, then a geometry problem-solving test and interviews were given to 6 students, namely two assimilator students, one converges, one accommodator, and two diverger students. The analysis is based on data from written test results and interviews. Then, time triangulation is carried out to obtain valid research data. The analysis was conducted based on data from written test results and interview results paired with video recordings. Then, triangulation of time is carried out to obtain valid research data. The results of the analysis showed that assimilator students and converger students were able to achieve at the global visual level, namely being able to carry out visual thinking activities well in solving problems, illustrate the problem correctly in geometric drawings/objects, represent problems in mathematical symbols precisely and can express relationships between images well. While accommodator and diverger students can only reach the local visual level, they have yet to be able to show every visual thinking activity well in solving geometry problems, illustrating problems in geometry drawings that could be more precise, and solving rudimentary geometry problems

Mathematics education reform in Trinidad and Tobago: the case of reasoning and proof in secondary school

In Trinidad and Tobago, there have been substantive efforts to reform mathematics education. Through the implementation of new policies, the reformers have promoted changes in mathematics curriculum and instruction. A focus of the reform has been that of increasing opportunities for students to engage in reasoning and proving. However, little is known about how these policies affect the opportunities for reasoning and proof in the written curriculum, the teaching of proof, and students' learning. Furthermore, we are yet to know how the high-stake assessment measures interact with these new policies to impact the teaching and learning of proof. In this dissertation, my overarching question asks: What are the implications of reform on the teaching and learning of secondary school mathematics in Trinidad and Tobago? To answer this question, I conducted three studies, which examined the content, teaching, and students’ conceptions. All the studies are situated in the teaching reasoning and proof when introducing geometry concepts. In the first study, I conduct a curriculum analysis focused on examining the opportunities for reasoning and proof in the three recommended secondary school textbooks. In the second study, I conduct classroom observations of teachers’ geometry instruction focusing on opportunities for engaging students in reasoning and proof. In the third study, I examine geometry students’ conceptions of proof. The three studies are intended to provide an overview of the impact of reform on instructional issues in relation to the dynamics between teachers, student, and content (Cohen, Raudenbush, & Ball, 2003). For the first study, I adapt a framework developed by Otten, Gilbertson, Males, and Clark (2014) to investigate the quality and quantity of the opportunities for students to engage in or reflect on reasoning and proof. The findings highlight some unique characteristics of the recommended textbooks such as, (a) the promotion of the explanatory role of proof through the affordances of what I define as the Geometric Calculation with Number and Explanation (GCNE) exercises, (b) the necessary scaffolding of proof construction through activities and exercises promoting pattern identification, conjecturing, and developing of informal non-proof arguments, and (c) the varying advocacy for Geometry as an area in the curriculum where students can experience the work of real mathematicians and see the intellectual of proof in their mathematical experiences. All these characteristics align with the reformers’ vision for the teaching and learning of reasoning and proof in secondary school mathematics. In the second study, I examine the nature of the teaching of reasoning and proof in secondary school. I use classroom observations along with pre- and post-observations interview data of three teachers to determine (a) the classroom microculture (i.e., classroom mathematical practices and sociomathematical norms), (b) teachers’ pedagogical decisions, and (c) teachers’ use of the Caribbean Secondary Examination Certificate (CSEC) examination materials and textbooks. I also determine whether the teachers’ instruction demonstrate the four characteristics of reform-based mathematics teaching (Hufferd-Ackles, Fuson, & Sherin, 2004). My analysis of classroom observations of the three teachers suggests that their instructional practices exhibit elements of reform-based instruction. These include teachers’ use of open-ended and direct questions to solicit students’ mathematical ideas and teachers’ consideration of students as the source of mathematical ideas. Each teacher established sociomathematical norms that governed how and when a student can ask questions. In this case, questioning helped students articulate their ideas when responding to questions and clarifying their understanding of other's ideas when they posed a question. Teachers also established sociomathematical norms that outlined what counts as a valid proof and what counts as an acceptable answer during instruction. The aforementioned norms supported the expectation that students must always provide explanations for their mathematical thinking, which is another characteristic of reform-based teaching. Teachers used group work and whole class discussions to offer opportunities for collaborative learning, which facilitated their creation of a social constructivist environment for learning reasoning and proof. Teachers used the reform-oriented curriculum materials to provide opportunities for construction of proofs. However, the textbooks and curriculum were limited in their support for proving some geometrical results. Overall, the teachers emphasized the making and testing of conjectures, which afforded students with authentic mathematical experiences that promoted the development of mathematical knowledge. In the third study, I use the six principles of proof understanding (McCrone & Martin, 2009) to examine 21 students’ conceptions of proof. I use semi-structured interviews to gather students’ perspectives of (a) the roles of proof, (b) structure and generality of proof, and (c) the opportunities for proof in the curriculum materials. The findings indicate that the students considered proof as serving the roles of explanation, verification, systemization, and appreciation in mathematics. The latter role helps students see the value and purpose of the mathematical results they learn (a) for applications during problem solving and (b) within the axiomatic system of Geometry results. The aforementioned roles also help students see the intellectual need for reasoning and proof in their mathematical experiences. Students’ talk suggests that, their teachers’ and the external examiners’ expectations of the structure, generality, and validity of proof influence their notions of what constitutes a proof. Students also consider the examination opportunities that require the development of reasoned explanations as possible opportunities to construct proof arguments. The combined findings of these three studies could help researchers understand the implications of the recent reform recommendations on the teaching and learning of proof in Trinidad and Tobago. Firstly, these findings can be useful to policy makers and education stakeholders in their future efforts for developing the national curriculum, revision or development of instructional policies, and recommendations of textbooks and instructional support materials. Secondly, these findings can help curriculum designers, examiners, and teachers in creating future opportunities in the national curriculum and CSEC mathematics syllabus to support students’ learning of proof in Trinidad and Tobago. Thirdly, these findings can help educational stakeholders understand the type of support that is needed for teachers’ future professional development and students’ competency with reasoning and proof on CSEC examinations. This international study is a case of the larger issues surrounding reform implications in a centralized governed educational system, which offers uniform prescriptive guidance for teaching and uniform curriculum support for learning. Furthermore this work potentially adds to the ongoing discussions in mathematics education about the interplay between policy, practice, and student learning

The Impact of van Hiele-based Geometry Instruction on Student Understanding

Developments over the last three decades provide momentum for revising high school geometry instruction as recommended by the van Hieles. Cognitive learning theories, brain research, multiple intelligence theories, revised national and state standards and computer technology-based tools all contribute to the rationale and the means to deliver instruction that enables students to construct knowledge and understanding through a sequential process of exploration, inductive and deductive reasoning. A Regents Geometry unit on quadrilaterals was developed based on these theories and techniques. Forty-three students enrolled in the high school Regents Geometry course received instruction using the newly developed materials. The results of these students showed improvement over the results of the previous year\u27s students under more traditional geometry instruction

Intuition and the Autonomy of Philosophy

The phenomenology of a priori intuition is explored at length (where a priori intuition is taken to be not a form of belief but rather a form of seeming, specifically intellectual as opposed to sensory seeming). Various reductive accounts of intuition are criticized, and Humean empiricism (which, unlike radical empiricism, does admit analyticity intuitions as evidence) is shown to be epistemically self-defeating. This paper also recapitulates the defense of the thesis of the Autonomy and Authority of Philosophy given in the author’s “A Priori Knowledge and the Scope of Philosophy” (Philosophical Studies, 1996)

A Model, Secondary Level, Mathematics Curriculum Developed in Alignment with Washington State Essential Academic Learning Requirements, Easton School District

The purpose ofthis project was to design and develop a model secondary level mathematics curriculum, in alignment with Washington State Essential Academic Learning Requirements, for the Easton School District in Washington. To accomplish this purpose, a review of current research and literature regarding Washington State Essential Academic Learning Requirements related to secondary mathematics was conducted. In addition, related information from selected sources was obtained and analyzed

Exploring grade 11 learners’ mathematical problem-solving skills using Polka’s model during the learning of Euclidean geometry

The skill of Problem-solving in Mathematics is very imperative. Poor performance by most South African learners in schools and international tests such as the Trends in International Mathematics and Science, calls for emphasis to be placed on problem-solving in the teaching and learning of Mathematics. Euclidean Geometry is perceived, especially by learners, to be one of the difficult components of Mathematics. Thus, the aim of this study was to explore and develop the mathematical problem-solving and geometric skills of Grade 11 learners in Euclidean Geometry. Polya’s model of problem-solving was employed in geometric skills development as a tool for intervention. The concepts Geometry and problem-solving formed the conceptual framework of the study, while the social cognitive theory constituted the theoretical framework. A case study was used as the main research method following a mixed method approach within an interpretivist paradigm. Purposive and convenience sampling methods were used in the selection of both the Mathematics class and the six learners whose work was further observed and analysed. Data about the geometric skills displayed by the learners was gathered using a moderated pre-intervention test; observations; document analysis; a moderated post-intervention test; and focus group interviews. Data was analysed quantitatively using descriptive statistics and qualitatively using thematic analysis. In the pre-intervention test, learners did not bring with them expected geometric skills to the classroom before they were introduced to grade11 Geometry content and when doing problem-solving during intervention, the four stages of model used were not necessarily following each other in a linear sequence with most of the learners not applying the fourth stage “look back”. In the post intervention test, the frequency of use and application of most geometric skills improved in comparison to the pre-intervention test; the frequency of correct and inappropriate application of the skills increased at the expense of incorrect application. Learners appreciated the four stages model and gave their views related to the challenged faced during the use of the four stages model and the challenges revealed include: practice related challenges, challenges specific to certain learners, concept related challenges, curriculum-related challenges, model application challenge, and context related challenges. The study concludes that the effective use of Polyas’ four stages model can yield great results in developing learners’ geometric and problem-solving skills. The study recommends that teachers give more attention to prior geometric knowledge, teaching of geometric theorems, teaching of geometric problem-solving, and the learning environment.Thesis (MEd) -- Faculty of Education, Secondary and Post School Education, 202

Realistic Mathematics Education as a lens to explore teachers’ use of students’ out-of-school experiences in the teaching of transformation geometry in Zimbabwe’s rural secondary schools

The study explores Mathematics educators’ use of students’ out-of-school experiences in the teaching of Transformation Geometry. This thesis focuses on an analysis of the extent to which students’ out-of-school experiences are reflected in the actual teaching, textbook tasks and national examination items set and other resources used. Teachers’ teaching practices are expected to support students’ learning of concepts in mathematics. Freudenthal (1991) argues that students develop their mathematical understanding by working from contexts that make sense to them, contexts that are grounded in realistic settings. ZIMSEC Examiners Reports (2010; 2011) reveal a low student performance in the topic of Transformation Geometry in Zimbabwe, yet, the topic has a close relationship with the environment in which students live (Purpura, Baroody & Lonigan, 2013). Thus, the main purpose of the study is to explore Mathematics teachers’ use of students’ out-of-school experiences in the teaching of Transformation Geometry at secondary school level. The investigation encompassed; (a) teacher perceptions about transformation geometry concepts that have a close link with students’ out-of-school experiences, (b) how teachers are teaching transformation geometry in Zimbabwe’s rural secondary schools, (c) the extent to which students’ out-of-school experiences are incorporated in Transformation Geometry tasks, and (d) the extent to which transformation geometry, as reflected in the official textbooks and suggested teaching models, is linked to students’ out-of-school experiences. Consistent with the interpretive qualitative research paradigm the transcendental phenomenology was used as the research design. Semi-structured interviews, Lesson observations, document analysis and a test were used as data gathering instruments. Data analysis, mainly for qualitative data, involved coding and categorising emerging themes from the different data sources. The key epistemological assumption was derived from the notion that knowing reality is through understanding the experiences of others found in a phenomenon of interest (Yuksel & Yildirim, 2015). In this study, the phenomenon of interest was the teaching of Transformation Geometry in rural secondary schools. In the same light, it meant observing teachers teaching the topic of Transformation Geometry, listening to their perceptions about the topic during interviews, and considering how they plan for their teaching as well as how students are assessed in transformation geometry. The research site included 3 selected rural secondary schools; one Mission boarding high school, a Council run secondary school and a Government rural day secondary school. Purposive sampling technique was used carefully to come up with 3 different types of schools in a typical rural Zimbabwe. Purposive sampling technique was also used to choose the teacher participants, whereas learners who sat for the test were randomly selected from the ordinary level classes. The main criterion for including teacher participants was if they were currently teaching an Ordinary Level Mathematics class and had gained more experience in teaching Transformation Geometry. In total, six teachers and forty-five students were selected to participate in the study. Results from the study reveal that some teachers have limited knowledge on transformation geometry concepts embedded in students’ out-of-school experience. Using Freudenthal’s (1968) RME Model to judge their effectiveness in teaching, the implication is teaching and learning would fail to utilise contexts familiar with the students and hence can hardly promote mastery of transformation geometry concepts. Data results also reveal some disconnect between teaching practices as espoused in curriculum documents and actual teaching practice. Although policy stipulates that concepts must be developed starting from concrete situations and moving to the abstract concepts, teachers seem to prefer starting with the formal Mathematics, giving students definitions and procedures for carrying out the different geometric transformations. On the other hand, tasks in Transformation Geometry both at school level and the national examinations focus on testing learner’s ability to define and use procedures for performing specific transformations at the expense of testing for real understanding of concepts. In view of these findings the study recommends the revision of the school Mathematics curriculum emphasising pre-service programmes for teacher professional knowledge to be built on features of contemporary learning theory, such as RME theory. Such as a revision can include the need to plan instruction so that students build models and representations rather than apply already developed ones.Curriculum and Instructional StudiesD. Ed. (Curriculum Studies

Predictive relationships of teacher efficacy, geometry knowledge for teaching, and the cognitive levels of teacher practice on student achievement.

This study explored the predictive relationships of teacher efficacy, teacher knowledge, and teacher practices with student achievement. More specifically, secondary mathematics teachers\u27 efficacy beliefs, geometry knowledge for teaching, and the cognitive complexity of the teachers\u27 classroom practices were examined for 72 teachers in both urban and rural districts across Kentucky, along with the student achievement data of their students. Teacher and student data were obtained from the NSF-funded Geometry Assessment for Secondary Teachers (GAST) project, which administered geometry teacher knowledge assessments at the beginning and end of the school year, and collected cognitive complexity data from lessons through three classroom observations. Student achievement was measured using a modified geometry end-of-course assessment with a geometry readiness test as a covariate. Teacher efficacy data was obtained from the same teachers through an online survey at the end of the GAST project. Correlation, multiple regression, and hierarchical linear modeling techniques were used to analyze the data. Results revealed that the cognitive level of teacher practices significantly predicted student achievement. This finding provides support for increasing teacher awareness of the importance of high cognitive instruction by helping them recognize the essential features of classroom activities that provide this instruction and assisting them to plan and implement high cognitive tasks in their classrooms