Communities in university mathematics

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

Teachers’ perspectives on collaboration with didacticians to create an inquiry community

This article was published in the journal, Research in Mathematics Education [Routledge © British Society for Research into Learning Mathematics]. The definitive version is available at: http://www.tandfonline.com/doi/abs/10.1080/14794800902732209A research and development project, Learning Communities in Mathematics (LCM) was designed to create opportunities for ‘co-learning inquiry’ between mathematics teachers in eight schools and didacticians in a university in Norway (UiA). The focus has been on improving mathematics teaching and learning at school levels from lower primary to upper secondary and on the developmental processes and partnerships involved. A central aim was to create a community of inquiry through which aspects of mathematics teaching and learning could be explored, and through which both teachers and didacticians could learn in practice. Theoretically, ‘Community of Inquiry’ derives from ‘Community of Practice’ as expounded by Lave andWenger, and particularlyWenger’s concept of ‘belonging’. The project included three, one-year phases of joint activity. At the end of Phase II, didacticians led focus group interviews with teacher teams to gain insights into schools’ and teachers’ perceptions of the project and its activity. We report on insights into how teachers thought about the activities of the project and what an inquiry community looks like in terms of the learning of those involved. We relate this back to the theoretical perspectives of communities of practice and inquiry

Teachers and didacticians: key stakeholders in the processes of developing mathematics teaching

This paper sets the scene for a special issue of ZDM-The International Journal on Mathematics Education-by tracing key elements of the fields of teacher and didactician/teacher-educator learning related to the development of opportunities for learners of mathematics in classrooms. It starts from the perspective that joint activity of these two groups (teachers and didacticians), in creation of classroom mathematics, leads to learning for both. We trace development through key areas of research, looking at forms of knowledge of teachers and didacticians in mathematics; ways in which teachers or didacticians in mathematics develop their professional knowledge and skill; and the use of theoretical perspectives relating to studying these areas of development. Reflective practice emerges as a principal goal for effective development and is linked to teachers' and didacticians' engagement with inquiry and research. While neither reflection nor inquiry are developmental panaceas, we see collaborative critical inquiry between teachers and didacticians emerging as a significant force for teaching development. We include a summary of the papers of the special issue which offer a state of the art perspective on developmental practice. © 2014 FIZ Karlsruhe

Mathematics teaching development as a human practice: identifying and drawing the threads

This article was published in the journal, ZDM Mathematics Education [© FIZ Karlsruhe] and the definitive version is available at: http://dx.doi.org/10.1007/s11858-012-0437-7The didactic triangle links mathematics, teachers and students in a consideration of teaching– learning interactions in mathematics classrooms. This paper focuses on teachers and teaching in the development of fruitful learning experiences for students with mathematics. It recognises primarily that teachers are humans with personal characteristics, subject to a range of influences through the communities of which they are a part, and considers aspects of teachers’ personhood, identity and agency in designing teaching for the benefit of their students. Teaching is seen as a developmental process in which inquiry plays a central role, both in doing mathematics in the classroom and in exploring teaching practice. The teacher-as-inquirer in collaboration with outsider researchers leads to growth of knowledge in teaching through development of identity and agency for both groups. The inclusion of the outsider researcher brings an additional node into the didactic triangle

Time as an operator/observable in nonrelativistic quantum mechanics

The nonrelativistic Schroedinger equation for motion of a structureless particle in four-dimensional space-time entails a well-known expression for the conserved four-vector field of local probability density and current that are associated with a quantum state solution to the equation. Under the physical assumption that each spatial, as well as the temporal, component of this current is observable, the position in time becomes an operator and an observable in that the weighted average value of the time of the particle's crossing of a complete hyperplane can be simply defined: ... When the space-time coordinates are (t,x,y,z), the paper analyzes in detail the case that the hyperplane is of the type z=constant. Particles can cross such a hyperplane in either direction, so it proves convenient to introduce an indefinite metric, and correspondingly a sesquilinear inner product with non-Hilbert space structure, for the space of quantum states on such a surface. >... A detailed formalism for computing average crossing times on a z=constant hyperplane, and average dwell times and delay times for a zone of interaction between a pair of z=constant hyperplanes, is presented.Comment: 31 pages, no figures. Differs from published version by minor corrections and additions, and two citation

Using tasks to explore teacher knowledge in situation-specific contexts

This article was published in the journal, Journal of Mathematics Teacher Education [© Springer] and the original publication is available at www.springerlink.comResearch often reports an overt discrepancy between theoretically/out-of context expressed teacher beliefs about mathematics and pedagogy and actual practice. In order to explore teacher knowledge in situation-specific contexts we have engaged mathematics teachers with classroom scenarios (Tasks) which: are hypothetical but grounded on learning and teaching issues that previous research and experience have highlighted as seminal; are likely to occur in actual practice; have purpose and utility; and, can be used both in (pre- and in-service) teacher education and research through generating access to teachers’ views and intended practices. The Tasks have the following structure: reflecting upon the learning objectives within a mathematical problem (and solving it); examining a flawed (fictional) student solution; and, describing, in writing, feedback to the student. Here we draw on the written responses to one Task (which involved reflecting on solutions of x+x−1=0 of 53 Greek in-service mathematics teachers in order to demonstrate the range of teacher knowledge (mathematical, didactical and pedagogical) that engagement with these tasks allows us to explore

A geometric interpretation of the spectral parameter for surfaces of constant mean curvature

Considering the kinematics of the moving frame associated with a constant mean curvature surface immersed in S^3 we derive a linear problem with the spectral parameter corresponding to elliptic sinh-Gordon equation. The spectral parameter is related to the radius R of the sphere S^3. The application of the Sym formula to this linear problem yields constant mean curvature surfaces in E^3. Independently, we show that the Sym formula itself can be derived by an appropriate limiting process R -> infinity.Comment: 12 page

Expression of ABCA4 in the retinal pigment epithelium and its implications for Stargardt macular degeneration.

Recessive Stargardt disease (STGD1) is an inherited blinding disorder caused by mutations in the Abca4 gene. ABCA4 is a flippase in photoreceptor outer segments (OS) that translocates retinaldehyde conjugated to phosphatidylethanolamine across OS disc membranes. Loss of ABCA4 in Abca4 -/- mice and STGD1 patients causes buildup of lipofuscin in the retinal pigment epithelium (RPE) and degeneration of photoreceptors, leading to blindness. No effective treatment currently exists for STGD1. Here we show by several approaches that ABCA4 is additionally expressed in RPE cells. (i) By in situ hybridization analysis and by RNA-sequencing analysis, we show the Abca4 mRNA is expressed in human and mouse RPE cells. (ii) By quantitative immunoblotting, we show that the level of ABCA4 protein in homogenates of wild-type mouse RPE is about 1% of the level in neural retina homogenates. (iii) ABCA4 immunofluorescence is present in RPE cells of wild-type and Mertk -/- but not Abca4 -/- mouse retina sections, where it colocalizes with endolysosomal proteins. To elucidate the role of ABCA4 in RPE cells, we generated a line of genetically modified mice that express ABCA4 in RPE cells but not in photoreceptors. Mice from this line on the Abca4 -/- background showed partial rescue of photoreceptor degeneration and decreased lipofuscin accumulation compared with nontransgenic Abca4 -/- mice. We propose that ABCA4 functions to recycle retinaldehyde released during proteolysis of rhodopsin in RPE endolysosomes following daily phagocytosis of distal photoreceptor OS. ABCA4 deficiency in the RPE may play a role in the pathogenesis of STGD1

Linear-response theory of the longitudinal spin Seebeck effect

We theoretically investigate the longitudinal spin Seebeck effect, in which the spin current is injected from a ferromagnet into an attached nonmagnetic metal in a direction parallel to the temperature gradient. Using the fact that the phonon heat current flows intensely into the attached nonmagnetic metal in this particular configuration, we show that the sign of the spin injection signal in the longitudinal spin Seebeck effect can be opposite to that in the conventional transverse spin Seebeck effect when the electron-phonon interaction in the nonmagnetic metal is sufficiently large. Our linear-response approach can explain the sign reversal of the spin injection signal recently observed in the longitudinal spin Seebeck effect.Comment: Proc. of ICM 2012 (Accepted for publication in J. Korean Phys. Soc.), typos correcte