Teaching and learning trigonometry with technology

Modern school classrooms have access to a range of potential technologies, ranging from calculators to computers to the Internet. This paper explores some of the potential for such technologies to affect the curriculum and teaching of trigonometry in the secondary school. We identify some of the ways in which the teaching of trigonometry might be supported by the availability of various forms of technology. We consider circular measures, graphs of functions, trigonometric identities, equations and statistical modeling and focus on activities that are not possible without the use of technology. Modern technology provides an excellent means of exploring many of the concepts associated with trigonometry, both trigonometric and circular functions. Many of these opportunities for learning were not available before technology development and access within schools we enjoy today. This paper suggests some of the avenues for exploration

Particle Resuspension in Turbulent Boundary Layers and the Influence of Non-Gaussian Removal Forces

The work described is concerned with the way micron-size particles attached to a surface are resuspended when exposed to a turbulent flow. An improved version of the Rock'n'Roll model (Reeks and Hall, 2001) is developed where this model employs a stochastic approach to resuspension involving the rocking and rolling of a particle about surface asperities arising from the moments of the fluctuating drag forces acting on the particle close to the surface. In this work, the model is improved by using values of both the streamwise fluid velocity andacceleration close to the wall obtained from Direct Numerical Simulation (DNS) of turbulentchannel flow. Using analysis and numerical calculations of the drag force on a sphere near a wall in shear flow (O'Neill (1968) and Lee and Balachandar (2010)) these values are used to obtain the joint distribution of the moments of the fluctuating drag force and its time derivative acting on a particle attached to a surface. In so doing the influence of highly non-Gaussian forces (associated with the sweeping and ejection events in a turbulent boundary layer) on short and long term resuspension rates is examined for a sparse monolayer coverage of particles, along with the dependence of the resuspension upon the timescale of the particle motion attached to the surface, the ratio of the rms/ mean of the removal force and the distribution of adhesive forces. Model predictions of the fraction resuspended are compared with experimental results.Comment: 31 pages 21 figure

Graphics calculator use in examinations: accident or design?

As graphics calculators become more available, interest will focus on how to incorporate them appropriately into curriculum structures, and particularly into examinations. We describe and exemplify a typology of use of graphics calculators in mathematics examinations, from the perspective of people designing examinations, together with some principles for the awarding of partial credit to student responses. This typology can be used to help design examinations in which students are permitted to use graphics calculators as well as to interrogate existing examination practice

Understanding what you are doing: A new angle on CAS?

Powerful Computer Algebra Systems (CAS) are often used only with reluctance in early undergraduate mathematics teaching, partly because of concerns that they may not encourage students to understand what they are doing. In this exploratory study, a version of a CAS that has been designed for secondary school students was used, with a view to considering the value of this sort of student learning support for first year undergraduate students enrolled in degree programs other than mathematics. Workshops were designed to help students understand aspects of elementary symbolic manipulation, through the use of the Algebra mode of an algebraic calculator, the Casio Algebra FX 2.0. The Algebra mode of this calculator allows a user to undertake elementary algebraic manipulation, routinely providing all intermediate results, in contrast to more powerful CAS software, which usually provides simplified results only. The students were volunteers from an introductory level unit, designed to provide a bridge between school and university studies of mathematics and with a focus on algebra and calculus. The two structured workshop sessions focussed respectively on the solution of linear equations and on relationships between factorising and expanding; attention focussed on using the calculators as personal learning devices. Following the workshops, structured interviews were used to systematically record student reactions to the experience. As a result of the study, the paper offers advice on the merits of using algebraic calculators in this sort of way

Symbolic manipulation on a TI-92: New threat or hidden treasures?

The availability of hand held devices that can undertake symbolic manipulation is a recent phenomenon, potentially of great significance for both the algebra and calculus curriculum in the secondary and lower undergraduate years. The significance to date of symbolic manipulation for mathematics is described, and parallels drawn with the significance of arithmetic skills for the primary school. It is suggested that, while symbolic manipulation is central to mathematics, many students develop only a restricted competence with the associated mathematical ideas. The Texas Instruments TI-92 is used to suggest some potential beneficial uses of technology that involves symbolic manipulation

Graphics calculators in the mathematics curriculum: Integration or differentiation?

Graphics calculators are examples of powerful technologies that we want our students to learn to use well. However if we use them in our courses only for learning, students will not regard them with due importance because they are not integrated into the assessment. On the other hand, if graphics calculators are integrated into both learning and assessment there are risks associated with students becoming calculator dependent, issues of equity arise associated with calculator access and there may be problems with setting an appropriate examination. We discuss this dilemma in the light of our experiences and the reactions of our students over the last two years

Graphics calculators and assessment

Graphics calculators are powerful tools for learning mathematics and we want our students to learn to use them effectively. The use of these hand held personal computers provides opportunities for learning in interactive and dynamic ways. However, it is not until their use is totally integrated into all aspects of the curriculum that students regard them with due importance. This includes their use in all kinds of assessment tasks such as assignments, tests and examinations as well as in activities and explorations aimed at developing students’ understanding. The incorporation of graphics calculators into assessment tasks requires careful construction of these tasks. In this paper, discuss issues of equity relating to calculator models, levels of calculator use and the purpose and design of appropriate tasks. We also describe a typology we have developed to assist in the design and wording of assessment tasks which encourage appropriate use of graphics calculators, but which do not compromise important course objectives

A model for the educational role of calculators

Calculators can be used effectively for mathematics education in a number of ways, although frequently they are regarded merely as devices for undertaking computations. In this analytic paper, we describe and illustrate a four-part model to understand more fully the potential role of calculators for learning mathematics. The four elements of the model include representation, computation, exploration and affirmation. Effective use of a calculator by students learning mathematics will often involve more than one of these four components. The model has been derived from analysis of educational materials developed to support rich calculator use

Research on oxygen toxicity at the cellular level Final report, 15 Apr. 1965 - 15 Jun. 1966

Oxygen toxicity at cellular level in manned spacecraf

Psychological adjustment of men with prostate cancer: A review of the literature

OBJECTIVE: Prostate cancer (PCA) is the most common malignancy and a major cause of death in men but, importantly, a substantial proportion will live for several years following diagnosis. However, they face the prospect of experiencing symptoms, side-effects of treatment and diminished quality of life. The patient's psychological adjustment is particularly complex, given the potential trajectory of the disease, from the point of diagnosis, with its immediate impact, to the phase of palliative care, with its attendant issue of facing mortality. Since a comprehensive review of the literature on psychological adjustment of men with PCA has not yet been done, we have documented relevant research, integrated findings and drawn conclusions, where possible, in order to map out clinical and research implications. METHOD: We searched 5 databases for the period 1994 - July 2006, during which most of the work in the field has been done. RESULTS: We found few studies of substance among the 60 we examined to draw conclusions about psychological adjustment to prostate cancer and its treatment. This is in marked contrast to the picture in breast cancer. While some patterns have emerged, many gaps remain to be filled. DISCUSSION: Aspects of methodology need attention, particularly longitudinal, prospective designs, incorporation of control groups and the use of valid and reliable measures. There is scope for qualitative studies as a complement to quantitative research