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

Minimal covers of Sn by abelian subgroups and maximal subsets of pairwise noncommuting elements, II

AbstractLet βn denote the minimum possible cardinality of a cover of the symmetric group Sn by abelian subgroups and let αn denote the maximum possible cardinality of a set of pairwise noncommuting elements of Sn. Then αn ≠ βn for all n ⩾ 15. This implies that the sequence (βnαn := 1, 2, …) takes on infinitely many distinct values and does not converge

RESEARCH STUDIES IN HIGHER EDUCATION: EDUCATING MULTICULTURAL COLLEGE STUDENTS-Ch 4

Book Chapte

A New Leader in Asian Free Trade Agreements: Chinese Style Global Trade: New Rules, No Labor Protections

In 2017, after the election of Donald Trump and his subsequent language and actions surrounding global trade, Chinese President Xi Jinping took the world stage at the World Economic Forum’s Annual Meeting in a moment that led many to say he assumed the mantle of world leader on globalism and global trade, particularly in Asia. Previously, President Obama noted that the TPP presented an opportunity for the U.S., along with its partners, to write the rules of international trade with Asia-Pacific countries. At the same time, China has been working to negotiate another trade agreement in the Asia-Pacific Region, the Regional Comprehensive Economic Partnership (RCEP). The RCEP aims to be the largest free-trade bloc in the world, comprising all ten ASEAN nations (Brunei, Cambodia, Indonesia, Laos, Malaysia, Myanmar, the Philippines, Singapore, Thailand, Vietnam) and the six other countries with which ASEAN already has free-trade agreements (FTAs)—China, India, Japan, South Korea, Australia, and New Zealand.With China ascending in global leadership on Asian trade at the same time the U.S. descends, a looming question arises—what is to be expected in terms of trade rules and labor protections? There is a history on both issues that is explored in this paper and which reveals there may be difficulties ahead for those looking for an even playing field in trade and attention to labor protections. To many, it will seem like RCEP is a green light for MNCs to further invest in their labor supply chains in the developing countries in Asia, much to the consternation of labor unions in the U.S. and the detriment to American and Asian workers. Further, there are outstanding questions as to the efficacy of any labor protections that arise in future agreements. The Asia-Pacific Region is one of the largest markets in the world, so answering these questions is critical. To arrive at a fair estimate of what to expect in terms of real labor law protections and their enforcement in Asian countries under the Chinese-influenced FTAs, this paper examines the social dimension provisions of the Chinese free trade agreements (FTAs) in Asia relating to labor

Eu-China Bit and FTA Enhance Labor Cooperation and Protection

An EU-China Free Trade Agreement (FTA) and Bilateral Investment Treaty (BIT) can stimulate new labor protections and collective labor union cooperation in Chinese and EU workplaces, benefitting individual workers. Not only will it bring possible substantive improvements provided by the sustainability provisions calling for compliance with ILO labor standards, but most importantly, the interface of EU trade unions with Chinese workers, employers, and the local ACFTUs, brought about by its accompanying labor cooperation provisions and activities will enhance worker rights. It will also provide a measure of certainty with rules to guide the evolving BRI across the New Silk Road. Perhaps it is time for the European Union (EU) and Chinese leaders to build on the existing EU–China 2020 Strategic Agenda for Cooperation, quickly conclude on-going negotiations on their EU-China Bilateral Investment Treaty (BIT), and begin substantive negotiations on an EU-China FTA? China is now the EU's second-biggest trading partner behind the United States and the EU is China's biggest trading partner. The resulting BIT and FTA could enhance trade and investment opportunities and contribute to employment opportunities and rising labor standards and bringing with it increased labor cooperation and worker protections. There are recent precedents indicating China is open to more substantive and cooperative labor provisions in their FTAs and BITs. This paper focuses on enhanced labor standards and collective labor cooperation under China and EU Members’ existing and proposed FTAs and BITs. Following the Introduction, Part II discusses the economic connections of trade and FDI between EU and China and the pathways to further the relationship; Part III provides a legal comparison of their different approaches on labor standards, dispute resolution, and labor cooperation in FTAs and BITs; Part IV provides analyses on the varying approaches of enhanced labor cooperation; Part V states the conclusion and suggests the EU-China corridor of the new Silk Road could and should be paved with standards consistent with international labor standards and emanating from a EU-China FTA and with its accompanying labor cooperation that exists within and behind the legal terms

Building Literacy Assessment into Gaming Programs for Libraries and Education

See presentation description

Up and down the Multinational Corporations' Global Labor Supply Chains: Making Remedies That Work in China

Today, multinational and domestic corporations in many industries are no longer self-contained vertical structures with permanent staff, but increasingly are horizontal organizations with fissured employment characteristics using outsourcing, franchising, and subcontracting with contractors and chains of subcontractors. Too often, the workers of the subcontractors suffer the consequences of the subcontractors’ cost cutting measures, work in unfavorable conditions, and have low wages and few benefits, all for the purpose of serving the interests and profitability of the primary corporation. This paper therefore focuses on domestic laws that provide workers with an additional avenue of remedy from an expanded employment relationship—a doctrine of joint employer liability that places obligations “up the chain” on the in-country originating contractor who benefits from the supply chain or operates it for the benefit of the offshore multinational corporation. Some form of this doctrine is already used to provide workers with wage remedies against Chinese construction companies and to provide dispatch workers wage and “employee” benefit remedies. Given China’s extensive role in multinational supply chains, this paper examines the doctrine of joint employer liability up the chain and evaluates whether it can be expanded in China to remedy labor law violations and protect workers in the labor supply chains