Teaching Undergraduate Mathematics on the Internet

This is Part 2 of a two-part study of how APOS theory may be used to provide cognitive explanations of how students and mathematicians might think about the concept of infinity. We discuss infinite processes, describe how the mental mechanisms of interiorization and encapsulation can be used to conceive of an infinite process as a completed totality, explain the relationship between infinite processes and the objects that may result from them, and apply our analyses to certain mathematical issues related to infinity.     

Towards a renaissance in urban parks

Public parks have been a familiar and popular feature of our towns and cities since their appearance throughout the 19th century. They arose out of social concern over public health and happiness and as a reaction to the squalid conditions endured by the masses. Civic pride determined that they were maintained to a very high standard as symbols of municipal power and excellence. Public parks were the first resort for local communities, especially children, to have fun and to relieve the pressures of modern life. As well as providing an urban pastoral they also provided for the exuberance of amatuer sport and lively play. They continued and strengthened the British fascination with horticultural magnificence ‐council apprenticeships provided the head gardeners for the National Trust and private estates as well as the men who competed with their peers in ever more imaginative and technically accomplished picture‐and carpet‐bedding displays which adorned the nation's premier public parks.

All this has sadly passed. The decline of parks can be traced back to the removal of railings for the War effort and consequent loss of sense of place, but the real damage became cumulative from the mid‐1970s. Local government reorganisation, political struggle between local and central government, privatisation of local services, year‐on‐year cuts in capital and revenue budgets and a shift of emphasis to foreign holidays and car‐borne countryside recreation all contributed to the downfall of urban parks into the dismal, neglected and vandalised landscapes which have become so familiar today. Responding to the concerns of voluntary and professional bodies, the Heritage Lottery Fund launched the Urban Parks Programme in 1996 to begin to address the issues. The Urban Parks Programme experience has highlighted how seriously underfunded parks have been and has committed far more money than intended to tackle the massive backlog of repairs to essential park infrastructure. It has also tried to address the causes of decline in partnership with local authorities and other bodies in the field ‐ loss of management structure and skills, lack of political support and understanding, and dearth of relevant data concerning parks.

This article traces the fortunes of public parks from their inception to their decline and documents the stirring of a potential renaissance as the government shows its concern with quality of life issues, with social exclusion, with multiple deprivation and with regeneration of the economic vitality and social coherence of urban areas. A government Select Committee has recently examined the state of the nation's parks and declared itself shocked and appalled at the extent of the problems that parks have faced in the last 30 years. There is now a chance that the long downward trend in the status and condition of urban parks can be reversed if a vigorous lead is given by government.