(A paradox is anything which offhand appears to be false, but is actually true; or which appears to be true but is actually false; or which is simply self-contradictory.)
Of many of the problems that occur within arithmetic and mathematics paradoxes are among the most appealing and instructive. Paradoxes abound in arithmetic and mathematics as evidenced by Eugene Northrop’s classic book (1). The equation I am considering at present is as follows: On one side the statement that each year the GCSE and GCE ‘A’ level mathematics results continue to get better; on the other side numerous reports proclaim declining standards in the subject in the country and most certainly when compared with other countries. Each year the government and its departments provoke a massive campaign of hype about the improving standards in mathematics and the increasing number of students taking GCE ‘A’ Mathematics and Further Mathematics. The number studying ‘A’ level has doubled in recent years to 77,000 with 11,600 taking Further Mathematics –BUT it must be said that the increase is from a woefully low base and when compared with many other countries is still very low both in terms of number and levels of achievement; but this is another example of how politicians manipulate statistics. So if you believe the evidence on this side of the equation all seems to be positive and rosy.
But on the other side of the equation there is a vast amount of quantitative and objective evidence from both national and international sources that articulates a very different picture. Couple this evidence with the continuing chorus from concerned employers, recruiting agencies, and college and university admissions tutors bemoan the parlous state of the mathematical competence of the majority of employees and entrants. Employers argue strongly that school/college leavers and university graduates lack mathematical skills and capability. This they argue is being more emphasised as a result of the rapid transitions in the workplace that increasingly require greater mathematical skills and understanding. One aspect of this can been seen in the changing nature of the workforce as the shift from manual and low- skilled jobs towards higher levels of skill continues and that requires mathematical capability, skills and knowledge. The level of understanding and ability of the majority of the population in the country to apply mathematical and numerical concepts is pitifully low and results an inability to cope with the requirements and challenges of the workplace of the future. These continuing concerns from the end-users of the education and training system are reinforced by a plethora of recent reports again restating the parlous and pathetic current state of mathematics in Britain and include the following:
- Is the UK an Outlier? Published by the Nuffield Foundation (1).
- Mathematical Needs. ACME. (2)
- Wolf Report. (3)
- Vorderman Report.
- UK Home Learning College’s ‘Welcome Back to Learning ‘campaign.
These reports come up with a series of well rehearsed conclusions and solutions. The OECD and Nuffield Reports identify that a lower proportion of students post-16 study mathematics i.e. <20% when compared with other developed countries. Scotland has a higher percentage namely 50%. The OECD and Nuffield data and accompanying commentaries highlight that these other countries see the strategic importance of mathematics in their economies and conclude that Britain except Scotland does not. In England, Wales and Northern Ireland just 13% study ‘A’ level mathematics compared with over 70% in Japan and Taiwan. Some figures provide an insight into the problem e.g. approximately a cohort of 700,000 students pass through the national system each year. The majority do not study any further mathematical subjects after 16 and almost 50% have not even achieved a grade C at GCSE and remember the ongoing debates about what percentages merits a grade C!
One issue highlighted in a number of reports is that mathematics is not compulsory after 16 and that in the majority of countries surveyed mathematics is required in general and vocational education. Many of the reports cited above advocate that mathematics must be compulsory post 16 although they differ on what kind of mathematics should be taught; a classical approach to debates on education in this country – world class in talking but not taking firm and meaningful action. So the paradox for me is: which of these two conflicting sets of evidence is true? Is it just the government manipulating and massaging the statistics for political capital or are there more serious issues like grade and credential inflation. But the reality surely is that the country has major problems, so for me the evidence from these recent reports and reports going back decades is overwhelming and convincing. So, how can the annual circus of GCSE and GCE results are justified. I feel sorry for the students many of whom work hard only to see and read the kind of comments I have been making. What is needed is a full and open debate about the problems and then a root and branch reform of the subjects establishing programmes and examinations that are relevant and fit for purpose. The mathematical content of the programmes must prepare the students to understand and apply the principles of mathematics and numerical concepts in their future work or study.
References:
(1) Northrop. E. P. Riddles in Mathematics. Pelican. 1944.
An International Comparison of Upper Secondary Mathematics by Ruddock. J. And Pepper. D. Of King’s College and Sturman. L and Ruddock. G. Of the NFER. December 2010.
‘Mathematical Needs.’ Two volumes: The Mathematical Needs of Learners and Mathematics in the Workplace and in HE. ACME.ISBN 978-0-85403-0 and 978-0-85403-895-4. June 2011.
‘Review of Vocational Education.’ Wolf Report. March 2011.
‘A world-class mathematics education for all our young people.’ Vorderman Report. August 2011.
UK Home Learning College’s ‘Welcome Back to Learning’ campaign. October 2011.
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