A very pleasing irony of writing this thoughtless and classless, “We’re all doomed” blog is that it has resulted in my being introduced to thoughtful and classy heroes like Tony Gardiner, mathematical stars who have been working tirelessly for decades that we not be doomed. The most recent introduction is to Edward Barbeau, a star of Canadian mathematics education. Tony Guttmann, AKA Mr. Very Big, alerted me to a long comment Ed contributed to a maths-ed discussion, on gifted education. Ed has kindly permitted me to reproduce his great comment here.
I do not see how anyone who has much dealing with children can deny that
there are essential differences in abilities that make some of them
superlative in their capacity to learn, understand and perform. However,
it is also the case that a large number of students can learn and perform
to a high level if the circumstances permit, and our schools should become
equipped to draw out this potential.
To be sure, for some students, the best approach may be to have separate
schools or at least special classes. In a very few cases it seems to me
that the gift that a child exhibits may be so entrenched in the
personality of that child that it is inconceivable for it to imagine any
other kind of life; this is what I pick up from some interviews with
leaders in certain fields such as sports, dance, arts. In this case, then
extraordinary and idiosyncratic approaches are needed.
But I do not think that this applied to all gifted students, however
interested they may be. Some may not realize that they have a gift for
mathematics until they have an exposure to the subject and find that
something clicks. Others could excel in different ways.
There are many possibilities for nurturing talented students, especially
these days when there is ready access to the net.
First of all, the ordinary syllabus should have some value added that
keeps students on board who may otherwise be bored by the standard fare
and approach. In the traditional curriculum, the Euclidean geometry served
this purpose. The deductions that students were called upon to solve
varied from the routine to quite challenging, and many deductions had
alternative arguments of varying elegance. I do not know how much mental
arithmetic occurs in the modern school (I suspect very little) but this is
another area where children can perform with more or less efficiency and
Even in basic algebra, word problems can vary from routine to challenging,
and some allow for different ways to set up and solve an equation.
Trigonometry is a dream topic for the secondary syllabus because it
touches on so many essentials of mathematical formulation and technique.
One test of a curriculum might be how often it calls upon students to make
So the first place I would look is to see how we can get teachers in front
of the children who have the mathematical experience to understand the
nature of what they are teaching, to assess where the kids are at and to
exploit the possibilities inherent in the syllabus. This is a political
question: how we allocate resources, recruit teachers who might otherwise
go into other fields, and foster and respect (especially when it comes to
salary and working conditions) the professionalism of teachers.
Another thing we need to look at is the range of extracurricular
activities in a school; a lot of the real learning (and setting students
on fire) comes from the activities they pursue after school. If the school
has a mathematics club, participates in mathematics competitions or math
fairs, this can go a long way to keeping good students interested and
progressing in mathematics. Of course, this is another thing that comes
down to the teaching corps who have to have the willingness and space to
be involved in such things. (Not if they have to find a second job to make
ends meet or not afford to live in easy commuting distance to the school.)
Then there are things that cut across schools that might be district wide.
One of the Toronto suburban boards was very good at this about forty years
ago. The school math heads met regularly during the year, they set up an
annual competition for schools in the board and had an annual weekend
“math camp” at a country property that the board opened. Politics again —
this particular board had a politically astute math coordinator that was
quite adept at squeezing the necessary fund from the trustees.
Then there is the outreach from universities. Back in the 1960s, Israel
Halperin at the University of Toronto established a correspondenc program
(the Gelfand Club) and started the Metro Math Club, a series of monthly
lectures. The University of Waterloo has a continuing programme of school
visitations, workshops for teachers and competitions. The Fields Institute
established Saturday morning sessions for secondary students.
Finally, we come to the net, where talented mathematics students all over
the world can make contact. More old-fashionedly, there are also a lot of
very good books that students can read, in particular, those by our
colleague Tony Gardiner.
My own predilection is to nurture students in their neighbourhood schools
while giving them access to resources and contacts of the outside world,
and our first job is allowing students to make these connections.
In my experience, most students who are good at mathematics are
multidimensional — they also excel in other areas and have a wide array
I find it hard to put my finger exactly on what has gone haywire in our
system of public education; there is a kind of desperation to improve it,
but somehow things do not quite work out. Twenty years ago, I would say
that the bulk of the students who rose to the top in competitions were in
the public system and I knew the teachers who made that possible. Now many
of them come from private schools.
The problem with the private or separate school option is first that it
can be ruinously expensive and secondly is can be a “caveat emptor”
situation; the parents are often not in a position to make a judgment and
you are not quite sure what you are getting. The other somewhat intangible
cost is that of taking students out of their home environment and into a
culture that may not be particularly copecetic.
The most persuasive example I have of the value of the public system is in
the field of music. The high school I went to had a sort of working
orchestra; we had a weekly assembly of the whole school which including
entry and exit numbers by the orchestra, which was also available for
school shows. My daughter went to a school that had an exceptionally fine
orchestra with a long tradition and among other things, I heard them give
a fine performance of a Shostakovitch symphony. She gained sufficient
proficiency on the cello that many years later, she decided to join her
community orchestra. Both her sons went to a middle school where virtually
every student was part of the music programme somewhere; one learned the
saxophone and other other the trumpet. The music director was
uncompromising on discipline, but his students gave a solid annual concert
and when he was recognized by the principal the cheering of the students
brought down the house — quite moving actually.
There was nothing special about this. I am sure that the kids involved
could have come from anywhere. What made the difference were the teachers
who were willing to invest their time and expertise to supporting these
kids. Again — the solution rests in politics and the insights and
priorities of the community at large.
Mathematical examples are rarer, but an outstanding situation is that of
Bruce While at Vincent Massey School in Windsor who ran several
mathematics clubs and produced a few team members for the International
Robin Pemantle pointed to the SEED program, which was a remarkable
undertaking. But it does beg the question as to whether this can be built
into public education.