APPENDIX 23
Memorandum submitted by the London Mathematical
Society
The London Mathematical Society (LMS) welcomes
the opportunity to submit evidence to the Inquiry. The LMS has
played an active role in the discussions on mathematics in the
schools for many years. Its most influential single contribution
was producing (in collaboration with the Institute of Mathematics
and its Applications and the Royal Statistical Society) the report
Tackling the Mathematics Problem in 1995. Members of the
Committee who have read this report and also those published by
the Engineering Council and the Institute of Physics[42]
will see that there is little or no difference between us and
our colleagues in science and engineering about the issues.
We strongly urge those members of the Committee
who have not yet read the reports to do so. They demonstrate clearly
that the problems that concern us are not merely problems for
mathematics and mathematicians, they affect all of science and
engineering in the UK.
The majority of the members of the Society are
engaged in university teaching. Much of this is to science and
engineering students, and the extent to which we are able to provide
what their home departments want is heavily dependent on what
they have covered at school. Moreover, we consider that students
who intend to read mathematics at university should follow the
same basic curriculum as those who intend to go on into science.
They may profit from enrichment along the way and some extra material
at the end, but their core needs are much the same. Following
essentially the same curriculum also allows students to delay
their decision about what to study at university until they are
applying for a place, or even until they actually begin their
university course.
In our response, therefore, while we focus on
mathematics in its role as part of the science curriculum, our
comments apply more broadly. We do not envisage a substantially
different curriculum for potential mathematicians. On the contrary,
we would consider this a mistake.
1. INTRODUCTION
1.1. The present system (it cannot really
be called a curriculum) based on A-levels evolved during a period
when only a very small proportion of the age cohort went on into
higher education. It may have been appropriate then; it certainly
is not well suited to the present situation. We strongly support
the concept of a curriculum for 14-19 year olds and we consider
that this requires the introduction of some sort of baccalaureate
or the certificate proposed by Dearing.
1.2. As Dearing proposed, mathematics should
be compulsory, but there should be a separate course for students
not intending to read mathematics or a subject that depends on
it.
1.3. We make no specific proposals here
about the material to be included. If the idea of a curriculum
is accepted, we would expect to discuss the details with the Qualifications
and Curriculum Agency (QCA). Much of what we would expect to see
in the curriculum is in fact already there; the question is how
much of it is taught and how and to whom. That is not to say that
we are content with what most students are being taught, but rather
that we do not see changing the official curriculum as the way
to improve the situation.
1.4. There should be no public examinations
during the period, only at the end, though in saying this we do
not mean to exclude the possibility of modular examinations.
1.5. As a matter of urgency, measures must
be put in hand to remedy the serious shortage of mathematics teachers
in the schools.
2. WHY THE
STRUCTURE MATTERS
2.1. Given the level that has been achieved
at age 14, the present content is, on the whole, not unreasonable,
though we are less satisfied about what is actually taught in
many classrooms. We are also concerned about the balance; in particular
there seems to us to be far too much time devoted to data handling,
In any case, we do not see how there could be a major change in
content unless there are also changes in what is taught up to
age 14, and we do not expect this to happen in the near future.
We would, however, hope that over the next few years there will
be a general raising of standards and some adjustments, but these
are matters to be discussed directly with the QCA. Such discussions
are already taking place and will, we assume, continue.
2.2. The devil, however, is in the detail,
for example in the materials provided to teachers and in the examinations.
It is also in the structure. That a topic is in the curriculum
does not necessarily mean it will be taught, still less that it
will be adequately covered. For example, the summary to the recent
Royal Society/Joint Mathematical Council report on the teaching
of geometry[43]
states, "Overall, for mathematics 11-16, we conclude that
the geometrical content of the new National Curriculum, with a
few adjustments, forms an appropriate basis for a good geometry
education." Yet the authors immediately go on to say, "In
order for this to be achieved, however, considerable changes are
needed in the way geometry is taught."
2.3. In the same way, while there is a reasonable
amount of algebra in the National Curriculum, that is all too
often not reflected in the classroom. Teachers have realised that
time spent on other parts of the curriculum is more likely to
increase the number of C grades at GCSE, and it is this on which
their school's performance is judged. Weakness in elementary algebraic
manipulation is one of the most serious problems reported by university
science departments and indeed mathematics departments as well.
2.4. We understand that the QCA has plans
to redesign the GCSE examination and remedy this problem. It would,
of course, automatically disappear if GCSE were to be discontinued,
as we understand is also being considered and which would be consistent
with a 14-19 curriculum. It remains, however, a clear example
of how merely putting something into the curriculum may not be
enough. If there are topics or indeed whole subjects that we consider
should be taught, the structure — including the assessment
— must be designed to ensure that they are.
2.5. Finally, we remind the Committee that
there is a serious shortage of qualified mathematics teachers
which on present form is likely to get worse, not better. If there
are not enough teachers with adequate qualifications in mathematics,
the subject will not be well taught, however well designed the
curriculum may be.
3. A BACCALAUREATE
3.1. It is not so long ago that only a small
proportion of the age cohort continued in full time education
after age 19. Now about 35 per cent go on to higher education,
and the Government intends that the figure will soon rise to 50
per cent. School is no longer the end of formal education for
most students. Instead, there is now an interface between school
and university in much the same way that there is an interface
between primary and secondary schools.
3.2. This has two very significant consequences.
First, an important consideration in designing the 14-19 curriculum
must be to prepare students for higher education, to give them
the knowledge and skills they will require for further study,
whatever subject they choose. Second, because almost all students
with two or more A-levels now go on to higher education, there
is ample opportunity for them to study a subject that interests
them but which is not available at A-level. It is therefore not
as necessary as it may have been in the past to offer a very large
menu of subjects from which they can choose "a la carte".
It is a luxury we can no longer afford, either in terms of the
cost of provision or in its effect on preparation for university
study.
3.3. Instead of the present system of a
more or less free choice among a very large number of subjects
at A-level, there should be some sort of structure leading to
a school leaving certificate like the French or International
Baccalaureate or the certificate proposed by Dearing. Indeed,
without such a structure, we cannot really speak of a curriculum.
3.4. There is a widely held view that the
universities can determine a de facto sixth form curriculum
by their admissions requirements. If, for example, science departments
demand A-level mathematics, then students will be obliged to study
mathematics at A-level. In fact, only a few institutions, and
a few subjects such as medicine, can do this. Most departments
in most universities operate in a buyers' market for places. This
has always been the case to some extent, but the ever-increasing
financial pressures on universities have made departments much
more vulnerable. Students can determine the entry requirements
simply by not applying in sufficient numbers with the appropriate
qualifications.
3.5. For example, while Further Mathematics
was once considered a prerequisite for most mathematics courses,
few, if any, departments now insist on it. That is not because
they decided that it was unnecessary, but because fewer and fewer
students were offering double mathematics. In its recent report,
the Institute of Physics writes that it is now considering a Physical
Sciences degree especially designed for students with low qualifications
in mathematics, largely because there are not enough applicants
for courses in physics who have adequate mathematical backgrounds.
3.6. The result, at least in mathematics
and science, is that universities are more and more being required
to teach what could have been done more cheaply and also more
effectively at school. (The pace and style of teaching at university
are necessarily quite different from at school, and the students
are older and more independent.) The problem is worse for students
who are reading science and engineering in universities than for
those reading mathematics. Because they do far less mathematics
as part of their course, any time wasted is a greater proportion
of the total that can be devoted to the subject.
3.7. There are strong reasons for making
mathematics compulsory for all students up to age 19. For example,
many UK students in courses in the life sciences now enter university
with no mathematics beyond GCSE, and with a two year gap since
they last studied the subject. These students find it very difficult
to learn the mathematics they need. As a result, many never acquire
the mathematical background that is standard for their contemporaries
in other countries.
3.8. Although mathematics should be compulsory
in the sixth form (ie in the last two years at school), there
should be different courses for different groups of students.
In particular, those who are not planning to study mathematics
or a subject dependent on mathematics should not be expected to
follow the same course as those who are, with only the stopping
point being different. They should have a course specifically
designed for their needs. It would also be better to have it continue
over two years, rather than being rushed through as the present
AS-level is.
3.9. Statistics would probably constitute
a larger proportion of the less demanding course than of the one
for students who intend to continue their mathematical studies
at university.
3.10. At present, applied mathematics at
A-level consists largely of mechanics. We would prefer to see
this broadened to reflect what applied mathematics has become
over the past 50 or more years.
3.11. The relatively small core and the
correspondingly large amount of optional material at A-level have
serious consequences for the teaching of mathematics at university.
Any material that is not in the core has to be taught at university,
and to a class most of whom have done it before. This is very
wasteful of resource and of students' time, and again the problem
is worse for students in science and engineering than for those
studying mathematics.
3.12. While there should be more than one
distinct mathematics course, we see no advantage in allowing choice
within the courses. It is hard to see how this choice really benefits
the student. Those who are going to study some mathematics at
university will be able to cover the topics later, while for the
others the choice is largely unimportant.
3.13. Allowing students within a school
to select different options also increases the amount of teaching
that has to be done. Even if we were convinced it was advantageous
we would still assign it a much lower priority than teaching mathematics
to more students and at a pace that they can cope with. The shortage
of qualified teachers is bound to limit what can be accomplished,
at least in the short and medium term, and that makes it all the
more important that we make the most effective use of the qualified
teachers there are.
4. PUBLIC EXAMINATIONS
4.1 If there is to be a real curriculum
for ages 14-19, we see no need for national testing during this
time. Suitable school leaving examinations will set the targets
that students will have to reach at age 19, and this, together
with the existing inspection process, should suffice to ensure
that what happens along the way is appropriate. Indeed, where
there is diversity in the students' achievements at the beginning
of a course, national tests at the half-way stage may make it
harder, not easier, for schools and colleges to reach the desired
end point. We have recently seen an example of this with AS-level
mathematics.
4.2. National tests take up a great deal
of time, not least the time devoted to preparing for them. They
tend to restrict what is taught to what can be easily examined.
They can lead both teachers and students to concentrate on preparation
for the tests rather than actually learning the subject. This
can make it difficult for many students to acquire one of the
valuable transferable skills that they are meant to acquire at
school and university, the ability to learn. It is disconcerting
to be asked by one's students at the very beginning of a course
for copies of previous examinations; more importantly it is a
clear indication of what they have been taught to see as the centre
of the learning process.
4.3. National tests are very expensive and
use resource that could otherwise be devoted to teaching and learning.
In our view they are necessary, but the policy should be to have
as few as are absolutely necessary to ensure that the required
material is being taught and to the right standard. A single set
of public examinations at the end of a four or five year course
should not distort teaching too much; to have examinations almost
every year is another matter altogether.
4.4. We do not mean to exclude modular examinations,
but if they are used they should be essentially final examinations
in stages, not national assessment throughout the whole period.
January 2002
42 Tackling the Mathematics Problem (London
Mathematical Society, 1995, ISBN 0-7044-16247); R Sutherland and
S Pozzi, The Changing Mathematical Background of Undergraduate
Engineers (The Engineering Council, 1995, ISBM 1-898126-65-8);
Measuring the Mathematics Problem (The Engineering Council,
2000); Physics-Building a Flourishing Future (Institute
of Physics, 2001, ISBN 0-7503-0830-3). Back
43
Teaching and Learning Geometry 11-19. (The Royal Society,
2001, ISBN 0-85403-563-X). Back
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