BAD MATHEMATICS

A couple days ago, I non-responded to the release of the 2023 PISA results (because, this). In a comment on that post, Tony Gardiner referred to a paper he had written for ICME10, held in 2004. In that paper, Tony claims, he “worked really hard … and totally failed” to make sense of “mathematical literacy”, one of the core nonsenses of PISA. Failure or otherwise, I think it’s a great paper, and Tony has kindly given permission to reproduce it here (and the PDF is here).

What is mathematical literacy?

Tony Gardiner

University of Birmingham, UK

Freedom is the freedom to say that two plus two makes four.
If that is granted, all else follows.           (George Orwell, 1984)

My goal is to explore the relationship between

Mathematics and “mathematical literacy”

in a way that might provoke a critical analysis of the many appealing, but often rather vague, claims made by the advocates of “mathematical literacy”. My hope is that this might help those in other countries to avoid some of the pitfalls that have characterized developments in England in the last 25 years, where a similar paradigm-shift occurred after the publication of the Cockcroft report (Cockcroft, 1982).

Mathematics is more permanent than perhaps any other human cultural activity. In contrast, in many western democracies concern about “mathematical literacy” (ML) is of relatively recent origin. Mathematics and mathematics education inhabit different worlds. The mathematical universe may be more permanent than that of mathematics education, but mathematicians have to accept that its principles and insights need to be mediated in various ways before they can become an effective part of the world of mathematics education. Similarly, if mathematics education in general – and ML in particular – are to benefit from their association in the public mind with the objective universe of “mathematics”, they are obliged to represent that universe faithfully when mediating its subject matter for the more pragmatic world of schools, of teachers and students, of politicians and bureaucrats, of curricula and examinations. Thus, in seeking to find improved approaches to elementary mathematics, we are not “free” to replace its objective character by something more “user-friendly”, but are obliged to respect the fundamental nature of the discipline.

In Peter Shaffer’s play Amadeus, Mozart represents God (or here Mathematics, the eternal), while Salieri represents Mammon (for us “mathematical literacy” – the transient). Salieri is “flavour of the month”, but his influence is short-term and superficial. In contrast, Mozart represents the nearest that Man can come to God. In Shaffer’s play Salieri understands this contrast perfectly well, and resents it bitterly:

“God needed Mozart to let himself into the world.
And Mozart needed me [Salieri] to get him worldly advancement.”

Salieri consistently exploits his temporal influence to cut the “God” Mozart down to size:

“What use, after all, is Man, if not to teach God his lessons?”

Salieri clearly understands the perfection of Mozart’s art and the relative crassness of the world’s judgement. After the first performance of The Marriage of Figaro he muses:

“Could one catch a realer moment? …
The disguises of opera had been invented for Mozart …
The final reconciliation melted sight.
Through my tears I saw the Emperor yawn.”
Emperor [coolly]: “Most ingenious Mozart. You are coming along nicely.”

Later Mozart, close to death, is reduced to repeating a childish tune and Salieri is jubilant:

“Reduce the man: reduce the God. Behold my vow fulfilled.
The profoundest voice in the world reduced to a nursery rhyme.”

Politicians (the Emperor) may misconstrue and trivialize mathematics; but mathematics educators should resist any temptation to take advantage of this distortion, and should never join with those who seek to replace this “universal heavenly music” by mere “tunes for the masses”. Mathematics teaching may be less effective than most of us would like; but we should hesitate before embracing the idea that school mathematics would automatically be more effective on a large scale if the curriculum focused first on “useful mathematics for all” (numeracy), with more formal, more abstract mathematics to follow for the few. Experience in England (and elsewhere) suggests that such a program may be possible if one is willing to restrict the initial focus to truly basic material (integers, fractions, decimals, proportion, word problems, algebra and geometry), and to teach it in a way which prepares the ground for subsequent developments; but one should not be surprised if such a program turns out to look strangely like what good mathematics teaching has always been.

There is no royal road to mathematics. Plausible-sounding reform rhetoric rarely translates into large scale improvement at the chalkface; so when faced with some new proposal, we have no choice but to exercise judgement in order to assess its likely efficacy. In the last decade, thousands of pages have been written on the general themes of “numeracy” and ML. So far, they remain largely on the level of rhetoric. We may all agree that, despite the artificiality of the school context, we should try to ensure that students experience school mathematics as something “useable”. But there is as yet little concrete evidence to suggest that a major shift of emphasis of the kind proposed by advocates of ML would lead to widespread improvement. Reforms in mathematics education (whether “new math”, or “problem-solving”, or “technology”, or “discrete math”, or “discovery learning”, or “back to basics”, or “constructivism”) sometimes take hold because political events – such as Sputnik, or the TIMSS and PISA results, or an official inquiry – create conditions in which a new magic fix is welcomed without subjecting it to the obvious critical analysis. In such conditions Snake-oil salesmen flourish, and even those with honest products to sell can get a little carried away.

“Mathematical literacy” and numeracy are often presented as though they were alternatives to traditional school mathematics, rather than by-products of effective instruction (like “literacy” or “maturity”). Politicians and employers naturally seize upon this suggestion that there may be a pragmatic-sounding alternative to “difficult” mathematics; bureaucrats then imagine that focusing on numeracy from the outset will deliver what they see as the required (utilitarian!) end-product more directly and more cheaply; and, sadly, some educationists see this paradigm shift as an opportunity to further undermine the idea of mathematics as the archetypal “objective” discipline. Thus it is important

• to look beneath the plausible-sounding surface to see whether the claims made for ML, and its siblings “numeracy” and “quantitative literacy” (QL), make sense
• to ask whether there is a real danger that the new emphasis on “mathematical literacy” may become the latest in a series of bandwagons whose negative effects will eventually outweigh any possible benefits
• to start work towards a possibly more useful interpretation of “mathematical literacy”.

As long as “success” in teaching mathematics to a mass audience remains elusive, we are obliged to explore alternative ways of teaching the subject at school level. However, experience suggests that there is no magic bullet: “success” may elude us because mathematics and mathematics teaching are simply hard! And if this is indeed the case, improvement may depend on painstaking didactical analysis, followed by design, piloting and planned implementation of modest changes to current practice, rather than on some brand new paradigm. So we should hesitate before embracing the latest substitute for traditional school mathematics, and be prepared to return to the mundane world of identifying central principles, and to the hard graft of devising and testing incremental improvements. In particular, we should avoid being carried along on a flood of overstated rhetoric – of which the following is just one of many examples (Steen, 2001):

“Unlike mathematics, which is primarily about a Platonic realm of abstract structures, numeracy is often anchored in data derived from and attached to, the empirical world … [and] does not so much lead upward in an ascending pursuit of abstraction as it moves outward toward an ever richer engagement with life’s diverse contexts and situations.”

Rather we should heed the warning of Hyman Bass (quoted in (Steen, 2004)), who describes with uncanny accuracy what happened in England in the late 1980s and 1990s:

“the main danger … is the impulse to convert a major part of the curriculum to this form of instruction. The resulting loss of learning of general (abstract) principles may then deprive the learner of the foundation necessary for recognizing how the same mathematics witnessed in one context in fact applies to many others.”

The origin of numeracy

The concept of numeracy emerged in the Crowther report (Crowther, 1959), where “numerate” is defined as

“a word to represent the mirror image of literacy … an understanding of the scientific approach to the study of phenomena – observation, hypothesis, experiment, verification [- and] the need in the modern world to think quantitatively.

Statistical ignorance and statistical fallacies are quite as widespread and quite as dangerous as the logical fallacies which come under the heading of illiteracy.”

The inclusion of “statistical ignorance” in this definition of “numeracy” is interesting, since at that time few mathematics teachers would have been in a position to do much about it. However, this claim that statistical ideas should be an integral part of basic numeracy resurfaced in the Cockcroft report (Cockcroft, 1982), where the topic was included as part of their Foundation list (para 458) of material for all students, despite the admission hidden away in para 774 that “surprisingly few of the submissions which we have received have made direct reference to the teaching of statistics”!

Ill-defined statistical examples are regularly used by advocates of “quantitative literacy”. But they never admit that achieving competence in probability and statistics is unrealistic – for these are subtle disciplines, which have fooled all of us more often than we would care to admit. Moreover, it is never acknowledged that statistical data as generally presented by politicians, in advertisements, or in the press, is often slanted to persuade, to mislead, or to impress, rather than to inform, so that experts also have trouble assessing the truth of what is claimed. It is clearly optimistic to imagine that one can educate John Doe to do better.

Data Handling is now one of the three content strands in the English National Curriculum; and after 15 years of repeated refinement, the structure, wording and interpretation of this strand still reveal the weakness of the “didactical analysis” on which it is based. Easy work on relative frequency and interpreting data certainly warrant attention within school mathematics; but devoting 20-30% of teaching time to Mickey Mouse statistics, as is now common in English high schools, has done much harm: it has reduced the time spent on important material, has failed to reduce the level of “statistical ignorance” in the general population, and may well have contributed to the apparent decrease in interest in statistics among mathematics undergraduates. Yet the dogma that “data handling” is more important for ordinary students than some of the traditional core techniques which it inevitably displaces remains apparently impervious to any criticism. For example, even when a recent government report (Smith, 2004) – written by a former President of the Royal Statistical Society – recommended unambiguously

“an immediate review of the future role and positioning of Statistics and Data Handling within the overall curriculum … informed by a recognition of the need to restore more time to the mathematics curriculum for the reinforcement of core skills, such as fluency in algebra …”,

nothing happened! Officials and vested interests simply closed ranks. It is all very curious.

The fabulous statistics continued to pour out of the telescreen.
(George Orwell, 1984)

We shall see that the levels of attainment in certain “advanced” countries – both in basic technique and in the ability to use the simplest mathematics – are currently so low as to make any concern about “statistical ignorance” largely irrelevant! Hence, rather than force school mathematics into the dead end of trying to teach “statistics” before the necessary mathematical and scientific groundwork has been laid, it would constitute a distinct improvement if we could achieve a sufficient level of numerical competence to produce a healthy scepticism towards anyone who appears to promise “something-for-nothing”!

Compared with many current advocates of QL, the Cockcroft report (Cockcroft, 1982) set its initial sights at a more realistic level, by starting out from the very basic notion that:

Numerate = “able to perform basic arithmetic operations”.

However, it then introduced (para 39) two hostages-to-fortune for which we are still paying the ransom:

“the word ‘numerate’ [implies] the possession of two attributes.

The first is an ‘at-homeness’ with numbers and an ability to make use of mathematical skills which enables an individual to cope with the practical demands of his everyday life. The second is an ability to have some appreciation and understanding of information which is presented in mathematical terms, for instance in graphs, charts or tables …”

Unfortunately, Cockcroft failed to understand that in the absence of mastery, there can be no such thing as ‘at-homeness’ with numbers, and that appreciation and understanding of information presented in mathematical form presupposes an ‘at-homeness’ with, and a mastery of, the relevant mathematical language.

One consequence of Cockcroft’s “two attributes” was that for 15 years the profession in England “talked the talk” of “at-homeness with numbers”, without ever sorting out what was needed to “walk the walk”. As a result

• tables and standard written algorithms were widely deemed to be optional
• simple word problems and multi-step exercises were neglected
• decimal arithmetic was delegated to the calculator and fractions were seen asoutmoded
• euclidean geometry, proportion and algebra were viewed as beyond most pupils, and
• the freed curriculum time was absorbed by such novelties as “data-handling”.

The Cockcroft report came at a time when England was struggling with the need to provide for “the bottom half” – a group which had traditionally been given a very raw deal. The report was also overtly utilitarian, focusing on (school) mathematics from the point of view of the needs of employment and adult life generally (their brief did include the needs of higher education, but these received less attention). Its attempt to address both issues at once was fatally flawed. In particular, its recommendation that the curriculum be designed from the bottom up (based on its “Foundation list”) was similar to “naïve ML”, in that it suggested that school mathematics be conceived in terms of a single curriculum “ladder” up which all students climb – at different speeds and to different heights – with pragmatic “numeracy-for-all” first, followed by “mathematics-for-those-who-insist” later. The following examples indicate that at the level of basic technique, this well-intentioned, but thoroughly misguided, utilitarian philosophy made it difficult for teachers and examiners to sustain the expectation that large numbers of pupils should be expected to master techniques which in other countries are routine. Later we shall present evidence that the ability to use simple mathematics has also suffered. This should serve as a lasting warning to us all.

TIMSS-R (1999) (Age 13-14)	International Average	England
7003 – 4028 A. 2035   B. 2975  C. 3005  D. 3925	74%	51%
4.722 – 1.935 = ?? A. 2.787   B. 2797   C. 2.887   D. 2.897	77%	58%
A. 0.515   B. 5.15   C. 51.5   D. 515   E. 5150	39%	16%
6/55 ÷ 3/25 = ??	23%	4%
Find the value of y if 12y – 10 = 6y + 32	44%	26%

Concern about the quality of those graduating from high school is not new. But the level of concern in recent years has been unprecedented. In 1995, after 15 years in which “reformers” had introduced many untested changes (for example, (i) that preoccupation with “mastery” of traditional content should give way to an emphasis on “understanding”; (ii) that there should be a shift in emphasis from “product” to “process”; (iii) that the availability of calculators replaced the need for everyone to “learn their tables”; (iv) that decimal calculator outputs could now replace the need to master the arithmetic of fractions; (v) that euclidean geometry should finally laid to rest with the ancient Greeks; (vi) that serious algebra should be re-classified as something which consenting adult enthusiasts might be allowed to engage in in private; and (vii) that mathematics classrooms should be guided by “discovery”, by children’s own “reasons”, and by “investigation”), mathematicians in the UK finally lost patience – declaring (LMS, 1995) that:

“Mathematics, science and engineering departments appear unanimous in their perception of a qualitative change in the mathematical preparedness of incoming students. Their criticisms … concentrate on three main areas.

4A Students enrolling on courses making heavy mathematical demands are hampered by a serious lack of essential technical facility – [lacking] fluency and reliability in numerical and algebraic manipulation.

4B There is a marked decline in students’ analytical powers when faced with simple two-step or multi-step problems.

4C Most students entering higher education no longer understand that mathematics is a precise discipline in which exact, reliable calculation, logical exposition and proof play essential roles”

Government at first tried to brush the evidence aside. But three weeks later the TIMSS results were published, and people were forced to sit up and take notice. The resulting Numeracy Strategy was an attempt to “plug the leak” at primary level. But the central problem has never been faced. Mathematics and other “hard” subjects (including physical sciences and languages) are in deep trouble. The pool of students is simply disappearing.

The number of undergraduates in England who major in mathematics, and the number who actually graduate, have declined only slightly; but this is because funding to universities is based on recruitment and retention: universities are faced with a stark choice: cheat (by passing those who should fail), or die. And many university mathematics (and science) courses and departments have in fact been closed.

The situation in many other countries is worse. In most western countries numbers staying on at school and proceeding to higher education have increased markedly; yet the quantity and quality of those specializing in mathematics at university have slumped. The reasons may be partly social; but mathematics education has not done enough to turn the tide.

The situation is serious. And when things get sufficiently bad, it becomes more tempting than ever to believe in the latest “magic fix”, and to throw out what is left of the mathematical “baby” along with the bathwater. Yet even those who are committed to the idea that high school graduates should be able to use elementary mathematics and make sense of simple quantitative information may find that the thousands of pages devoted to reports on “mathematical literacy” and “quantitative literacy” make depressing reading. For most contributors write as though they do not know

• that mathematics and mathematics teaching are simply hard
• that there is no “cheap alternative” to facing the fact that abstraction is a crucial part of elementary mathematics – almost from the outset
• that countries like England have already tried versions of what is now being proposed elsewhere, and have paid the price
• that current abysmal levels of achievement indicate the need for hard work andincremental improvement, rather than the launch of yet another bandwagon.

My own attempts to pin down “mathematical literacy” have called to mind nothing so much as Lewis Carroll’s classic parable The hunting of the Snark. Mathematical literacy, like most other brands of educational “Snake-Oil”, would seem to be a fiction, or “Snark”. And those who sacrifice school mathematics to such a fiction may eventually be obliged to admit that the “Snark is a Boojum”! So I hope I will be excused for sharing extracts from

The hunting of the Snark
An agony in Eight Fits
by Lewis Carroll

“Just the place for a Snark!” the Bellman cried,
As he landed his crew with care;
Supporting each man on the top of the tide
By a finger entwined in his hair.

“Just the place for a Snark! I have said it twice:
That alone should encourage the crew.
Just the place for a Snark! I have said it thrice:
What I tell you three times is true.”

He had bought a large map representing the sea,
Without the least vestige of land:
And the crew were much pleased when they found it to be
A map they could all understand.

“What’s the good of Mercator’s North Poles and Equators,
Tropics, Zones and Meridian Lines?”
So the Bellman would cry: and the crew would reply
“They are merely conventional signs!”

“Other maps are such shapes, with their islands and capes!
But we’ve got our brave captain to thank”
(So the crew would protest) “that he’s bought us the best —–
A perfect and absolute blank!”

This was charming no doubt: but they shortly found out
That the Captain they trusted so well
Had only one notion for crossing the ocean,
And that was to tingle his bell.

But the central moral is that, once one sets out in pursuit an ill-defined quarry, the “Snark” that one is busy hunting may turn out to be a Boojum – and with horrible consequences!

“It’s a Snark!” was the sound that first came to their ears,
And seemed almost too good to be true.
Then followed a torrent of laughter and cheers:
Then the ominous words “It’s a Boo—”

Then silence. Some fancied they heard in the air
A weary and wandering sigh
That sounded like “—jum!” but the others declare
It was only a breeze that went by.

In the midst of the word he was trying to say
In the midst of his laughter and glee,
He had softly and suddenly vanished away —-
For the Snark was a Boojum, you see.

In mathematics education as in life, mistakes are unavoidable. But it is time we learned openly from these mistakes. So let us hope that in future, when a “reform” promises much and delivers little, it will no longer be allowed to “softly and suddenly vanish away”, but will be openly analyzed in a way that might contribute to a cumulative professional wisdom.

Nothing is nobler than this endless struggle between the truth of today
and the truth of yesterday.                                            (George Sarton)

Failure is instructive. The person who really thinks learns quite as much
  from his failures as from his successes.                             (John Dewey)

In the space that remains I shall move tentatively toward what I hope may be a more useful interpretation of “mathematical literacy”. But first we have to answer the question as to whether “numeracy” and “mathematical literacy” are essentially adult competencies (like “maturity”), or whether “numeracy” and “mathematical literacy” are things one can genuinely teach and assess at school level.

“Maturity” is highly desirable; but no-one would claim that it should be “taught” or assessed at school level: it is rather an elusive by-product of an extended nurturing process, which includes detailed attention to a variety of specific disciplines and activities. In the same way it would seem that that numeracy and mathematical literacy are desirable by-products of school mathematics. If this is correct, then it may make more sense to interpret “numeracy”, or “quantitative literacy”, as a basic

willingness to engage effectively with quantitative information in simple settings,

and to preserve the term “mathematical literacy” to denote

a more subtle, long-term aspiration (involving simple insights
into the nature of elementary mathematics), which cannot be
centrally assessed, and which is likely to be trivialized if one tries.

Before we proceed on this basis we should admit that this modest interpretation contrasts starkly with “mathematics literacy” as defined by PISA (Program for International Student Assessment, OECD), which rolls everything into a single “capacity”:

“Mathematics literacy is an individual’s capacity
– to identify and understand the role that mathematics plays in the world,
– to make well-founded mathematical judgements and
– to engage in mathematics,
in ways that meet the needs of that individual’s current and future life
as a constructive, concerned and reflective citizen.”

This elusive notion PISA then claims to “measure”!

The PISA “definition” has a certain appeal. But the examples below show that it is far too pretentious to lead to reliable assessment even in a single classroom, let alone nationally or internationally. If one tries, the problems used are bound to be culturally biased; the marking schemes are likely to be artificial, and their implementation less than robust! This may help to explain why a country like England, where basic technique is so weak, showed up so well on (the admittedly small scale) PISA 2000. Sadly, the whole PISA process (problems, methodology, implementation, interpretation, etc.) remains largely “hidden”; for example, in the UK the details are buried in the Office for National Statistics, which is not in the habit of engaging in open academic debate! The neutral observers I know who have tried to make an assessment of PISA have all come to the uncomfortable conclusion that there is something seriously amiss at almost all levels of the PISA program.

The following problems illustrate students’ ability to use elementary mathematics. They are much simpler than those used in PISA. Yet the weaknesses they reveal are such as to make more complex assessment tasks seem inappropriate. These simple tasks reveal basic obstructions to any attempt to “engage effectively with quantitative information in simple settings”; and their simplicity makes the observed outcomes more telling. Moreover, they illustrate what has happened in a country which embraced “numeracy” 20 years before those who are now in headlong pursuit of this educational phantasm. Their message is clear: those who prefer Salieri to Mozart risk landing up with nothing worth having.

TIMSS (1995: large sample of 13 and 14 year olds – 13 years after Cockcroft!)

P13. A person’s heart is beating 72 times a minute. At this rate, about how many times does it beat in one hour?

A. 420 000    B. 42 000    C. 4 200    D. 420

Highest scoring country    86.8%          Average    63.8%          England    44.6%

(2003: 83 1^st year honours mathematics students aged 18, at a good English university)

Q1. Two cyclists Two cyclists 42km apart, are heading towards each other. They set out at 8am. At 11am they pass each other. One travels at an average speed of 7.5 km/h. What is the average speed of the other cyclist?

Successful 67%

Q2. Tom, Dick and Harry Tom and Dick take 2 hours to complete a job; Dick and Harry take 3 hours to do the same job; Harry and Tom take 4 hours for the job. How long would all three of them take for the job – working together?

Successful 0%

In 2004 the success rate of a similar class on the last two problems was slightly higher. But in both classes 80% of the students – who are among the most successful English high school graduates in mathematics – tried to “solve” Q2 by writing “T + D = 2, etc.” and so revealed that they had never learned the most basic lesson of all in school algebra!

Mathematics instruction is never as effective as one would like. In countries with strict social expectations and carefully structured teaching, large numbers of students master basic methods rather well (e.g. the highest scoring countries achieved impressive success rates on the TIMSS problems); yet their students often declare a “distaste” for Mathematics. In countries with a more relaxed social structure, students may say they “like” mathematics, yet perform the most basic procedures in a way which obliges one to ask what such a claim can possibly mean. Nevertheless, the examples I have given are definitely trying to tell us something, and it is time the advocates of numeracy, “situated learning”, and “Realistic Mathematics Education” tuned in and started to listen. The English infrastructure may be weak, but such examples cannot be simply shrugged off. They illustrate basic failures in students’ ability to use mathematics, which do not arise because the importance of “numeracy” has been neglected: indeed for 20 years England has had little else.

Even in countries where the links between curriculum, texts and training have been handled more carefully than in England (as I believe in the Netherlands) the reports are worrying:

• “realistic mathematics education” at school does not appear to encourage students to pursue the study of Mathematics at university; and
• those (few) students who do move on to highly numerate courses at university seem to know very little, and know what they know remarkably inflexibly.

Something has clearly gone very wrong – and perhaps not only in England. But what? Any summary is bound to be inadequate, but the root of the problem in England (and maybe elsewhere) would seem to be that we have lost sight of what constitute the key ideas and methods of elementary mathematics, and have “forgotten” how much time and effort is needed to lay foundations and to develop fluency, precision and flexibility with these ideas.

In struggling to be a little more precise we enlist an unlikely helper! Antonio Gramsci (1891-1937) was a radical, but enlightened Italian communist between the wars. As a result of his activities, Gramsci spent long periods in prison, and used much of this time for study and writing. His diaries are extensive and fascinating. In his diary for 1932 we read:

“The new concept of schools is in its romantic phase,
in which the replacement of “mechanical” by “natural” methods has become unhealthily exaggerated. […]
Previously pupils at least acquired a certain baggage of concrete facts.
Now there will no longer be any baggage to put in order […]
The most paradoxical aspect of it all is that this new type of school is advocated as
being democratic, while in fact it is destined not merely to perpetuate social differences
but to crystallize them in Chinese complexities.”

In “turn of the century” England this “naturalist” fallacy took many forms, such as:

• a touching faith in “students’ own methods”, even where this hinders their progress
• a distaste for doing the groundwork of establishing a robust fluency in basic technique
• claims that it is “unnatural not to encourage the use of calculators”
• the idea that one can learn to “estimate” without linking this to exact calculation
• the confusion of “algebra” with subjective “pattern spotting”
• the view that “abstraction” is off-putting and can be avoided
• an insistence on embedding problems in “fake contexts”, despite the irrelevant “noise”and confusion which then undermines the claimed purpose of the task
• a “behaviourist” belief that mathematics can be reduced to simple “outcomes”, which can then be taught and assessed one at a time – forgetting that the hardest, and most rewarding, aspect of learning mathematics is the challenge to integrate simple stepsinto effective wholes.

The overall situation is also complicated by social changes, among which I single out

• the impact of mass education, and
• the fact that within a single generation, the hard-won humano-scientific Enlightenmentideal – that way of knowing and living which struggles to combine respect for truth with respect for humanity – has been almost swept away by a breathtakingly crass “consumer democracy”.

In place of the traditional image of life as an “honest, but rewarding, upward struggle”, we have embraced the delusion of an endless, painless downhill “free ride”.

In this Humpty Dumpty world, the consumer (the student!) is always right – no matter how ignorant s/he may be, and everyone is encouraged to have their own opinion, and to use the power of modern communications to express it – a process which converges inexorably to some lowest common denominator. Leadership, and right and wrong, have vanished; in their place we now have a potent, but ultimately vacuous, combination of political opportunism and bureaucratic control through accountability. In such a world there is no such thing as traditional education, and there can be no such thing as “mathematics”.

The mental universe of mathematics

Returning to the more restricted world of mathematics education, the biggest mistake has perhaps been that we have lost sight of the most basic fact of all – namely that

the world of mathematics is a mental universe.

So if we are to open the minds of ordinary students to the power and flexibility of genuine mathematics we must

• rediscover the fact that mathematics is in some ways inescapably abstract from the very beginning, and that effective mathematics teaching has to reflect this fact (in a sensitive way), and
• concentrate on identifying – and achieving mastery of – core techniques, which are routinely linked flexibly into multi-step wholes to solve extended exercises and problems.

Motivation using suitably chosen “real” problems, and the application of learned techniques to everyday problems are important. But to achieve the kind of mastery and “at-homeness” needed to make mathematical sense of the simplest “real” problems, students must be able to move around freely inside the mental universe which constitutes “elementary mathematics”. Thus,

if we wish students to be ‘at-home’ in this mental universe
we must ensure they come to experience it, and the associated mental operations, as “real”.

As with “literacy”, any sensible interpretation of the terms “numeracy”, “quantitative literacy”, and “mathematical literacy” must see them as desirable by-products of this process. To be sure, they are by-products which have to be planned and worked for: they are not “automatic”. But the English experience demonstrates that it is a mistake to imagine that more will be achieved if traditional school mathematics is replaced by some more palatable “real-world” alternative.

Imagination, literacy and the three Rs

Once we accept the “mental” nature of the “mathematical universe”, this has a truly liberating consequence (as long as one remains faithful to the elusive, but undeniably “objective”, character of the discipline) – namely that

elementary mathematics becomes accessible to anyone with a mind.

And once students’ imagination is released, everything becomes possible.

The Ey’s confined, the Body’s pent
In narrow Room; Lims are of small Extent,
But thoughts are always free.
And as they’re best
So can they even in the brest
Rove ore the World with Libertie;
Can enter Ages, Present be
In any Kingdam, into Bosoms see.
Thoughts, Thoughts can come to Things and view
What Bodies can’t approach unto.
They know no Bar, Denial, Limit, Wall:
But have a Libertie to look on all.             (Traherne, Thoughts, I)

1. Our concern here is to identify those key ingredients of this “mental universe of mathematics”
   which are relevant on the simplest level of numeracy, and
2. which might represent a modest outline of the kind of features which one could include in a more appropriate interpretation of mathematical literacy as a desirable adult by-product of school mathematics.

Our tentative ingredients are deliberately simple, and avoid epistemological considerations. However, for even such apparently low level objectives to be achievable, what is taught and learned must respect what Semadeni (Semadeni 2004) has called “the triple nature of mathematics”. That is, the underlying philosophy needs to go beyond the surface features of familiar notation and techniques, and to reflect the way mathematics sifts out deep ideas (which are quite different from the “Big Ideas” beloved of many educationists), and transforms them into systems within which we can think and calculate.

What constitutes numeracy?

As suggested earlier, we do not try to define “numeracy”, but simply take it to mean:

a willingness to engage effectively with quantitative information in simple settings.

We list four components which would seem to be necessary in any program to achieve such a “willingness to engage effectively with quantitative information in simple settings”. While these include competence in handling certain basic techniques, “numeracy” cannot be reduced to a detailed syllabus; thus the list included in the third of our four components is deliberately short, and the other three components are in many ways more important.

1. The (not so traditional) three Rs: The first component of our attempt to pin down what might be meant by “basic numeracy” is the deliberately provocative Trinity of Obligations:

1. to remember (i.e. to learn – if necessary by “rote”)
2. to “reckon” (i.e. to calculate accurately)
3. to reason (i.e. to think mathematically).

2. Mathematics as the science of exact calculation: The second component – which is essential if the other three components are to be effective – is the suggestion that all students should absorb and appreciate the spirit of the statement that, in contrast to the messy “real world”, mathematical calculations (whether numerical, symbolic, logical or geometrical) are “special”, in that they concern “ideal” rather than real objects, and so are exact.

3. Techniques: The third component is that students should be expected to achieve mastery of a limited core of basic techniques (some large subset of: multiplication tables, place value and decimals, measures, fractions and ratio, negative numbers, triangles and circles, Pythagoras’ theorem, basic trigonometry and similarity, coordinates, linear and quadratic equations, how to handle formulae, straight line graphs and linear functions), which techniques constitute the “background” material in terms of which the other three components can be interpreted.

4. Applications: Whatever list of content and techniques is adopted, these should be used regularly and routinely to handle problems and situations which systematically cultivate the notion that, despite its “ideal” character, one important aspect of elementary mathematics (integer and decimal arithmetic; simple and compound measures; fractions, ratio and simple proportion; the simplest geometrical representations; etc.) is that it is useful.

The real test of these four components is whether they can help us to devise strategies which achieve more than we do at present for the majority of students. But though any such strategy will be judged in terms of the outcomes for its target audience, it is worth stressing that these components are more than merely “utilitarian”, and should be seen as part of a profoundly humane education. To remind us that even the first component is included in this spirit, it may be worth sharing three quotations from the writings of George Steiner.

Remembering: “A cultivation of trained, shared remembrance sets a society in natural touch with its own past [and] safeguards the core of individuality.What is committed to memory and susceptible of recall constitutes the ballast of the self. The pressures of political exaction, the detergent tide of social conformity, cannot tear it from us.”

Reckoning: “By virtue of mathematics, the stars move out of mythology and into the astronomer’s table. And as mathematics settles into the marrow of a science, the concepts of that science, its habits of invention and understanding, become steadily less reducible to those of common language. The notion of essential literacy is still rooted in classic values, in a sense of discourse, rhetoric and poetics. But this is ignorance or sloth of imagination. … All evidence suggests that the shapes of reality are mathematical, that integral and differential calculus are the alphabet of just perception.”

Reasoning: “How to stick to principle or social aim while facing facts as they are is the peculiar problem for human intelligence in a democratic culture. … Anybody can take sides when things are labeled “revolutionary”, fascist”, “progressive”, or “democratic”. But what is it we are asked to believe, to consent to, to support ? What value is there in opinions that flow from us like the saliva in Pavlov’s dogs, at the ringing of a bell?”

What is mathematical literacy?

In seeking to make a useful distinction between the terms “numeracy”, “quantitative literacy” and “mathematical literacy”, we have proposed at one extreme (following Cockcroft) to interpret “numeracy” as meaning a basic “willingness to engage effectively with quantitative information in simple settings”. At the other extreme, echoing the spirit of the familiar notions of “maturity” and “higher literacy”, we proposed an interpretation of “mathematical literacy” in more elusive terms as: a subtle, long-term aspiration involving important insights into the nature of elementary mathematics and its utility.

(While we shall not discuss “quantitative literacy” here, it might then naturally be interpreted either as a more ambitious “college” version of “numeracy”, or as a numerically-oriented version of what we have labelled “mathematical literacy”.)

Thus we suggest that the term “mathematical literacy” should be reserved for something distinctively different from “numeracy”, namely that it should refer to a deeper adult residue of students’ experience of school mathematics. Naturally, different adults will emerge from school with such a residue in different measures, but we restrict attention here to those residual insights and competencies which one could reasonably expect – to varying degrees – from a substantial percentage (say 30%-60%) of each cohort.

Some aspects of this “residue” make sense only in the context of specific learning (with the depth of this learning varying from one adult to another). Nevertheless, insofar as it makes sense to embrace “mathematical literacy” as a desirable goal for large numbers of adults, it should include the expectation that they:

• know at first hand the most important parts of elementary mathematics
• have achieved complete fluency, automaticity and robust mastery of the basic processes of school mathematics.

However, they should not only have mastered, remembered, and be able to use some core of mathematical techniques; they should also have reflected (the fourth “R”!) to some extent on the distinctive character of mathematics. It is therefore important that they

• have had an in depth experience of at least one “rich” area in mathematics, and
• have extensive experience of grappling with, and solving, simple multi-step problems.

In parallel with this experience of specific content and activity we also need to include those residual “impressions”, or general principles, which one would like large numbers of adults to absorb (often unconsciously) from their experience of school mathematics, including:

• a clear distinction between serious mathematics and mere tests
• a recognition of, and respect for, the logical character of all mathematics
• an insight into why elementary mathematics is inevitably “abstract” (in some sense)
• a sense of mathematics as “the science of exact calculation”, and how the basic methods of exact calculation can be modified to approximate effectively
• an insistence on “meaning”, and hence a recognition of the importance of simplification
• a recognition that real mathematics begins with multi-step problems
• a recognition of the importance of connections between apparently different topics, and that much of the power of mathematics stems arises when a simple method from one domain is used in a very different context.

Conclusion

While society has changed dramatically in the course of 5000 or so years of civilization, it would appear that the human mind has evolved rather slowly. The art of introducing young minds to the delights and frustrations of elementary mathematics has also changed far less than we are often led to believe. In the modern era of mass education, a nagging awareness that “success” in mathematics remains elusive for most pupils therefore compels us to re- consider the assumptions on which current approaches to the teaching of elementary mathematics are based, and to ask whether there might not be some simple alternatives which would allow a larger number of students to taste a little more ‘success’.

At the same time, one has to remember that the most likely reason why success remains elusive is that mathematics and mathematics teaching are simply hard – in which case the probability that there is some simple, more effective alternative is likely to be rather small. Unfortunately, in any domain where success proves consistently elusive, magic ‘solutions’ and Snake-oil salesmen emerge from time to time, and may even be believed (for a while).

In this, mathematics education is no exception: the last 40 years have witnessed a succession of proposed paradigm shifts which, we were assured, would lead to marked improvements, though in almost all cases, despite interesting exploratory work at the ‘local’ level (for particular groups of pupils and teachers), the eventual outcome has proved disappointing.

The most recent “alternatives” to traditional school mathematics are reflected in the widespread current use of the terms numeracy, quantitative literacy and mathematical literacy. These expressions now pervade much of the current mathematics education debate – especially in western-style democracies, where politicians, administrators and certain educationists exploit them in their different ways to deconstruct the ‘elitist’ (i.e. hard, and hence politically inconvenient) character of traditional elementary mathematics. Some countries have already changed their curricula by replacing the traditional label “Mathematics” by “Mathematical Literacy”. Given these pressures, it is important for those in the wider mathematical community – including mathematicians and those who work in mathematics education – to examine critically the claims underlying this global trend to avoid the disappointment of yet another “false dawn”.

This paper has sought to sound a warning about the “hype” surrounding the current use of the terms “numeracy”, “quantitative literacy” and “mathematical literacy”, and to attempt an initial analysis which might help us accommodate the important notions of “numeracy” and of “mathematical literacy” within the broader goals of serious mathematics education.

References

Cockcroft. 1982. Mathematics counts. HMSO London (UK).

Crowther. 1959. 15 to 18: Report of the Central Advisory Council for Education (England). HMSO London (UK).

Semadeni, A. 2004. The triple nature of mathematics: deep ideas, surface representations, formal models. Proceedings of 10^th International Congress of Mathematics Education, Copenhagen

Smith. 2004. Making mathematics count. HMSO London (UK).

Steen, Lynn A. (ed.) 2001. Mathematics and democracy, the case for quantitative literacy. National Council for Education and the Disciplines.

Steen, Lynn A. (ed.) 2004. Achieving quantitative literacy. Mathematical Association of America. LMS. 1995. Tackling the mathematics problem. London Mathematical Society, Institute of Mathematics and its Applications, Royal Statistical Society; London (UK).