More than half a century ago, Piaget concluded from an investigation of children’s representational thinking about the nature of space that the development of children’s representational thought is topological before it is Euclidean. This conclusion, commonly referred to as the “topological primacy thesis,” has essentially been rejected. By giving emphasis to the ideas that develop rather than the order in which they develop, this work set out to develop a new form of non-metric geometry from young children’s early and intuitive topological, or at least non-metric, ideas. I conducted an eighteen-week teaching experiment with two children, ages six and seven. I developed a new dynamic geometry environment called Configure that I used in tandem with clinical interviews in each of the episodes of the experiment to elicit these children’s non-metric conceptions and subsequently support their development. I found that these children developed significant and authentic forms of geometric reasoning. It is these findings, which I refer to as qualitative geometry, that have implications for the teaching of geometry and for research into students’ mathematical reasoning. / text

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Teacher educators use children's thinking activities as a means to prepare prospective teachers to teach mathematics. Research in methods courses and student teaching practica has shown these types of activities help prospective teachers deepen their own mathematical knowledge as well as better understand how children think. This study investigates prospective teachers' perceptions in a mathematics for teaching course when using children's thinking activities. Specifically, this study documents how four case study students perceived of the activities, their own and children's mathematical thinking, and the role of language in mathematics teaching and learning.Observations, interviews, and written class work were analyzed through the generation of themes within and across cases. It was found that all of the participants perceived the children's thinking activities as beneficial to providing insight into how children think mathematically. In contrast, two of the four participants did not perceive the activities as supportive of their own mathematical learning. However, the participants' reflections, both on children's mathematical thinking and their own, showed that all participants grappled with important mathematical concepts. Specifically, through consideration of their own thinking, children's strategies, and children's understanding, participants discussed the fractional whole, the meaning of operations, and connections among strategies, among other topics. This difference between how the participants perceived the use of children's thinking activities and what was evidenced through their reflections suggests that participants might have felt a stigma in studying elementary mathematics.While the participants did not often reflect on the role of language in teaching and learning mathematics, they did articulate various ideas regarding language and mathematics, in particular regarding the instruction of English Language Learners (ELLs). They discussed ways to help ELLs acquire mathematical language, which three participants saw as more than just vocabulary. All participants also thought it was important for children to express their mathematical understanding. However, two participants seemed to conceive of a unique way to express that understanding. This finding suggests that prospective teachers need to be critically exposed to literature regarding mathematics instruction of language-minority children.

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In an attempt to show how rational explanation of human and animal behaviour has a place in the scientific explanation of our physical world, Fodor advances the language of thought hypothesis. The purpose of this dissertation is to argue that, contrary to the language of thought hypothesis, we need not possess a linguistic internal representational system distinct from any natural language to serve as the medium of thinking. I accept that we have an internal representational system, but by analyzing Fodor's theory of content, I show Fodor's argument that the internal system must be as expressive as any natural language, which he uses in arguing that the internal system is the linguistic medium of thought, is unsound. Distinguishing an informational theory of content from a causal theory of content, which Fodor conflates, I argue that internal representations, whose content is determined by information they carry, cannot be related in a way that corresponds to semantic associations between terms in natural languages, reflecting actual associations of items in the world. Furthermore, provided certain animal cognition, which is homogeneous with human cognition, can be explained without requiring that the internal system possess anything corresponding to the logical connectives, the internal system need not possess anything corresponding to the logical connectives. I give such an explanation of animal cognition by developing an approach to content in the Rylean/Dennettian tradition, based on the notion of embodied cognition, in which animals embody the hypotheses they entertain in virtue of their total dispositional state, rather than explicitly representing them. It follows that there are two features of natural languages, semantic associations of terms and possessing logical connectives, that the internal system need not have. Hence a rational interpretation of linguistic behaviour need not be derived from an intentional interpretation of the transformations on int

In the literature on reflection in teaching, authors frequently lament the lack of clarity in understandings of this concept, despite its wide acceptance as a phenomenon beneficial to teaching and learning. This dissertation reports a study of this literature that attempts to clarify the meaning of reflection and to establish a methodology for examining such a complex concept. Three analyses, each intended to explore the literature on reflection from a different perspective, comprise the study. The first is an analysis of the literature on reflection in three professional communities---continuing professional development, higher education and teacher education---to establish general themes in this literature. The second analysis examines definitions of reflection from the three communities, focusing in particular on processes and rationales of reflection. The third analysis explores a variety of critiques of reflection to determine predominant epistemologies and recurring themes in the literature. The merging of the results of the three analyses leads to a framework for understanding reflection. This integrative framework highlights the importance of underlying epistemologies as the bases for different understandings of reflection and shows the intricate interrelationships among four major themes in the literature: the processes involved in reflection, the rationales behind it, the context in which it occurs, and its connection to action. The framework also points to the link between the self and the reflective context, the possibilities of reflection in-, on-, for-, and as-action, the unclear connection between the cognitive and affective processes and the movement from internal to external rationales. The study contributes both conceptually and methodologically by making sense of the range of ways reflection has been understood and by providing a possible model for exploring a complex concept. It provides a consistent language for discussing reflection, demonstrates the complexities of the concept and the interrelationships of the themes contained in the literature, allows for the situating of individual works within the literature, increases understanding of the connection of reflection and action, and helps to position the concept of reflection within broader theories of cognition and social practice.

This thesis explores how the University of Canterbury has responded to the Tertiary Education Strategy's (2002-2007) concerns vis-à-vis declining Maori participation and unsatisfactory rates of retention and completion in mainstream universities. This research is based on the qualitative method of in-depth taped interviews with twenty-five participants enrolled as 'Maori' at Canterbury in 2004. Notwithstanding increased recognition of biculturalism at Canterbury, issues relating to entrenched monoculturalism identified by Grennell (1990), Clothier (2000) and Phillips (2003) appear to be largely unresolved. Participants confirm the Ministry of Education's (2001) contention that Personal and Family Issues, Financial Difficulties, Negative Schooling Experiences, Inadequate Secondary Qualifications, Transitional Difficulties, Isolation, Unwelcoming Tertiary Environments and Inappropriate Support Structures are barriers to Maori success. However, testimonies reflect that these barriers represent exogenous factors derived from state and institutional policies and practices, not endogenous factors attributable to Maori genes, cultural socioeconomic status or engagement with the system. The Tertiary Education Strategy's (2002-2007) devolution of responsibility to institutions to address ethnic disparities in human capital imposes the same structural constraints on Maori that undermine achievement in the compulsory sector. The types of support structures participants identify as conducive to addressing deficit cultural capital and fostering academic achievement are Maori-centred initiatives, devoid of the deficit ideology that underpins mainstream assimilationist interventions; and or institutional provisions that incorporate greater stakeholder input with improved accountability and monitoring mechanisms that safeguard against recourse to deficit rationalizations for underachievement. Maori parity in engagement with the tertiary education sector is contingent upon the state and its institutions redressing the cumulative effects of the colonial and neo-colonial marginalization of Maori in Aotearoa/New Zealand.

Critical thinking skills have been identified as learning outcomes expected of students for most courses of tertiary education in many countries including Malaysia. One of the courses where critical thinking is required is academic writing. Producing academic writing which is well argued, insightful, thought-provoking, characterised by evidence and wide reading is a challenge for undergraduate students. Not only do the students need to have a good command of the language, they also need to be critical as they examine viewpoints, facts and arguments and synthesise them. This thesis explores several approaches to developing critical thinking skills in an academic writing course for undergraduate students. The use of a rubric or a checklist and discussion with peers were identified in the study to support the development of critical thinking. Their potency was explored in a quasi- experimental study involving undergraduate students taking English for Academic Writing course. The three treatments groups were: peer review where students used a checklist and discussed their ideas; peer evaluation where students used the rubric and discussed their ideas and evaluations; self-evaluation where students used the rubric but did not discuss their ideas. The level of critical thinking for each groups and a control group who received no treatment, was measured before and after learning interventions using two instruments: the Cornell Critical Thinking Test Level X (CCTT-X) and the English for Academic Writing term paper. In addition, students’ and instructors’ perspectives on the learning activities were elicited by means of questionnaires and interviews. Classroom observations were also carried out. The rubric which was used in the peer evaluation and self-evaluation activities is called the Critical Thinking for Academic Writing Analytical Rubric (CAWAR). It contains 12 criteria with descriptions of the two ends of performance domains i.e. the best and the weakest points and a commentary space. The checklist used in the peer review activity, on the other hand, replicates the CAWAR except that it does not have the grading element. The study found that all treatments showed some potential for fostering the development of critical thinking skills. Theoretically, it is argued that peer evaluation has the greatest potential of the three treatments provided that both teachers and students understand the value of collaborative learning and the importance of giving sufficient time for discussion. The introduction of either the rubric or checklist or promoting peer discussion has promoted critical thinking in an academic writing course.