Mathematics teaching and learning on Outcomes Based Education and Curriculum 2005

Student Number : 9110316P - M Ed research report - School of Education - Faculty of HumanitiesThis study seeks to establish if teaching Grade 7 Algebra accords with Outcomes Based Education [OBE] in a sample of three state primary schools in a province, South Africa. Following the methods of illuminative evaluation the researcher looked for ‘matches’ and ‘mismatches’ between what was planned in an OBE text with what ‘actually happens’ in classroom teaching to gauge if the shift to outcomes has taken place in teaching Mathematics in these schools, and make recommendations to improve. Data was collected using document analysis to establish how percentages was planned to be taught by teachers and using naturalistic observations with follow-up probing interviews to establish how this teaching actually took place in classrooms. The data was checked by questionnaire data seeking the views of educators doing this teaching. The data showed 5 Patterns in this teaching, one only according with planned OBE teaching, 2 other Patters where teaching was more-or-less as intended, and 2 further Patterns where teaching failed to accord with the OBE text. Just over half the teachers or 58% of the sample seemed to have shifted to OBE, and less than half or 42% of educators seem not to have done so. Primary amongst the findings is that educators failed to teach Mathematics conceptually first as planned, preferring in a variety of ways to omit conceptual explanations by way of introduction to lessons in favour of ‘guiding examples’, ‘group work’ and ‘report back’, ‘teacher and learner assessment’ and ‘concluding exercises’, the six categories which emerged for teaching in these lessons. The study recommends primarily that educators re-claim teaching Mathematics conceptually first, and prior to completing examples and giving exercises to learners. It concludes that fewer educators than expected seem to have shifted towards OBE teaching in these Mathematics classrooms, 6 years into the national innovation, C 2005

Brain functors: A mathematical model for intentional perception and action

Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics—with adjunctions being the primary lens. If adjunctions are so important in mathematics, then perhaps they will isolate concepts of some importance in the empirical sciences. But the applications of adjunctions have been hampered by an overly restrictive formulation that avoids heteromorphisms or hets. By reformulating an adjunction using hets, it is split into two parts, a left and a right semiadjunction. Semiadjunctions (essentially a formulation of universal mapping properties using hets) can then be combined in a new way to define the notion of a brain functor that provides an abstract model of the intentionality of perception and action (as opposed to the passive reception of sense-data or the reflex generation of behavior)

Exploring the Development of Core Teaching Practices in the Context of Inquiry-based Science Instruction: An Interpretive Case Study

This paper describes our reflection on a clinical-based teacher preparation program. We examined a context in which novice pre-service teachers and a mentor teacher implemented inquiry-based science instruction to help students make sense of genetic engineering. We utilized developmental models of professional practice that outline the complexity inherent in professional knowledge as a conceptual framework to analyze teacher practice. Drawing on our analysis, we developed a typography of understandings of inquiry-based science instruction that teachers in our cohort held and generated a two dimensional model characterizing pathways through which teachers develop core teaching practices supporting inquiry-based science instruction

Categories without structures

The popular view according to which Category theory provides a support for Mathematical Structuralism is erroneous. Category-theoretic foundations of mathematics require a different philosophy of mathematics. While structural mathematics studies invariant forms (Awodey) categorical mathematics studies covariant transformations which, generally, don t have any invariants. In this paper I develop a non-structuralist interpretation of categorical mathematics and show its consequences for history of mathematics and mathematics education.Comment: 28 page

Hard, Harder, and the Hardest Problem: The Society of Cognitive Selves

The hard problem of consciousness is explicating how moving matter becomes thinking matter. Harder yet is the problem of spelling out the mutual determinations of individual experiences and the experiencing self. Determining how the collective social consciousness influences and is influenced by the individual selves constituting the society is the hardest problem. Drawing parallels between individual cognition and the collective knowing of mathematical science, here we present a conceptualization of the cognitive dimension of the self. Our abstraction of the relations between the physical world, biological brain, mind, intuition, consciousness, cognitive self, and the society can facilitate the construction of the conceptual repertoire required for an explicit science of the self within human society