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Standards and Quality in Secondary Schools 1995-1999: Mathematics

1. THE CURRICULUM

Courses

The pattern of provision of mathematics was very similar in all secondary schools. For S1/S2 there was a two-year course covering aspects of information handling, number, money and measurement; shape, position and movement; and problem-solving and enquiry. Most courses were intended to lead pupils towards appropriate levels of attainment as defined in the 5 - 14 National Guidelines in Mathematics. Although some departments had mixed-ability classes throughout S1/S2 the majority of departments inspected had classes set by ability in S2. This was increasingly the case over the four year period of this survey.

In S3/S4, pupils followed Standard Grade courses. Departments inspected employed some form of setting or broad-banding at these stages. Pupils were placed in class groups according to their likely attainment of awards, for example, Foundation/General, General/Credit, or Credit.

The curriculum in S5/S6 consisted of courses leading to Scottish Certificate of Education (SCE) Higher mathematics, Certificate of Sixth Year Studies (CSYS) and programmes of National Certificate (NC) modules. Pupils obtaining a Credit Award at Standard Grade typically followed a one-year Higher course in S5. Some of these pupils proceeded to CSYS study in S6, most taking only the core Paper 1 but some also attempting one or more of the four specialist papers available. Another group, usually consisting of pupils who had narrowly achieved a Credit Award together with pupils who had comfortably achieved a General Award, took a two-year Higher course. In many schools this course included several NC modules in S5, most commonly Core Mathematics 4, followed by Analysis/Algebra 1 and Calculus 1. A third group of pupils took a programme of NC modules.

1

The strengths of very good courses in S1/S2 included the following:

  • well balanced coverage of subject content and the inclusion of regular investigative work;
  • frequent opportunities for pupils to tackle challenging problems and think for themselves;
  • appropriate periods of revision and consolidation;
  • programmes which took full account of pupils' achievements in primary school and encouraged progress at a good rate towards higher levels of attainment from the start of S1;
  • carefully planned transition to Standard Grade courses allowing more able pupils to cover a significant amount of level F work in S2;
  • an effective response to the recommendations of Improving Mathematics Education 5-14 including: good development of pupils' skills in algebra and mental calculation; class organisation to enable pupils to make accelerated progress; and
  • guidelines for teachers on aspects of teaching and learning such as problem solving, appropriate use of calculators and differentiation.

Quality of courses

Many S1/S2 courses were not well planned. Only 10% of courses were very good and 50% had more strengths than weaknesses. The situation was better in S3/S4 where 10% of courses were very good and 60% had more strengths than weaknesses. They were better still in S5/S6 where 80% of courses were either very good or had more strengths than weaknesses. The strengths of very good courses in S1/S2 are set out in Box 1.

Although most departments had made efforts to bring their S1/S2 courses into line with national guidance, few courses were judged to be very good and too many displayed major weaknesses. Most still gave too little attention to mental calculation or problem solving and enquiry skills. Some did not give pupils enough experience of conducting surveys in information handling and using data bases and spreadsheets. Courses judged to be only fair were generally not well matched to 5-14 National Guidelines. Many of these courses covered much of what was necessary but did not phase the work in a way which allowed pupils' progress through 5-14 levels to be monitored. Some relied too heavily on a commercial scheme, others consisted of an uncoordinated mixture of class lessons and individualised programmes of work. This often led to pupils repeating familiar work and working below their potential, particularly in S1.

In S3/S4, most Standard Grade courses were well designed. Departments often offered separate Credit/General and General/Foundation courses in S3 with common elements, particularly early in S3 in order to allow pupils to transfer between courses as necessary. In S4, courses commonly focused on preparing pupils for a particular level of award. The strengths of very good courses in S3/S4 are set out in Box 2.

Those S3/S4 courses judged to have some important weaknesses generally needed to include raised expectations of pupils, with better pace and more emphasis on problem solving. Common weaknesses included:

Following consultation, the Scottish Qualifications Authority (SQA) has removed investigating as an assessable element in the Standard Grade course and has advised schools to give pupils greater experience of extended problems through coursework tasks. All schools should respond positively to this advice.

In S5/S6, most Higher and CSYS courses were well designed and prepared pupils well for the external examination. The best Higher courses gave pupils the opportunity to attempt more than the minimum of one investigation as required for certification. The new Higher course introduced from 1999 under the Higher Still Programme does not include an assessed investigation but, as at Standard Grade, mathematics departments should let pupils attempt sufficient of the SQA's published coursework tasks, or their equivalent, so as to develop skills in coping with extended and non-routine problems.

2

The strengths of very good courses in S3/S4 included:

  • well balanced coverage of all the elements with regular attention to investigative work which helped pupils to develop skills in problem solving progressively throughout S3 and S4;
  • challenging work from the outset of S3 giving pupils experience of solving non-routine problems throughout the course;
  • Credit Level courses which recognised that pupils had already covered some Credit content in S2;
  • sufficient time at the end of the course for revision and consolidation;
  • good emphasis at Credit Level on algebra and constructing and interpreting graphs, thereby providing a sound basis for Higher mathematics;
  • Foundation/General Level course design which allowed all pupils to aim for a General level Award for as long as possible, including good preparations to take the examination at General Level; and
  • guidelines for teachers on the pace of work expected and specific learning/teaching approaches to key topics.

though overall most S5/S6 courses were judged to be very good or to have more strengths than weaknesses, NC modular courses and two-year Higher courses were less well designed than one-year courses at Higher and CSYS levels. Common weaknesses included:

The new courses being introduced progressively from 1999 under the Higher Still Programme are intended to help schools provide better structure and progression for all pupils at all levels.

3

The improvements needed in mathematics courses are:

S1/S2

  • better opportunities for pupils to progress at a faster pace starting from where they have reached in primary school and, where appropriate, leading to level F by the end of S2;
  • increased use of direct teaching of attainment groups; and
  • more emphasis on mental calculation, algebra and problem solving.

S3/S4

  • better phasing,with less time spent on easy topics earlier in the course and more time for revision and consolidation towards the end;
  • more emphasis on solving non-routine and more challenging problems;
  • better knowledge of algebra for pupils following a Credit course in preparation for Higher work; and
  • a better preparation for the General level examination for pupils following Foundation/General courses.

S5/S6

  • more emphasis on direct teaching, greater opportunities for consolidation and revision and less reliance on individualised approaches; and
  • better phasing of work in two-year Higher courses.

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