A quick search of the World Wide Web reveals more than 500 entries on this
emerging academic field. Among these, the International Study Group on
Ethnomathematics (http:// www.cohums.ohio-state.edu/comp/isgem.htm)
promotes the recognition of the mathematical achievements of non-European
peoples, claiming that their contributions are generally ignored. It points out,
for example, that some central ideas of differential calculus, such as infinite
series, were known to the 15th-century Indian mathematician Madhava of Kerala
long before Newton and Leibnitz. Indeed, Ubiratan D’Ambrosio, the ISGEm
president, coined the term ethnomathematics to describe the maths practices of
various cultural groups.

So ethnomathematicians advocate the introduction of “culturally responsive”
teaching methods that challenge what they call Eurocentrism in maths education.
By teaching students in a culturally and historically meaningful way—such
as the study of spatial designs and practical construction techniques used in
indigenous societies—and placing less emphasis on getting the right
answer, they claim that almost anyone could be good at maths.

Not so, reply the members of Mathematically Correct, an organisation of
parents and citizens concerned about maths education in San Diego, California
(http://ourworld.compuserve.com:80/homepages/mathman). Worried at what
they see as a by-product of ethnomaths, that mathematical rigour and correctness
are relative concepts, this group has mounted a counterattack on the “soft” New
Age approach that has gained momentum in some American states. Its members
prefer traditional teaching methods to the relaxed maths-must-be-fun
movement.

Advocates of ethnomaths may have a point when they denounce the cultural bias
in the way the history of maths is taught in Western schools and universities.
But history depends on who writes it, and they are certainly free to write their
own. Anyone may claim to possess the “historical truth”—the hard part is
to make others believe it.

The essence of maths is its search for universal truths regarding quantity,
shapes and patterns in the world around us. Although opinions differ as to
whether such truths are discovered or created by the mind, there is no doubt
that they lie at the heart of modern science and technology. They have been
laboriously acquired over the ages thanks to the efforts and ingenuity of men
and women from virtually every culture and civilisation. It’s best to leave who
did what and when to historians to determine.

Rather than engage in divisive and sterile debates (“this is our theorem, not
yours”), why not take a constructive approach and share the good things that
maths has to offer all of us? Mathematical knowledge is available to everyone.
There is no copyright on Pythagoras’s theorem, Fourier series, the mathematics
of chaos or quantum mechanics. Doing and applying maths is a supranational
activity that requires only a tool every individual in every culture comes
equipped with—a human brain.

In a recent interview, AndrĂŠ Joyal of the University of Quebec
explained the value of teaching students how to carry out mathematical proofs
(Le Devoir, 17 November 1997). “Demonstrating a truth by a logical
argument rather than imposing it is at the heart of democracy . . . Whoever
claims that a certain proposition is true has an obligation to prove it, if he
or she expects others to be convinced. Mathematics is democratic in the sense
that it convinces without resorting to force or to an authority argument.”

For the 19th-century German mathematician Karl Jacobi, the sole design of
mathematics is “the honour of the human mind”. One may disagree with Jacobi’s
definition, but could hardly deny the fact that it applies equally well to
people in Tonga, Dublin, or Kuala Lumpur. Is mathematics culturally biased?
Strongly so—in favour of human culture.

In The Pleasures of Counting (Cambridge University Press,
ÂŁ17.95/$34.95, ISBN 0 521 56823 4), Korner takes us on a guided
tour of the magnificent palace of mathematics, aided by frequent quotations from
many of those who contributed to its splendours. The book is intended as “maths
for mathematicians”—from beginners to professionals—and for those
who “value mathematics without fearing it”. This is real mathematics, not the
watered down versions served in “maths for liberal arts” courses.

The section titles are as intriguing as the choice of topics is original:
“Why are we not all called Smith?” (populations and probability). I especially
enjoyed the chapters on the role of operations research and other mathematical
methods in helping to win two world wars.

The text is sprinkled with thought-provoking “exercises”, many of which might
be beyond the educated general reader, but will surely be appreciated by those
more mathematically equipped.

Not just maths lovers, but a much larger audience, should enjoy Korner’s
witty prose and enlightening comments, numerous historical passages and
anecdotes. I would particularly recommend the book to all maths teachers. They
should end up better prepared to face the dreaded question—what is the use
of mathematics?

A legal battle is on the cards. An anthropology professor is considering
filing a lawsuit to reclaim the bones on the grounds that the skeleton, which
has Caucasian features, is not that of a Native American.

Not long ago, Haim Watzman reported a much more violent dispute over who can
do what with the contents of ancient tombs, as ultra-orthodox Jews clashed with
Israel’s archaeologists and historians (Forum, 1 June, p 49). Unfortunately for
the frustrated archaeologists on both continents, science no longer enjoys the
public respect it commanded in happier days. A recent survey in France (
L’Express, 13 June) revealed that 71 per cent of those polled fear the
consequences of unchecked scientific research (particularly in genetics and
nuclear energy), while 73 per cent think that the work of scientists should be
further controlled. They may have good reasons to worry. The book Why Things
Bite Back: Technology and the Revenge of Unintended Consequences by
historian Edward Tenner (Fourth Estate, 1996) is a catalogue of scientific and
technological plans gone thoroughly awry. For instance, when pesticides were
sprayed in the southern US to eradicate the South American fire-ant in the 1950s
and 1960s, the poisons ended up wiping out their predators, actually increasing
the feared ant population.

One could argue that the knowledge generated by “soft” sciences such as
anthropology and history hardly leads to ecological disasters. But consider
this. The geographer Dominique Franche claims in a recent article in Le
Monde (12 November) that the tragic conflict between Hutus and Tutsis in
Rwanda might have been exacerbated by sloppy scientific work. Franche explains
that the two peoples have been living together for centuries, belong to the same
ethnic group and share a common language and religion. Their alleged
morphological differences (in height, facial traits, and so on) have been
exaggerated by European anthropologists who compared Hutu peasants with Tutsi
royalty, rather than with ordinary Tutsis. And the domination of the Hutu
majority by the Tutsi minority might have been amplified—if not
fabricated—by historians early in the century, and afterwards easily
imposed in a country which had no written history.

Some have even questioned whether the pursuit of science is not simply the
creation of a new religion. The French mathematician RenĂŠ Thom remarked
in La Magie Contemporaine that though scientists wanted to rid
themselves of the ancients’ magical and metaphysical entities, they have
introduced myriad objects and subatomic particles of all kinds which cannot be
observed by ordinary people and are increasingly difficult to imagine. Science
is supposed to be about knowledge, not belief, but the border between the two is
a fuzzy one. No one can check everything out, not even scientists, and no
knowledge is possible without a certain number of well-established beliefs. The
Earth revolves around the Sun, water is made up of two atoms of hydrogen for one
atom of oxygen. Is it knowledge or belief? Both, some say. But the French
philosopher AndrĂŠ Comte-Sponville reckons it is the question of how
well-established the belief is that distinguishes science from religion.

Some may see the conflicts over ancient remains not as a clash between the
advancement of knowledge and spiritual beliefs, but as driven by the pursuit of
fame and research grants by the scientists. But spiritual matters too are
conducted by humans; and besides dealing in the divine, religion may also be
about Earthly power. Seen in this less-than-noble light, the digging quarrels
merely oppose two groups of people, each defending its own interests.

Watzman’s article on the Israeli tombs suggests that the curiosity of
scientists does not justify the violation of religious sensibilities. In that
case, archaeologists are seeking a blanket licence to dig thousands of tombs,
and as Watzman says, they are “Zionists” and have a political agenda alongside
the scientific one.

But the American dispute is only about one particular piece of evidence, a
single message from humanity’s distant past. One of the oldest and most complete
skeletons ever found in the Pacific Northwest is no ordinary corpse. I wonder if
in this case, a better solution can be found in place of the court battles and
confrontation seen in Israel. It is surely in the interest of all humanity to be
given a chance to gain a better understanding of itself and its evolution. Why
can’t science and religion complete, rather than oppose, each other?

This is just one example of the current commitment to the preservation of
biodiversity—partly for its own sake, but also because saving other
species from extinction might hold the key to our own survival. In his recent
book The Value of Life: Biological Diversity and Human Society,
American biologist Stephen Kellert suggests that our dependence on such
diversity is not just physical but also emotional and intellectual.

But while most people agree that the diversity of life forms is a good
thing—with the exception, perhaps, of deadly viruses—other sorts of
multiplicity are much less popular.

Take, for example, followers of political or scientific doctrines. They tend
to be convinced that they have found Truth and have no use for different views.
And in a commercial arena, monopoly and protectionism are artificial ways of
restraining the choice of products and services. Religious diversity does not
enjoy widespread support either, even if the coexistence of a variety of beliefs
may be the prerequisite for a peaceful world.

On a more mundane front, culinary diversity is seriously threatened by
international fast-food chains. From New York to Lima to Vladivostok, millions
of customers are proudly offered identical ingredients, cooked in the same way
to guarantee exactly the same taste.

And yet, evidence from several fields indicates that there can be a net
advantage to diversification, beginning with the proverbial warning against
carrying all our eggs in one basket. Wide diversity has been observed to develop
in the DNA of lymphocyte cells, which need to evolve rapidly to improve the
body’s immune response. And palaeontologists have found in fossil records a
correlation between rapid evolution and the rapid formation of species
(speciation) recently isolated from each other by genetic barriers.

The American physicist Freeman Dyson has proposed a similar explanation to
account for the diversity of human languages. He argues that linguistic variety
was not simply a historical accident but nature’s way of making us evolve
rapidly. “Just as speciation gave life freedom to experiment with a diversity of
form and function, the differentiation of languages gave humanity freedom to
experiment with diversity of social and cultural traditions,” writes Dyson in
his autobiography Disturbing the Universe. Would life be simpler if we
all spoke the same language? Possibly, but it would certainly be richer if we
all spoke each other’s language.

Even those studying the past ought to welcome, not discourage, a diversity of
interpretations. The pursuit of a single and definite “historical truth” may not
be such a realistic—or desirable—exercise after all. Finding truth
is one thing, the hard part is making others believe it. So why not let people
be happy with their favourite version of events? When Mikhail Gorbachev, the
former Soviet leader, was asked if he expected history’s verdict to be kinder on
him than current public opinion in Russia, he replied: “History is a capricious
thing. It depends on who writes it.”

To accelerate scientific progress, the philosopher of science Paul Feyerabend
recommended a proliferation of theories—even faulty ones, because these
may grow and improve. Those responsible for research policy in Canada disagree.
To promote “excellence”, they favour giving more money to fewer scientists, in
effect stifling a plurality of views. In his provocative book, Against
Method, Feyerabend says: “Unanimity of opinion may be fitting for the
victims of some myth or the followers of some tyrant. Variety of opinion is
necessary for objective knowledge.” But he might go a bit too far by advocating
that proliferation must be enforced from the outside, and “by political means”,
whenever parts of science become hardened and intolerant.

With the advent of the Internet, plurality is now disseminated at electronic
speed, threatening to become so overwhelming that “objective knowledge” (if such
a thing exists) may be drowned in an ocean of prejudice and irrelevance. Perhaps
so, but you can never have too much diversity of opinion, even if it means
sacrificing some “objectivity”. In a recent ruling against the restriction of
free speech, the American judge Stewart Dalzell likened the Internet to a
never-ending worldwide conversation that deserves protection from government
intrusion. “Just as the strength of the Internet is chaos,” he wrote, “so the
strength of our liberty depends upon the chaos and cacophony of the unfettered
speech the First Amendment protects.”

Dyson draws a parallel between biological evolution and humanity’s future. He
explains in his book that a “clone” is a single population of genetically
identical individuals, slow to adapt and slow to evolve. On the other hand, a
“clade” (from a Greek word meaning a branch of a tree) is a rapid multiplication
and diversification of small populations, the stuff of which great evolutionary
leaps forward are made.

And then he asks: “Are we to be a clade or a clone?”—a question which
hardly offers a variety of sensible answers.

The case of X, who prefers to remain anonymous, is significant. X belongs to a small group of scientists who challenge the cosmological doctrine known as the big bang. For the past ten years, he has worked on an alternative theory, but has yet to publish a single paper because he believes that his work is still incomplete. He has not applied for grants either – with nothing to show for his efforts, and working on the “wrong” theory, he would have no chance anyway.

And so, despite ten years’ work on a subject as important as the origins and evolution of the Universe, he does not get any recognition from his university administration. No more, in fact, than if he had spent his time surfing, moonlighting or doing something else irrelevant to science. However, at the time X decided to change his field of interest from pure mathematics to cosmology, he was a full professor and had tenure, so his job situation was well secured. But how many young scientists can afford to spend ten years working on a controversial theory or a difficult problem where results may be hard to come by? Not just research funds and advancement, but their very jobs would be in jeopardy if they did not pay regular tribute, in the form of publications, to the bureaucratic establishment.

X’s case might be dismissed as that of a mediocre scientist who is getting nowhere with his research. In science as in sports, honours only come after the achievement, just trying is not good enough. This is what happened to Andrew Wiles of Princeton University, who in 1994 solved the most famous problem in mathematics by proving Fermat’s Last theorem. “Wiles would have been treated very poorly by the [Canadian] system,” admits a member of a grant awards committee, “not because he was working on a problem whose probability of success would have been judged low, but because he worked alone and went seven years with no publications.” If Wiles did not bother publishing, it is probably because he was too busy trying to solve Fermat’s problem.

There is a story about a young mathematician from a developing country who was visiting a renowned European institute. During an informal gathering, one of the institute’s permanent members asked the visitor how many papers he had published, clearly expecting to embarrass him in front of his colleagues. “Two more than Riemann,” was the visitor’s quick reply. He was referring to Bernhard Riemann, the 19th-century German mathematician whose works rank among the most original contributions to modern mathematics and which Albert Einstein was later to use in his theory of relativity.

Once research has been completed it should be communicated – and the author(s) should receive due credit. Publication in a scholarly journal is still the preferred way of communicating results. Respectable journals subject articles to a rigorous review by peers: experts in the field who advise the editor on the merits of the research. Although this process in not flawless, and some journals are more prestigious than others, publication of an article generally entails recognition of its worth. And so a long list of publications can go a long way to impressing the members of funding and promotion committees, who are not always experts in the field. Hence the proliferation of third-rate papers, or repeated versions of the same one with only superficial changes.

Perhaps authors should be required to rate their own papers as a condition for publication. This could be done on a scale of 1 to 10, say, according to how far the article advances knowledge in its field (some famous papers deserving 9 or 10 could be cited by way of example). Many authors will think twice before submitting a paper deserving a meagre 2 or 3, and few would dare to overrate themselves for fear of the consequences.

The unreasonable importance ascribed to the paper output of scientists also has its brighter side. The publishing of scholarly journals is a flourishing business, judging from their number and the fact that their average price has increased twelvefold since 1965. University libraries cannot find the money to keep up with all these price hikes, let alone subscribe to the new journals cropping up almost every month.

But the days of paper publishing may be numbered. We are entering the era of electronic journals, when a trip to the library will be replaced by reaching for the keyboard. Scientific American says there were 440 electronic journals and newsletters in 1994, of which about 100 are peer-reviewed. One of the most conspicuous is the computer archive created four years ago by Paul Ginsparg at the Los Alamos National Laboratory, New Mexico. The archive receives 1000 abstracts a month in 20 disciplines and serves more than 25 000 subscribers.

Electronic publishing has a brilliant future, judging by the enthusiasm surrounding the Internet and the obvious advantages of wider dissemination and faster accessibility at low cost. To these advantages must be added the possibility of interaction (comments and corrections may be annexed to papers by interested readers), not to mention hypertext links. There is no reason why electronic papers should not become as prestigious as printed articles today, provided the standards demanded for publication remain as high. Just how these high standards can be safeguarded and who should pay or get paid for what is an ongoing debate (“Storming the barricades”, New ĐÓ°ÉÔ­´´, 17 June).

Whether they appear on paper or on screen, academic papers will probably continue to be the favoured yardstick for evaluating the average scientist, even if their number is not necessarily proportional to the author’s talent, nor their absence a proof of scientific lethargy. But the importance presently ascribed to publications may actually hinder original and serious inquiry, as academics stick to safe avenues of research or to obscure, hyperspecialised fields where publishable material is easier to find. If scientists could be freed from the publish-or-perish law, more of them might be willing to tackle truly relevant and challenging problems, resulting perhaps in fewer publications but more significant knowledge.

In Canada, thousands of academics apply each year for grants from the Natural Sciences and Engineering Research Council (NSERC). The C$400 million (about ÂŁ200 million) handed out by the agency helps to pay for laboratory equipment and materials, grants for assistants (mostly graduate students), conference travel and other research expenses.

Like many of his fellow scientists, Kok-Keong Tan, a mathematician at Dalhousie University, Halifax, was disappointed to learn that his research grant had been cut. But when his appeal was unsuccessful, he did what none of his colleagues had ever done before: he filed an application for judicial review before the Federal Court of Canada. His lawyers argue that the NSERC’s appeal procedure denies him natural justice – by permitting persons involved in the initial decision to participate in the appeal process.

Whatever its outcome, this unprecedented court action has put the cake-slicers in the spotlight. Meanwhile, the Canadian Association for Responsible Research Funding (CARRF), a newly formed coalition of some 150 scientists, is pushing for major changes in current funding policies. The NSERC’s cake-slicing method is based on the peer review principle. Committees of scientists decide on the merits of research proposals submitted by other scientists (from the same or related disciplines) and determine how much support each project deserves, if any.

Peer review has been widely touted as the fairest system, but it has lately come under attack (Forum, 11 February). In the latest twist of the Canadian debate, the Federal Court gave CARRF a mandate to address the “inherent bias and lack of objectivity in the peer review system as operated by NSERC”.

The main targets of CARRF’s criticism are the anonymity of the reports given about funding decisions and the fact that scientists serving on the awards committees are themselves often recipients of sizable grants. It might be argued that if committee members also happen to be the largest grant holders, this is because they are the best qualified in their fields.

While those serving on the committees have the power to select winners and losers in the race for grants, the real power to direct research is political and is wielded by the bureaucrats who create the programmes, allocate budgets, appoint the committees and impose the rules of the funding game. Funding agencies are supposed to provide support for the research community, but they often act as if they have a mandate to direct the way research is conducted. An example is the Collaborative Grants Program which NSERC created in Canada last year, against strong opposition from the awards committees, to fund interdisciplinary teams of researchers.

At a meeting in Western Canada between the directors of the NSERC and some 170 researchers and graduate students, a young physicist asked members of the audience to give their opinion about collaborative grants by a show of hands. The result was 4 in favour and 160 against. There is no reason to believe that the vote would be significantly different in the rest of the country. Nevertheless, the programme is being implemented and its budget increased from C$10 million to C$25 million.

CARRF also denounces the perverse effect of splitting the money among too few researchers and giving none at all to about half of them – a policy allegedly intended to promote “excellence” but in reality compelling people awarded grants to stick to “safe” avenues of research to please the judges and guarantee continuous funding.

This extreme method of selection results in a relative overfunding of mainstream research, producing mostly incremental knowledge (and lots of publications) in well-established fields. At the same time, claims CARRF, scientists with unorthodox views or truly innovative, if untested, ideas stand little chance of being funded – an allegation that is hard to prove and even harder to refute. So, a system that pretends to foster excellence may end up perpetuating mediocrity.

Beyond a certain limit, the critics in CARRF argue, more dollars poured into the same research field do not automatically translate into more quality research. As they point out, even if he had four times the money, Albert Einstein would not have produced three more theories of relativity. Supporters of current policy reply that, in general, the extra money pays for more postdoctoral fellows and graduate students thus encouraging research to the greatest extent possible.

To stimulate quality and innovation, a viable alternative would be to fund individual researchers instead of, or in addition to, funding projects. Scientific breakthroughs are often the result of unexpected discoveries, hardly the kind of stuff you can include in a research proposal. How would Michael Faraday have described his “project”: “To conduct a series of experiments with the purpose of discovering the induction of an electric current”?

CARRF proposes a basic operating grant for all active researchers and direct support for graduate students. But this idea is at odds with the trend towards funding interdisciplinary and collaborative work. In 1986, 49 per cent of the NSERC funding went to individual grants. Last year, that proportion dropped below 40 per cent and the NSERC’s authorities seem to favour its gradual reduction to zero. Do all scientists become more productive when forced to work in groups? Where is the evidence? It is ironic that the agency responsible for funding scientific and engineering research should be so unscientific in its own policies.

The NSERC authorities, who are not elected but appointed by the politicians, can impose their own agenda. They add and delete programmes, and in general make policies without the agreement of the researchers themselves, who are, at best, merely “consulted”. He who pays the fiddler … except that this fiddler is paid with the taxpayers’ money. Inevitably, there will always be conflicting views on how best to slice the research funds cake. But it seems only fair that the scientists for whom the cake was baked should have a greater say in the way it is cut especially now that its size keeps shrinking.

The problem with referendums (the Swiss have one every other week) is that
they are so clear cut. The answer is either yes or no, black or white. There
is no room for ambivalence or compromise. This provokes an artificial
polarisation of opinions. “Yes, but”, “perhaps” and “rather not” are
unacceptable answers. And yet, many people have mixed feelings and
hesitations. They are not absolutely for or against something but something in
between the two extremes.

Working as I do in fuzzy logic – a mathematical technique for dealing with
vagueness and ambiguity – it appeared to me only natural that people should
express, rather than suppress, their uncertainty. This could be achieved
through the use of a fuzzy vote.

Besides the usual question, a fuzzy ballot contains two circles labelled
Yes and No and joined by a horizontal line. The elector votes by inscribing a
vertical stroke, either on the line or inside one of the circles. Partisans of
the Yes or the No would simply mark the appropriate circle, just as in a
traditional referendum. Those preferring a more biased answer would place
their stroke on the line, more or less closer to Yes (or to No) depending on
the strength of their conviction. Besides black and white, the fuzzy ballot
offers voters the choice of every shade of grey.

Counting the fuzzy ballots would be relatively simple. A mark inside the
Yes circle would count as 1 vote for Yes and 0 vote for No. A mark on the line
would give a fraction of the vote to each one of the options. For instance, a
mark closer to Yes and three-quarters down the line would count as 0.75 vote
for Yes and 0.25 vote for No. Even those put off by figures or percentages
will have nothing to fear. All they must do is enter a mark, the position of
which will reflect their support for each alternative. A computer connected to
a scanner will perform the necessary calculations. By quantifying the “degree
of conviction” of each ballot, the will of the people should be better
expressed and democracy better served.

What could happen if the fuzzy vote was adopted? As an example, suppose
that the outcome of a conventional referendum was 56 per cent to 44 per cent
in favour of No. If one-quarter of those who voted No could have given a O.6
No/0.4 Yes answer (instead of 1 No/0 Yes), the No would still have carried it,
but only by 5O.4 per cent to 49.6 per cent – a difference of less than one
percentage point. But if, in addition, one-eighth of the Nos had voted No in a
slightly less categorical way, say O.8 No/0.2 Yes, then the outcome would have
been reversed, with the Yes winning by 51 per cent to 49 per cent.

The above assumes that all those voting Yes do so without reservation,
while only about two-thirds of the Nos are totally convinced of their choice
(the rest having split their vote). Of course, many other distributions of the
uncertainty are possible.

Fuzzy voting has its predecessor in the multiple-choice opinion poll
question of the type: “On the proposal to lower the voting age in national
referendums to 16 years, are you: (a) strongly in favour; (b) somewhat in
favour; (c) undecided; (d) somewhat against; (e) strongly against”? But this
approach makes it difficult to quantify the answers, for example, how do you
interpret “somewhat in favour”?

A variant of the above goes in the right direction: “Use a number from 0 to
5 to estimate your degree of support of … (0 = strongly against, 5
absolutely in favour)”. The fuzzy vote goes all the way and gives a continuum
of choice. At the same time, it permits a smoother evaluation of the answer by
offering a geometric (or analog) format instead of a digital one.

Surely the fuzzy vote could not be adopted without the support of the
population. And what better way to sound out the people on the issue than by
holding a referendum?

The rules of the game are simple. The players must go about their usual
business – teaching, writing, speaking and so on – until someone cries ‘politically
incorrect’. At this point, all players within hearing distance must plead
guilty, repent and promise never to do it again.

Things only become complicated when someone does not play by the rules
(I forgot to mention that everybody is supposed to play, whether they like
it or not). Take Harold Lief, professor emeritus of psychiatry at the University
of Pennsylvania. Far from admitting mea culpa, he carried on speaking very
politically incorrectly at McGill University in Montreal last November.
His opponents finally forced him to stop with their jeering and whistling.

The subject of the professor’s talk was the controversial false memory
syndrome, or FMS: recollections of childhood sexual abuse uncovered in adults
during therapy that are later found to be false. Neither Lief nor any other
academic who has studied FMS questions the reality of sexual abuse. But
because the existence of FMS would cast doubts on the beliefs of tens of
thousands of adults, women for the most part, opposition to even discussing
the idea has been heated.

Attempts by the chairman of McGill’s psychiatry department to salvage
Lief’s talk by turning it into a question-and-answer session bought some
time. But 20 minutes later, livid therapists and support-group volunteers
brought the evening to a shambling halt (a show of hands had indicated earlier
that some 180 people out of the 200 present wanted to hear Lief out). ‘The
issue is one of academic freedom,’ the dean of medicine at McGill told protesters.
‘You are in fact suppressing an idea.’

The problem with accepting the principle that some theories should not
even be discussed is: who decides which ideas to proscribe? The Church?
Popular vote? A commission of experts? Or the level of noise made by those
opposing the idea, as was the case at McGill?

Not everyone has been won over by the PC game, and some bad losers are
trying to change the rules. An example of this lack of sportsmanship is
the petition signed by a group of professors at Trent University in Peterborough,
Ontario. ‘We defend,’ their petition says, ‘the right to certain types of
speech and academic expression which, in fact, we do not condone, and in
some cases deplore. This includes the right to offend one another . . .
It also includes the right to make others uncomfortable.’ The academics
were reacting to a policy order from Ontario’s Department of Education and
Training, proclaiming ‘zero tolerance of harassment and discrimination’.

To eradicate harassment and discrimination is a noble goal which few,
if anyone, would refuse to endorse. But while any good dictionary defines
‘to harass’ as ‘to make repeated attacks on’, the bureaucrats see harassment
as anything that is known or might reasonably be known to be offensive,
hostile or inappropriate, or that might create a negative environment.

The zero-tolerance policy applies not just to the classroom, but everywhere
teachers and students meet, including off-campus bars and restaurants. It
covers visiting speakers as well as university staff. Those who consider
themselves victims of harassment or discrimination can lodge a complaint
which must be investigated by the university. In a nutshell, turning the
other cheek is out, thou shalt not tolerate is in.

Few would disagree with limiting freedom of expression in certain extreme
cases in order to protect the feelings of those who could be offended or
provoked – and to protect social peace at the same time. But once we start
down the slope of protectiveness, where do we stop?

Should we protect individuals against words, works of art or scientific
theories that might be hostile, annoying or merely displeasing to them?
Writings in favour of law and order might irritate supporters of anarchy.
Reporting an unhappy event such as an earthquake, a stock exchange crash
or even bad weather will produce some sorrow or disappointment, so newscasters
beware. And what about the cosmologists teaching that the Universe was created
by the big bang? Are they not upsetting people who believe in more divine
causes?

Society cannot protect us against hurt feelings and disagreement any
more than it can protect us against foolishness or bad taste. Attempting
to impose respect of others by restricting what people can say or write
aims at the wrong target. Just as beauty is in the eye of the beholder,
isn’t the offence in the mind of the recipient? And above all, the popularity
of the PC game notwithstanding, it is in everybody’s interest to preserve
a right we all need to exercise so often: the right to be wrong.

Arturo Sangalli is a mathematician at Champlain Regional College in
Quebec.

Besides the teachers, whose prime concern is how best to get their message
across, there were those preoccupied with what to teach – the curriculum
developers – and even some pondering why teach mathematics at all. They
came to seek or to offer answers to these questions – answers that are
usually modest, at times grandiose but seldom definitive.

The reason that so many people study maths is simply that the subject
appears in almost every curriculum, from primary school to university. Whatever
the reason for this, the challenge of maths education has become one of
mass education. A great deal of research in maths education is devoted to
trying to find out why so many people fail at maths.

There is no shortage of suspects: inadequate teaching methods, outdated
curricula, poorly trained teachers and lack of resources are the most obvious.
The pupils themselves are almost above suspicion. And so a young person’s
inability to learn differential calculus – a topic that not so long ago
was taught only to university students in science and engineering – is considered
as a kind of disease that must be treated and can be cured.

No technological advance or pedagogical trick will be spared in the
attempt to force-feed mathematical wisdom to an often uncooperative student.
The arsenal of teaching aids is impressive, at least in the affluent countries,
from books that explain the minutest details to the latest videos, computer
graphics and software for calculation and experimentation. Not to mention
the support teachers get from in-service training and student advisers.
One is left wondering, if all this does not do the trick, what will?

On the use of calculators and computers, the question is no longer if
but how the arrival of such powerful tools as graphic calculators, symbolic
manipulation packages and interactive software will affect teaching methods
and curricula. Papers carrying titles such as ‘Is algebra dead?’, ‘Should
students learn integration rules?’ or ‘Do we still need to teach fractions?’
illustrate the kind of questioning going on. But the real revolution will
come when all that computing power – and more – will be as cheap and portable
as an ordinary pocket calculator.

But even if having lots of calculating power is all right, one must
know what to use it on. The latest fashion is ‘problem-solving’, an approach
that is supposed to motivate the students while showing them how mathematics
can be applied to the real world. The idea of teaching problem-solving is
appealing, but the real problem is how many people can teach – let alone
learn – intuition, imagination and insight, to mention just three of the
tools in every problem-solver’s kit.

When it is time to measure how much the student has learned, there are
plenty of imaginative alternatives to the dreaded written examination. A
novel assessment method asks the students to summarise what they have learned
in a creative form: cartoon strip, game, story, poem and so on. An excerpt
from a young pupil’s creation: ‘She teaches us about quadratic equations.
Do they do this in other nations?’

But what happens (or does not happen) in the classroom was not the only
preoccupation. Concerns about social and political issues marked the congress
from the opening ceremonies. Miguel de Guzman, president of the International
Commission on Mathematical Instruction, made a passionate appeal for a ‘spirit
of solidarity’ and called for more personal involvement to help bridge the
gap between rich and poor countries. ‘Improving educational opportunities
by sharing resources,’ he said, ‘should be viewed as an act not of charity
but of justice.’

The question of the popularisation and the public image of mathematics
engaged both professional mathematicians and educators, for the way mathematics
is perceived influences students’ attitudes and expectations. Besides
the contests, exhibitions and television shows, maths trails are becoming
a favourite form of popularisation. A maths trail is a recreational activity
for people of all ages that combines elementary mathematics with an awareness
of the environment. The idea was created in Australia by Dudley Blane of
Monash University. The participants follow a path through the country or
city and must answer simple mathematical questions based on the features
(architectural, topographical and so on) that they come upon along the way.

David Wheeler, chairman of the Congress Programme Committee, reminded
the participants that maths education has still some way to go before being
recognised as a ‘scientific discipline’ in training colleges. No matter.
Until they meet again in Seville in 1996, the educators’ faith in the ultimate
goal of mathematical literacy for everyone will, like the Olympic flame,
keep burning on.

Arturo Sangalli is in the Department of Mathematics at Champlain Regional
College, Lennoxville, Quebec, Canada.