Yet more maths problems.
This article considers several key problems facing the teaching and learning of mathematics at secondary level. In particular, it studies the need for better defined aims for mathematics teaching; the standards currently being attained by students viewed from an international perspective; the validity of many of the assessment procedures on which the government places such emphasis; and the underlying problem, that of an insufficiency of well-qualified mathematics teachers -- a problem that, for several decades, governments have chosen to ignore. Suggestions are made on how the various aims of mathematics education might be better met.
'Does "Mathematics for all" mean "No mathematics for all"?' Title of a lecture given by J. de Lange in 1983.
'A calculator,..., a friend or an independent financial advisor can substitute for an education in mathematics for instrumental purposes' (Bramall, 2000).
'Q. I would like to know the rote of inflation for the years since 1987 to the present time to work out the true value of my savings. Can you help?
A. Certainly. Since 1987 the cost of living has gone up by 70 per cent. So [pounds sterling]1 today is worth the equivalent of only 30p then.' Reader's question and financial expert's answer in 'Your money', Saga Magazine, April 2001.
The teaching of mathematics in English secondary schools is far from satisfactory. Concerns arise regarding how the needs of different types of students are being met, the standards attained, the manner in which students' attainments are assessed, and the provision of adequately qualified teachers. We shall look briefly at each of these aspects, before proposing possible ways ahead.
Mathematics for all
The coming of comprehensive education in many western countries raised important issues concerning curriculum design. Just what did 'mathematics for all' mean? Nowadays the phrase is often taken to mean that mathematics was not to be found in the old English secondary modern schools -- that only arithmetic was taught in them (an assumption contradicted by the data given in the following section). (1) Of course, there were clear distinctions to be made between the aims of mathematics teaching in the grammar schools and in the bulk of technical and secondary modern schools. Put baldly, in the former, students were prepared for further academic study: in other schools, for taking their place within society. This was perhaps most clearly typified in the teaching of geometry. For the academic students there was Euclid-style theorem and proof; for the others practical geometry: the classification of shapes and solids, mensuration and, to varying extents (depending on the nature of the school), elementary scale and technical drawing.
The introduction in 1965 of the CSE qualification was not intended to threaten this dichotomy of aims. However, comprehensive schools contained all types of students and their early differentiation would have defeated the aims of comprehensive schooling and produced only multilateral schools. (2) Yet postponing differentiation meant postponing streaming for CSE or GCE O-level and this, in turn, resulted in the former becoming a watered down version of the latter. The Cockcroft report of 1982 proposed to rectify this by a bottom-up approach to curriculum design that concentrated first on the needs of the lower attainers. Accordingly it drew up a 'foundation list': what it saw as a basic mathematical kit for all school leavers. That, in its turn, was supplanted in the late 1980s by the National Curriculum (NC) which appeared to be designed not so much to meet the needs of students but rather those of an untried and ambitious assessment scheme. All students were now to follow the same curriculum -- but at their own rate. A hierarchy of levels was defined which were to be followed by students at varying speeds, but the question of depth of treatment within a level was ignored. No one, apart from the highest attainers, had any fixed curriculum goal -- students had simply to swallow as much of the curriculum as they could before the age of sixteen. No heed was taken of the wise words of the 1947 Hamilton-Fyfe report:
'Whatever be the values of the "subject" carried to its full term in university study, they cannot be achieved for the child of 16 by simply snipping off a certain length of the "subject" like a piece of tape.... Every course must have its own unity and completeness and a proper realism requires that content and methods alike be so regulated as to reach their objective within the time available.'
The 'piece of tape' mentality still persists and forces weaker students to learn algebraic techniques (see some of the items from the KS3 tests quoted below) which they will never develop into usable knowledge. Of course such students are no longer being 'denied' the opportunity to learn algebra -- but instead are simply forced to learn algebraic techniques that might conceivably (but with a fairly low probability) lead to something more useful and valuable. It is difficult to see exactly what the aims of the present curriculum are. There is an attempt to please everyone and do everything, at the expense of a focusing on clear aims and the provision of sound and secure learning. (3) That, after eleven years of compulsory mathematics, it should be felt necessary to institute post-16 courses and tests in key skills for sixth-formers illustrates the problems.
Such curricular considerations led recently to the publication of a collection of essays entitled Why Learn Maths?, edited by two philosophers of education, Bramall and White, which questioned the arguments put forward in the defence of teaching mathematics and its status as a compulsory subject within the national curriculum pre-16. Although reported in newspapers as a polemic, the monograph contained contributions that genuinely merit consideration. The present mathematics curriculum cannot be justified solely by the repetition of pious cliches or such contenders for 'Pseuds' Corner' as the NC's claims of mathematics' promoting 'spiritual development through ... helping pupils obtain an insight into the infinite', or 'moral development through...helping them learn the value of mathematical truth'.
Essentially, all the contributors accepted that every student should learn the mathematics that is 'commonly useful' -- basic arithmetic and mensuration -- but 'beyond that the case for inclusion is not so clear-cut'. Some contributors still argued for the teaching of mathematics for non-utilitarian reasons, e.g., training of the mind or the intrinsic delights of the subject and its place in human culture, but the editors themselves saw no justification for compulsory mathematics post-14. (4) Indeed, White argued, 'by the year 2002, 75 per cent of British [sic] children will have reached [the current] (5) Level 4 ... by the age of 11. That will provide them with the basic arithmetic they need to get by'. One wonders if White is fully aware of the arithmetical demands of Level 4. Some of the contents of Level 5 should cause us to doubt the validity of his argument: 'to multiply and divide whole numbers and decimals by 10, 100 and 1000; to order...negative numbers in context [e.g. temperatures], to solve simple problems involving ratio and proportion,...calculate fractional or percentage parts of quantities...'. No! Level 4 plus a few odds and ends of 'civic arithmetic' will not suffice for an educated citizen.
On the other hand, are the needs of the future mathematician being met? Certainly the percentage of the age cohort which opts to study mathematics at A-level remains disappointingly small, as does that which goes on to read mathematics at university (see below). And what of the standards attained? Here we need only quite from a recent report, Measuring the Mathematics Problem (Engineering Council, 2000), which presented evidence of a marked decline in university entrants' mathematical skills: 'This decline is well established and affects students at all levels'.
Clearly, problems still remain concerning the definition of attainable aims for secondary school mathematics within today's educational system. What more, however, can be said about the standards our students are currently attaining, both with respect to their predecessors in England and to students elsewhere?
Standards and the four mathematics studies of the International Association for the Evaluation of Educational Achievement (IEA)
As in many other aspects of life, it is valuable to attempt to compare the standards currently being attained in England with those to be found elsewhere. The IEA was established in the 1960s with the aim of carrying out international comparative studies and since then it has mounted four such in mathematics: the First and Second International Mathematics Studies (FIMS and SIMS) in 1964 and 1981 respectively, the Third International Mathematics and Science Study (TIMSS or TIMSS-95) in 1995 and TIMSS-R (or TIMSS-99), a partial replication of TIMSS, in 1999. England was one of the few countries to participate in all four studies, hut did this only at the Y9 (age 13+) level. (6)
Making comparisons of relative movements between countries is not easy, for the sets of participating countries have changed greatly over the years. However, some clear points emerge concerning both relative and absolute standards. Thus, ten countries participated in both FIMS and SIMS and thirty or so items used in 1964 were used again in 1981. Overall a few countries lost some ground in the period 1964-81 on these items, but England's average mark dropped from 52 per cent to 44 per cent and we were the only country to have recorded lower levels of attainment in all the four divisions: Arithmetic, Algebra, Geometry and Measurement, and Statistics. Some reasons can be advanced for this poor performance (see Howson, 1989), notably the under-representation of independent and boys' grammar schools in our sample schools (54 per cent of the students sampled were girls), but even those did not allow such a falling-off to be ignored. Here it must be noted that the 1964 testing took place in the much denigrated multi partite educational system. It is also important to bear in mind the following observation based on graphs showing the percentage of students from England, Japan and West Germany obtaining a particular mark in FIMS: 'Note that England and Wales exhibited a greater spread of ability than did Japan and West Germany. We had a greater percentage of "high attainers" than these countries, but this was more than counter-balanced by the weak performance of our low-attainers' (Howson, 1989). Since that time, it has often been argued that the weakness of English mathematics education lies in our 'long tail' of under-achieving students, but that our better students are the equal of those in any other country. We shall investigate later whether such a belief is still valid.
England's weak performance in SIMS clearly influenced the government in its decision to introduce a national curriculum. It is important, then, to see how changes since the NC was implemented in the early 1990s have affected our performances in the later studies. Here, I shall quote data from a study by Robitaille and Taylor (forthcoming). They considered the results from the twelve educational systems which participated in SIMS-1981, TIMSS-95 and TIMSS-99: Belgium (Flanders), Canada (British Columbia and Ontario), England, Hong Kong, Hungary, Israel, Japan, the Netherlands, New Zealand, Thailand and the USA. First they took the aggregated raw scores for these countries in the whole test and in the various sub-tests, and then calculated the mean score for the twelve systems and the standard deviation. Countries with means more than half a standard deviation greater than the overall mean were classified as high-attaining, and those with means more than half a standard deviation less than the overall mean as lo w-attaining. In SIMS (despite the questionable sampling) England achieved a 'middle' position. However, in TIMSS-95 it was relegated to 'low-attaining' and was ranked with Ontario, New Zealand and the USA. By TIMSS-99, Ontario and the USA had been promoted and England now had Israel and Thailand as classmates. England was, however, now 'top of the third division'. Similar analyses were carried out for the various sub-tests and England's performances on these are shown in table 1.
Such results make one wonder how the efforts of the last ten years and the money spent on implementing the NC are to be justified.
In fact, looking at items in detail and at other data can prove even more discouraging. For example, one item repeated in three tests was: 'If P = LW and if P = 12 and L = 3, then W is equal to (A) .75, (B) 3, (C) 4, (D) 12, (E) 36'. In 1964 this was answered correctly by 64 per cent of English Y9 students, in 1981 by 56 per cent, and in 1995 by 53 per cent. (7) This item was replaced in TIMSS-99 by a still unreleased (and slightly more difficult) item: a question similar to 'Which of the following equations do the values P = 12, L = 3 and W = 4 satisfy? (A) P = L/W, (B) P = W/L, (C) ...'. England's score was 46.6 per cent, the international mean 58 per cent (this was from all participating countries, 39 per cent of which were developing ones with per capita GNPs of less than $2,995).
Reference has already been made to the belief that English high-attainers compare well with those from Japan and elsewhere and evidence to support this belief quoted from FIMS. The reports for TIMSS-95 and TIMSS-99 showed the percentage of Y9 students for each country who earned a place in the top 10 per cent and top 25 per cent of students drawn from all countries. Some specimen figures are shown in table 2.
We note that, as a result of there being a considerably greater percentage of less-developed countries in TIMSS-99 than in its predecessor (39 per cent as against 22 per cent), the higher attainers in the developed countries have made progress between TIMSS-95 and TIMSS-99. However, the comparatively low level of improvement shown by the English high-attainers, and the relatively low percentage of the age cohort that fit into these two categories, are disappointing and raise further serious questions.
Although extremely valuable, the IEA studies have their problems. Curricula and curricular emphases vary between countries and the balance and quality of the test items are on occasions questionable. Essentially, then, one must penetrate beyond the 'horse-race' results quoted here and look at individual items, asking whether or not these test what we wish our students to learn and, if they do, whether the results indicate that we are as successful in meeting our aims as are other countries. Such a study has, however, yielded little in the way of consolation for England. (8)
Recent governments have made great play of their use of increased external assessment of learning in schools. Tests have been introduced at more and more stages of education, in a manner that is unmatched in any other country, and we are constantly told how standards are improving. For example, by 2000, the percentage of A-level mathematics candidates obtaining A or B grades had reached 48, yet in the same year the report, Measuring the Mathematics Problem, presented its evidence of a marked decline in university entrants' mathematical cal skills. Is this, then, another case of 'Never mind the quality, count the numbers'?
External assessment should be judged upon three factors: its validity, reliability and beneficence. Does it test what we wish to test, is the marking consistent, and does it lead to a desirable backwash in schools? Here we note a major problem, reliability. With so many tests and candidates for examinations there is a demand for more and more examiners: nearly every teacher is called upon to serve. (In 2001 there were reported shortage of examiners at GCSE level. One Board was reportedly offering financial incentives to those existing examiners who could bring along a new one!) The effect of having such a large, virtually untrained army is that marking schemes must be made rigid and the only way of doing this effectively is to break questions down into snippets that teachers can mark as either correct or incorrect. Here it is essential to look at an actual example. I have selected one on algebra from the KS 3 extension paper - intended for our brightest and best at age 14. The choice was dictated by the fact that it was one of the few algebra questions that actually used algebra to solve a problem rather than to create one (e.g., items such as 'simplify 7+ 2t + 3t; b + 7 + 2b + 10' which appeared on the papers for low- and average-attainers), was intended for high-attainers, and raises questions concerning curriculum construction that we shall wish to consider later.
"The lowest of four consecutive integers is n.
(a) Write the sum of the four consecutive integers as simply as possible in terms of n.
(b) Write the product of the lowest and the highest of the four consecutive integers in terms of n.
(c) Use your answers to parts (a) and (b) to show that there are only two sets of four consecutive integers whose sum is equal to the product of the lowest and the highest integers.
(d) Write down the two sets of consecutive integers.
The marking scheme assumed, incidentally, that in (c), students would make use of the formula for solving quadratic equations (supplied in the test booklet), rather than by a simple foactorisation. Such a question is, I believe, unsuitable for the students for which it was set. It possesses little validity, if our aim is to produce students who an reason and determine themselves what mathematical knowledge they must call upon to solve a problem. The item would, to me, have greater validity if recast as:
Find all sets of four consecutive integers satisfying the condition: 'the sum of the four consecutive integers is equal to the product of the lowest and the highest'. How do you know you have found all such sets?
It can now be solved without the use of algebra, but then the last part demands good reasoning ability. However, if algebra is used, then the last part is obvious -- and, indeed, the power of algebra is demonstrated. The recast question is more difficult and might be more appropriately asked at a later stage, but should not high-attainers be encouraged to develop the powers of dealing with difficult questions, rather than merely learning to respond to the step-by-step promptings of examiners? Might it not be assumed that such students should be capable of either factorising a simple quadratic, spotting the two solutions by inspection, or even, if all else fails, remembering the formula? Of course, the reliability of marking would decrease greatly with the second question: marks would have to be given for various partial solutions and producing a mark scheme for a mass of untrained examiners would be a nightmare. But what is the backwash on schools and students of such examining as we have now? (9) Is this to some extent to blame for the decline in standards of our better students?
This fragmentisation of mathematics and its learning is to be found not only within examination questions, but now, on a larger scale, in the modularisation of examinations. There is no doubt that the trend towards the use of modules, both in GCSE and AS/A2 courses, has done much to increase the confidence and motivation of weaker students. However, this has been achieved at the cost of fragmenting mathematics much further, of encouraging the view that modules are things to be learned, passed and forgotten, of discouraging students from attempting challenging problems that call upon mathematical knowledge gained from the study of apparently disparate modules, and of increasing administrative problems within schools. A serious study of the pros and cons of modularisation as it affects different types of students is required, together with, if necessary, proposals for how current procedures might be improved.
Questions have been raised on many occasions concerning the effect of having a number of commercial enterprises responsible for examining at GCSE and A-level. That number has been reduced in recent years, but the question remains: on what grounds can these bodies compete other than their generosity? (Evidence was given to the committee that produced the report, Tackling the Mathematics Problem, London Mathematical Society, 1995, of heads of department who were ordered by their schools to change examination boards in order to obtain better results.) Some other populous countries do not set the same external examinations for all students, but those countries with which I am familiar determine which examinations a school should take according to region and also retain greater public control over the actual examining.
What, however, must be realised is that external assessment is not necessarily good in itself and its effects can be far from beneficial if it is overdone and not of a high standard. (10) Governments must recognise, and respond to, this.
Teacher shortages and teacher qualifications
The shortage of qualified mathematics teachers is not something that has suddenly occurred in the last few years -- and here we must firmly distinguish between the shortage of qualified teachers and the now growing and more readily apparent shortage of teachers who may be required to take mathematics classes. The problem has been increasing for several decades, but has largely been concealed. Indeed, in 1958 the International Commission on Mathematical Instruction (ICMI) prepared an enquiry into 'la penurie (scarcity -- not penury!) des professeurs': for a contemporary UNESCO study had shown that about half of the seventy nations consulted (many of which might have been developing ones) had reported a shortage of mathematics teachers in their schools, colleges and universities. Yet, already by 1961, Bryan Thwaites argued that from then on England would experience 'possibly... a steady and irretrievable decline, in the proportionate numbers of adequately-qualified teachers'. The ICMI enquiry proved too ambitio us, for it asked largely unanswerable questions, but which were, nevertheless, much to the point. Was the shortage of teachers specific to mathematics? If there was a shortage, was it possible to identify causes? What was the effect of industrial and commercial changes upon the recruitment of mathematicians; the status of teaching compared to other professions on a similar intellectual level; and the teacher's load in terms of hours spent teaching per week, in preparation, in administration, in out-of-school and other non-mathematical activities? On the other hand, what were seen as the advantages of a teaching position? And was there a flow of students to higher education that was capable of meeting the educational systems' demands for mathematics teachers?
As it happened, the UK government's acceptance of the Robbins report, and the general expansion of the universities and student numbers in the 1960s, created a larger pool from which to recruit mathematics teachers. This eased the problem of numbers, but concealed that arising from the quality of teachers entering the profession. A 1963 Mathematical Association report, The Supply and Training of Teachers of Mathematics, referred to the way in which the rapidly expanding university sector was absorbing many of the ablest young mathematical talents, while the colleges of education (then lengthening their courses) were recruiting experienced, able teachers. Moreover, whereas 'in 1938 well over 75 per cent of newly qualified mathematics graduates entered the teaching profession ... industry now absorbs a comparable percentage ... including half of those with honours'. (11) The demand for mathematicians was threatening the well-being of the system that supplied them. Yet, this was still a time when a friend, a Cam bridge First in mathematics, followed medical advice and left his post in industry for a less stressful one in school teaching.
He went to teach in a grammar school, which not only gave him both greater status than would a post in a secondary modern school, but also the opportunity to teach mathematics from 11+ to A-level Further Mathematics and university entrance scholarship level. What has happened in the intervening years cannot be fully described here. However, the status of the teacher in general has declined, particularly that of the former grammar school teacher, i.e. the one most likely to have the better academic qualifications. Anyone with a particular desire to teach mathematics at an advanced level is drawn towards the independent schools or those sixth-form colleges that can still offer a Further Mathematics course. As a result, the majority of teaching in KS3 (11-14 age range) appears to be being given by teachers whose degrees were not in mathematics but, at best, in a mathematically-related subject. (12) There is no need here to write of current discipline problems in secondary schools, nor of the extra administrative workloads that have been placed on teachers in the last decade along, often, with the implication that they have the solution to, or are responsible for, many of society's ills. Responding to the initial NC proposals, I argued (Howson, 1989) that 'good teaching depends upon good teachers - nor good syllabuses, materials or assessment procedures, although these can assist, and may sometimes compensate, in small part, for a weak teacher force. There is now a great shortage of teachers, and it is essential to ask what effect the current proposals [the original NC] will have in attracting or deterring people from entering the profession.' One asked, but no response was forthcoming. Recently, the government has taken some steps to attract teachers through the use of financial incentives, but other major problems remain. Here it must be emphasised that there are still well-qualified mathematics teachers to be found in schools and that some less well-qualified ones may still be outstanding teachers. However, in the words of the 1912 Staff Inspector for Mathematics (for the creation of state secondary schools created problems in those days also):
'The efficiency of teachers [cannot]... be measured by ... academic qualifications. None the less when the question is not of an individual or of a small group, but of a large number, it remains true that the lack of good qualifications must seriously limit the efficiency of teaching.'
Before ending this sorry tale it is necessary to warn of yet another impending teacher shortage - within the universities. University lecturing has now little financial or social status and lecturers have also had a ridiculous assessment load imposed upon them. That profession, too, is in danger of losing its appeal. (13)
The shortage of well-qualified mathematics teachers has a serious effect upon the curriculum that schools can offer, and upon the effectiveness with which this can be done. It is vital, then, to understand the context within which consideration of school curricula and the effectiveness of teaching must take place.
The varied aspects of mathematics
Clearly, a prime aim of school mathematics must be to provide all students with that mathematics required by today's thinking citizen. What exactly, though, is that? Two recent attempts to define this merit a mention. One was in a section of TIMSS in which England did not participate. It was a test on mathematical and scientific literacy set to students in their last year of secondary school whether or not they were still studying mathematics. The items were all posed in real life contexts and covered topics on arithmetic (including estimation), data handling (including graphical representation), geometry (including mensuration), and (informal) probability. The resulting data were of considerable interest in indicating the extent to which countries had prepared their students to deal with the kind of mathematics they would meet in the street or the press.
Another significant offering is a report, Mathematics and Democracy: the Case for Quantitative Literacy, produced by the US National Council on Education and the Disciplines (Steen, 2001), that seeks a complete reorientation of the traditional US school mathematics syllabus. Steen distinguishes between what he terms quantitative literacy, which stresses the use of those mathematical and logical tools needed to solve common problems (e.g. percentages and mensuration), and mathematical literacy, which emphasises the traditional tools and vocabulary of mathematics (e.g. formal algebra and, later, the calculus). (14) As one might expect, there is some variation in what the contributors to the report believe quantitative literacy to mean. At one extreme we find 'mathematics and quantitative literacy are not the same thing... mathematics is more formal, more abstract, more symbolic than quantitative literacy, which is contextual, intuitive and integrated'. However, another view is that there is no essential dichoto my between formal mathematics and context-rich quantitative literacy. (15) This latter view is one I share -- provided that the need for an increased emphasis on reasoning and the ability to deal with complex problems is recognised. I have no doubt that, for example, the correspondent and adviser in the Saga Magazine exchange quoted earlier would have happily tackled the questions on percentages to be found on our SATs and GCSE papers. It was the extra complexity of having to determine which figure had risen by 70 per cent that threw them. Teaching and examinations must prepare students to answer complex as well as one-step, so called, real-life problems. That ability to reason, which can be built up on percentages just as well as on circles, chords and tangents, is what both users of mathematics and budding mathematicians require. Neither is it the case that such a course need be mechanical, boring and unimaginative.
Essentially, then, I believe that it would be possible to develop a GCSE course for all that would prove more valuable and would motivate more students than the present one if it were focused more specifically on the mathematics of citizenship, culture, other curricular subjects, personal finance and health,.... This would still leave us with a fund of worthwhile mathematics to teach in arithmetic, geometry, data-handling, simple probability and the use of algebraic formulae. (16) Of course, where appropriate (not merely where there is an opportunity), computer software and calculators would be used. Moreover, providing emphasis were given to extended reasoning, coping with complexity, and arousing the students' involvement and interest, such a course could still prove a firm foundation for the further study of mathematics.
Yet, the high-attaining young mathematician would benefit from a course that offered more than this and concentrated less on mathematics' role as 'a servant'. I see, then, the need for both a compulsory literacy-oriented mathematics GCSE course that need not be be-devilled by tiering, (17) and also an optional two-year one which introduced students more explicitly to proof and rigorous mathematical thought. The two would have a relationship similar to that between 'English' and 'English Literature'. The great aim of the new course would be to introduce students to a wider view of mathematics and provide the intellectual challenges that are so frequently missing in today's GCSEs.
There would be some problems in introducing such a course. An obvious one is that of teacher supply and of the numbers of teachers who are adequately prepared and could be spared to teach such a course. Also, it would seem essential that the new examination were not seen as a necessary preliminary to A-level mathematics, (18) and did not degenerate into an early exposure to A-level work, as was the fate of the old 16+ (GCSE) 'Additional Mathematics'. We cannot afford any falling off in the already insufficient numbers who enrol for A-level. (19) Moreover, one must ask how, in the present league-table obsessed climate, schools would react to such an innovation. The current situation is so bad that one must beware of any untested initiative that might make it worse. (20)
Such a dichotomy of pedagogical aims could also be advantageously employed at A-level. Single-subject mathematics must almost of necessity be largely concerned with the preparation for, and the teaching of, calculus, i.e. it emphasises the 'service' or 'literacy' aspect of mathematics. However, 'Further Mathematics' should, perhaps, be recast to ensure that students must include within it modules that give more emphasis to mathematical thought, e.g. a module on geometry as is proposed in the recent RS/JMC report (see footnote 3).
Some new initiatives
It would be wrong merely to stress the dark side of mathematics education. There are some brighter signs.
First, the National Numeracy Strategy and the Improving Primary Mathematics (Barking and Dagenham) Scheme appear to have achieved a considerable degree of success. The evidence I base this remark on is not so much improved student achievement in the national SATs (significant though that is), but on the remarks of those teaching Y7 in secondary schools.
The effect of the numeracy strategy has been such (notwithstanding its critics) that the temptation to extend it to KS3 must have been irresistible. For at this level there was also the need to provide a smoother transition from primary to secondary school. However, I believe that here doubts arise concerning the validity of transferring the methods employed in the primary school. Certainly, the newly published, Framework for teaching mathematics: years 7, 8 and 9, provides clear indications of what is required to meet the requirements of the National Curriculum and, through its many examples (the vast majority good), indicates desirable levels of attainment. However, as one proceeds upwards through the school, problems of differentiation increase as do those of engaging the attention of, and motivating, students. There is, then, concern that insufficient experience-based advice is being provided or sought on the treatment of students of differing abilities and attainments. It is also unclear how teachers wil l motivate much of the mathematics the framework contains. It is one thing to exemplify the hoops through which one would like students to be able to jump: but a more difficult matter altogether to motivate the jumping. Certainly, there is little indication how, for example, a teacher will motivate and demonstrate the value of, say, operations on fractions. The treatment of the latter also illustrates what many would see as a great weakness of the NC and, indeed, of a tradition that has grown up within England. That is, the too early introduction of too many topics, followed by their frequent repetition with little progress made on each occasion. (21) Indeed students will have barely time to draw breath before moving on. One wonders what, if anything, weaker students then carry forward to their next encounter with a topic. (22) Indeed, is this the root cause of so many of our low-attaining students' problems?
Even if one were to accept these aspects of the framework there would still be key questions left to answer. This framework, although not intended to be rigid, certainly invites the hack author to produce books that adhere rigidly to it and the ill-qualified or unsure teacher to use them. Moreover, what encouragement does it provide committed professionals to develop their own schemes (to be tediously written up and explained to visiting inspectors), or to write texts for that diminishing band of teachers who would consider stepping outside the 'national' line? (23) Is, in fact, the arrival of such a framework to be taken as official recognition of the fact that we can no longer recruit sufficient teachers who are qualified and committed enough to exercise any real degree of autonomy? (Autonomy over curricular matters went a decade or more ago; is that on teaching to follow?) What will be the backwash of this apparent denial of professionalism?
Here, also, it must be pointed out that the framework, like its primary school predecessor, is mathematically backward looking -- which is not necessarily bad in itself. There are some nods in the direction of computers; some good, some questionable and many that I suspect will prove impracticable in classroom terms. Yet the main drive, overall, is to improve teaching methods by a return to older methods, perhaps slightly modified as a result of experiences in other countries. However, how is mathematical curriculum development going to occur, and from what quarter are new ideas to emerge? Nothing significant in this direction has happened since the imposition of the national curriculum and no mechanism for encouraging developments appears to exist. The issuing of fiats from the Qualifications and Curriculum Authority (QCA), if they have not been based upon well-conducted and extended pilot studies will simply result in yet more chaos within the educational system. Where is that freedom that permits and encou rages innovation and presents the committed and mathematically adventurous teacher with a challenge? Are students in state schools to be provided solely with a diet which is, at best, bureaucratically competent but which lacks flair?
This paper has concentrated on the position in England and on its comparative showing in the IEA studies. However, it would be wrong to leave the impression that ours is the only country with problems. Other countries, too, frequently experience difficulties relating to teacher quality and recruitment. Also, a study of the TIMSS data for specialist final-year students of mathematics (the equivalent of our A-level students) from various western countries (see Howson, forthcoming) has revealed similar problems as are to be experienced here. In particular, much mathematics teaching appears to be centred on getting students to jump through technical hoops. Sometimes this is done with considerable success. However, when students are tested on an understanding of fundamental concepts or on the ability to deaf with multi-step problems and those where the mathematics to be employed is not obvious, there is a considerable dropping off in the success rate. Paradoxically, students are being trained to perform those oper ations that can now be dealt with using suitably chosen software, but all too often students share the computer's inability to analyse a problem and to reason. One wonders to what extent students have been empowered to use the mathematics they have been taught in new contexts, rather than merely to answer stock examination questions on it. The need in many other countries for a clearer definition of goals for school mathematics would seem required. We have already pointed out the need for such a reappraisal of aims in England, particularly within the age range 11-19. This, however, must mean more than setting up committees to express their misgivings about what is currently being achieved in algebra, geometry, and so on. Sensible mathematics curricula can only be devised within a wider context, encompassing general educational aims and the national curriculum as a whole. Moreover, in any rethinking of the curriculum the nature and individuality of subjects must be recognised.
Again, the comparative data on high-attaining students provided in this paper must cause us to question current policies. Are these students simply being encouraged to scamper through the various levels and the accompanying tests, rather than to develop their reasoning powers, their problem-solving abilities, and their grasp of fundamentals?
Any serious reconsideration of the NC will need to take time. Moreover, as we have seen, it is essential also to ensure that external evaluation assesses valid curricular aims, rather than imposing its own. The effects of league tables and university entrance procedures based on the simplistic notion that all subjects are equal must be re-assessed to see to what extent they lead to a drift away from those subjects which, although important for national well-being, are seen by students as more challenging.
Yet, this will not solve the pressing problem of teacher recruitment and retention. Two quotations appearing within days of each other illustrate both the nature of the problem and an apparent lack of understanding on the part of the government's Teacher Training Agency:
'With improved pay structures, more classroom assistance, increased use of technology, and improved standards, teaching today is a competitive, rewarding and purposeful profession. There really has never been a better time to be a teacher.' Mary Doherty, Head of Teacher Recruitment and Supply, Teacher Training Agency, The Independent, 26 July, 2001.
'The worrying situation [regarding teacher recruitment] has been compounded by the number of teachers leaving the profession because the job is now regrettably much less attractive than when they joined [it].' J. Dunford, General Secretary, Secondary Heads Association, The Independent, 30 July, 2001.
It is true that improved pay structures have recently made teaching more financially rewarding. The impossible administrative demands made of teachers following the imposition of, for example, OFSTED inspections and the testing accompanying the NC, have also been eased and are now merely excessive. Yet, what, say, is the effect on teachers of increased use of technology? The provision of technology in the classroom is not like that of an automatic gearbox in a motor car. None of the teacher's knowledge and skills is made redundant; simply more is asked of him/her. Administrative problems are increased and, most importantly, the teacher is asked to exercise judgement -- with scant or conflicting evidence to provide guidance -- on the extent to which such technology should be used, and with what specific educational aims in mind. This last point is well-illustrated by the way in which 'official' advice given to teachers over the last two decades on the use of calculators in mathematics lessons has vacillated, a nd the practices of examination boards shifted.
Simply improving pay will not meet the problems of teacher retention -- and retaining experienced and well-qualified teachers is equally if not more important than attracting less-well qualified entrants to the profession. Attention must be paid to teachers' other problems. Some causes for the loss of status cannot be remedied. They result from changing economic and commercial circumstances. Some, indeed, resulted from the actions of a minority of teachers in the 1970s, and the situation has not always been helped by the actions of trade union activists. However, others have been allowed to develop. How can this process be reversed?
We have already noted the teacher's loss of autonomy. Yet not only has autonomy virtually disappeared, but so also, in both the classroom and in society, has much of a teacher's authority -- the power derived from office, or character, or prestige. Again, some causes could not be remedied, yet others have been allowed to develop and governments have been slow to realise the consequences. Finding a way in which to restore some of that authority, the key to solving the problem of classroom indiscipline, is a major problem for which a solution must be found.
Ameliorating the present situation will not be easy. Some initiatives have already been taken. Many others are required. However, of one thing we can be certain: there is no short-term solution to our problems. A succession of knee-jerk reactions will be unlikely to produce much of value. Clear medium-term goals are required, along with a willingness openly to recognise and discuss existing problems and constraints.
(*.) Emeritus Professor, Faculty of Mathematical Studies. University of Southampton (correspondence: firstname.lastname@example.org).
(1.) The Newsom report (1963) stated that 31 per cent of girls' secondary modern schools sampled taught only arithmetic. However, in the late 1950s, Mathematics Teaching carried articles illustrating that 'more and more mathematics [was being included] in the curricula ... and with freedom from external constraints, more secondary modern schools are doing interesting work of high standard and quality, many including calculus'. Perhaps 'many' might have been better replaced by 'some', yet it must be remarked that calculus has now disappeared from all pre-16 syllabuses.
(2.) Multilateral schools (those in which students are separated into different curricular streams within the same school -- as are to be found in the Netherlands) were considered by the 1938 Spens committee but rejected on a number of grounds -- the first being their size. Spens suggested that students gained more from being in a school of not more than 800 pupils -- also the 'modern' stream would have the unsuitable grammar school curriculum forced upon them. It is interesting to see how the Spens committee foresaw so many of the problems that comprehensive schools brought in their wake.
(3.) This same tendency is very evident in the recent report on geometry teaching produced by the Royal Society and the Joint Mathematical Council. Yet more topics and emphases are suggested, but there is no hint of what should give way to accommodate these.
(4.) This is a position diametrically opposed to that found in the recommendation of the 1968 Dainton Report: namely, 'in our view all pupils should study mathematics until they leave school' [i.e. including time spent in the sixth form]. Teacher-supply problems made this a pipe-dream in 1968. The position now is worse, and the problems arising from the implementation of the recent key skills and AS initiatives would not encourage any moves in that direction. Improving mathematics education pre-16 must take priority over introducing yet more palliative measures post-16.
(5.) The national curriculum for mathematics has been amended three times in its brief lifetime.
(6.) England also participated at 17+ in FIMS and SIMS, but not in TIMSS, and at Y4, Y5, and Y8 in TIMSS-95. Only the equivalent of Y9 was tested in TIMSS-99.
(7.) Some time ago, I asked a Physics teacher how he coped with teaching Ohm's Law to students who appeared to have little idea how to work with formulae. He said there was no problem: 'We tell them to draw a triangle and put V in the top half and I and R under it. Now you cover up with your finger whichever one you want to calculate and then work out the resulting multiplication or division.' The curse of mathematics teaching: rules without reasons!
(8.) Providing one or two further depressing examples might simply attract the defence that these were hand-picked to make a point and were not truly representative. It is necessary, then, to consider a much wider range of items and, in particular, to see how our students performed on those topics, e.g. probability, which receive greater curricular emphasis in England than in many other countries. This I do in a paper to appear in Robitaille and Beaton.
(9.) Those who would like to consider this question further should obtain a copy of, say, GCSE Mathematics Intermediate Level: The Revision Guide, Combination Group Publications. This is a remarkable collection of 'rules without reasons'. There is an implicit assumption that examination questions will be stereotyped and, in the main, trivial, and that successful candidates will not be expected to have much understanding of what they are doing or be equipped to make any real use of any mathematical knowledge they possess. One is left pondering: 'What in this gallimaufry is likely to be of lasting value, and to whom?'
(10.) Examples of conceptual errors contained in public examination questions on probability are given in Howson (1989 and 2002). The June 2001 issue of the Mathematical Association News contains a letter sent to the Northern Ireland Board complaining of errors in statistics questions on three of the A-level papers set in 2000. In the same issue the MA Council complain that 'some exam boards have suspended their monitoring committees of teachers': university superintendence of examinations ceased years ago. In addition, recent changes have seen the powers of the QCA A-level assessors curbed.
(11.) By 1974 the percentage of mathematics graduates entering teaching had shrunk to 20 (see Royal Society, 1974) and is currently less than 10. Between 1966 and 1973 the annual output of graduate mathematicians increased from 1754 to 2832 (Royal Society, op. Cit.). However, the later expansion of universities did not have so marked an effect on the number of mathematics graduates produced which has continued to hover above the 3000 mark. Various other reports on teacher shortages have appeared since 1974, including a 1988 Engineering Council study (Smithers and Robinson) which concluded that the position (if by that we mean supplying enough trained and competent mathematics teachers) is irremediable. It is also the case that the graduate mathematics teacher is unlikely to have as good a class of degree as his/her counterparts in most other disciplines.
(12.) Since no accurate data are available on the mathematical qualifications of teachers in schools one can only judge by anecdotal evidence, supported by the findings of such studies as Smithers and Robinson (1988).
(13.) The vast majority of the university appointments reported in the first six issues in 2001 of the Newsletter of the London Mathematical Society have been of mathematicians seemingly from overseas -- many from the former USSR. This is not a complaint about the employment of lecturers from abroad -- I personally gained too much from the pre-war refugees from Germany to deny today's students such help from what, I am sure, are outstanding research mathematicians. But anecdotal evidence from colleagues suggests that many bright young English mathematicians are no longer seeking academic employment. Can we rely on there being such an overseas pool of academics available when those recruited during the expansionist years of the late 1960s and early 1970s retire?
(14.) In the 1990s Japanese upper secondary school (USS) curriculum, two mathematics courses were offered (corresponding to our Advanced and Further Mathematics courses) stressing, respectively, mathematical literacy and mathematical thought. Students hoping to study mathematics, engineering or physics at university were encouraged to take both. 'Mathematical literacy' had the meaning ascribed to it by Steen. All students had to take mathematical literacy in their first year (YII) in USS. In the latest revision of the curriculum, all students must still continue to study mathematics in the first year of the USS, but now they may take the one-year 'Fundamentals of mathematics' option, which is very much 'quantitative literacy' -- i.e. mathematics for citizenship, or begin the three year 'Science mathematics' course (i.e. mathematical literacy leading to the calculus) or, in addition to that, the three year 'Mathematics' course (including, e.g., some formal Euclidean geometry).
(15.) Again there are parallels in other countries. For some years now the Netherlands has stressed 'realistic' mathematics and we note how well its students performed in TIMSS (including the mathematical literacy test). However, there were warning signs (as in England) about 13+ students' ability to handle the 'mathematical reasoning and problem solving' aspects of a test devoted to the solution of more open and extended items. (Students from both countries were willing to attempt problems -- but they frequently lacked the ability to complete them.) Moreover, Dutch university colleagues have complained for some years now of their students' inability to 'prove', an aspect of mathematics given scant recognition in realistic or, indeed, mathematical literacy courses.
(16.) Such a course should not necessarily be based on a subset of the present NC. There are other topics that would merit serious consideration for inclusion, e.g. simple combinatorics (the multiplicative aspects of strings of choices) and various topics in practical geometry.
(17.) In an attempt to move away from the 'piece of string' curriculum, QCA recently recast the NC mathematics curriculum in KS4 in two tiers. However, the Ministry wished to protect schools from 'too many changes' and so examination boards are still to examine the two-tier curriculum in the three tiers developed in the 1980s to meet earlier curricular demands -- a Gilbertian state of affairs. A quantitative literacy curriculum could be examined either without tiers, or, if this were found impracticable, with a lower tier devoted to quantitative literacy and a higher one which added topics of mathematical literacy, such as more algebraic work.
(18.) This is not the place to indicate in detail a suitable syllabus but possibilities would include a more formal study of geometry (either Euclid-style or transformation), simple number theory, set theory and logic, aspects of combinatorics, and graph theory and applications. It would seem essential to offer teachers some curricular choice -- let them select the mathematics that particularly interests them in the hope that they will pass on this enthusiasm. Such a GCSE could be administered by only one examination board but made available through all boards.
(19.) Early (but unconfirmed) evidence suggests that the creation of AS Mathematics has only led to a decrease in the numbers opting for the main A-level course.
(20.) It is essential that governments should curb their enthusiasm for announcing their intention to implement educational changes before 'the details are worked out'. Only when the details are made explicit and these are implemented in schools are the true problems revealed. Those responsible for the non-stop flow of educational initiatives might pause to read Trollope and consider the Duke of St. Bungay's counsel: 'I have never been a friend of great measures, knowing that when they come fast, one after another, more is broken in the rattle than is repaired by the reform'.
(21.) See, e.g., Howson et al. (1999), which is the report of a QCA study of primary school mathematics in ten countries, for a comparison of the number of topics introduced in primary schools and the comparative ages of introduction.
(22.) This is not a problem confined to either GCSE level or weaker students as is indicated by a letter from a boy hoping for very good A-level grades (Independent, 10 May, 2001): 'It's the speed at which [mathematics] teachers go through things. You have a week to grasp a concept and then you're on to something else. By that time you've forgotten what you learnt the week before.'
(23.) The Mathematical Association has, again, drawn attention to the danger of "narrowly focused textbooks [which] do not encourage good teaching". One also notes with alarm the growing role of the examination boards as producers of texts designed to steer students through the examination courses which they, too, have designed.
Bramall, S. (2000) 'Rethinking the place of mathematical knowledge in the curriculum', in Bramall, S. and White, J. (eds), Why Learn Maths?, London, Institute of Education.
Cockcroft Report (1982), Mathematics Counts, London, HMSO.
Dainton Report (1968). Enquiry into the Flow of Candidates in Science and Technology into Higher Education, London, HMSO.
Engineering Council, Institute of Mathematics and its Applications, London Mathematical Society (2000), Measuring the Mathematics Problem, London, Engineering Council.
Hamilton Fyfe Report (1947), Secondary Education, Edinburgh, HMSO.
Howson, A.G. (1989), Maths Problem: Can More Pupils Reach Higher Standards?, London, Centre for Policy Studies.
-- (2002), 'Questions on probability', Teaching Statistics, London, Royal Statistical Society, pp. 17-21.
Howson, A.G., Harries, T. and Sutherland, R. (1999), Primary School Mathematics Textbooks: an International Study Summary, London, QCA.
London Mathematical Society, Institute of Mathematics and its Applications, Royal Statistical Society (1995), Tackling the Mathematics Problem, London, London Mathematical Society.
Newsom Report (1963), Half our Future, London, HMSO.
Robitaille, D.F. and Beaton, A.E. (eds) (forthcoming), Secondary Analysis of the TIMSS Results: a Synthesis of Current Research, Dordrecht, Kluwer.
Robitaille, D.F. and Taylor, A.R. (forthcoming), 'From SIMS to TIMSS (1995 and 1999)', in Robitaille and Beaton (forthcoming).
Royal Society (1974), The Training and Professional Life of Teachers of Mathematics, London, Royal Society.
Royal Society, Joint Mathematical Council (2001), Teaching and Learning Geometry, 11-19, London, Royal Society.
Smithers, A. and Robinson, P. (1988), The Shortage of Mathematics and Physics Teachers, Manchester University.
Spens Report (1938), Secondary Education with Special Reference to Grammar Schools and Technical High Schools, London, HMSO.
Steen, L.A. (ed.) (2001), Mathematics and Democracy: the Case for Quantitative Literacy, Princeton, N.J., National Council on Education and its Disciplines.
Thwaites, B. (1961), 'Education: divisible or indivisible?', Inaugural Lecture, Southampton University.
Table 1. Comparative sub-test scores Area SIMS TIMSS-95 TIMSS-99 Algebra L L L Geometry M L L Number and number sense L L L Measurement M M M Data analysis M M M Note: M denotes 'middle-ranking' and L 'low-ranking'. Table 2. Performance of high attainers Top 10% Top 25% TIMSS-95 TIMSS-99 TIMSS-99 TIMSS-99 Singapore 45 46 74 75 Japan 32 33 58 64 Netherlands 10 14 30 45 Canada 7 12 25 38 USA 5 9 18 28 England 7 7 20 24
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|Title Annotation:||evaluation of secondary math education in the United Kingdom|
|Publication:||National Institute Economic Review|
|Article Type:||Statistical Data Included|
|Date:||Jan 1, 2002|
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