A REVIEW AND COMPARISON OF THE MATHEMATICS GCSE SYLLABUSES OFFERED FOR EXAMINATION IN 1988
by PETER DAVID MORRIS B.Sc.
A Master's Dissertation submitted in partial fulfilment of the requirements for the award
of the degree of M.Sc. in Mathematical
Education of the Loughborough University
of Technology, 1988.
Supervisor: Mr. P. K. Armstrong B.Sc. M.Sc.
~ by PETER DAVID MORRIS 1988 ABSTRACT
In September 1986 the first groups of children began courses
which will culminate in the award of the General Certificate of
Secondary Education. From November 1987 (January 1988 in the case
of London University Examinations) the General Certificate of
Education and the Certificate of Secondary Education are no longer available.
The Department of Education and Science and the Welsh Office have published
GCSE: The National Criteria - General Criteria (9)
in which the rules and regulations for the development of any
GCSE courses are detailed. This includes a commitment to
'Criteria Referencing' as opposed to 'Norm Referencing' for the award of GCSE grades.
The seven grades defined will eventually be awarded for
reaching 'pre-determined standards of performance specific to the subject concerned'.
The publication:
GCSE: The National Criteria - Mathematics (10)
sets out the essential requirements which must be satisfied by
all syllabuses for examinations entitled 'Mathematics'. Within
the criteria specified the Examining Groups are responsible for
devising their own syllabuses and techniques for assessment.
The Subject Criteria for Mathematics specifies fifteen
~ims' and seventeen 'Assessment Objectives' which should be adopted by syllabuses on offer. The mathematical content is defined by List 1 which has to be included in ~ syllabuses; by
List 2 which has to be included in all syllabuses on which a candidate can be awarded a grade 'C'. Grades 'A' and 'B' will
only be awarded to candidates who have completed a syllabus well
in excess of that contained in List 1 and List 2.
The dissertation studies the background to GCSE and the
Subject Criteria specific to Mathematics.
Methods of differentiation are discussed together with a general section concerning coursework.
The main part of the the work details the schemes of assessment specified by the Examining Groups; each has submitted a scheme including a school based component and a scheme examined entirely by end of course written papers. In addition to these syllabuses the dissertation includes details of other nationally available schemes, namely SMP, KMP and MEI, together with all of the Mode 2 and Mode 3 syllabuses (none from LEAG and NEA).
Schemes which are based. on 100% coursework are also reviewed:- ATM/SEG, MEG and GAIM (not available in 1988).
Syllabuses which are undertaken in one year, usually on offer to 'mature' students, are also reViewed.
The dissertation ends by posing questions about GCSE and
Mathematics which have become prominent since its inception. ACKNOWLEDGEMENTS
I would like to express my thanks to the following people:
My tutor, Hr Peter Armstrong, for his invaluable help,
8lidance and constructive critism during the writing of this dissertation,
my wife, Geraldine, who typed and corrected the manuscript and whose help was indispensible. DECLARATION
I, Peter David Morris, declare that the contents of this dissertat1on, 'A Review and Compariscn of the Mathematics GCSE
Syllabuses offered for Examination in 1988' are entirely my own work. CONTENTS
Chapter
Abstract
1. Introduction 1
2. The Mode 1 Syllabuses 12 3. Nationally Available Mcde 1 Syllabuses 26
4. Mode 2 Syllabuses 31 5. Mode 3 Syllabuses 39
6. Mature Syllabuses 51 7. Other Schemes 56 8. Conclusion 66
Bibliography 75
Appendices 78 CHAPTER 1
Introduction
1 1.1 General Introduction.
The summer of 1988 will see the first staging of a single
system of examining pupils at 16+. The G.C.E. O-level and C.S.E.
examination dual system will be replaced by the General
Certificate of Secondary Education (GCSE). The main aims of the GCSE are:
to improve the quality of education to raise standards of attainment
to produce a system that is fairer to candidates to motivate teachers and pupils
to enhance the esteem in which examinations are held
to promote improvements in the secondary school curriculum
to remove the need for schools to enter candidates for
O-level and C.S.E. in the same subject.' (8)
The GCSE is to be administered by six Examining Groups:
The London and East Anglian Group (L.E.A.G.)
The Midland Examining Group (M.E.G.)
The Northern Examining Association (N.E.A.)
The Southern Examining Group (S.E.G.)
The Welsh Joint Education Committee (W.J.E.C.)
The Northern Ireland Schools Examinations Council (N.I.E.C.)
The Examining Group's work is to be monitored by the
Secondary Examinations Council (S.E.C.) and in particular the
S.E.C. have the right to sanction or veto any syllabuses submitted according to whether the National Criteria and the
2 Subject Specific Criteria have been followed.
The Examining Groups have to abide by the National Criteria, published in March 1985, when they are :-
drawing up examination syllabuses
devising assessment procedures
conducting examinations
awarding grades and issuing certificates.
The National Criteria are set out in two parts, General and
Subject Specific.
1. The General Criteria set out rules and procedures which govern the practices of the Examining Groups over all assessment matters. The General Criteria lay down guidelines for all syllabuses in all subjects.
2. The Subject Specific Criteria determine a framework for each of twenty (approximately 85% of total entries are made in these) subjects in terms of aims, objectives, content and assessment procedures.
The change from the dual system of G.C.E. and C.S.E •• to the single system of GCSE has been prompted for a variety of reasons which the aims of the GCSE mirror.
A criticism aimed at a Joint 'a' level and C.S.E. system concerns the proliferation of examination titles and syllabuses, the number of which has increased year by year. The Examining
Boards, of which there are 20, had by 1984 developed almost
20,000 different syllabusus. It was not unusual for neighbouring schools to have been undertaking different syllabuses in mathematics. Many schools were also dealing with many different
Examining Boards because different departments would be attracted
3 to syllabuses from different boards for a variety of reasons.
Confusion was apparent amongst pupils, teachers, parents and
employers. Difficulties arose when pupils transferred schools and
employers found difficulty in understanding why one syllabus
would include an item which they considered important whereas
pupils from another local school, following a different syllabus, would not have covered the same ground.
1.2 The Need for Change.
The majority of schools are now comprehensive and it seems logical for all comprehensives schools to follow a single examination system. The '0' level, introduced in 1951, was aimed at the population who attended grammar schools although many pupils who entered secondary modern schools gained considerable success in the examination.
The C.S.E. examinations, introduced in 1965, were meant to be nationally available for all pupils and the 'average' attainment target was grade 4 with grade 1 being perceived as equivalent to a G.C.E. '0' level grade C.
Many people, parents, pupils, teachers and employers never really gave the C.S.E. examination the credit it was due. Quite often it was more difficult to achieve a grade 1 at C.S.E. than it was to achieve a grade C at '0' level in the same subject.
C.S.E. examinations also introduced the idea of coursework in examinations so that the final grade awarded did not rest on the candidates performance. in one or two final timed written examination papers. The performance of a pupil over an extended
4 period, usually two years, was assessed. However, in mathematics,
coursework was often a misnomer. , Coursework , consisted of a
series of timed written, formal tests.
Pupils who were neither truly '0' level standard or truly
C.S.E. standard were placed in the difficult position, as were
the teachers, of having to enter two examinations. More often
than not they would be expected to follow an '0' level course in
preparation for both '0' level and C.S.E. examinations.
A single system of examinations, therefore, with a seven
point (eight really if. one includes ungraded) attainment scheme
should cater for the majority of pupils in a comprehensive
school.
1.3 GCSE and Mathematics.
In the introduction to The National Criteria for Mathematics
(10) it states that the 'criteria should be so interpreted that any scheme of assessment will:-
(1) assess not only the performance of skills and techniques but also pupils understanding of mathematical processes in the solution of problems and their ability to reason ma thematically;
(ii) offer differentiated examination papers so that, by choosing papers at an appropriate level, pupils are enabled to demonstrate what they know and can do rather than what they cannot do;
(iii) encourage and support the provision of courses which enable pupils to develop their knowledge and understanding of mathematics to the full extent of their capabilities, to have
5 experience of mathematics as a means of solving practical problems and to develop confidence in their use of mathematics;
(iv) have regard to the need for examination tasks to relate, where appropriate, to the use of mathematics in everyday situations.'
The National Criteria of Mathematics document lays down fifteen Aims (see Appendix A) and seventeen Assessment Objectives
(see Appendix B).
The Syllabus Content is divided into List 1 and List 2 (see
Appendix C).
The content of List 1 has to be included in all syllabuses called Mathematics and candidates expected to achieve grades
E,F,G will not be expected to know more.
List 1 and List 2 content must be included in schemes which can lead to grade C or higher.
Those candidates expected to gain grade A or B should 'have covered a syllabus well in excess of that contained in List 1 and
List 2'. (10)
The 'Techniques of Assessment' state that the written paper(s) should account for at least 50% of the total marks and that the coursework element (not compulsory until 1991) should account for at least 20% of the total marks available.
The National Criteria Mathematics document gives 'Grade
Descriptions' (see Appendix D) which are to give a general indication of the standards of achievements likely to have been shown by candidates awarded particular grades.
6 ~------!
1.4 Approaches to Differentiation
The S.E.C. has put out working papers, namely Differentiated
Assessment in GCSE (39), to advise about methods of ensuring that
single examinations are suitable and fair for the range of
ability entered.
In order to achieve fairness 'All examinations must be
designed in such a way as to ensure proper discrimination so that
candidates across the ability range are given opportunities to
demonstrate their knowledge, abilities and achievements - that
is, to show what they know, understand and can do. Differentiated
papers or differentiated questions within papers will be required
accordingly in all subjects.' (34)
The techniques suggested by S.E.C. are as follows:-
1. Differentiation by outcome - candidates would be given a
'neutral stimulus' and their attainment judged by their response
to that stimulus. In English the prompt may be 'a personal
experience, a picture or an argument'. In Mathematics the
stimulus may form the beginning of a problem to be solved or a
situation to be investigated.
2. Structured questions - the parts of a question are to be
in ascending order of difficulty so that less able pupils will
succeed with the earlier parts of the question and the more able
will succeed with more parts. This is fairly easy to achieve in
Mathematical questions but there are drawbacks to the techniques
which need to be considered.
3. Stepped papers - the questions are arranged in ascending
order of difficulty. Once again less able candidates should
achieve well on earlier questions and more able candidates should
7 be able to answer more questions. Similar drawbacks to those associated with structured questions are apparent with this approach.
4. Papers with differentiated sections - more difficult sections are to be attempted only by the more able candidates.
1.5 Differentiation in Mathematics
In Mathematics differentiated papers are compulsory. This is specified in the Subject Specific Criteria for Mathematics (10) and at least three levels of assessment must be provided. All of the new mathematics syllabuses which are aimed at the whole ability range include three levels of assessment named in a variety of ways - Foundation, Intermediate, Higher, Level 1,
Level 2, Level 3, or more obscurely Level P, Level Q, Level R (N.E.A.).
The most common way of providing three assessment levels is to offer four separate examination papers and for students at each of the three levels of entry to take a pair of papers.
Candidates at the lowest level of entry would sit papers and 2, at the middle level of entry wou19 sit papers 2 and 3 and at the highest level of entry would sit papers 3 and 4. (see below)
8 ------~---- Increased difficulty ------> Paper 1 Paper 2 Paper 3 Paper 4
Foundation Intermediate Higher
List 1 content List 1 and List 1 and
List 2 (paper3) List 2 plus
additional content:
Grades G CD E F E @ C D C ® A Available (target grade for level ringed) ------M.E.G.'s scheme of examination papers differs from that
above in that they are to set six papers i.e. three distinct
pairs one for each level.
The 'four-in-line' model has been adopted by the other
Examining Groups.
1.6 Coursework in Mathematics
The introduction of a compulsory coursework component in
Mathematics has been delayed until 1991 and at the moment 'all
Examining Groups must provide at least one scheme which includes
some elements of these two objectives. From 1991 these objectives must be realised fully in all SChemes'. (10)
The two objectives in question, stated in (10), are:
'3.16 respond orally to questions about mathematics,
discuss mathematical ideas and carry out mental
calculations' and
'3.17 carry out practical and investigational work
and undertake extended pieces of work'.
9 The emphasis in G.C.E. and (most) C.S.E. examination schemes
of assessment has been on end-of-course timed written examination
papers. 'The limits of both what pupils learn and how they learn
it; many aspects of subjects, and many skills and competences,
cannot be tested in written examinations - and so they go
unrewarded. Furthermore, because they are not tested, they are
not developed. In the hectic build up to fifth-year examinations
there is no room in the overloaded curriculum for activities that
the teacher may know 'are valuable, but which will not contribute
to the candidates final grades'. (28)
There is always the danger of the scheme of assessment dictating to the curriculum. An aim of GCSE is to widen the range of skills and competencies developed in children.
A list of skills and abilities which might be assessed using
coursework is shown below:- * research skills, including the ability to collect, order and process information and to
consider and weigh evidence; * interactive skills, including interaction with teachers, other pupils and adults, information
sources, materials and tools; * the ability to work in groups, including the ability to lead, to cooperate and to negotiate; * the ability to make and record accurate observations; * the ability to plan a piece of work of an extended nature;
10 • the ability to sustain an extended piece of work
from its conception and planning through its
execution to its conclusion; • to demonstrate perseverence; • motor skills, including processing materials and using and manipulating apparatus and machinery; • oral skills; • aural work, understanding abstractions; • investigative work of an open-sided nature (preferable to open~ended); • problem solving, selecting the right approach, adjusting methods in the light of feedback; * mathematical modelling; * the ability to produce good scale drawings.
If a work scheme includes items of this nature it may be fuat fewer children will leave school without the confidence to tackle even simple mathematical problems.
It is yet to be seen whether the GCSE syllabuses on offer help to develop any competencies, attributes or abilities listed above.
11 CHAPTER 2
The Mode 1 Syllabuses
12 2.1 A General View.
The examining bodies have each produced a syllabus with a school based assessment component and a syllabus without this coursework component. This means that objectives:-
'3.16 respond orally to questions about mathematics, discuss mathematical ideas and carry out mental calculations' and
'3.17 carry out practical and investigational work, and undertake extended pieces of work' (10) will not be fully assessed for every candidate until 1991.
The two assessment objectives can only really be achieved by undertaking some form of coursework or continuous assessment rather than by means of a time-limited written paper in a formal examination.
So during the period 1988 to 1990 'all Examining Groups must provide at least one scheme which includes some elements of these two objectives'.(10)
The examining groups have undertaken to do this and have approached the problem in different ways.
S.E.G. These two syllabuses are issued in separate booklets.
(i) Mathematics (without Centre Based Assessment) (36).
90% of the marks awarded for timed written papers.
10% of the marks awarded for aural tests set by
the Group but administered by the Centre.
(ii)Mathematics (with Centre Based Assessment) (37).
40% of the marks awarded for the Centre Based
Assessment.
10% of the marks awarded for aural tests set by the
group but administered by the Centre.
13 50% of the marks awarded on Group assessed timed·
written papers.
L.E.A.G. These two syllabuses are offered in separate booklets.
(i) Mathematics A (Without a School-Based Assessment). (17)
100% of marks awarded on Group assessed timed written
papers.
(ii)Mathematics B (with a School Based Assessment). (18)
25% of marks awarded for 'Tasks'.
5% of marks awarded on a Mental Test. 70% of marks awarded on Group assessed timed
written papers.
M.E.G. These two schemes are offered in a single booklet.
Scheme 1.(Code 1650) (25)
100% of marks awarded on Group assessed timed written
papers. Scheme 2.(Code 1651) (24)
25% of marks awarded on coursework assignments.
75% of marks awarded on Group assessed timed written
papers. N.E.A. Syllabus A.(29) (These two schemes are offered in a
single booklet)
Scheme 1.
no% of marks awarded on Group assessed timed written
papers.
Scheme 2.
25% of marks awarded on a coursework component.
14 75% of ' marks awarded on Group assessed timed written
papers.
N.E.A. Syllabus B.(30) (These two schemes are offered in a
single booklet)
This differs from N.E.A. Syllabus A in that the target group are those pupils who have followed courses based on the
'traditional' texts published by the School Mathematics Project.
W.J.E.C. (43) These two"schemes are offere~ in a single booklet.
Assessment Scheme A.
70 out of 270 marks for Centre based element.
Assessment Scheme B.
100% of the marks awarded on Group assessed timed written
papers.
2.2 Schemes of Assessment.
2.2.1 S.E.G. Mathematics (without Centre Based Assessment.)
(This syllabus is also appropriate for external candidates.)
There are three overlapping levels specified:-
Level 1 - with target grade F and,grades available E, F
and G.
Level 2 - with target grade D and grades available C, D,
E, and F.
Level 3 - with target grade B and grades available A, B,
C, and D.
The target grade at each level is awarded for approximately two thirds of the marks available and the grade below for approximately half the marks available. Each candidate will take
15 two written papers and two aural tests. The papers numbers 1· to 4 are arranged four-in-line and each paper carries 45% of the total marks available.
The aural tests, numbered 5, 6, 7 and 8 are arranged four- in-line; each consists of 15 questions 'requiring single responses'.
SE.G. have put forward an Assessment Objective Grid (shown below) to illustrate the breakdown of their three levels.
: Knowledge Skills Application Problem Total : Solving
Level 1 25 25 25 25 100 Level 2 20 25 25 30 100
Level 3 17 23 27 33 100
It can be seen from the grid that whereas each of the subdivisions are equally weighted at Level 1, those at level 2 and Level 3 Applications and Problem Solving are more heavily weighted than Knowledge and Skills.
2.2.2 S.E.G. Mathematics (with Centre Based Assessment).
The levels 1 to 3, their target grades and grades available are identical to S.E.G.'s other syllabus (see page 15)
However because of the presence of a coursework element worth 40%, the four-in-line written papers are each weighted at
25% of the marks available and the four-in-line aural tests, of which the candidate does two, contribute the remaining 10%.
16 The Centre Based Assessment is to be composed of three units, each unit being either a single task or a series of short assignments. The choice of tasks or assignments is left up to the teacher
'so that they may be relevant to the needs of individual pupils'.
(37). They should not be completed under formal conditions but should form 'an integral part of a unified course'.(37)
Each of the three units of coursework together with an oral unit are marked out of twelve and then weighted to produce an overall mark out of 120. The oral unit is designed to be a series of discussions about the candidates work under the following headings:
a) Starting points and assumptions.
b) Method of recording and reporting.
c) Results.
d) Checking.
e) False leads.
f) Extensions.
g) Conclusion.
The Centre Based component is to be marked by the candidates teacher using a scheme with headings:
1. Comprehension of task.
2. Planning. 3. Carrying out the task.
4. Communication.
Finally, the moderation of the coursework will be by consensus, by groups of teachers from various centres meeting to
17 discuss the work of each others candidates.
2.2.3 L.E.A.G. Mathematics A. Mathematics A uses four papers in a four':'in-line arrangement to examine the three levels X, Y and Z. Each paper 1s weighted
50% of the total mark.
Level X target grade F, grades available E, F and G.
Level Y target grade 0 , grades available C, 0 , E and F.
Level Z target grade B, grades available A, B, C and o. Candidates achieving a mark below the minimum for the award of the lowest grade at any level will be ungraded.
2.2.4 L.E.A.G. Mathematics B. The Scheme of Assessment in terms of arrangement of written papers is identical to that shown above. The difference being that each of the four papers contributes 35% of the total marks.
The remaining 30% is contributed by School Based Assessment comprising of:
A. Mental Test (5%)
B. Tasks.
The mental tests are given at the three levels X, Y and Z and are administered after the centre has made its entries.
L.E.A.G. candidates are expected to submit five coursework tasks which may be of their own choice. Alternatively they may choose to do five of the seven which are set by the Group. Three of these will be tasks suitable for candidates at all levels of entry and the remaining two will be suitable for either grade X and Y or Y and Z.
18 Each of the tasks are to be assessed by the candidates own teacher under the headings:
i) Identification of the task and selection of strategy.
ii) Implementation of the chosen strategy and reasoning.
iii) Interpretation and communication.
The work of selected candidates will be moderated by an ecternal assessor. This may be extended to the work of all candidates at a centre.
2.2.5 M.E.G. Mathematics (1650).
M.E.G. have opted to use the three pairs model for their six external papers. There are three levels offered, namely,
Foundation, Intermediate and Higher. Their target grades are shown below:
------~------: Papers Level Intended Target Group Grades Available
1 and 4 Foundation E, F, G. E, F, G.
2 and 5 Intermediate C, D, E. C,D,E,F.
3 and 6 Higher A, B, C. A, B, C, D.
If a candidates work does not reach the standard required for the lowest grade available at any given level, then he/she will be unclassified.
Each paper contributes 50% of the marks available.
19 2.2.6 M.E.G. Mathematics (1651).
In this scheme, which includes a coursework component, the arrangement of written papers is identical to that shown for
1650. The weighting allocated to the papers differs from the previous scheme.
Component Level Weighting
Paper 1 Foundation 50~
Paper 2 Intermediate 50~
Paper 3 Higher 50~
Paper 4 Foundation 25~
Paper 5 Intermediate 25~
Paper 6 Higher 25~
The remaining 25~ is obtainable by submitting five OQursework assignments, two of which have to be carried out during the last twelve months of the course.
The assignments which are to be teacher assessed and then externally moderated, are to be chosen one from each category shown below: 1. Practical Geometry.
2. An Everyday Application of Mathematics.
3. statistics and/or Probability.
4. An Investigation.
5. A Centre Approved Topic.
The assignments have to be marked under the headings:
1. Overall Design and strategy.
20 2. Mathematical Content.
3. Accuracy.
4. Clarity of Argument and Presentation.
5. Controlled Element. The Controlled Element, which illustrates a difference in
approach between M.E.G. and the other Groups, has to be
undertaken individually by the candidates. They may take one of
several forms: 'a timed or untimed written test;
an oral exchange between the candidate and the teacher;
a parallel investigation or piece of work; a parallel piece of practical work, or practical test
including a record of results;
a written summary or account.' (25)
1I1e intention is to provide a check on the candidates ability in Mathematics·together with the authenticity of the work completed.
2.2.7 N.E.A. Syllabus A Scheme 1 and Syllabus B Scheme 1
The arrangement of papers, four-in-line, the weighting given to each paper and the grades available at each level are the same for both of the above syllabuses.
21 ------: Level component Weighting Candidates limited to grades : ------Level P Paper 50% E, F, G.
: Paper 2 50%
Level Q Paper 2 45% C, D, E, F, G. Paper 3 50%
Level R Paper 3 45% A, B, C, D, E, F, G. Paper 4 50% ------
Candidates can be unclassified if they fail to reach the standard for grade G.
At the time of writing, the principle of a candidate being able to 'slide down' the grades at a particular level had not been accepted by the Secondary Examination Council.
The approach, favoured by N.E.A., is fundamentally different from that adopted by the other boards whose assessment schemes do not offer to candidates this allowance.
2.2.8 N.E.A. Syllabus A Scheme 2 and Syllabus B Scheme 2
Scheme 2 differs from Scheme 1 in that it includes a coursework element. The arrangement of papers, four-in-line, the weighting given to each paper and the grades available at each level are the same for both of the above syllabuses.
22 ------: Level Component Weighting Candidates limited to grades :
Level P Paper 1 37.5% E, F, G. Paper 2 37.5% Coursework 25% Level Q Paper 2 33% C,D,E,F,G. Paper 3 42% Coursework 25%
Level R Paper 3 33% A, B, C, D, E, F, G. Paper 4 42% Coursework 25%
Another unresolved issue between N.E.A. and S.E.C. concerns the relationship between the grades attained on coursework, on written papers and the resulting final grade. N.E.A. feel that
'if the grade thus obtained by candidates offering the coursework option is less than the grade that would have been obtained on the written paper alone, the higher grade will be awarded'. (29,30) The coursework component is very much up to the Centres to organise including decisions on the nature of tasks and teachers are free to devise their own assignments for assessment purposes.
Teachers have to complete a Candidate Internal Assessment Form commencing with a recording of any initial discussion about choice of coursework and planning of work.
Candidates submissions can take various forms and are to be assessed under the headings:-
23 a) Practical Work.
b) Investigational Work. c) Assimilation (to include recall of knowledge and oral
communication concerning the subject matter).
N.E.A. suggest moderation using the performance in the written examination as the moderating instrument and is carried
out statistically rather than by subjective comparison of
candidates work. 2.2.9 W.J.E.C. Mathematics Scheme A. Candidates for Scheme A have to sit two of the four-in-line
papers set as well as completing two coursework tasks. The same
written examination papers are used in Scheme A and Scheme B
(see 2.2.10). Target Grades are as follows:-
Level 1 Target grade F, grades available E, F and G.
Level 2 Target grade D, grades available C,D and E.
Level 3 Target grade B, grades available A, Band C. Each paper at a level is worth 100 marks and the coursework
component carries 70 marks. W.J.E.C. discriminate between levels when specifying the
coursework requirements. Candidates at Level 1 will be required: 'i) to undertake an investigation of a particular theme
using various mathematical techniques;
ii) to demonstrate and apply certain practical
skills' .(43) Whereas those at Levels 2 and 3 have to undertake:
'i) a practical investigation; ii) a.problem solving investigation.' (43)
24 W.J.E.C. supply two of each type for Levels 2 and 3 and also for (i) at Level 1. In (ii) at Level 1, however, the Council provides four practical exercises of which the candidate must do three. Centres may choose their topics.
Assessment of the assignments falls into four categories:
1. Understanding the Problem.
2. strategy. 3. Mathematical Content and Development. 4. Communication.
The scheme includes an oral and aural component. The aural test consists of a mental test worth 10 out of the 70 marks available. Moderation will be done internally initially and if necessary samples of work will be externally moderated.
2.2.10 W.J.E.C. Mathematics Scheme B.
This scheme uses the same written papers as Scheme A (see
2.2.9). The four-in-line papers, of which the candidate sits two, are each worth 50% of the marks available.
The target grades and grades available at each level(1, 2 md 3) are identical to those specified in' 2.2.9 for Scheme A.
25 ------EXAM. GROUP LEAG MEG NEA ------SEG WJEC Proportion of 25% for 5 tasks. 25% for 5 tasks. total marks 25% in three 40% for three 26% for two Oral assessment Oral assessment areas : allocated to not specifically units : tasks may be One fifth of this to coursework included. included. 9/30 fol' oral and oral formal. assessment.
------Proportion of 5% for mental None None total marks test. 10% for two 10/70 of the allocated to aural tests marks for aural aural tests. based on practical investigations. ------Arrangement of 5 tasks from 3 5 tasks, one from assignments, Coursework not 3 tasks one of 2 assignments categories each of:- specified. oral and aural which to be an a practical i) investigation i) prac. geometry Subdivided into tests. 11) problems extended piece of investigation and: 11) everyday practical work, work. i11) practical applications a problem investigational 8% for oral solVing invest. Set by LEAG or iii) stats/prob. work and their own. Oral assessment of Practical iv) investigation assimilation (to coursework. work not v) centre investigation include oral 10% for aural. includes aural specifically : approved topic : communication) included. No oral test but test. Oral may be included assessment fol' in task. each assignment ------
THE EXAMINING GROUPS ARRANGEMENTS FOR COURSEWORK CHAPTER 3
Nationally Available Mode 1 Syllabuses
26 3.1 A General View.
Whereas it is likely that a school within the area of a particular Examination Group will be encouraged to enter
candidates for the scheme their Home Group provides, they can
look elsewhere for schemes and enter candidates under other
Groups. The school will more than likely find that the L.E.A. is
not very co-operative, however, when it comes to releasing
members of staff for GCSE training days or workshops if the
school chooses a syllabus offered by an alternative Group.
There are two other Mode 1 syllabuses which are available
nationally which can be entered via the school's Home Board.
These are Mathematics (S.M.P.)(40) and Mathematics (S.M.P. 11-16)
(42) which will be examined by M.E.G. in 1988 in both the summer
and autumn series in the case of S.M.P., but only summer for
S.M.P. 11-16.
3.2 M.E.G. Mathematics (S.M.P.) Syllabus Code 1652.
~ Mode 2 scheme based on the syllabus and containing a school based element is also available, see chapter 4.]
This syllabus and scheme of examination has been designed for schools who are using S.M.P. textbooks from the lettered and numbered series. Although not all topics covered in the texts are to be examined, the syllabus and specimen papers retain the
"flavour" of the S.M.P. approach.
The scheme of assessment allows for entry at three separate levels, namely Foundation, Intermediate and Higher. These have target grades indicated in the table below.
27 ------Level Intended Target Group Grades Available ~ ------F Grades E, F, G. E, F, G.
I Grades C, D, E. C, D, E, F.
H Grades A, B, C. A, B, C, D. ------
Candidates enter for either papers 1 and 2 (Foundation), 2 and 3 (Intermediate) or 3 and 4 (Higher). Each paper of this
four-in-line system ,is worth 50% of the marks available. Unlike
other examinations running four-in-line papers, papers 1 and 3,
and 2 and 4 will probably be timetabled simultaneously.
3.2 Mathematics (S.M.P.11-16) Syllabus Code 1653.
This scheme and syllabus is offered by M.E.G. and is
available nationally. It has been designed for schools who are using the S.M.P. 11-16 materials, although any school following any scheme may enter. L.E.A.G. are assisting M.E.G. with this examination which has achieved National Curriculum Project status.
The assessment scheme has component parts of written papers, mental tests and coursework tasks. There are three levels of assessment, namely Foundation, Intermediate and Higher and these are targeted at the grades shown below.
28 Level Intended Target Group
Foundation Grades E, F, G.
Intermediate Grades C, D, E.
Higher Grades A, B, C.
The syllabus states that 'the great majority of candidates entered at the Foundation Level are expected to be awarded grades
E, For G; in exceptional circumstances grade D may be awarded'.
Similar statements are made about the Intermediate Level where some candidates may be awarded F and the Higher Level where some candidates may be awarded grade D.
The components of the scheme are:- i) Written Papers;
ii) Coursework;
and iii) Orally given Tests counting for 70%, 25% and 5% of
available marks respectively.
The four written papers are arranged four-in-line and candidates take either papers 1 and 2 (Foundation), papers 2 and
3 (Intermediate) or papers 3 and 4 (Higher)'. Each paper is worth
35% of total marks available.
The Coursework component is assessed through six coursework tasks (eight in future) which are of one of three types:-
ST - to be done in school, under supervision with a time
limit of one hour.
S - to be done in school, under supervision but with no
time limit.
29 H - to be done out of school.
The coursework tasks are specified by S.M.P. who also
provide marking schemes. The marking is undertaken by the
candidate's teacher, is internally moderated and also externally
moderated by a team led by a Chief Coursework Moderator.
In future examinations, schools may be allowed to submit
their own or their pupils assignments in the place of two of the sets of tasks.
The orally given tests are administered by the school in the latter half of the examination cohorts final year. Each candidate is required to take two oral tests.
30 CHAPTER 4
Mode 2 Syllabuses
31 4.1 A General View
Mode 2 syllabuses are permitted under the GCSE provided
that: - they conform to the requirements of the General
Criteria;
- they conform to the requirements of Subject Specific Criteria (for such subjects where Subject
Specific Criteria exist).
Mode 2 syllabuses are offered by an Examining Group having
been developed by an individual school or group of schools. The
examinations are conducted by the Examining Group.
Mode 2 syllabuses have been submitted by:-
The Schools Mathematics Project (S.M.P.) (41)
The Kent Mathematics Project (K.M.P.) (13)
The Mathematics in Education and Industry Schools
Project (M.E.I).(22)
The S.M.P. scheme has been done independently of the
S.M.P.(11-16) assessment development and includes a Mental
Mathematics Test, an Oral Test and a folder of Extended Pieces of
Work. The K.M.P. scheme is based on the K.M.P. approach to
Mathematics and has 50% of available marks awarded by internal
assessment using the K.M.P. materials. The M.E.I. scheme differs from the others in that it offers
~ur levels of entry whereas all the other schemes offer only three. It includes a coursework element, mental calculation tests and written papers.
32 4.2 M.E.G. Mathematics (S.M.P.) Mode 2.
The S.M.P. Mode 2 proposals differ from other schemes in a variety of ways. They maintain that syllabuses content should be reduced, if necessary, to allow time for coursework and its assessment.
The scheme has four component parts:
a mental mathematics test;
an oral test;
a folder of extended pieces of work (EPW);
and a timed written examination.
The mental mathematics test is to be orally administered given to a class or group together. There will be a limited time given for a response and answers are to be written on a prepared answer sheet.
The oral test is to be conducted as a one-to-one interview probably with a teacher from the same school but who does not teach the pupil mathematics. The teacher has a script which he/she must use during the test.
The oral test consists of three separate items:-
a) a stage in which the candidate is able to discuss
with the interviewer an EPW that he/she has
completed. This is not marked as a separate item but
will help to give an overall view of the candidates
ability to communicate mathematically,
b) an item selected by the candidate prior to the test
and hence prepared,
c) an unprepared item.
Marks are awarded on items b) and c) and on the overall
33 impression.
1be extended pieces of work will be prompted by the use of starting points from the categories:
Investigational
Applied
Practical
Numerical
Spatial
Candidates have to submit their best four items from at least six done. An EPW is expected to take up about two weeks of mathematics lessons and associated homeworks.
The starting points will be issued by the Examining Group and will be a single sheet of paper posing specific questions about a situation but then allowing the candidate to develop the situation in any chosen direction.
Moderation of the EPW will be carried out initially by the teachers in each school who have already marked the work and a coursework moderator who will have overall responsibility for the
EPW in participating schools.
The marks awarded are shown below:
Mental 5:t
Oral 5:t
EPW ( 4 at 7:t) 28:t
Timed, written papers 62%
The written element consists of 4 papers: Foundation Level, papers 1 and 2, Intermediate Level, papers 2 and 3 and Higher
Level, papers 3 and 4 ••
34 This scheme appears to be the only one which contains a
genuine oral test and allows the student to discuss mathematics.
4.3 K.M.P. Mathematics This syllabus has been designed for students who have been
prepared for the GCSE using the K.M.P. system. The marks are divided into 50% for externally assessed timed written papers and
50% for work of an investigational nature, practical nature
together with work requiring candidates to collect and select the
appropriate data from the real world. 'Classroom activities are
important to the development of mathematical ideas and it is
possible to include in such activities particular tasks which are
better tested whilst in progress, that is in the formative stage,
rather than as a summative assessment'. (16)
There are three overlapping examinationn levels:-
Examination Level 1 Target Grade F
Grades available E, F and G
Examination Level 2 Target Grade 0
Grades available C, 0, E and F
Examination Level 3 Target Grade B
Grades available A, B, C and 0
The target grade will be awarded for approximatly two-thirds
of the marks available for both the internal and external assessments. The formative assessment for K.M.P. GCSE combines two types of approach, the Discussion Investigation Practical (D.I.P.) component and one based on content and the matrix tests.
Every task in each of the K.M.P. levels has been analysed
35 to see if it involves Discussion, Investigation and Practica~ work • For example, Level 1 has 94 tasks of which 29 are
Investigational and 66 Practical; Level 6 has 90 tasks of which 51 are Investigational and 46 Practical. All tasks afford opportunity for discussion.
The teacher assesses the D.I.P. component as either having been passed or failed but cannot be marked off as being completed until the teacher is satisfied that the criteria set out under
Discussion, Investigation, Practical have been met.
After having completed the tasks, the pupil takes a matrix test. The result is converted from a raw mean mathematics level to a corrected mathematics level which is used as a measure of continuous assessment.
Please refer to K.M.P. and GCSE A Guide for Teachers for more information.
4.4 M.E.I. Mathematics
The Mathematics in Education and Industry Schools Project
GCSE scheme differs from all the other schemes on offer in that it allows candidates to be entered at one 'of four levels. Other schemes allow three levels of entry.
The components of the scheme are similar to those of other schemes i.e.:-
Timed written papers;
Mental tests;
Set tasks.
36 The timed written papers are set at four levels:-
Level 1 Basic Test + Paper 1 Grades E, F, G.
Level 2 Paper 1 + Paper 2 Grades C, 0, E, F.
Level 3 Paper 2 + Paper 3 Grades B, C, 0, E. Level 4 Paper 3 + Paper 4 * Grades A, B, C, D. * Paper 5 is available for those candidates who can expect a comfortable Grade A from Papers 3 and 4.
The target grade at any level can be achieved by gaining 65%
of available marks. The highest grade at each level can be obtained by gaining 80%+ of the marks available.
At each level the two written papers both contribute 37% of all marks available leaving 26% for the coursework component.
At level 1 candidates are required to attempt six 30 minute papers on a limited syllabus. Their best five results are totalled in order for the award to be made. The basic tests are set on a series of 'themes' which will be made known to participating schools in the previous September.
The coursework element can be divided into 1) set tasks and
2) mental calculation tests.
The four set tasks have to be completed by the spring term of the examination year and are designated Shorter Set Tasks and
Extended Set Tasks. The candidate has to undertake three Shorter
Set Tasks (about 4 hours/one weeks mathematics time) and one
Extended Set Task (about 8 hours spread over 5 weeks).The work in the Shorter Set Tasks is authenticated by a short test and that in the Extended Task by a one to one situation where the candidate will answer oral questions.
37 The mental calculation tests are straightforward mental
tests given at either the Lower Level (levels 1 and 2) or at the Higher Level (levels 3 and 4).
Marking of the mental tests and set tasks is completed internally and moderated internally. Samples of marking have to be sent to the Examination Group for moderation.
M.E.I.'s submission has been made through M.E.G. and S.E.G ••
38 CHAPTER 5
Mode 3 Syllabuses
39 5.1 A General View.
Mode 3 syllabuses differ from Mode 1 and Mode 2 syllabuses in that they are often designed and written by an individual school or group of schools. The candidates work is also assessed by those schools and the Groups involvement is only one of moderation and guidance.
Mode 3 syllabuses have to conform both to the requirements for the General Criteria and to the Subject Specific Criteria.
5.2 Mode 3 Syllabuses Overseen by M.E.G.
5.2.1 Leicestershire Schools Mode 3 Syllabus A (21)
Names of Centres:-
The Robert Smyth School
Earl Shilton Community College
English Martyrs R.C. School
The City of Leicester School
Crown Hills Community College
Judge Meadow Community College
Sir Jonathan North Community College
The John Cleveland College
Lancaster BOys School
King Edward VII Upper School
De Lisle R.C. School
Leicester High School
John Ellis Community College
40 The syllabus recognises three overlapping levels:-
Level 1 Target Grade F Grades Available E, F, G.
Level 2 Target Grade D Grades Available C, D, E, F.
Level 3 Target Grade B Grades Available A, B, C, D. The syllabus has three components:-
a) Coursework Folio (30%) b) Coursework Tests (20%)
c) Written Examination (50%)
1be Coursework Folio is designed to encompass
Investigational Work, Problem Solving and Practical Work. These will comprise 15%, 10% and 5% respectively of the final mark awarded.
The investigations are to include short and long and should total more than five (3 long and 2 short).
The problem solving is couched in mathematical modelling terms and should include three good examples.
The practical work is expected to amount to about three
~UN.
The coursework tests are of two kinds. The first are Topic Tests which are short tests taken under examination conditions soon after the teaching of a topic. The candidates best nine results (out of 12) are used for assessment purposes. The Topics to be tested are designated by the Consortium and the tests are set by the Consortium, although schools may produce their own
(Consortium approved) versions.
The mental mathematics tests are provided by the Consortium and may be either aural tests or timed written papers. The candidates final best ten should contain 3 of each type. ,
41 The written examination consists of six papers with two
papers to each level. The target grade can be attained by
achieving a mark of around 65%.
5.2.2 Leicestershire Schools Mode 3 Syllabus B (21)
(Administrative Centre - The Longsdale College)
Main Consortium:-
Beauchamp College, Oadby
Babington College, Leicestershire
Beaumont Leys School, Leicestershire
Bosworth College, Desford
Burleigh College, Loughborough
Groby College, Groby
Longsdale College, Birstall
The scheme of assessment includes written examination papers
and a Coursework Folio. The three levels of assessment have target grades F, D and B which can be achieved by scoring approximatly 70% of marks available.
The written examination papers are in two parts:
i) Section A - an aural test
ii) Section B - a written examination.
Unlike all other schemes the Leicestershire scheme only has one substantial examination paper per level.
The Coursework Folio is expected to include practical work, investigations and problem solving as well as an area of mathematics specified as:-
42 Level 1 Practical applications based on simple locus properties.
Level 2 Loci in 2-D and 3-D. Intersection of loci.
Level 3 Matrices and transformations, matrices for combined transformations.
The rest of the Folio is to be made up of any work deemed
suitable and should cover a balanced, broad range of work.
5.2.3 Henry Fanshawe School (Limited Grade) (14) This GCSE syllabus appears to be the only one which is
designed to offer only a limited number of grades. In this case the grades available are E,F and G. The centre is The Henry
Fanshawe School, Dronfield, Sheffield.
The scheme uses 'continuous, periodic and terminal a3sessment'. (14)
The scheme of assessment comprises five written tests and four problem solving assignments including an oral assessment.
Written periodic tests with
mental calculations (Tests 1,2,3,4) 40%
Terminal written examination (Test 5) 30%
Problem solving assignments 1,2,3,4
and oral assessment 30%
The oral aseessment is meant to take place over the two
~rs of the course. The problem solving component is classified as:
P1 An investigation
P2 A statistical problem
P3 A problem from everyday life
43 p4 A geometric problem.
Each assignment consists of a number of tasks on the given
theme and the best mark is counted for the contribution from that assignment.
Marks are awarded under the headings:-
1) Understanding
2) The solution 3) Accuracy 4) Presentation
5) Conclusions and Evaluation. The end of course examination lasts for 90 minutes and covers all sections of the specified syllabus content.
5.3 Mode 3 syllabuses overseen by S.E.G.
5.3.1 Avon Trent Consortiums (2)
This is a cluster group syllabus for Further Education
Colleges for a limited range of grades A-D. The consortium members are as follows:-
South Warwickshire College of Further Education
East Warwickshire College of Further Education
North Warwickshire College of Further Education Grantham College of Further Education
Harrow College of Further Education
North Hertfordshire College of Further Education
1I1e students are expected to cover the common core (based on
List 1 and List 2 of the GCSE National Criteria (10)) together with at least four of the option modules.
44 The assessment scheme is as follows:-
1. Formal Examination 30%
2. 2 Supervised Module Tests 20%
3. 2 Unsupervised Module Assessments 20%
4. Aural (Mental) Test 5%
5. 3 Assignments - Written Reports 20%
- Oral 5% Assignments are marked under the headings:-
1) Appreciation of the Task
2) Procedure
3) Communications.
The oral assessment will be 'undertaken as a question and answer session on discussion with a tutor, possibly after the student has presented his assignment work to his fellow students'
(2). It will be based on one of the candidates assignments.
5.3.2 st. John Rigby College (39) 'Purpose: To provide a course and a scheme of examination
suited to the needs of sixth formers who will, in
the main, have obtained grades 2 to 4 in C.S.E.
(subsequently comparable grades in GCSE)'.(39)
The scheme of assessment is shown below:
Assessment Grades C/D/E
(J:lmponen t Actual
Examination Structured Questions of variable ) Examination 2 lengths without choice.
Mental Test Mental Arithmetic and Data Response
Short Assignments 10 of them
45 Class Tests 10 of them
Projects 3 (to include oral)
N.B. 1 ) Assignments are completed out of class.
2) Assignments and class tests will include
specifically arithmetical and calculator work.
3) Projects to be open ended and include investigational, practical and oral work.
The candidate aiming for grades AIBIC have to follow a
amilar course of ~ssessment sitting Examination 2 (see Grades
C/D/E) and Examination 3. Projects are marked under the general headings:-
Comprehension and Planning (10 marks)
Performance of Task (10 marks)
Interpretation and Communication (10 marks)
Cardinal Wiseman School - Mathematics Mode 3 (5)
The intended target group is Grades C,D,E,F and G at GCSE and the syllabus is divided into two option areas,
1) Coursework only 2) Coursework with projects.
Assessment of Option 1:- Continuous assessment 60%
School based course tests 30%
Aural tests (2) 10%
Assessment of Option 2:- Continuous assessment 40%
Project assessments (3) 20%
School based course tests 30%
Aural tests (2) 10%
46 The scheme is based on four courses:-
1) Money Management 1
2) Money Management 2
3) General Mathematics
4) statistics.
The optional projects (Option 2 only) are based on courses
1), 2) and 4).
~e course is generally assessed under the following headings:-
1. Task Comprehension
2. Planning
3. Task 4. Communication.
5.4 Mode 2/3 syllabuses overseen by W.J.E.C.
5.4.1 Newport and District Mode 2 (32) The target grade is grade F with grades available being E,F and G. Candidates will be entered for two written papers and a school based element. The weighting being 200:100 respectively.
The two written papers are set on:-
1 - Basic Mathematics and
2 - Home, Business and Trade, ,Sport Leisure and Travel.
They will be externally set and marked.
The school based element is divided into three parts where candidates have to undertake:-
(i) an investigation
(ii) an oral test
47 (iii) an aural (mental) test.
The investigation is marked under three stages:-
stage 1 Collection and Representation of Data 15 marks stage 2 Relevent analysis using Tabular Data 15 marks
stage 3 Extension 15 marks 5 marks are allowed for .. Presentation.
Within each stage the marks are subdivided under five headings: i) Understanding
a) What is involved and what is expected b) Independence
11) strategy
iii) Content and Development
iv) Communication.
~e proposed oral/practical tests are by far the most
comprehensive of all the schemes reviewed. They are designed to
be carried out on a one to one basis with each pupil and may last 30 minutes.
5.4.2 Mid Glamorgan County Council Mode 2 (27)
The course is intended for candidates able to achieve grades
E,F and G in the GCSE. The scheme is composed of five modular assessments, an oral examination and a final written examination.
The final examination paper is that set by W.J.E.C. as Paper 1 of its Mode 1 scheme.
The five modules and assessment schemes are shown below:-
48 Module 1 Use of a calculator and basic number properties .
Aural Test
Written Test
Investigation
Module 2 Maps and Plans Aural Test
Written Test
Practical
Module 3 Opinion Polls, Market Research and the Media
Aural Test
Written Test
Survey
Module 4 Sport, Leisure and Travel Aural Test
Written Test
Practical
Module 5 Earnings and Spending Aural Test
Written Test
Practical
The oral test comprises 1) Interpretation of Chart, Graph or
Table
2) Real Life Maths 3) Spatial Relationships.
The written tests for each module are one hour.
49 5.4.3 Clwyd Scheme Mode 3 (7) The scheme is targeted at those canidates who may achieve grades E,F or G at GCSE. The scheme of assessment is outlined below:-
Written Assessment on Unit 1 5%
Written Assessment on Unit 2 5%
Written Assessment on Unit 3 5%
Written Assessment on Unit 4 5% Two Investigations 20%
Practical S1(111s and Applications 5%
Aural 5%
Final Examination Paper 1 25% Paper 2 25%
The investigations are to be assessed under the general
headings:-
1. Understanding
2. strategy 3. content and Development
4. Communication
5. Methods used and Accuracy.
5.5 Mode 3 offered through L.E.A.G. L.E.A.G. were unable to inform the author of any Mode 3
submissions in their area.
5.6 Mode 3 offered through N.E.A.
N.E.A. refused the author access to any information
regarding Mode 3 syllabuses in their area.
50 CHAPTER 6
Mature Syllabuses
51 6.1 General. There are many situations in schools whereby candidates wish to take one year courses of study. One year courses may be used to extend general education of sixth formers and supplement the curriculum. They may contribute to pre-vocational and vocational programmes of study. They may provide an opportunity for improvement of a GCSE grade from a previous examination. External candidates involved in Adult Education classes generally follow a one year course which sometimes leads to a public examination.
GCSE (Mature) syllabuses are available in Mathematics and many other subjects and are sanctioned by paragraph 15 of the
General Criteria (9). 'Examining Groups will be free to design syllabuses for more mature students as well as secondary school pupils, provided that they comply with the National Criteria.'
6.2.1 M.E.G. Mathematics [Mature] 1654.(26) The syllabus has been designed to serve the needs of 17+ candidates and more mature students and is aimed predominantly at two groups:- '(a) Candidates who require a higher GCSE grade as an
entrance qualification for employment or for a course
of further study. The majority of these are likely to
have taken a GCSE examination recently.
~) Candidates who hope to take 'A' Level Mathematics at a
later date or who are already taking other subjects at
'A' level and require more Mathematics in support.'
Assessment is provided by an aural test (10%) and written
52 papers (90%) and there is no coursework component whatsoever.
The written papers, six in all, are aimed at the three
levels Foundation, Intermediate and Higher. The first paper at each level is worth 40% of available marks and the second 50%.
There are three different aural tests 91, 92 and 93, one for each level.
In order to reflect the interests of the likely candidates the syllabus has been changed slightly from that of M.E.G.
Mathematics 1650 (25). Transfo:mation, geometry, tessellations, nets and number patters are deleted. Money management, kinematics and statistics are increased. At the higher level some additional algebra has been included.
6.3 L.E.A.G. Mathematics (M) Syllabus No's 5375, 5376, 5377
and 5378(19) This syllabus is offered at only the two levels,
Intermediate (5375 and 5376) and Higher (5377 and 5378), The content is grouped into modules of which each candidate will have to have studied three for the Intermediate Level examination and five for the Higher Level examination.
The modules are listed below:-
1. Design and Pattern
2. Collection, Presentation and Analysis of Data
3. Industrial Mathematics
4. Commercial Mathematics
5. Decision Mathematics
6. Transformation Geometry
53 7. statistics
8. Calculator Mathematics. At the Intermediate Level candidates have to answer questions on Modules 1 and 2 and either 3 (Syllabus 5375) or 4
(Syllabus 5376). The two papers are aimed at Grade D but Grades
C, D, E and F are available. At the Higher Level candidates have to answer questions on
1, 2, 3 and two on 5, 6, 7, 8 (syllabus 5377) or 1, 2, 4 and two on 5, 6, 7, 8 (syllabus 5378). The two papers are aimed at grade
B but grades A, B, C and D are available.
There is a coursework component which is weighted at 30% of available marks. Candidates are required to complete two coursework tasks each lasting approximatly 15 hours. The broad areas of activity i) investigations, ii) problem solving, iii) practical, are suggested.
6.4 N.E.A. Mathematics Syllabus C (31).
The scheme of assessment'· consists of two written papers
05%) and a compulsory coursework component (25%) at each of the three levels P,Q and R.
Level P is aime,d at grade F, Level Q at grade D and Level R at grade B. The scheme of assessment is very similar to that of
N.E.A. Syllabus A Scheme 2 and N.E.A. Syllabus B Scheme 2. The coursework is meant to be an integral part of the course and is to be assessed under the headings:
a) Practical Work
b) Investigational Work
c) Assimilation.
54 6.5 S.E.G. Mathematics without Centre Based Assessment.
This is the same scheme which was reviewed in Chapter 2, The
Mode 1 Syllabuses.
6.6 S.E.G. Mathematics Mature Modular Syllabus. The following modules are available:- a. Core Mathematics; Alternative A
b. Core Mathematics; Alternative B
c. Money Management
d. Statistics
e. Surveying.
~rtification is available for candidates who offer either a,c and d or b,c and e. The modules are available for assessment at a General Level
(grades C,D,E) and at a Higher Level (grades A,B,C).
The scheme of assessment comprises:
Written Papers on modules a,b,c,d,e,
Aural Assessment on modules a,b,
Centre Based Assessment of coursework on modules
a,b,c,d,e, An oral assessment component is included in the coursework assessment of all modules.
55 CHAPTER 7
Other Schemes
56 7.1 General There is a feeling amongst many teachers and schools that the Mode 1 syllabuses available from the Examining Groups do not match their needs. They feel that the changes in teaching styles brought about by documents such as the Cockroft Report (6) and
'Mathematics from 5 to 16' (11) need to be matched by schemes of asessment which are not heavily biased towards timed written examinations. There are three schemes of assessment which do not incorporate a final written examination. These are:-
i) A.T.M./S.E.G. GCSE ii) M.E.G. Project on Assessment of Graded
Objectives
iii) G.A.I.M. Project.
7.2 A.T.M./S.E.G. GCSE (1) The assessment of this scheme is done completely through coursework which is determined by the school. Each candidate will submit for final grading a folio of work which has been completed during the course. The folio should contain evidence of Extended
Pieces of Work, Practical Work, Self Evaluation and Reports of
Oral Interviews. Written Tests may be included.
Assessment of each candidates folio is completed by the teacher with reference to nine Assessment Objectives namely:
1. Communication 2. Preparation and Approach
3. Implementation
4. Mathematical Knowledge
5. Interpretation
57 6. Mathematical Attitude
7. Autonomy
8. Self Evaluation
9. Working in Groups.
~ere are guidelines set out in the syllabus for the assessment with criteria related to specific grades e.g.
'7.3 Implementation Under this heading the candidates ability to use
suitable mathem~tical t~chniques and skills is
judged.
7.3.1 Grade F To be awarded this grade a candidate will be able to
make appropriate calculations, compare data and
generate simple plans and diagrams.
7.3.2 Grade C To be awarded this grade a candidate will be able to
interpret calculations, analyse data and derive
simple formula.
7.3.3 Grade A To be awarded this grade a candidate will be able to
critically analyse calculations and data and will be
able to derive, test and use formulae.' (1)
The onus is on the school to devise a teaching syllabus designed to allow students to achieve the stated Assessment
Objectives. Each student's folio will be cross checked against lists 1 and 2 of the National Criteria (10) to conform with the National
58 Criteria.
7.3 The Midland Examining Group Project on Assessment of
Graded Objectives. (23)
The project aims to divide the secondary schools mathematics curriculum into a series of levels, each level demanding work of a more advanced nature than the previous one. The assessment of
these levels should be integrated with the learning process and should 'be compatible with the teaching, learning and assessment strategies of the school. Equally it has been thought important that the views expressed in Mathematics Counts (6), Mathematics
5-16 (11), The National Criteria (10) and The S.E.C. Draft Grade
Criteria (33) should be reflected in the scheme'. (23)
The objectives at each level are represented within a
~riculum framework of Process Domains and Content Topic Areas illustrated by the diagram below:-
59 N AL SP ST PROCESS DOMAINS NUMERICAL ALGEBRAIC SPATIAL STATISTICAL CONTEXTS
1\
Interpret and Some objectives related to' particular MATHEMATICAL Communicate content, others independent. information , B
Select Appropriate Objectives largely independent of EVERYDAY Technique or content. strategy AND·
C . , Apply Knowledge, Almost all objectives related to OTHER Skills. and particular content'. Techniques .
D
Check, Verify Objectives largely independent of SUBJECTS Generalise and content •.. Prove
The project suggests nine levels of achievement and
proposes that the levels be linked to GCSE grades in the
following way:
Graded Objective Level GCSE Grade , 1 - 2 - 3 G
4 F
5 E
6 D 7 c
8 B
9 A
60 Assessment of the pupils is carried out on units of wor~, which may last two to three weeks, and is related to the Process
Domains and content Topic Areas outlined above. The unit of work may be class lessons, a project, group
W)rk, investigative work etc. but must include a balance of the following activities:
'~ Communicating Mathematics by appropriate means
(written, oral, practical).
Write about, discuss and/or model a situation in
mathematics using appropriate mathematical language.
PW. Practical Work. Use mathematical knowledge, skills and techniques
in practical situations.
EW. Extended Work. Develop a piece of work beyond that originally
envisaged, or complete an extended piece of work in
mathematics.
PS. Problem Solving. Select, or devise, and then implement suitable
strategies to solve problems.•
IM. Investigating Mathematics.
Investigate an area of mathematics in which there
is not necessarily a pre-determined outcome.
ST. Skills/Techniques (written, mental).
Use mathematical knowledge, skills and techniques
in both mental and written work.' (23)
1I1e teacher has to complete a Teachers Record Sheet which details each task set, the types of activity and domains it is
61 possible to cover given successful completion.
The teacher has to complete a Pupil Record Sheet deciding
~ich activities and domains have been successfully demonstrated by the pupil. It is important that the tasks selected allow the pupil to achieve the required balance across the domains and activities described.
7.4 Graded Assessment in Mathematics (G.A.I.M.) (13).
7.4.1 Outline of G.A.I.M. Scheme. G.A.I.M. is an assessment scheme being developed jointly by the I.L.E.A., L.E.A.G. and Kings College London which is being supported by the Nuffield Foundation. It claims to be a system of assessment which provides both the basis for a GCSE award, where appropriate, and a Record of Achievement for a pupil.
The scheme is based on two components:-
1) Topic Criteria
2) Coursework Activities.
Topic Criteria are a bank of 'can do' statements which are designed to describe what the pupil knows, understands and can apply. These criteria are organised into 15· levels of difficulty across 6 topic areas, namely:- Logic
Measurement
Number
Space
Statistics Algebra (levels 9 and above only).
62 Examples of the Topic Criteria are shown below:
'Can predict what to do to reverse the effect of two
familiar reversible operations.' (Logic Level 4)
'Can estimate the lengths or heights of familiar objects or objects which are present, using the appropriate metric units:
a) millimetres,
b) centimetres,
c) metres.' (Measurement Level 6)
'Can read and write numbers up to 100 in figures' (Number Level 1)
'Can - a)shape a cuboid (with or without isometric paper),
and b)make it by first drawing it's net'.
(Space Level 7)
'Can play an active part in deciding to choose a sample
outside the school, selecting a method of gathering data
and collecting data systematically.' (Statistics Level 8)
Pupils are not expected to start at level 1 and work up. The majority of pupils in year 1 of a comprehensive would be likely to start on level 5 or 6. Progress should be in terms of about one level per year. The second component of G.A.I.M consists of a bank of open ended coursework activities divided into 'Investigations' and
'Practical Problems'. Examples given in 'Newsletter 5' include:- • investigating the different symmetry patterns obtained by shading squares on a grid;
63 • investigating the number of ways of giving change fo~ different sums of money; • planning the layout of newspaper advertisements; • booking guests into appropriate rooms in a hotel; • finding out the ages and origins of local cars; • designing a wardrobe or bookcase. The teachers notes for these activities include guidance on what kind of responses are likely to be given by pupils working at each of the 15 G.A.I.M. levels. An indication of the topic criteria mastered in the course of an activity by the pupils is also provided.
7.4.2 G.A.I.M. and GCSE. The London and East Anglian Group submitted a proposal to the S.E.C. in January 1987 for a GCSE based on the G.A.I.M. assessment scheme. G.A.I.M. levels 9-15 have been designed to be equivalent to grades G to A in GCSE.
GCSE grade: G FED C B A
G.A.I.M. level: 9 10 11 12 13 14 15
The original submission, which was accepted in February
1987, was for a limited grade (G-D) award starting in 1989 for pilot schools only. The S.E.C. granted a waiver in relation to paragraph 6.3 of the National Criteria (10) which requires an end of course timed written examination to count for 50% of the marks available on the assessment scheme.
64 7.4.3 G.A.I.M. and Records of Achievement.
The Department of Education and Science have expressed an intention that all secondary students should have Records of
Achievement before 1990. These will take the form of detailed formative records describing what students are achieving as they progress through a school, together with a final summary document. The G.A.I.M. Topic Criteria should be used as profile statements, for example see 7.4.1. The G.A.I.M. 'Statement of
Achievement' could be used directly as part of a Record of
Achievement. The 'Statement of Achievement' is only awarded at a particular level if the student has to, achieve at that level (as above) for their work on each
of 10 G.A.I.M. activities (at least 4 Practical Problems
and at least 4 Investigations),
satisfy all the Topic Criteria at that level.
It may be that the G.A.I.M. project enables teachers to have at their disposal a single scheme which provides formative and summative profiles and the facility for an appropriate GCSE grade to be awarded.
65 CHAPTER 8
Conclusion
66 Conclusion It will not be possible to determine whether all of the
stated aims for the GCSE have been achieved in Mathematics for some time. There is no doubt, however, that the GCSE together with 'Mathematics from 5 to 16' (11) and the Cockroft report (6) have prompted a remarkable period of change in the way
Mathematics is taught and assessed. This change has provoked many discussions and raised many questions such as those listed by
Burghes (3).
'1. What are the main problems faced by school mathematics
departments in implementing GCSE? How can they be
overcome?
2. Will the GCSE (with school based assessment) provide a
fairer assessment of pupil ability?
3. What will be the problems of pupils continuing from
GCSE to 'A' level mathematics? How can they be
overcome?
4. Should the content/assessment of A level mathematics
change in light of the GCSE mathematics implementation?
5. Will GCSE mathematics provide suitable challenges for
able pupils?
6. How will schools choose the appropriate entry level for
candidates? Will parents be involved?
7. The GCSE encourages an investigational/problem solving
attitude to teaching mathematics - do you welcome this?
8. Are there sufficient resources available for GCSE
teaching? If not, what is needed?
9. Has there been sufficient in service training? If not,
67 what type of training is needed?
10. Should mathematics be an optional topic for years 4/5?
One problem faced by departments involving GCSE concerns
choice of syllabuses. Whereas with G.C.E. and C.S.E. schools were
able to choose any syllabus which they thought was well thought
out and relevent, this is not now the case. Even though the
S.E.C. suggests that 'there is scope for teachers who want to
begin coursework assessment now to choose their own scheme which matches their sense of their own present ability to embark on work of this kind' (42), this is not the case. In many areas the
L.E.A. will only support, in terms of in service training,
departments who choose the 'local' syllabus unless it is a
nationally available scheme. So here is a restriction in choice
of assessment scheme.
If departments were allowed a 'free' choice of scheme then,
as can be seen from the previous chapters, there is a multitude
of schemes available.
If one does not include the Mode 3 syllabuses detailed
earlier then there are seventeen (O.A.I.M. is not available as
GCSE this year) schemes of assessment avaiiable.
Full details of all Mode 3 syllabuses have not been included
because of the unavailability of information from L.E.A.G. and
N.E.A. on this subject.
The schemes differ greatly in their choice of the content to
supplement List 1 and List 2, their approaches to coursework and
their approaches to oral work (non-existent in most cases).
Many syllabuses allow a freedom of choice when it comes to
68 coursework. Candidates may compile a 'folio' of their work over the two years of the course. On the other hand S.M.P. 11-16 prescribe their coursework 'tasks' and provide detailed markschemes. This has drawn criticism from some quarters for being too much like a series of tests. It has the advantage of giving some structure to coursework, however, for teachers who want to move more cautiously on coursework and not to commence an
'open' scheme. By 1991 it is likely that all schemes will have to be more open and consequently the problems of assessing and moderating many, many different pieces of work in one school and across schools will grow.
The schemes of assessment range from those which include no coursework component at all to those which are completely reliant upon coursework and with no end of course timed written examination. This ·range of choice will be restricted in 1991, however, as all courses will then have to have a coursework component. There is no doubt that in terms of reliability - the consistency with which the·y test - end of course examinations have been successful. Their validity - the. extent to which they test what they are intended to test - is open to some question.
There are important parts of Mathematics Education for which an alternative scheme of assessment is needed e.g. an ability to undertake an extended piece of work. It is in these areas that the teacher should welcome the opportunity to be involved with the assessment of the pupils.
There are drawbacks with the completion of coursework just as there are with assessment by examination, so will the GCSE
69 really prove to be a fairer system of assessment?
The assessment of coursework produces many new problems:-
1) where an extended piece of work is being completed,
either in school or out of school, the teacher needs to
ascertain the amount of help sought by each pupil;
2) the pupils background may not be one of encouragement;
the facilities for work may be poor or even nonexistent;
3) internal moderation, to ensure reliability within a
school, is time consuming and 'good practice' may
deteriorate;
4) external moderation, to ensure reliability across
schools, will prove to be an extremely complex affair
given the incredible range of mathematical activity
which is likely to be undertaken by candidates and presented for assessment.
, If everyone is doing something different~ how do we mark it?
How do we assess such qualities as independence, creativity,
insight, perseverence?
If we encourage pupils to work co-ope:atively in groups, is it right that we then mark them individually?'(12)
Approaches to Mathematics which include oral work, aural work, investigational work, problem solVing, discussion and practical work require two important commodities - 1) suitably qualified mathematics teachers and 2) time.
It is apparent that there is a shortage of suitably qualified mathematics teachers. 1984 figures suggest that 28% of maths teaching (the work of about 6000 teachers) was in the hands
70 of unsuitably qualified staff.
The learning activities noted in the previous paragraph will prove to be more demanding on the teacher:-
'a) mathematically because the teacher must be able to
perceive, to some extent at least, the consequences of
the different approach on which pupils are embarked;
b) pedagogically because the variety of approaches going on
in the classroom will be much wider and
c) personally, because the teacher must have the confidence
to accept that he will not know all the" answers' (4)
[These comments were related to mathematical modelling by
Burkhardt but are just as relevent to Investigations,
Problem Solving etc •• ]
Oral approaches to Mathematics have been almost completely neglected by the majority of the schemes of assessment. Although some use 'mental' tests and others aural work few actually encourage the candidates to talk about and to discuss
Mathematics. The S.M.P. Mode 2 scheme has probably the best approach with W.J.E.C. Mode 1 scheme also incorporating an 'oral' component.
In the light of proposals for the National Curriculum it seems likely that most schools will be required to reduce the amount ot time spent in the mathematics classroom. Whereas the time allocated to mathematics historically has been around about
13% of one weeks lessons, the National Curriculum will bring pressure to bear on this allocation. The core subjects:
Mathematics, Science and English together with the foundation
71 subjects: History, Geography, Technology, Music, Art, P.E. and a
Modern Language, were originally to take up 80-90% of the time
available. This has now been reduced to 70%.
A reduction in time available for mathematics is likely to
pressurise teachers to shy away from enlightened mathematical
activity and 'teaching to the syllabus' and 'transmitting
knowledge' may become predominant.
Similar pressures could result from schools becoming full or
associate members of T.V.E.I.
The majority of the schemes included in this dissertation
follow the award of grades shown below:------Level Target Grade Grades Available ------Higher B A,B,e,D.
Intermediate D e, D, E.
Foundation F D,E,F,G. ------In order to achieve the target grade, approximately two thirds of the marks available have to be accumulated (call this
65%). The highest grade at a level may be attained by achieving approximately 80% or better and the lower grade by
50%-65%.
The grades and required percentages are illustrated in the table below:-
72 ------Higher Intermediate Foundation ------A 80%+ B 65%-79% C 50%-64% 80%+ D (40%-49%)7 65%-75% (90%-100%)7 E 50%-64% 80%
F 65%-79% G 50%-64% :------: The figures in the brackets are based on supposition. From
the table above I would suggest that a borderline
Higher/Intermediate candidate would be served better (in terms of final grading) by being entered for the Higher Level. To score
50%-64% on a more difficult paper is probably more likely than scoring 80% on two easier papers. This would seem to contradict the 'positive achievement' ethos of the GCSE examination.
The candidate runs the risk of being ungraded completely but a grade D can still be achieved by scoring less than half of the marks available at the Higher Level.
If the DES insist that all 16 year olds will have to sit a
Mathematics examination the Foundation Level of the GCSE will have to be adjusted. It may be possible for candidates to be awarded a GCSE grade by completing a continuous assessment course which does not end with a timed, written examination (e.g. the
S.M.? 11-16 Graduated Assessment Scheme).
The effects on future 'A' level candidates will also have to be monitored. There is no doubt that GCSE schemes do not have the
73 algebraic content of a traditional 'Q' level. Some examining groups are offering an extension paper for GCSE which will assist prospective 'A' level candidates. This produces furthe prssure on both the teacher's and the candidate's time.
There is no doubt that the GCSE, not only in Mathematics but across the curriculum, has given rise to a multitude of questions and will surely do so for several years to come.
It may be that in the future the plethora of syllabuses will be reduced by:- a) amalgamating the Examining Groups into one National
Examining Body,
b) conforming the Mathematics Syllabuses taught
in schools to a National Curriculum Model.
74 BIBLIOGRAPHY
1. The Association of Teachers of Mathematics/The Southern
Examining Group GCSE Syllabus.
~ Avon Trent Consortium Mode 3 Syllabus, South Warwickshire
College of Further Education.
~ Burghes D. (1987) Experience in Mathematics Assessment
14-16. Report of the Symposium held at Central London
Polytechnic September 17th 1986. Teaching Mathematics and
its Applications Vol 6 No. 3 pp 93-94.
4. Burkhardt H. (1984) 'Modelling in the Classroom - How can we
get it to happen?' in Teaching and Applying Mathematical
Modelling, ed. Berry et al, Ellis Horwood.
5. Cardinal Wiseman School Birmingham Mode 3 Syllabus.
6. Cockroft Dr.D.W. (1981) Report of the Committee of Inquiry
into the Teaching of Mathematics in Schools, London: HMSO.
7. Clwyd Mode 3 Scheme for GCSE.
8. DES (1985) General Certificate of Secondary Education: A
General Introduction, London HMSO.
9. DES (1985) General Certificate of secondary Education: The
National Criteria - General Criteria, ~ondon: HMSO.
10. DES (1985) General Certificate of Secondary Education: The
National Criteria - Mathematics, London: HMSO.
11. DES (1985) Mathematics from 5 to 16, London: HMSO.
12. Evans J. (1987) Investigations the State of the Art.
Mathematics in Schools, Vol 16 No.l Jan 1987 pp 28-30.
13. G.A.I.M.: Graduated Assessment in Mathematics, Kings College
London.
14. Henry Fanshawe School Derbyshire Limited Grade Mode 3.
75 15. Isaacson Z. (1987) Teaching GCSE Mathematics, Hodder and
Stoughton, London.
~. Kent Mathematics Project and GCSE A Guide for Teachers. 11. London and East Anglian Group GCSE Syllabus A (without
coursework). 18. London and East Anglian Group GCSE Syllabus (with coursework).
19. London and East Anglian Group Mature GCSE Syllabus
(5375, 5316, 5377, 5388). 20. Leicestershire Schools Mode 3 GCSE Syllabus A.
21. Leicestershire Schools Mode 3 GCSE Syllabus B. 22. Mathematics in Education and Industry Schools Project.
23. Midland Examining Group Project on Assessment of Graded
Objectives. 24. Midland Examining Group GCSE Syllabus Mathematics (1651).
25. Midland Examining Group GCSE Syllabus Mathematics (1650). 26. Midland Examining Group GCSE Syllabus Mathematics (Mature)
(1654). 27. Mid Glamorgan County Council Mode 2. 28. Mobley M. et al (1986) All about GCSE - A Clear and Concise Summary of all the Basic Information about GCSE, Heinemann
Educational Books, London.
29. Northern Examining Association Syllabus A.
30. Northern Examining Association Syllabus B.
31. Northern Examining Association Syllabus C (Mature).
32. Newport and District Mode 2.
76 33. Secondary Examinations Council (1985) Report of Working
Party, Mathematics: Draft Grade Criteria, S.E.C.
~. Secondary Examination Council (1985) W?rking Paper 1:
Differentiated Assessment in GCSE, S.E.C.
35. Secondary Examination Council News (1987) GCSE Mathematics,
Autumn, No. 7. 36. Southern Examining Group GCSE Syllabus (without coursework). 37. Southern Examining Group GCSE Syllabus (with coursework).
38. Southern Examining Group GCSE Mature Modular Syllabus.
39. st. John Rigby College Mode 3 GCSE.
40. S.M.P. Mathematics Mode 1 Syllabus (M.E.G. 1652).
41. S.M.P. Mode 2 GCSE Syllabus. 42. S.M.P. (11-16) Mode 1 GCSE Syllabus (M.E.G. 1653).
43. Welsh Joint Education Committee GCSE Syllabus Mathematics.
44. Welsh Joint Education Committee GCSE Syl.labus Mathematics
(Mature).
77 APPENDIX !
GCSE - THE NATIONAL CRITERIA - MATHEMATICS AIMS
The statement which follows sets out ideal 2,7 educational aims for all those following recognise when and how a situation courses in Mathematics which lead to GCSE may be represented mathematically, examinations. Some of these aims refer to the identify and interpret relevant factors development of attributes and qualities and, where necessary, select an which it might not be possible, or desirable, appropriate mathematical method to to assess directly. ,'" "' •• :-....,' :::', " solve the problem; 2.8 All courses should enable pupils to: : i ' use mathematics as a means of "":',.:::: communication with emphasis on the 2.1 develop thelir mathematical ' " use,of clear expression; knowledge and oral, written and .' 2.9 practical skills in a manner which' develop an ability to apply encourages confidence; : ,,'.', mathematics in other subjects, '. . .' . '~" .• r .' particularly science and technology; '2.2 . read mathematics, and write and talk, 2.10 , about the subject in a variety of ways; develop the abilities t6 reason . .' ': ',': ";. ,. I , logically, to classify, to generalise and to prove; 2.3 develop a feel for number, carry out ., . ""-- --'calculations and understand the . significance of the results obtained; 2.11 appreciate patterns and relationships in mathematics; '. 2.4 , apply mathematics in everyday 2.12 , ' situations and develop an . produce and appreciate imaginative understanding of the part which and creative work arising from mathematics plays in the world mathematical ideas; around them; I '... '" . 2.13 develop their mathematical abilites 2.6 by considering problems and solve problems, present the solutions conducting individual and clearly, check and interpret the results; cooperative enquiry and experiment, including extended pieces of work of a practical and investigative kind; 2.6 develop an understanding of • mathematical principles; 2.14 appreciate the interdependence of different branches of mathematics; 2.16, acquire a foundation appropriate to their further study of mathematics and of other disciplines.
78 APPENDIX 11.
GCSE - THE NATIONAL CRITERIA - MATHEMATICS ASSESSMENT OBJECTIVES
The objectives which follow set out essential 3.10 interpret, transform and make mathematical processes in which candidates' attainment will be assessed. They form a appropriate use of mathematical statements expressed in words or minimum list of qualities, abilities and skills. symbols; The weight attached to each of these objectives may vary for different levels of 3.11 assessment within a differentiated system. recognise and use spatial relationships in two and three Any scheme of assessment will test the ability dimensions, particula~ly in solving of candidates to: problems; 3.12 3.1 recall, apply and interpret analyse a problem, select a suitable mathematical knowledge in the . strategy and apply an appropriate context of everyday situations; . technique to obtain its solution; '- 3.13 3.2 set out mathematical work, including apply combinations of mathematical skills and techniques in problem the solution of problems, in a logical solving; . and clear form using appropriate symbols and terminology; 3.14 ._- -.. _- -,._.- -.-~~ --- _._---- make logical deductions from given 3.3 organise, interpret and present . mathematical data; imformation accurately in written, tabular, graphical and diagrammatic 3.15 respond to a problem relating to a forms; relatively unstructured situation by translating it into an.appropriately , i 3.4 perform calculations by suitable structured form. methods; Two further assessment objectives 3.5 use an electronic calculator; can be fully realised only by assessing work carried out by 3.6 understand systems of meqsurement ' candidates' in addition to time-limited in everyday use and make use of written examinations. From 1988 to them in the solution of problems; 1990 all Examining Groups must provide at least one scheme which • 3.7 estimate, approximate and work to includes some elements of these two. degrees of accuracy appropriate to objectives. From 1991 these the context; objectives must be realised fully in all schemes. 3.8 use mathematical and other instruments to measure and to draw 3.16 respond orally to questions about to an acceptable degree of accuracy; mathematics, discuss mathematical ideas and carry out mental 3.9 recognise patterns and structures in a I __ . calculations; variety ?f si!uations, and form .• 3.17 generaiIsatlons; carry out practical and investigational . work, and undertake extended pieces of work. .
79 APPENDIX Q GCSE - THE NATIONAL CRITERIA - MATHEMATICS CONTENT
List 1 List 2 List 1 continued List 2 continued Whole numbers: odd, Natural numbers, Personal and even, prime, square. integers, rational and household finance, Factors, multiples, irrational numbers. including hire idea of square root. Square roots. purchase, interest, Directed numbers in Common factors, taxation, discount, practical situations. common multiples. loans, ,?,ages and Vulgar and decimal Conversion between salaries. fractions and vulgar and decimal Proft and loss, VAT. percentages; fractions and Reading of clocks'and equivalences percentages. dials. between these forms Standard form .. Use of tables and in simple cases; charts. conversion from Mathematical vulgar to decimal language used in the fractions with the help media. of a calculator. Simple change of units including foreign Estimation. Approximation to a ·"currency. given number of Average speed. significant figures or decimal places. Approximation to Appropriate limits of Cartesian Constructing tables of obtain reasonable accuracy., coordinates. values ~or given answers. Interpretation 'and use functions which of graphs in practical include expressions of The four rules applied The four rules applied situations including the form: ax + b, ax·, to whole numbers and to vulgar (and mixed) . travel graphs and a/x (x {. 0) where a decimal fractions. fractions. conversion graphs. and b are integral Drawing graphs from constants. Drawing Language and given data. , and interpretation of notation of simple related graphs; idea vulgar fractions in of gradient. appropriate c.ontexts, including addition and • subtraction of vulgar The use ofletters for (and mixed) fractions . Transformation of generalised numbers. simple formulae. with simple Substitution of denominators. numbers for words Elementary ideas and and letters in Expression of one formulae. notation of ratio. quantity as Percentage of a sum of percentage of Basic arithmetic money. another. processes expressed Scales, including map Percentage change. algebraically. scales. Proportional division. Directed numbers. Elementary ideas and Use of brackets and applications of direct extraction of common and inverse factors. proportion. Positive and negative Common measures of integral indices. . rate. (continued overleaf)
80 . I I I I I
APPENDIX .9...... _.. _._._- .. ---_._------
List 1 continued List 2 continued Vocabuldry of Properties of polygons triangles, quadrilat directly related to erals and circles; their symmetries. properties of these . Angle in a semi-circle; figures directl y angle between related to their tangent and radius of symmelries. a circle. Simple linear Angle properties of Angle properties of . equations in one triangles and quadri-· regular polygons. unknown. laterals. The geometrical Congruence. terms: point, line, Simple solid figures. parallel, bearing, right angle, acute and Use of drawing Practical applications obtuse angles, instruments. based on simple locus . perpendicular, Reading and making properties. similarity. of scale drawings; Perimeter and area of Area of rectangle and parallelogram. Measurement of lines Angles formed within triangle. Area of circle. and angles. parallel lines. Circumference of Volume of a cylinder. Angles at a point. circle. Enlargement. Volume of cuboid. Efficient use of an Results of Pythagoras. electronic calculator; Sine, cosine and application of tangent for acute appropriate checks of angles. accuracy. Application of these to calculation of a side or Measures of weight, an angle of a right length, area, volume angled triangle. and capacity in current units. Collection, classifica Histogram with equal Time: 24 hour and 12 tion and tabulation of intervals. hour clock. statistical data. Construction and use Money, including the Reading, interpreting of pie-charts. use of foreign • and drawing simple Construction and use currencies. inferences from tables of simple frequency and statistical distributions. diagrams. Construction of bar charts and pictograms. Measures of average . . and the purposes for which they are used.
Probability involving Simple combined only one event. probabilities.
81 APPENDIX .Q. GCSE - THE NATIONAL CRITERIA - MATHEMATICS GRADE DESCRIPTIONS Assessment GradeF GradeC Objective examples examples
3.3 Extract.information from simple Construct a pie chart from simple data. timetables. Tabulate numerical data to Plot the graph of a linear function. find the frequency of given scores. Draw a bar chart. Plot given points. Read a travel graph.
3.4 Add, subtract and multiply integers. Apply the four rules of number to Add and subtract money and simple integers and vulgar and decimal fractions without a calculator. fractions without a calculator. Calculate a simple percentage of a Calculate percentage change. given sum of money. 3.5 Perform the four rules on positive Perform calculations involving several integers and decimal fractions (one operations, including negative operation only). Convert a fraction to a numbers. decimal. 3.6 Measure length, weight and capacity Use area and volume units. using metric units. Understand relationships between mm, cm, m, km; g,kg.
3.7 Perform a money calculation with a Give a reasonable approximation to a calculator and express the answer to calculator calculation involving the the nearest penny. four rules.
3.8 Draw a triangle given three sides. Use a scale drawing to solve a two Measure a given angle. dimensional problem.
3.9 Continue a straightforward pattern or Recognise, and in simple cases number sequence. formulate, rules for generating a , pattern or sequence.
3.10 Use simple formulae, ego gross Solve simple linear equations. wage = wage per hour x number of Transform simple formulae. Substitute hours worked, and use of A = I x b to numbers in a formula and evaluate the find the area of a rectangle. remaining term. 3.11 Recognise and name simple plane Calculate the length of the third side of figures and common solid shapes. Find a right-angled triangle. Find the angle the perimeter and area of a rectangle. in a right-angled triangle, given two Find the volume of a cuboid. sides. . In schemes of assessment where the objective is applicable: 3.17 Carry out a simple survey; obtain Investigate and describe the' straightforward results from the relationship between the surface area information obtained. and volume of a selection of solid shapes.
82