73 research outputs found

    ‘Warrant’ revisited: Integrating mathematics teachers’ pedagogical and epistemological considerations into Toulmin’s model for argumentation

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    In this paper, we propose an approach to analysing teacher arguments that takes into account field dependence—namely, in Toulmin’s sense, the dependence of warrants deployed in an argument on the field of activity to which the argument relates. Freeman, to circumvent issues that emerge when we attempt to determine the field(s) that an argument relates to, proposed a classification of warrants (a priori, empirical, institutional and evaluative). Our approach to analysing teacher arguments proposes an adaptation of Freeman’s classification that distinguishes between: epistemological and pedagogical a priori warrants, professional and personal empirical warrants, epistemological and curricular institutional warrants, and evaluative warrants. Our proposition emerged from analyses conducted in the course of a written response and interview study that engages secondary mathematics teachers with classroom scenarios from the mathematical areas of analysis and algebra. The scenarios are hypothetical, grounded on seminal learning and teaching issues, and likely to occur in actual practice. To illustrate our proposed approach to analysing teacher arguments here, we draw on the data we collected through the use of one such scenario, the Tangent Task. We demonstrate how teacher arguments, not analysed for their mathematical accuracy only, can be reconsidered, arguably more productively, in the light of other teacher considerations and priorities: pedagogical, curricular, professional and personal

    Communities in university mathematics

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    This paper concerns communities of learners and teachers that are formed, develop and interact in university mathematics environments through the theoretical lens of Communities of Practice. From this perspective, learning is described as a process of participation and reification in a community in which individuals belong and form their identity through engagement, imagination and alignment. In addition, when inquiry is considered as a fundamental mode of participation, through critical alignment, the community becomes a Community of Inquiry. We discuss these theoretical underpinnings with examples of their application in research in university mathematics education and, in more detail, in two Research Cases which focus on mathematics students' and teachers' perspectives on proof and on engineering students' conceptual understanding of mathematics. The paper concludes with a critical reflection on the theorising of the role of communities in university level teaching and learning and a consideration of ways forward for future research

    Students’ Evolving Meaning About Tangent Line with the Mediation of a Dynamic Geometry Environment and an Instructional Example Space

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    In this paper I report a lengthy episode from a teaching experiment in which fifteen Year 12 Greek students negotiated their definitions of tangent line to a function graph. The experiment was designed for the purpose of introducing students to the notion of derivative and to the general case of tangent to a function graph. Its design was based on previous research results on students’ perspectives on tangency, especially in their transition from Geometry to Analysis. In this experiment an instructional example space of functions was used in an electronic environment utilising Dynamic Geometry software with Function Grapher tools. Following the Vygotskian approach according to which students’ knowledge develops in specific social and cultural contexts, students’ construction of the meaning of tangent line was observed in the classroom throughout the experiment. The analysis of the classroom data collected during the experiment focused on the evolution of students’ personal meanings about tangent line of function graph in relation to: the electronic environment; the pre-prepared as well as spontaneous examples; students’ engagement in classroom discussion; and, the role of researcher as a teacher. The analysis indicated that the evolution of students’ meanings towards a more sophisticated understanding of tangency was not linear. Also it was interrelated with the evolution of the meaning they had about the inscriptions in the electronic environment; the instructional example space; the classroom discussion; and, the role of the teacher

    Invariant Slot Attention: Object Discovery with Slot-Centric Reference Frames

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    Automatically discovering composable abstractions from raw perceptual data is a long-standing challenge in machine learning. Recent slot-based neural networks that learn about objects in a self-supervised manner have made exciting progress in this direction. However, they typically fall short at adequately capturing spatial symmetries present in the visual world, which leads to sample inefficiency, such as when entangling object appearance and pose. In this paper, we present a simple yet highly effective method for incorporating spatial symmetries via slot-centric reference frames. We incorporate equivariance to per-object pose transformations into the attention and generation mechanism of Slot Attention by translating, scaling, and rotating position encodings. These changes result in little computational overhead, are easy to implement, and can result in large gains in terms of data efficiency and overall improvements to object discovery. We evaluate our method on a wide range of synthetic object discovery benchmarks namely CLEVR, Tetrominoes, CLEVRTex, Objects Room and MultiShapeNet, and show promising improvements on the challenging real-world Waymo Open dataset.Comment: Accepted at ICML 2023. Project page: https://invariantsa.github.io

    Student perspectives on the relationship between a curve and its tangent in the transition from Euclidean Geometry to Analysis

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    The tangent line is a central concept in many mathematics and science courses. In this paper we describe a model of students’ thinking – concept images as well as ability in symbolic manipulation – about the tangent line of a curve as it has developed through students’ experiences in Euclidean Geometry and Analysis courses. Data was collected through a questionnaire administered to 196 Year 12 students. Through Latent Class Analysis, the participants were classified in three hierarchical groups representing the transition from a Geometrical Global perspective on the tangent line to an Analytical Local perspective. In the light of this classification, and through qualitative explanations of the students’ responses, we describe students’ thinking about tangents in terms of seven factors. We confirm the model constituted by these seven factors through Confirmatory Factor Analysis

    Conceptually driven and visually rich tasks in texts and teaching practice: the case of infinite series

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    The study we report here examines parts of what Chevallard calls the institutional dimension of the students’ learning experience of a relatively under-researched, yet crucial, concept in Analysis, the concept of infinite series. In particular, we examine how the concept is introduced to students in texts and in teaching practice. To this purpose, we employ Duval's Theory of Registers of Semiotic Representation towards the analysis of 22 texts used in Canada and UK post-compulsory courses. We also draw on interviews with in-service teachers and university lecturers in order to discuss briefly teaching practice and some of their teaching suggestions. Our analysis of the texts highlights that the presentation of the concept is largely a-historical, with few graphical representations, few opportunities to work across different registers (algebraic, graphical, verbal), few applications or intra-mathematical references to the concept's significance and few conceptually driven tasks that go beyond practising with the application of convergence tests and prepare students for the complex topics in which the concept of series is implicated. Our preliminary analysis of the teacher interviews suggests that pedagogical practice often reflects the tendencies in the texts. Furthermore, the interviews with the university lecturers point at the pedagogical potential of: illustrative examples and evocative visual representations in teaching; and, student engagement with systematic guesswork and writing explanatory accounts of their choices and applications of convergence tests

    Where form and substance meet: using the narrative approach of re-storying to generate research findings and community rapprochement in (university) mathematics education

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    Storytelling is an engaging way through which lived experience can be shared and reflected upon, and a tool through which difference, diversity—and even conflict—can be acknowledged and elaborated upon. Narrative approaches to research bring the richness and vibrancy of storytelling into how data is collected and interpretations of it shared. In this paper, I demonstrate the potency of the narrative approach of re-storying for a certain type of university mathematics education research (non-deficit, non-prescriptive, context-specific, example-centred and mathematically focused) conducted at the interface of two communities: mathematics education and mathematics. I do so through reference to Amongst Mathematicians (Nardi, 2008), a study carried out in collaboration with 20 university mathematicians from six UK mathematics departments. The study deployed re-storying to present data and analyses in the form of a dialogue between two fictional, yet entirely data-grounded, characters—M, mathematician, and RME, researcher in mathematics education. In the dialogues, the typically conflicting epistemologies—and mutual perceptions of such epistemologies—of the two communities come to the fore as do the feasibility-of, benefits-from, obstacles-in and conditions-for collaboration between these communities. First, I outline the use of narrative approaches in mathematics education research. Then, I introduce the study and its use of re-storying, illustrating this with an example: the construction of a dialogue from interview data in which the participating mathematicians discuss the potentialities and pitfalls of visualisation in university mathematics teaching. I conclude by outlining re-storying as a vehicle for community rapprochement achieved through generating and sharing research findings—the substance of research—in forms that reflect the fundamental principles and aims that underpin this research. My conclusions resonate with sociocultural constructs that view mathematics teacher education as contemporary praxis and the aforementioned inter-community discussion as taking place within a third space
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