A Distant Mirror
Mathematics Education in South Africa
Mark Saul
Professor Mark Saul visited South Africa from the USA to lead a mathematics
summer camp for South African teachers and learners. This is his impression of
how mathematics education has evolved against the backdrop of Apartheid and how
mathematics continues to grow under the new government.
You don’t see the problems of the country when you first
get off the plane in Cape Town. The airport could be one in southern California,
with its modern terminal, busy newsstands, and screaming advertisements. The
people could be American, except that they have continental accents and drive on
the left.
But a mile down the six-lane freeway you begin to see
Africa. Someone has thrown a network of plastic tarps over a stand of trees,
like a construction of tent caterpillars. These are the dwellings of newly
arrived squatters from the rural East Cape province. Further on you see more
permanent structures of wood or corrugated iron. Area lights poke out high above
the squalor, and outhouses line the riverbank.
Further still the huts have acquired electric lines and
lost their outhouses. Some, built by the government, are made of concrete. Under
apartheid, their occupants had no legal right to live where they could get work,
and squatters’ camps, shadow cities to each South African metropolis, sprang
up. These are the “townships”, of which Soweto is the most famous. Long
excluded from the country’s economy, their populations are restive and
anxious. The new government has the task of making these settlements into
livable neighborhoods.
The progression continues. Some houses now have two
stories, and cars are parked outside. Bars cover the windows: there are things
inside that others covet. Next come the older, more stable, “coloured”
neighborhoods. This label was given by the architects of apartheid to the large
Cape Town community of mixed race, people who speak English or Afrikaans and who
lost much as the structures of separatism were raised. Before 1948,
they had constituted a solid lower middle class.
After this drive through history I reached my destination,
the suburban home of my host, John Webb. John’s family has lived in South
Africa for the last century. He got his Ph.D. in mathematics at Cambridge and
now teaches at the University of Cape Town. John helps to run a system of
national competitions in mathematics, leading to the selection of a team to
represent the country in the International Mathematical Olympiad. This system
has been quite successful: the Olympiad team has scored higher and higher over
the years, and participation in preliminary talent searches has increased
steadily. It is December, summer vacation in the southern hemisphere, and I have
come to help out in a week-long summer camp to develop the students’
competition experience and also to work with teachers from all over this diverse
country. (My work was being supported by the Anglo-American Chairman’s Fund
and the D. G. Murray Trust.)
John’s suburban neighborhood resembles San Diego more
than it does either Africa or Europe. But a few miles away, the shantytowns are
growing, as people arrive on buses to join the cash economy of the city. It is
as if the lava from an erupting volcano had hardened into a crust just before it
engulfed the neighborhood. But someplace far off, the volcano is still
smoldering. How do you cap this eruption?
Part of the answer walked into John’s living room the
next day. Theo Mokgatlhe is a fifteen-year-old high school student from the
Orange Free State. He had worked his way up in the competitions and been invited
to the summer camp.
“First I lived in a township near Johannesburg,” Theo
explained, “where I started school. Then, when apartheid ended, my mom moved
us to Welkom,a small city further to the south. I now go to a private
school. We speak Sotho at home.” Theo speaks and writes a refined continental
English. Because he grew up speaking an African language, he would have been
classified under apartheid as “black”, not “coloured”, and would have
had available to him only the most meager of the country’s educational
resources.
“The trip here took sixteen hours. But it was fun. There
were other kids on the bus. One of them told me that the Bronx is the toughest
neighborhood in America. You grew up there? I would never have guessed.”
The full fury of apartheid was directed at children like
Theo. Denied an education, they could not form a professional class, could not
threaten the hegemony of the South Africans of European ancestry, could not
upset the state of affairs these South Africans had so carefully constructed to
insulate themselves against the rest of the continent they lived on. In the new
South Africa, Theo will prosper. But will the prosperity he will enjoy spread
far enough and quickly enough to forestall the social unrest that the
country’s dreadful inequities foretell?
If John Webb has his way, it will. Like many farsighted
South Africans, he early saw the untenable nature of apartheid. He waited
patiently through the repressive 1970s and 1980s and worked quietly to undermine
the system that was taking its toll on the country. People with views such as
his forced the integration of local public schools
and of the dormitories at the University of Cape Town. Only after the
crumbling of apartheid, however, were these efforts appreciated by the outside
world. For apartheid reserved one of its subtlest torments for its opponents of
European descent. When they traveled abroad, they were often put in the position
of representing to foreigners the very system they despised and worked to
undermine.
Cape Town enjoys a stunning natural environment. The
climate is Mediterranean, and a drive about the city reveals majestic vistas of
mountain and sea. But I didn’t have long to enjoy the sights, as the camp I
had come to work at was held outside of the city, in Stellenbosch. This is a
university town across the Cape Flats (a wide plain) from Cape Town, in the
center of a wine-growing region. The town is all white stucco, and the language is mostly Afrikaans. It was from this university
that many of the leaders of apartheid graduated. Now it strives to serve a more
diverse population.
My work with the students was not very different from the
work I’ve done countless times in the U.S. These students were hungry for
mathematics and good at it. I gave a talk each afternoon, and they couldn’t
get enough. Indeed, they devoured one of the talks so quickly that I had to
supplement it with the next day’s material, then was forced to invent yet
another lecture. The work was enjoyable, and the only exotic touch was the
students’ accent, the way they said “anticlockwise” instead of
“counterclockwise” (to describe the way the water drained in the sink), and
the cricket bat they gave me as a parting gift.
The rest of the day I worked with black teachers from towns
throughout the country. My work with these teachers also had a familiar quality.
These were highly motivated volunteers who had come during
the first week of their vacation to learn some mathematics.
I was a little concerned that I would pitch my material too
high or too low. Also, I would be working in English, a second or third language
for most of the teachers. But I quickly found that the level appropriate for
these teachers was more or less the same as the level for many American teachers
I had worked with. And in fact the same classroom techniques also worked. I went
through several modules using games to explore mathematics, and they took to the
work quickly.
With one important exception. It seems that they hadn’t
had much of an idea before this workshop that mathematics was something to
“do”. Rather, they seemed to look on mathematics as a body of knowledge one
rehearsed and performed. They all had good presentation skills. (I was told that
their typical class had forty or more students.)
And they could think: with varying degrees of success, they
could solve the problems I set them. But somehow the notion that the solving of
problems actually was the mathematics did not quite click. Why not?
I met with Wilmot James, vice chancellor of Cape Town
University. A tall, soft-spoken man, he described a bit of the experience of his
family, classified “coloured” under apartheid. “Things have changed since
the rationalization of the educational system. Despite other professional
aspirations, my father was sent to a vocational school, studied metalwork, then
lectured on this in a college. Education for us usually meant a vocational
school. Nonetheless, we all managed to get an education, even under
apartheid.…We are working towards equity, but more than that, towards
instilling values in the next generation.” I was suddenly reminded not just of
the extraordinary violence used to enforce the inequities of the old regime, but
of the extraordinary courage, restraint, and moral commitment of those who
eventually triumphed over that regime. Later, a moving visit to Robben Island,
the notorious prison for foes of apartheid, put into perspective the remarkable
history of this country.
Back in Stellenbosch my hosts filled in for me the picture
of “Bantu” education under apartheid. More than 90 percent of the state’s
education budget went to “white” education. With three other groups (“coloured”,
“Asian”, and “black”) to share the remaining 10 percent and with
“black” students constituting the majority of the student population, there
simply wasn’t much to go around.
In the view of some there is evidence that apartheid could
not continue, even for the white population whom it ostensibly benefited. Years
after the Bantu Education Act but before the breakdown of the whole system, it
became clear that South Africa was suffering from a lack of skilled workers, of
programmers and other professionals. And this was not just the result of a brain
drain. It was also the result of the starvation of the minds of five-sixths of
the population. This shortage was beginning to take its economic toll. Too, the
exclusion of so much of the population from the economy deprived the country of
an internal market and made the international boycott much more than an
inconvenience. So quite apart from the political and moral failure of apartheid,
the system contained the seeds of its own destruction. The failure of the
educational system was part of the failure of the larger system that spawned it.
John Webb spoke more about the budget. “We talk a lot
about introducing computers into the schools. But for schools serving the black
population, you must first talk about introducing electricity.
The attitude of the apartheid regime towards ‘Bantu’
education was that these people were going to be drawers of water and hewers of
wood [an allusion to a famous use by H. F. Verwoerd of a biblical phrase] and
needed little mathematics.”
So they got little. What they got, it seems—I was dealing
with the successes of this system—was a mastery of mechanical methods, the
“syntax” of mathematics, without any insight into what the syntax could
express.
For example, I gave the teachers in my class a problem:
Find the smallest positive number that leaves a remainder of 3 when divided by 7
and also by 9. (No one gave the answer 3, so this version of the problem was
safe.)
One of the students, let’s call her Anita, got the
answer, so I asked her to explain to the others how she did the problem. Poised
and articulate, she came up to the blackboard and said, “I followed the method
of Pólya. I read the problem. Then I looked for a way to represent the number.
I could use and also
.” Anita didn’t notice that she needed a second variable.
But no matter. In the midst of her formal explanation,
Anita suddenly changed course and gave a good description of how she
“really” solved the problem, unrelated to her earlier speech about Pólya
and problem solving. She explained that if she subtracted 3 from the number she
wanted, she would get a number divisible by 7 and also by 9. The smallest such
is 63, so the number we want is 66.
Anita and others clearly understood the problem. They were
able to give me the next such number and a bit later to write a general
expression for all such numbers. This example was typical. The teachers had
clearly learned from Pólya, but had mistaken a formal description of the
thought process for the thought process itself.
I ran into this difficulty again and again. One day I gave
the teachers a problem involving the game Tic-Tac-Toe. It turned out that they
weren’t familiar with the usual version of the game, so I had to describe it
to them. In the midst of my description they all began to mumble something,
which turned out to be the word mrabaraba, the name of a game that they all knew
and of which my description had reminded them. I quickly withdrew my own
explanation and asked them to tell me about their game.
Mrabaraba is a version of the game known in England as
“Nine Man Morris”, which has been analyzed mathematically in some detail
[1]. We played it, then posed several mathematical problems based on the game.
On encountering one of these problems, the teachers would often first stare at
it, trying to fit it into some category of familiar knowledge. I had to give
them some sort of tacit or verbal permission to begin to experiment. When they
did start to think, they quickly reached important insights and often solved the
problem.
Nic Heideman earned his Ph.D. in mathematics at Washington
University in Saint Louis. He teaches at Rhodes University, in the East Cape
province, and, supported by Old Mutual (an insurance company), he works with the
teachers in the Queenstown region, a vast rural area with three hundred high
school mathematics teachers who have virtually no other professional support.
He came to southern Africa from Holland when he was eleven
years old, and he speaks English, Afrikaans, and some Xhosa.
One day we were looking at a map, and Nic described to me
where the teachers live. One lives near the border with Lesotho, an independent
country surrounded by South Africa. It had been a British protectorate, but
never got absorbed into the old Union of South Africa, and so its population
never suffered the experiences of apartheid. But they did, and do, suffer
grinding poverty and other effects of underdevelopment. It is a poor and
mountainous country, inhabited mostly by Sotho-speaking people.
I mentioned to Nic that one of the teachers, Posetso
Sekotlo, was a Sotho speaker and told me that he was sent to Lesotho to finish
his schooling when the tightening noose of apartheid denied him a good education
in South Africa.
Nic’s comment was, “It shows, doesn’t it?” And in
fact it did. Mr. Sekotlo had no trouble digging into the problems, thought well,
and did not have to wait for the extra “boost” that many others needed. He
and a few other teachers formed an “advanced” group, to whom I constantly
had to feed extra problems.
After class one day Mr. Sekotlo approached me and said,
“I learned that if you want to multiply some number, say 25476, by 11, all you
have to do is write down 6, add 6+7, write down 3, carry 1…”. He described a
computational trick for multiplying a number by 11. Essentially, if the number
was N, he was adding the numeral for N to the numeral for 10N in a nonstandard
way.
Then Mr. Sekotlo asked me, “Why does it work?” I showed
him why it worked, and he understood immediately. But in this case the question
was more significant than the answer. It is the essence of mathematics to try to
understand why a mechanical technique works and not just to practice it. Mr.
Sekotlo knew to ask why. He understood this in a way that the others didn’t.
He had not been educated under apartheid.
This is, I think, an important lesson. The curriculum of
skill, without concepts, is a curriculum of oppression, a dead end. It is a
lesson we in the United States might well take into our own classrooms.
After school one day I was invited to attend a braaivleis,
a South African barbecue. While we chatted, Nic Heidemann turned to one of the
teachers. “Veronica, do you still remember how to carry objects on your
head?”
Veronica Xhantini replied, “Of course” and proceeded to
demonstrate. We helped her pick up a pail of water so heavy that I had trouble
maneuvering it with two hands and placed it on her head. She proceeded to walk
around the campgrounds, balancing the pail without using her hands at all.
“Amazing!” I cried. “How did you learn that?”
Ms. Xhantini answered, “It’s like learning to ride a
bicycle. We learn when we are young, from our mothers. And we don’t forget.
Your women cannot do this, can they? When I was younger, we had to carry water
like this for ten kilometers. But now I have a pipe and a faucet in my front
yard.”
And perhaps ten kilometers from her front yard, just over
the horizon, people are living a lifestyle similar to that of suburban America.
South Africa is a land of contrasts. But America too has
its diversity, and other countries can help us to explore how to work within
this diversity and even to make it work for us.
While we don’t have people totally excluded from the
economy, we have many with whom its bounty is not shared. While we don’t have
schools without electricity, we have many without computers. Perhaps most
important, we have students who are learning the dead-end mathematics of skill
without concept. We have much to share with our neighbors in South Africa, and
perhaps each of us has much to learn from the other.
Acknowledgment
The author is grateful for the help of Magen Govender of
the South African Consulate in New York.
This article courtesy of the Notices of the American
Mathematical society (AMS). AMS
Mark Saul is an associate editor of the Notices and teaches
in the Bronxville School District. His e-mail address is MSaul@compuserve.com
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