Developing geometrical reasoning in the secondary school: outcomes of trialling teaching activities in classrooms, a report to the QCA

This report presents the findings of the Southampton/Hampshire Group of mathematicians and mathematics educators sponsored by the Qualifications and Curriculum Authority (QCA) to develop and trial some teaching/learning materials for use in schools that focus on the development of geometrical reasoning at the secondary school level. The project ran from October 2002 to November 2003. An interim report was presented to the QCA in March 2003. 1. The Southampton/Hampshire Group consisted of five University mathematicians and mathematics educators, a local authority inspector, and five secondary school teachers of mathematics. The remit of the group was to develop and report on teaching ideas that focus on the development of geometrical reasoning at the secondary school level. 2. In reviewing the existing geometry curriculum, the group endorsed the RS/ JMC working group conclusion (RS/ JMC geometry report, 2001) that the current mathematics curriculum for England contains sufficient scope for the development of geometrical reasoning, but that it would benefit from some clarification in respect of this aspect of geometry education. Such clarification would be especially helpful in resolving the very odd separation, in the programme of study for mathematics, of ‘geometrical reasoning’ from ‘transformations and co-ordinates’, as if transformations, for example, cannot be used in geometrical reasoning. 3. The group formulated a rationale for designing and developing suitable teaching materials that support the teaching and learning of geometrical reasoning. The group suggests the following as guiding principles: • Geometrical situations selected for use in the classroom should, as far as possible, be chosen to be useful, interesting and/or surprising to pupils; • Activities should expect pupils to explain, justify or reason and provide opportunities for pupils to be critical of their own, and their peers’, explanations; • Activities should provide opportunities for pupils to develop problem solving skills and to engage in problem posing; • The forms of reasoning expected should be examples of local deduction, where pupils can utilise any geometrical properties that they know to deduce or explain other facts or results. • To build on pupils’ prior experience, activities should involve the properties of 2D and 3D shapes, aspects of position and direction, and the use of transformation-based arguments that are about the geometrical situation being studied (rather than being about transformations per se); • The generating of data or the use of measurements, while playing important parts in mathematics, and sometimes assisting with the building of conjectures, should not be an end point to pupils’ mathematical activity. Indeed, where sensible, in order to build geometric reasoning and discourage over-reliance on empirical verification, many classroom activities might use contexts where measurements or other forms of data are not generated. 4. In designing and trialling suitable classroom material, the group found that the issue of how much structure to provide in a task is an important factor in maximising the opportunity for geometrical reasoning to take place. The group also found that the role of the teacher is vital in helping pupils to progress beyond straightforward descriptions of geometrical observations to encompass the reasoning that justifies those observations. Teacher knowledge in the area of geometry is therefore important. 5. The group found that pupils benefit from working collaboratively in groups with the kind of discussion and argumentation that has to be used to articulate their geometrical reasoning. This form of organisation creates both the need and the forum for argumentation that can lead to mathematical explanation. Such development to mathematical explanation, and the forms for collaborative working that support it, do not, however, necessarily occur spontaneously. Such things need careful planning and teaching. 6. Whilst pupils can demonstrate their reasoning ability orally, either as part of group discussion or through presentation of group work to a class, the transition to individual recording of reasoned argument causes significant problems. Several methods have been used successfully in this project to support this transition, including 'fact cards' and 'writing frames', but more research is needed into ways of helping written communication of geometrical reasoning to develop. 7. It was found possible in this study to enable pupils from all ages and attainments within the lower secondary (Key Stage 3) curriculum to participate in mathematical reasoning, given appropriate tasks, teaching and classroom culture. Given the finding of the project that many pupils know more about geometrical reasoning than they can demonstrate in writing, the emphasis in assessment on individual written response does not capture the reasoning skills which pupils are able to develop and exercise. Sufficient time is needed for pupils to engage in reasoning through a variety of activities; skills of reasoning and communication are unlikely to be absorbed quickly by many students. 8. The study suggests that it is appropriate for all teachers to aim to develop the geometrical reasoning of all pupils, but equally that this is a non-trivial task. Obstacles that need to be overcome are likely to include uncertainty about the nature of mathematical reasoning and about what is expected to be taught in this area among many teachers, lack of exemplars of good practice (although we have tried to address this by lesson descriptions in this report), especially in using transformational arguments, lack of time and freedom in the curriculum to properly develop work in this area, an assessment system which does not recognise students’ oral powers of reasoning, and a lack of appreciation of the value of geometry as a vehicle for broadening the curriculum for high attainers, as well as developing reasoning and communication skills for all students. 9. Areas for further work include future work in the area of geometrical reasoning, include the need for longitudinal studies of how geometrical reasoning develops through time given a sustained programme of activities (in this project we were conscious that the timescale on which we were working only enabled us to present 'snapshots'), studies and evaluation of published materials on geometrical reasoning, a study of 'critical experiences' which influence the development of geometrical reasoning, an analysis of the characteristics of successful and unsuccessful tasks for geometrical reasoning, a study of the transition from verbal reasoning to written reasoning, how overall perceptions of geometrical figures ('gestalt') develops as a component of geometrical reasoning (including how to create the links which facilitate this), and the use of dynamic geometry software in any (or all) of the above.10. As this group was one of six which could form a model for part of the work of regional centres set up like the IREMs in France, it seems worth recording that the constitution of the group worked very well, especially after members had got to know each other by working in smaller groups on specific topics. The balance of differing expertise was right, and we all felt that we learned a great deal from other group members during the experience. Overall, being involved in this type of research and development project was a powerful form of professional development for all those concerned. In retrospect, the group could have benefited from some longer full-day meetings to jointly develop ideas and analyse the resulting classroom material and experience rather than the pattern of after-school meetings that did not always allow sufficient time to do full justice to the complexity of many of the issues the group was tackling

NASA/DoD Aerospace Knowledge Diffusion Research Project. Paper 31: The information-seeking behavior of engineers

Engineers are an extraordinarily diverse group of professionals, but an attribute common to all engineers is their use of information. Engineering can be conceptualized as an information processing system that must deal with work-related uncertainty through patterns of technical communications. Throughout the process, data, information, and tacit knowledge are being acquired, produced, transferred, and utilized. While acknowledging that other models exist, we have chosen to view the information-seeking behavior of engineers within a conceptual framework of the engineer as an information processor. This article uses the chosen framework to discuss information-seeking behavior of engineers, reviewing selected literature and empirical studies from library and information science, management, communications, and sociology. The article concludes by proposing a research agenda designed to extend our current, limited knowledge of the way engineers process information

Vision Statements and Road-Map Methodology for Knowledge Management Adoption

The present paper describes the strategy of introducing future knowledge management system at farms. The FUTUREFARM and PREZEM projects strive to apply new knowledge management methods in arable farming where they guarantee an easy adaptation of the farming sector to the everchanging conditions in short, middle and long-term perspective. The knowledge management methods have to be put into practice on strategic, tactic and operational planning levels. Based on the project analysis and workshops with farmers, the paper brings an outline of the main goals and obstacles for new knowledge management methods adoption and furthermore defines the target groups and relevant methods of dealing with them.Knowledge management, adaptation, arable farming, Farm Management, GA, IN,

Problem-Solving Behaviours, Organisational Forms and the Complexity of Tasks

--

A Scientific Roadmap for Antibiotic Discovery: A Sustained and Robust Pipeline of New Antibacterial Drugs and Therapies is Critical to Preserve Public Health

In recent decades, the discovery and development of new antibiotics have slowed dramatically as scientific barriers to drug discovery, regulatory challenges, and diminishing returns on investment have led major drug companies to scale back or abandon their antibiotic research. Consequently, antibiotic discovery—which peaked in the 1950s—has dropped precipitously. Of greater concern is the fact that nearly all antibiotics brought to market over the past 30 years have been variations on existing drugs. Every currently available antibiotic is a derivative of a class discovered between the early 1900s and 1984.At the same time, the emergence of antibiotic-resistant pathogens has accelerated, giving rise to life-threatening infections that will not respond to available antibiotic treatment. Inevitably, the more that antibiotics are used, the more that bacteria develop resistance—rendering the drugs less effective and leading public health authorities worldwide to flag antibiotic resistance as an urgent and growing public health threat

Beyond written computation

This collection of papers based on research into aspects of number is a result of a writing conference held on Rottnest Island, near Perth, Western Australia. The concept of the conference emanated from Alistair Mcintosh and Len Sparrow and was based on two similar meetings organised by Cal Irons and Bob Reys. All papers in this book were discussed at the Rottnest conference and subsequent changes were made by the authors based on comments and recommendations from the peer group who attended the conference

Remedial Education for Black Children in Rural South Africa: An Exploration of Success Using Evolutionary Innovation Theory

remedial, South Africa, rural