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Memorandum submitted by Dr. A.
Gardiner (
0: EXECUTIVE SUMMARY 0.1 The submission is restricted to mathematics. The inadequate training of mathematics teachers deserves attention in its own right. It may also highlight broader failings within the system. 0.2 The main weakness of mathematics teaching in 0.3 Mathematics teaching is hard. The best teachers learn their craft the hard way over many years - by working to master their subject, by struggling to understand their pupils' difficulties, by reflecting critically on the effect of their teaching, and by developing approaches which prove to be effective. Thanks to the bureaucratic desire to certify teachers as "qualified" immediately after Initial Teacher Training, most teachers never begin this extended process: their ITT courses are rushed, and most of what is covered has no soil in which to take root - so is washed away in the deluge of their first years in teaching. So instead of mastering a serious and important craft, most adopt various substitute "strategies for survival". 0.4 Initial Teacher Training needs to be re-conceived. Existing ITT need not change dramatically, and might still serve to prepare potential teachers: but such ITT short-courses should constitute only the initial part of a more extended programme, with "qualified status" being earned after 3-5 years, on completion of components designed to strengthen subject knowledge (to a basic level) and to develop didactical reflection. 0.5 This would steadily improve the overall quality of mathematics teaching. But there is also a critical shortage of high quality mathematics teachers, and hence an urgent need to encourage schemes that might nurture the creation of a critical mass of such "potential subject leaders". (The National Mathematics Teachers' Summer School is an example of precisely such a scheme. It costs peanuts, has been successful beyond its organisers' wildest dreams, but cannot find a home under any of the current agencies, and may well not survive.) 1: BACKGROUND 1.1 Despite being officially a research mathematician, much of my time for the last 40 years has been spent working with and for teachers and pupils at school level - both nationally and internationally. I started (and for 10 years ran) the UK Mathematical Challenges, which now involve 650 000 pupils each year. I was President of the Mathematical Association 1997-8 and was Chair of the Education Committee of the European Mathematical Society (2000-2004). But my comments are rooted - in working closely with teachers and pupils over many years (I am currently travelling, working with teachers and pupils in Singapore, in Australia, in Hong Kong, and in Thailand, and apologise that I only have primitive access to e-mail), - in producing material for schools (15 books currently in print), - in curriculum development projects, and - in engaging with the educational process on a wider front. 1.2 My comments are restricted to mathematics teaching. Mathematics is one of the two most important subjects taught in schools, and is recognised as being a key indicator of the effectiveness of any national school system - so negative features which become clear for mathematics teaching are not only important in their own right, but may be indicative of wider failings. 1.3 It is customary to claim that the main problem in teaching mathematics has its roots in primary schools. There is certainly considerable scope for improvement in primary schools; however, the teaching in our secondary schools should give us much greater pause for thought - as the recent Ofsted report Mathematics: understanding the score (2008) shows! Primary mathematics presents its own problems, but it is relatively simple: the basic ideas (of number and measures) connect naturally with everyday experience, and they recur throughout the primary years - giving them time to take root. Such material is taught the world over (often very effectively) by suitably trained generalists. In contrast, secondary mathematics quickly becomes more abstract (fractions, negative numbers, algebra, ratio, functions, trigonometry), with new ideas depending on appropriate mastery of previous material in ways that new teachers do not realise - and many never fully appreciate (as is made plain in Mathematics: understanding the score). Given that many teachers have such a flimsy grasp of their subject, the pressures for schools "to perform" leave such teachers with only one cheap option: rather than working to ensure that new material is properly understood, they use the shortcut of giving "rules", in the hope that this will deliver short-term success on official tests - even if it almost guarantees failure in the longer term. 1.4 In short, the evidence is clear (and is
usefully encapsulated, in an remarkably frank way in the aforementioned Ofsted
report): the most serious failings of mathematics teaching in Fortunately this makes them easier to address: there is no obvious reason why we could not design and implement a permanent system of in-service support that markedly strengthens a significant fraction of the 50 000 or so secondary mathematics teachers within any 10 year cycle. 2: DETAILS 2.1 There are serious weaknesses in the present training regime for mathematics teachers. The greatest weakness is perhaps the belief that teachers can be adequately prepared by a short initial training programme, after which they are more-or-less abandoned. This makes it almost impossible for NQTs to develop any embryonic insights into the subtleties of mathematics teaching which they may have gleaned during ITT; instead they are overwhelmed by the deluge of reporting, classifying pupils' "levels", league tables, teacher absences, overwork, etc, with no time for reflection - and no notion that such reflection is part of their professional remit. One serious issue is therefore how to use the first few years in teaching more effectively. Another is how to encourage professional development in the longer term. (This latter question might, in principle, be addressed by the new MTL structure - though current indications are not encouraging.) The 1999 ITT National Curriculum in Mathematics (which constituted a relatively modest requirement) was drafted under the clear understanding that it could not be completed and audited by the end of a 1 year PGCE or 3 year B.Ed. course, but that the given framework should be viewed as material to be mastered and audited by the end of a 3-5 year initiation programme. This understanding was over-ridden (by the TDA or DfES?) and the resulting scheme was imposed without the intended caveat. It duly failed, and was withdrawn in short order! Hence there are broad issues which warrant serious attention. But they deserve attention from more experienced commentators than me. 2.2 My comments here focus on a much more serious failing: namely our consistent failure to recognise the need to cultivate an elite - a critical mass of unusually competent mathematics teachers, - who might be spread throughout the system (by using suitable incentives), - who might give a lead in writing texts and developing materials, in examining, in advising LAs and central agencies, and - who might provide incidental professional development in hundreds of undocumented ways. 2.2.1 Any large organisation needs to do more (much more!) than merely "fill vacancies". A business needs shop-floor workers: these may not arrive ready-made, but can often be trained on the job. However, the long-term success of the organisation requires that considerable attention be paid to nurturing middle managers, and to recruiting and developing competent technicians and executives. An army needs squaddies - who may be recruited, trained and rewarded in an appropriate fashion; but its overall quality depends on the quality and ethos of its officers and NCOs. 2.2.2 Yet 2.2.3 Mathematics teaching is hard, and the
qualities required to teach well are rare - at present especially in In preparing his report Making mathematics count (2004) Adrian Smith discovered that DFES had given up collecting data on "mathematics teachers' qualifications". Partly as a result of his complaints, the DFES commissioned two surveys which were published recently (NFER 2008). Sadly their "findings" blatantly contradict the experience of everyone "at the chalkface", and attempts to find out how the authors arrived at their figures have failed to clarify where the error lies. (One report claims that 42% of secondary mathematics teachers "have mathematics degrees"; the other claims 47%. The true figure is closer to 30%, and in 11-16 schools is very much lower.) So these reports should be treated with caution. 2.2.4 Any successful school system requires a critical mass of unusually competent teachers - who understand the subject they profess to teach; - who appreciate how the hierarchical character of elementary mathematics not only constrains the order in which topics are introduced, but also determines the form in which each topic must ultimately be mastered (in order to make possible subsequent material for which that topic needs to be mastered in a specific form); - who are sensitive to the difficulties with which pupils are confronted; - who develop, and continually refine, ways of introducing, mediating, and linking together the abstract ideas of elementary mathematics so that they can be internalised naturally by beginners. 2.2.5 The seriousness of the current situation in 2.2.6 How has this situation arisen? The growth of state schooling after the war
was accompanied by a remarkable intake of talented and idealistic teachers, who
entered teaching in the late 1940s and early 50s after having had their
idealism awakened by the challenge of building a 2.2.7 Instead of cultivating and supporting potentially excellent mathematics teachers, most effort in recent years has focused on "emergency plumbing" for some of those who officially "teach mathematics", but who lack the basic mathematics required to function. Many recent initiatives have driven many excellent mathematics teachers into other employment or early retirement, or have removed them from the classroom to join the growing army of administrators in initiative-driven agencies. 2.2.8 In the past this vacuum might have been filled by independent agencies (such as the Nuffield Foundation, or the School Mathematics Project); but most such agencies have been driven out of business by the stranglehold exerted by centralisation, which has left little scope for them to operate. Thus voluntary initiatives struggle to survive. One such is the National Mathematics Teachers' Summer School (NMTSS) - an intensive residential course, which selects 60-80 of our best young teachers each year (nominated by their schools), and opens their eyes to the wider world of elementary mathematics - sending them back to the chalkface with a new sense of the magical and subtle profession they belong to. They arrive thinking of themselves as good teachers - even the best; they leave much more self-critical, realising how much there is yet to learn. At present there is no mechanism to ensure a critical mass of high quality teachers. If we could provide this experience for 200 or more teachers each year, we could reach 5-10% of those entering the profession. But though the annual cost is trivial, and despite rave reviews, we have failed to find any agency willing to support NMTSS in the longer term and it may well not survive.
(I have asked that a copy of the report National
Mathematics Teachers' Summer School: One year on be sent separately. If it were helpful, I could forward an
electronic copy on my return to the January 2009 [1] Not published on CSF Committee Website. |