A Further Response to the Post 14 Inquiry

Proposal for a Dual GCSE in Mathematics

The purpose of this short paper is to expand on the proposals already submitted to the Smith Enquiry, for a dual award in mathematics available to most students from age 14.

Rationale:
1.
Whatever structure is adopted, it is imperative that for effective implementation there is sufficient professional training available for teachers: we cannot afford to get it wrong.
2.
The TCT 2001 report 'Engaging Mathematics'(1) demonstrates the need for effective strategies to engage students' interests in mathematics if we are to reverse the present large attrition rates in qualification and attitude. Tinkering around the edges of the present system is not enough, although any significant changes must be based on substantial and medium-term trials.
3.
Students need to be educated not only for knowledge and technical mastery, but for organisation and research skills, oral and other communication skills, breadth of application and mathematical insight.
4.
Any successful curriculum must address the promotion of students' spiritual, moral, social and cultural development, the key skills of communication, IT, working with others, developing as an independent learner, and problem-solving, as well as generic thinking skills, financial capability, enterprise and entrepreneurial skills, and work-related learning.
5.
At present anecdotal evidence suggests that the work involved in Mathematics GCSE is comparable to that of other, dual award, courses (English Language/English Literature, Double Science). Students should be rewarded for that effort with a dual award in mathematics, and there would be a consequent change in status for the subject.

Proposal:
q An underlying feature for both awards would be that achievement of a particular grade should require 80% facility of the appropriate material. This represents a change in nature of current assessment practices, which require low command of a wide-ranging syllabus. The evidence is that this is of limited value both to prospective users of the qualification and for mathematical progression; it also undermines fluency in using those skills and external confidence in the qualifications. The two awards between them would cover roughly the material at present required for a single Mathematics GCSE; it might be that without further time devoted to the subject area, the quantity of material required to be covered for a dual award at grade C, for example, would be significantly less that the present Intermediate Level specification, in order to meet the requirement of mastery. The precise split of the present content remains a matter for further discussion, but it is proposed that it is split along the lines indicated below:

1.1 Use of Mathematics would be, with English, Science and ICT, one of four core subjects at KS4, with summative assessment being largely external or externally moderated, and would be available at entry Level and at Levels 1 to 3 by age 18. It would include numeracy for citizenship and quantitative literacy, with the emphasis transferring to the latter at higher levels.
1.2 Mathematical literacy should be interpreted in such a way as to expose all learners to the power and beauty of mathematics, as well as to its utilitarian and immediate application.
1.3 Core generic skills which the Use of Mathematics course would be expected to develop, include - integrated mathematics and IT skills
-an ability to create a formula (when using a spreadsheet for example)
-calculating and estimating (quickly and mentally)
-proportional reasoning
-calculating and understanding and interpreting percentages correctly
-multi-step problem solving
-a sense of complex modelling, including understanding thresholds and constraints
-use of extrapolation
-recognising anomalous effects and erroneous answers when monitoring systems
-an ability to perform commonly used paper and pencil calculations and mental calculations as well as calculating correctly with a calculator
-communicating mathematics to other users and interpreting the mathematics of other users
-an ability to cope with the unexpected
Many of these aspects of mathematics were identified by the report 'Mathematical Skills in the Workplace' (2) as being significant to employers, and underdeveloped in many employees.
1.4 The present AS level Use of Mathematics framework is seen as a successful one where it is employed, but its present use is limited by the perception that GCSE Mathematics is the standard recognised qualification. We should like to see the present structure of FSMUs adopted, whereby students study principles and develop applications of specific aspects of mathematics to some depth, drawing upon and enhancing other areas of their work, studies or interests, and are assessed by equally weighted elements of portfolio evidence and written examination. We believe this structure mirrors good pedagogical practice and is most likely to result in the desired outcomes outlined above.
1.5 There is a lot to recommend the present FSMU areas of study, namely:
-at level 1, managing money/working in 2- and 3- dimensions/making sense of data;
-at level 2, calculating finances/solving problems in shape and space/handling and interpreting data/making connections in mathematics/using functions and graphs (with some of these areas requiring quite sophisticated thought for top grades) We would prefer not to see this implemented as a modular course, though, because of the fragmentation in teaching and learning that ensues, despite apparently 'better' results.
1.6 It is hoped that Level 2 would be achieved by 60% at age 16 and 80% at 18.It should give smooth transition to a Level 3 qualification which would be an entry qualification to Foundation degrees of a substantially quantitative nature as well as providing a good background for a wider range of employment and further study as well as courses outside a quantitative sphere.

2.1 Mathematics would be a non-core course at age 14. We see no reason why there should not be opportunities for internal summative assessment as well as external validation.
2.2 Alongside content appropriate for progression, it would develop problem-solving skills for both open and closed problems and contain some elements of mathematical literacy.
2.3 It is envisaged that the course would have a substantial take-up at Level 2, comparable to that of English Literature, with some students following level 2 Mathematics post-16.
2.4 Mathematics at Level 2 would give a smooth transition to Level 3 mathematics which would be to the same depth as the present AGCE Mathematics but with the amount of content being determined by the overall 16-19 structure. This would be the route taken by those wishing to progress to quantitative degrees where the mathematical demand was comparable to that for a B.Eng (See original paper for Further Mathematics proposals),as well as those who wish to pursue mathematics for its own sake.
2.5 'Mathematics' would have its roots in mathematical problem-solving. It would require the development of abstract algebraic and geometric reasoning in particular, as well as such applications as the use of fractions in probability, and the application of formulae to basic mechanical or other scientific embodiments. Rigour and fluency would be encouraged by the requirement for 80% mastery on written papers for any given grade, but there would also be a substantial element of sustained coursework eligible for assessment in at least some specifications: despite the well-known problems associated with the fair implementation of coursework assessment, we feel that these are essential skills which should be accredited.
2.6 Within the course, the most able would have suitable provision as outlined in the paper 'Making better use of mathematical talent', but this would be embedded in an inclusive pedagogy developed and enhanced by the National Academy. Thinking skills and cooperative learning would be an integral part of the mathematics classroom because the problem-solving nature of the course would demand group discussion and interdependence.

3.1 It is important that both 'Use of Mathematics' and 'Mathematics' should make significant use of ICT as an integral part of the course, for example for practising and consolidating skills, for developing skills in mathematical modelling, experimenting with, framing hypotheses from, exploring, discussing and explaining patterns and behaviour in shape and space and their links with algebra, developing logical thinking and modifying strategies and assumptions, making connections across and within areas of mathematics, working with realistic, and large, sets of data, and exploring, describing and explaining patterns and relationships in sequences, tables and transformations. It is important that our students develop these transferable skills, and learn to work independently as well as cooperatively. Constructive use of ICT facilitates this. Students should have frequent access to graphical calculators, as well as larger, more powerful machines.

In conclusion, we would support a dual award of this nature at GCSE as a means of revitalising perceptions of the subject, making the course more meaningful to students as well as the qualification more transparent, and giving GCSE mathematics a status commensurate with the effort involved. Draft specifications and assessment materials could be provided if required.

References
1. Oldknow, A and Taylor: Engaging Mathematics (Technology Colleges Trust, 2001)
2. Hoyles, C, Kent, P, Molyneux-Hodgson, S and Wolf, A: Mathematical Skills in the Workplace (Institute of Education, London and Science, Technology and Mathematics Council, 2002)

17.6.03.