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

Understanding Relationships with Attributes in Entity-Relationship Diagrams

Conceptual modeling is an important task undertaken during the systems development process to build a representation of those features of an application domain that are important to stakeholders. In spite of its importance, however, substantial evidence exists to show that it is not done well. Designers often provide incomplete, inaccurate, or inconsistent representations of domain features in the conceptual models they prepare. Users often have difficulty understanding the meaning inherent in a conceptual model. In this paper, we investigate the proposition that part of the difficulties that stakeholders experience with conceptual modeling arises when a conceptual modeling grammar or a representation produced using the grammar lacks ontological clarity. Lack of ontological clarity arises when a one-one mapping does not exist between conceptual modeling constructs and real-world constructs. For example, the grammatical construct of an entity is used to represent both things and events in the real world. Specifically, we focus on the grammatical construct of a relationship with attributes, which is often used in entity-relationship modeling. We argue that use of this construct produces ontologically unclear representations of a domain. We also report results from an experiment we undertook where we investigated the impact of using relationships with attributes in conceptual modeling representations on the problem-solving performance of users of these representations. Consistent with our predictions, we found that using relationships with attributes undermined problem-solving performance in unfamiliar domains. Contrary to our predictions, however, their use did not undermine problem-solving performance in familiar domains

Neighbourhoods, Households and Income Dynamics: A Semi-Parametric Investigation of Neighbourhood Effects

Using a unique dataset, we present evidence on income trajectories of people living in micro neighbourhoods. We place bounds on the influence of neighbourhood making as few parametric assumptions as possible. The paper offers a number of advances. We exploit a dataset that is large, representative, longitudinal with very local neighbourhoods. We analyse income growth over one, five- and ten-year windows. We analyse the whole distribution of income growth and track large gainers and losers as well as average outcomes. We consider the appropriate definition of neighbourhood. We find little evidence of a negative relationship between neighbourhood and subsequent income growth.neighbourhood effects, income dynamics, small scale neighbourhoods

Predicting the Brexit vote: getting the geography right (more or less)

In April 2016 in two contributions to this blog Ron Johnston, Kelvyn Jones and David Manley predicted the likely geography of support for Brexit in the EU referendum. In this concluding piece they compare their predictions to the result. The general pattern of their predictions turned out to be very accurate, but regional differences were more pronounced than anticipated, with variations in both late electoral registrations and turnout introducing unexpected impacts on the geography of the outcome

The Growing Spatial Polarization of Presidential Voting in the United States, 1992-2012:Myth or Reality?

ABSTRACTThere has been considerable debate regarding a hypothesis that the American electorate has become spatially more polarized over recent decades. Using a new method for measuring polarization, this paper evaluates that hypothesis regarding voting for the Democratic party’s presidential candidates at six elections since 1992, at three separate spatial scales. The findings are unambiguous: polarization has increased substantially across the country’s nine census divisions, across the 49 states within those divisions, and across the 3,077 counties within the states—with the most significant change at the finest of those three scales.</jats:p

Local Neighbourhood and Mental Health: Evidence from the UK

Using a very local definition of neighbourhood, and characterising that neighbourhood along five relatively orthogonal dimensions based on the socio-economic characteristics of the population of the neighbourhood, this paper examines the association between neighbourhood and levels and changes in mental health. We find that the extent of association between neighbourhood and both levels and changes in mental health is limited. While there are some individuals whose mental health is statistically associated with their neighbourhood composition, the importance of these differences is not large. What appears to be important for levels of mental health are the characteristics of individuals and their households, not place. Changes in mental health are not even associated with the characteristics that predict levels of health.mental health, neighbourhood, multilevel modelling

Can we really not predict who will vote for Brexit, and where?

In a recent Guardian article, Simon Jenkins suggested that voter decisions regarding the EU referendum will be made on the basis of gut instinct alone, and that personal characteristics and previous party support provide no guide. Using a new modelling strategy applied to a large body of YouGov opinion poll data, Ron Johnston, Kelvyn Jones and David Manley address Jenkins’ claim, and find it wanting. The young and the well-qualified appear much more inclined to Remain than the elderly, and voting intentions to Leave seem to match, not surprisingly, support for UKIP