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

Analysis of a worker-based participatory action research approach to the identification of selected occupational health and safety problems in Canada using mapping

There are limitations to conventional occupational health and safety research approaches and practices and numerous barriers to overcome in order to achieve progress. Occupational health and safety is impacted by the broader social-political environment. Corporatism affects the directions, ideas and practice of regulators, educators, the labour movement, scientists, medical professionals, and society as a whole, thus inhibiting workers' power to influence change. The thesis therefore explores both the wider influences and barriers to occupational health and safety advances, focusing particularly on the Canadian situation, through the general research questions: What has influenced occupational health and safety policies and practices, especially in Canada? What are some of the limitations of conventional occupational health and safety research and practices? To what extent can participatory action research and mapping address identified limitations? These questions are explored from the perspective of the population potentially at risk. New theories and approaches to occupational health and safety research are then applied in this thesis in order to explore a more specific multi-part research question: Can mapping within worker-based participatory action research be used to explore occupational health and safety conditions? In particular, can mapping contribute to occupational health and safety improvements at a local level and beyond; establish workers' previous exposures for compensation purposes; support efforts to bring about justice through compensation for workers affected by unsafe working conditions; and raise worker and public awareness of health and safety? These questions are explored through two different case studies, which examine, in depth, occupational health and safety action and possible remedies. Casino gaming workers in Windsor, Ontario, Canada undertook a collaborative study to investigate and improve current health and safety conditions. Former Holmes foundry and asbestos insulation workers in Sarnia, Ontario, Canada undertook a collaborative study to provide evidence of exposures and ensuing health problems to support claims for compensation. The outcomes of the case studies shed light on the bigger Canadian health and safety picture and demonstrate that mapping as a data collection method used within a participatory action research approach can accomplish a broad range of objectives. Mapping can raise workers' awareness, facilitate communication, build solidarity and cohesiveness, foster community support, mobilise workers to take action to reduce hazards or win compensation, in turn influencing employers, the compensation board and government agencies. The case studies accomplished the shared objective of raising worker and public awareness. The casino workers also gained occupational health and safety improvements and the Holmes workers were successful in gaining compensation

A Study of Rural-Urban Migration Trends In Manitoba

Report: iii, 37 pp., Digital file.This paper focuses on the nature of the population shifts in Manitoba in the last few decades and the changes in the structure of the economy. It also deals with the repercussions which rural-urban migration have for the rural communities and the Metropolitan Area of Winnipeg

Introduction: Critical Race Praxis and Legal Scholarship

The publication of this symposium issue is an occasion for three distinct and yet related celebrations. First, we honor the Western Law Teachers of Color, whose sixth annual meeting on the sublime Oregon Coast in 1998 provided the occasion for organizing the papers published here. Dean Strickland\u27s preface, as well as Professors Linda Greene\u27s and Jim Jones\u27s essays examine the historical significance of this occasion in greater detail. Second, we engage in a festschrift of a particular member of this group-Professor Eric K. Yamamoto -whose publication of a book this year is a significant capstone to fifteen years of scholarship on racial justice. The articles in this symposium issue address one of Yamamoto\u27s many path-breaking concepts: critical race praxis. Finally, the various pieces published here form a testament to the growing maturity of legal scholarship on race and law, as well as-sadly-the still highly contested legitimacy of this kind of scholarship within the mainstream legal academy as an editorial board of one of the Western Law Teachers participating law schools\u27 law reviews decided against publication despite an earlier commitment. The very fact that there was a politicized dispute elsewhere over the articles published here demonstrates the on-going nature of racial struggle inside the walls of law schools, as well as the strategic importance of law students committed to the principle of racial justice. Thus our obligatory first footnote, which thanks those on the editorial board of the Michigan Journal of Race & Law, does not begin to convey the complexity of the interracial dynamics-both alliances and fractures-that undergird this particular legal scholarship project

Cultivating Democracy at One High School Intervention Program for Latinos at Risk of Dropping Out

In California, where this study takes place, it is estimated that 85,000 students drop out of high school annually. Consequences are often linked to economic and social issues including long term economic costs to the state and the likelihood of lesser participation in voting and civic engagement (Rumberger, 2012). This account documents one high school’s alternative intervention program that includes online academic credit recovery and socio-emotional guidance leading to graduation for Latino students who are at risk of dropping out. Findings highlight the program’s support for these students in gaining confidence in self, envisioning themselves in the community and, for some, finding validation of their role in society through a neighborhood civic engagement project

How Do Experienced Architects Use Architecture Development Methods?

Software architecture methods play a central role in the development of large enterprise computer systems. However, the extent to which individual experienced IT architects employ a software architecture method is largely unknown. In this paper, we surveyed a group of experienced architects, and set up an “in practice” field study of some of these architects to explore the way they applied a well documented software architecture development method. Among the findings to emerge are that architects modify their method more regularly than previously recognised, and that tools for visual communication are the most commonly used tools by these architects

Ares I-X Flight Test--The Future Begins Here

In less than two years, the National Aeronautics and Space Administration (NASA) will launch the Ares I-X mission. This will be the first flight of the Ares I crew launch vehicle, which, together with the Ares V cargo launch vehicle, will eventually send humans to the Moon, Mars, and beyond. As the countdown to this first Ares mission continues, personnel from across the Ares I-X Mission Management Office (MMO) are finalizing designs and fabricating vehicle hardware for a 2009 launch. This paper will discuss the hardware and programmatic progress of the Ares I-X mission