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

Mathematics meaning-making and its relation to design of teaching

This paper addresses the design of teaching to promote engineering studentsâ conceptual understanding of mathematics, and its outcomes for mathematical meaning-making. Within a developmental research approach, inquiry-based tasks have been designed and evaluated, through the use of competencies proposed for their potential to promote conceptual learning. A sociocultural frame draws attention to interactions between different cultural elements to address challenges to teaching related to student perspectives and the mathematical meanings they develop. The paper recognizes tensions between design of inquiry-based practice and the outcomes of that practice, and demonstrates the need for new research to address mathematical meanings of a student community within a sociocultural frame

Problematic advice from suicide prevention experts

Based on a 10-year systematic review of suicide prevention strategies, 29 suicide prevention experts from 17 European countries recommend four allegedly evidence-based strategies to be included in national suicide prevention programs. One of the recommended strategies is pharmacological treatment of depression. This recommendation is deeply problematic for several reasons. First, it is based on a biased selection and interpretation of available evidence. Second, the authors have failed to take into consideration the widespread corruption in the research on antidepressants. Third, the many and serious side effects of antidepressants are not considered. Thus, the recommendation may have deleterious consequences for countless numbers of people, and, in fact, contribute to an increase in the suicide rate rather than a decrease

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

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

Sphingolipid Long-Chain Base Hydroxylation Is Important for Growth and Regulation of Sphingolipid Content and Composition in \u3ci\u3eArabidopsis\u3c/i\u3e

Sphingolipids are structural components of endomembranes and function through their metabolites as bioactive regulators of cellular processes such as programmed cell death. A characteristic feature of plant sphingolipids is their high content of trihydroxy long-chain bases (LCBs) that are produced by the LCB C-4 hydroxylase. To determine the functional significance of trihydroxy LCBs in plants, T-DNA double mutants and RNA interference suppression lines were generated for the two Arabidopsis thaliana LCB C-4 hydroxylase genes Sphingoid Base Hydroxylase1 (SBH1) and SBH2. These plants displayed reductions in growth that were dependent on the content of trihydroxy LCBs in sphingolipids. Double sbh1 sbh2 mutants, which completely lacked trihydroxy LCBs, were severely dwarfed, did not progress from vegetative to reproductive growth, and had enhanced expression of programmed cell death associated–genes. Furthermore, the total content of sphingolipids on a dry weight basis increased as the relative amounts of trihydroxy LCBs decreased. In trihydroxy LCB–null mutants, sphingolipid content was ~2.5-fold higher than that in wild-type plants. Increases in sphingolipid content resulted from the accumulation of molecular species with C16 fatty acids rather than with very-long-chain fatty acids, which are more commonly enriched in plant sphingolipids, and were accompanied by decreases in amounts of C16-containing species of chloroplast lipids. Overall, these results indicate that trihydroxy LCB synthesis plays a central role in maintaining growth and mediating the total content and fatty acid composition of sphingolipids in plants

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

Числовий метод знаходження ефективного показника заломлення пористих композиційних матеріалів на основі мікрорівневих моделей

The application of the developed numerical method for finding an effective refractive index of porous nanocomposites is shown. The numerical method of finding an effective refractive index of porous composites is developed on the basis of the use of the micro-level cellular structure model, the method of generation of random fibrous inclusions with the help of Bezier curves and micro-level cellular models. The cellular models are used in this paper for generation of porous composites structural models. They describe composite structure by representative volume elements that contains big amount of regular voxel cells that can be simultaneously used as finite element discretization. Voxel cells contain scalar intensities in diapason from 0 to 1. This enables the description of nanostructural heterogeneity of material within a model, and its direct use as a regular finite-element discretization. This method allows considering complex structural inhomogeneities of the material within the framework of a similar model and to synthesize the corresponding refractive index on the basis of numerical simulation of the electrostatic field. The method of finding an effective index of refraction of porous composite structures described in this paper was programmed in C++ 11 algorithmic language using OpenCL version 1.2 and Qt SDK version 5.4.1. The proposed implementation is simpler and requires less computation poser and resources comparing to similar analytical methods. Due to the regular structure, the obtained micro-level model can be used directly as finite-element sampling, since the use of Bezier curves enables the pores to be modeled taking into account nanostructural heterogeneities. The proposed method was tested by comparing with existing analytical models for finding an effective refractive index, such as Maxwell-Garnett model, Bruggeman model and Drude (Silberstein) model. Based on the estimation of the upper bound of the finite element method approximation error, the obtained results indicate greater accuracy compared to the Drude (Silberstein) analytical model.Розглянуто застосування розробленого числового методу знаходження ефективного показника заломлення для випадку пористих нанокомпозитів. На основі використання мікрорівневої коміркової моделі структури, методу генерування випадкових волокнистих включень з допомогою кривих Без'є та мікрорівневих коміркових моделей структури розвинено числовий метод знаходження ефективного показника заломлення пористих композитів, що дає змогу в рамках однотипної моделі розглядати складні структурні неоднорідності матеріалу та синтезувати відповідний показник заломлення на основі числового моделювання електростатичного поля. Така реалізація є простішою та потребує меншої кількості обчислень та ресурсів порівняно з аналогічними аналітичними методами. Завдяки регулярній структурі отриману мікрорівневу модель можна використовувати безпосередньо як скінченно-елементну дискретизацію, оскільки використання кривих Без'є дає змогу моделювати пори з урахуванням наноструктурних неоднорідностей. Запропонований метод було перевірено шляхом порівняння з наявними аналітичними моделями знаходження ефективного показника заломлення, такими як: Максвелла-Гарнета, моделлю Брюгемана та моделлю Друде (Сільберштейна). Спираючись на оцінку верхньої границі похибки апроксимації використаного методу скінченних елементів, отримані результати свідчать про більшу точність порівняно з аналітичною моделлю Друде (Сільберштейна)

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