Mathematicians are renowned for hair-splitting; but they are not naturally contentious animals. After the unusual exertions of recent months, it is the hard to avoid the impression that everyone wants just to “get back to normal”.

Many colleagues assume that matters are now decided. But if the future is to learn from the past, it may be necessary to take a deep breath and to look at some of the urgent tasks that confront the Society in the immediate future.

The LMS is principally concerned with supporting the professional environment within which academic mathematicans work. But its charitable commitment to mathematics in the UK also obliges the Society to work with like-minded Associations and agencies to monitor developments in other areas – such as education (at university and at school level) – and where appropriate to seek to influence policy for the common good. This involvement includes the way education policy and practice affects the supply of mathematically competent manpower at all levels.

So as part of a broader attempt to look ahead, there may be some value in an occasional comment on one or more of the educational issues currently lurking in the undergrowth.

We live in an age where bureaucracies are busy learning how to annexe democracy to their own purposes. “Consultations” – whether for European treaties, or for changes to A level mathematics – invite us all to “register our views”. But they are often timed and packaged, and the questions posed are often loaded, so as to make it difficult for respondents to deliver anything but “the right answer”.

The recent QCA “Level 3 Mathematics Consultation” emerged in the Spring – apparently without prior public discussion. It rolled together five related, but very different changes, and sought to treat the whole as a single package.

There was no mention of the facts:

- that the 2003 emergency changes to A level Mathematics (in response to the calamitous “Curriculum 2000”) have recently led (?) to a modest but steady increase in numbers for A level mathematics (from a UK-wide low of 54K in 2002 to 64K in 2008 – not quite back to the 66Klevel of 2001);
- that the MEI-led Further Maths Network initiative has given rise to a doubling of numbers taking Further Mathematics A level (from less than5K in 2002 to more than 9K in 2008);
- that this burgeoning of Further Mathematics has “coincided with” (I hesitate to say “led to” because the proof is not to hand, but theconnection is almost self-evident) an annual 10+% increase in undergraduate Mathematics applications each year from 2003-2007 with a curious 6% dip in 2008; at the same time admissions numbers have risen by 3-5% each year from 2003-2007).

One may not understand the detailed dynamics of such trends; but it is remarkable that there was no official concern about the potential dangers of “rocking the boat” at such a time.

It seemed that the motivation for the QCA package was largely bureaucratic:

Every other A level (we were told) has 4 modules rather than 6, so Mathematics A level should be forced into line. (Curiously Further Mathematics is to remain at 6 modules. And the “new kids on the block” – “Use of Mathematics” and “Use of Statistics” were also to have 6 modules!)

In principle there may be good reasons to favour fewer modules: indeed some of us are old enough to appreciate the value of having just one module – with an exam at the end of the course! But the decision is being imposed top-down, without discussion. And it has other features and consequences: such as

(i) the abolition of the non-calculator paper; and

(ii) inexplicable changes/omissions in the detailed content list (which force one to infer that the proposal was drafted without external monitoring, and – in the context of the complete complicated package – one suspects most respondents to the consultation failed to notice).

More worrying was the ex Cathedra manner of the attempt to claim that the proposed changes to applied modules were being “demanded by universities”. The truth would seem to be that

- there are all sorts of reasons to criticise A level mathematics; but
- as regards structure, universities argued for, and seem broadly satisfied with, a large common core of mathematical methods (C1-C4).

I am unaware of any clear critical views at university level of the current arrangement whereby A level students can choose their two applied modules (from mechanics, statistics and discrete) in a number of different combinations, choosing either two introductory modules, or two modules of the same kind (e.g. M1 and M2).

The current situation may not be ideal; but removing this choice would only constitute an improvement if all end-users in universities (in science, engineering, social sciences, etc.) could agree on what A level applied modules were needed, and if all schools and colleges were equipped to teach a more tightly prescribed programme. Neither requirement is satisfied; so the current compromise may be the best we can hope for.

And the proposed alternative may well be worse. Despite the proposed switch to 4 modules, the two applied modules wiill be half the size of the two pure modules; so each applied module will still be one sixth of the whole A level. And the proposal is to make each of these small modules “half statistics and half mechanics” – which seems likely to result in a superficial treatment (encouraging selective exam-

question spotting), that fails to give students the chance to engage seriously with any one application.

Attributing this proposed change to “demands emanating from universities” would seem to be wholly false. (When we asked mathematicians, scientists, and engineers in my own university, we found an astonishing unanimity in favour of “doing something reasonably well” rather than “everyone doing the same superficial

mix”.)

Someone somewhere clearly wants to force all A level Mathematics courses to take a single specific form, and to eliminate the current English fudge, whereby some modules may form part of either Mathematics or Further Mathematics. One can understand the bureaucratic preference for a tidier structure. But it is harder to

understand the failure to assess the likely effect – on candidates and on entry numbers. (I leave others to assess whether this may also be linked to future moves towards a single, centralised Awarding Body.)

The proposed changes to A level Mathematics have knock-on effects on Further Mathematics. These changes were judged by MEI (“Mathematics in Education and Industry” – an independent curriculum development group that initiated and managed the scheme which has led to the remarkable revival in Further Mathematics entry numbers) as being likely to undo much of their good work <http://www.mei.org.uk/files/pdf/MEI_A-level_position_statement_Final.pdf>. Yet those agencies and bodies whose job it is to represent the wider profession in the corridors of power found it convenient not to listen.

The LMS certainly submitted a response to the QCA consultation. But the response does not seem to be available to members. It seems almost too obvious to say that Society responses should routinely be made available to members, and should probably be posted on the website. (I was shown a draft “in confidence”, and can report that it left me bewildered. The temptation to please different factions

in education by sitting on the fence merely allows any competent bureaucrat to draw whatever conclusions he or she wishes.)

If these proposed changes to Mathematics and to Further Mathematics had been presented as a limited package, universities and professional bodies might have managed to debate the details and to respond intelligently. But they were in fact wrapped up with other changes so as to create an interrelated and indigestible bundle. As a result an official from QCA had to circulate an emergency e-mail a

couple of weeks before the consultation closed, pleading for responses from HEIs. It remains unclear how many universities managed to drum up such a response in the first week of July, how representative of the community these responses were, or how well considered they may have been!

Meanwhile our professional neglect of these important issues was about to be upstaged by a remarkable campaign which targeted a completely different component of the QCA package of proposed changes. The resulting “media event” opened up all sorts of interesting possibilities for the future, if only we have the courage to consider them. It also highlighted the urgent need for a those who sit on Councils and Committees to “represent” the interests of the wider community to re-

connect with the grass-roots.

In the last 18 months the Think Tank “Reform” has focused attention on mathematics. Their interest is rooted in economic concerns; but those involved are intelligent souls who, given a degree of guidance, have brought a certain naive freshness to the professional debate. In 2008 they produced a report on the economic impact of school mathematics; in 2009 they targeted A level reform.

With the enthusiasm of innocents, and the media-savvy of a Think Tank, they seized on – and managed to hijack – the “QCA Level 3 Consultation” in the media. Instead of worrying about what *we* might see as the central changes (to Mathematics and to Further Mathematics), *they* targeted the proposal to introduce a new maths A level: “Use of Mathematics” (UoM).

Although the proposed new A level had been operating as a pilot for a couple

of years, the QCA Consultation failed to draw attention to any of the consequent evidence (e.g. in the form of syllabuses, asessment details, exam papers, etc.).

In reality, UoM is very different from A level – with a large measure of portfolio assessment, a strong emphasis on “data-sheets” and calculators, and little by way of formal mathematics. Until one works with students who have completed the course it is hard to be sure; but experience suggests that undergraduates (e.g. in

engineering) who have been admitted in the past with similar qualifications have been singularly ill-prepared for the transition to an “academic” approach.

Thus, whatever its other merits may be, UoM would appear to be unsuitable as a preparation for traditional university courses in any subject. An official, alternative maths A level, that is very different from existing A level Mathematics, but which to the uninitiated looks and sounds like “an A level in mathematics”, would seem like a recipe for confusion

(i) for students, parents and employers,

(ii) for schools and colleges, and maybe also

(iii) for some admissions tutors.

Moreover, since it is designed to serve a group of students who currently cannot handle A level Mathematics, it seems inevitable that it will be “easier” (in the sense that students who take both are likely to score a higher grade on UoM); so it offers a rather obvious wheeze for school Principals who can only staff one “Level 3”

course in mathematics, or who are under pressure to improve their average grades!

Curiously, no-one seems to have linked this QCA proposal to the fact:

- that the government has recently adjusted (upwards) its target for “A level Mathematics entry numbers”.

In one sense “Use of Mathematics” was but a small part of the package of proposals, and one suspects that universities would normally have paid it scant attention. But “Reform” used its resources to mobilise (at very short notice in early July!) an impressive group of opponents in academia – quickly managing to boast of support from “64 mathematics professsors” <http://reform.co.uk/>. They then used their media contacts to launch a campaign – including extensive slots on Today, on Radio 5, on Channel 4 (where Professor Franco Vivaldi put up a remarkable performance on behalf of mathematics).

Not all views expressed in the course of this campaign were equally well-informed. But they were evidently genuine and demonstrated a significant level of concern. (More interestingly, the list of “64 professors” on the “Reform” website

<http://www.reform.co.uk/EfR/EducatorsforReform/UseofMathematics/tabid

/152/Default.aspx> reveals a grass-roots mix of pure and applied colleagues which should gladden the hearts of all of us who judge closer cooperation to be in everyone’s interests.)

Instead of welcoming this “spontaneous” professional interest in serious matters, our official representatives immediately “closed ranks”. (The naive observer may by now be beginning to see the hidden thrust of this extended parable.)

On 10 July the LMS and IMA issued a joint Press Release with the grand title: “UK Mathematical Societies’ support for new Maths A level” (see <http://www.lms.ac.uk>, go to Policy, then to Press releases). As far as one can tell, this seems to have involved “rather limited” consultation. Was the draft, one wonders, discussed and agreed by the Education Committee of each society? And there are no signs that any moves were made to try to understand, to accommodate, or to enlighten those – often eminent colleagues – who had expressed serious misgivings.

In one sense the LMS-IMA Press Release is correct. Currently around 12% of the age group take A level maths. But there are two additional groups who currently cannot manage A level maths, but who ideally need some mathematics in Years 12 and 13.

(a) There are “academic” pupils who find maths hard, but who should ideally not give up maths at age 16 (perhaps 15-20% of the cohort).

(b) There are also “vocational” pupils who need a single “general purpose” KS5 programme to strengthen their mathematical foundations in a way that will support whatever trade or craft they may pursue. (These constitute perhaps 20% of the cohort).

However, if one pretends to meet these “non-A level” needs within the current “National Qualifications Framework”, the resulting qualilfications have to be classed as “Level 3” and so are bureaucratically equivalent to A level Mathematics (e.g. when awarding UCAS points, or league table positions), and will therefore lead inevitably to the kind of confusion indicated above.

So before we can sensibly address these other needs, the qualifications framework needs to be challenged and adjusted to avoid the obvious confusion and abuses. (We also need to train many more maths teachers.) To suggest – as does the LMS-IMA Press Release – that any pitfalls can be avoided by giving pupils “appropriate advice

and guidance” contradicts all our experience (e.g. the same cop-out has been routinely used to argue that premature use of calculators need not be harmful).

The LMS-IMA were not alone. ACME and JMC were also rattled; but instead of taking stock (ACME failed to consult their own “ready-made” Think Tank – the ACME Outer Circle), they sent a public knee-jerk response to the Secretary of State seeking to repudiate the misgivings expressed about UoM.

It may be that the doubters have got it wrong. But what is crystal clear is that, in a democratic community, the case for the QCA proposal has clearly not been understood in the wider mathematical community. ACME-JMC derive their authority from the support of all sections of that wider community. Yet they went out on a limb in declaring this as an urgent, make-or-break issue: “we strongly urge you [as Secretary of State] to have the courage to introduce a full A level in Use of Mathematics”; and the doubters position was declared “deeply harmful to mathematics education in England”.

The most striking feature about the “Reform” campaign was that it gave a voice to colleagues – some of them remarkably lucid – who might have had much to contribute, had theior professional bodies managed to interact more effectively with the wider mathematical commmunity. It also demonstrated (what has often been doubted) that there are effective lobbyists out there who care deeply about mathematics education, even if their grasp of the issues can sometimes be rather crude.

Clearly the future LMS needs to cooperate with all sorts of groups. But it had never occurred to me before the “Reform” campaign that we should actively seek to “inform” groups such as Think Tanks, who may lack experience of the detailed issues, and who surely have their own agenda, but with whom we might sometimes profitably make common cause.

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