Mathematics in Multicultural Classrooms: Uthongathi Students'
Voices
Norma C. Presmeg
The Florida State University
It is now recognized that mathematics, the one subject in
the school curriculum which was thought by many to be culture-
free, is in fact a cultural product (Bishop, 1988). However, the
practical implications of this view for the mathematics classroom
have barely begun to be explored. In this paper a qualitative
research project in a multicultural school, Uthongathi, in
KwaZulu-Natal, South Africa, is described. The issues which are
identified transcend national borders. Even countries as cultur-
ally homogeneous as traditional Sweden now have large immigrant
populations which contribute to the multiculturalization of the
classroom. Certainly in the U.S. it is usual to have many nation-
alities, languages and cultural groups represented in the same
classroom. But the richness of this resource of cultural diversi-
ty often goes unrecognized, particularly in mathematics class-
rooms.
Culture may be defined as "a complex of shared understand-
ings which serves as a medium through which individual human
minds interact in communication with one another" (Stenhouse,
1967, p. 16). In a constructivist paradigm, in order to highlight
the contextual nature of these understandings, it is preferable
to describe them as taken-as-shared. We cannot know what under-
standings another individual has constructed, but the indexicali-
ty of these understandings (Leiter, 1980) provides a basis, in a
particular context, for the assumption that we share these con-
structions. Thus the context is an essential backdrop for
interaction and communication. This complex of taken-as-shared
understandings may be viewed on a macro scale, comprising
national cultures, or on a micro scale, yielding for instance,
the culture of a single mathematics classroom. In this paper, the
interplay of the macro- and micro-cultures is a focus.
The multicultural research project from which illustrations
are drawn, was exploratory in nature, thus a qualitative method-
ology was appropriate; in order to preserve the voices of the
students who were interviewed their actual words are quoted
whenever possible.
Uthongathi
South African schools are increasingly opening their doors
to children from all racial backgrounds. What makes Uthongathi
(Zulu for "a place of importance") special, is that this school
was the first of the fully multiracial New Era Schools Trust
(NEST) schools, which were built and funded by business concerns
which provided generous scholarships so that no student was
turned away on financial grounds. Uthongathi opened in January,
1987. The school maintains a balance in numbers between students
from various cultural groups, and between day scholars and board-
ers. The 25 students in the mathematics project, who were inter-
viewed in 1988 by the four researchers (Manjul Beharie, Yanum
Naidoo, Anita Frank and project leader Norma Presmeg), were from
Indian, Black, and White race groups, in grades 7, 8, and 9.
Thirteen of these students were boys and twelve were girls;
fourteen were boarders and eleven were day scholars. The inter-
views took place after school once a week over an eight month
period; each student was interviewed by the same interviewer each
time. One of the purposes of the interviews was to investigate
the interplay between students' cultural home background and
their learning of mathematics. Another purpose was to try to
understand what it feels like to be a student in a mathematics
classroom in a school such as Uthongathi.
The interviews were semi-structured around the following
five themes:
1. Getting to know you."
2. "Matchstick problems." These were mathematical problems of
varying levels of difficulty, all of which involved the use
of matchsticks as manipulatives so that the need for verbal
facility would be minimized.
3. "Word problems." Six problems from Presmeg's (1985) test for
mathematical visuality were given to students to solve
aloud. Students had the choice of reading the problems in
Zulu or English or both languages. Where language of
instruction is not the student's mother tongue, imagery and
diagrams may provide a bridge; these problems were designed
to investigate preferences for visual ways of solving
mathematical problems.
4. "School problems." Students solved problems from their
school mathematics textbooks, after discussion about the
mathematics they were doing in class.
5. "Mathematics in home activities." Students were questioned
about their perceptions of the uses of mathematics in their
daily lives, both in and out of school.
Language
The issue of language enters inevitably in any research on
culture. It is beyond the scope of this paper to address this issue
in any depth, but a brief comment is necessary. At Uthongathi
English is the language of instruction, which immediately places at
a disadvantage those students whose home language is not English.
Berry (1985) identified two types of mathematical learning
difficulties associated with language, and both of these types were
evident in the Uthongathi interviews. The first type is a fluency
difficulty which is relatively easy to alleviate by improving
fluency in the language of instruction. The second type results
from the `distance' between cognitive structures natural to the
student and those assumed by the teacher, and requires a
restructuring of the curriculum and methodology to build on the
learner's natural modes of cognition. This may be no easy task in
a multicultural classroom: KwaZulu-Natal is a region where at least
25 different home languages are spoken. However, once fluency is
sufficient to allow free communication, the taken-as-shared
understandings which constitute the microculture of the classroom
may facilitate the rich interweaving of elements of the
macrocultures of which the languages are an integral part. The
following protocols are illustrative.
NOMBU (grade 9, home languages Zulu/English): We
are
doing
word
probl
ems
and
I'm
not
enjoy
ing
it.
INTERVIEWER: Why not?
NOMBU: I don't understand what the sentence means.
Sometimes I mix it up or misunderstand the
sentence.
Nombu is describing type A difficulties. Xolani (grade 7, home
language Zulu), helped us to understand that type A difficulties
are inseparable from type B, which may underlie problems of
translation from one language to another. He read the first
problem in the "word problems" interview, in English and in Zulu,
for four minutes, then pointed out that the Zulu wording did not
mean exactly the same as the English. He explained what the Zulu
wording was saying:
XOLANI: John miss one day, then go, misses one, then go.
Peter misses two days, and then go, two days then
go. After four days? ... No.
The problem, in English, reads, "One day John and Peter visit a
library together. After that, John visits the library regularly
every two days, at noon. Peter visits the library every three days,
also at noon. If the library is open every day, how many days after
the first visit will it be before they are, once again, in the
library together?" Phyllis Zungu (lecturer in the Zulu Department,
University of Durban-Westville), who did the translation, confirmed
the difficulty in translating the problems, pointing out that it
was necessary sometimes to "talk around" English mathematical terms
when translating them into Zulu, either because a direct
translation was not possible or because the Zulu terms were not
well known even to Zulu speakers.
Fluency improved as students continued at Uthongathi. Thami
(grade 9) found that learning was more "enjoyable, because last
year I learned my English background" (the language), "and this
year I understand the teachers better." Tebo (grade 8) commented
that "Mr B-, especially, tends to emphasize on Black students
reading books every day so that they become fluent with the
English language, but I think this is also good."
Beyond the desirability of students becoming fluent enough
in the language of instruction not to have to translate back to
their home languages, what stood out in the Uthongathi research
was that the cultural groups formed amongst students did not
follow language or racial lines to any great extent. Students
named individuals from all cultures among their friends, but
there was a tendency for their best friend to belong to their own
race group. Bhavna (grade 8, Indian) expressed the position
articulately:
BHAVNA: When it comes to speaking and mixing we do that
very well and there is no racial barrier there,
but you would find in close friends we generally
stick to the same race group. That's not done
intentionally; in my opinion it's done because we
come from similar backgrounds and environs and you
enjoy certain things in life and you feel
comfortable.
Some students explained that rather than division into race
groups at the school, there was a division into boarders and day
scholars:
TAMMY (grade 8, White, boarder): There is quite a big
division with some of
them. I found that at
first. This was before I
had day pupil friends,
now I don't find it very
much. We're sort of like
a bees' nest. We're
inside the nest. The
boarders are the bees,
busy having their
activities, and when the
day pupils come, they're
like stray wasps, and,
you know, we get cross,
and when they come in our
dorm, it's our home and
we don't like them
imposing on us. It's like
a dog's territory; you
feel it's yours.
Although some students commented that they had been lonely
when first adjusting to the school, all expressed positive feel-
ings about attending Uthongathi. The following comments were
typical:
THAMI (grade 9, Black): It's good for us to study together,
because it helps the other race
groups to find out about other
people. Maybe we think that they
speak this other language, and that
they are different from me, so I
shouldn't bother about them; they
have different cultures. But as you
get to know them you see how they
work out things and they see how
you work out things and you can
help each other.
MARC (grade 9, Indian): I have learned how to communicate.
At first it was quite hard
interacting with other race groups;
it wasn't difficult but most people
were wary - like if you said
something you were not sure how the
other person will think or feel;
you may offend them. But after a
while we got used to it.
CRAIG (grade 8, White): It doesn't bother me at all now; I
work with others just as if they
were the same.
Other students commented on the "friendly, relaxed atmos-
phere" at Uthongathi. What all these students are describing is
the forging of a common culture in their school, which transcends
race and language differences. In fact, in the mathematics class-
room, too, the researchers concluded that there were sufficient
elements of a common culture to make a common mathematics curric-
ulum viable. In the school a subject called "Comparative Reli-
gion" is taught. One student referred to this subject as follows:
TEBOGA (grade 8, Black): Here we compare other religions
with our own. We see that some
things are similar and others
different; for example the Hindu's
Trinity is God the Creator, God the
Preserver, and God the Destroyer.
Comparative Religion provides a precedent for the develop-
ment of a classroom mathematics curriculum which takes the back-
ground cultural practices - the macrocultures - of the students
into account. Such a subject might be called "Comparative Ethno-
mathematics". No such subject was taught at Uthongathi, where the
mathematics curriculum was a traditional one such as the writer
has observed, with minor variations, also in Sweden and the
U.S.A. What follows, then, is of necessity speculative; but the
Uthongathi research leads the writer to believe that in any
classroom where several macrocultures are represented, Compara-
tive Ethnomathematics is a viable option.
Multiculture: rich resources for teachers
Bishop (1988) identified six "environmental activities"
which are present in all macrocultures, viz., counting, locating,
measuring, designing, playing and explaining. Some or all of
these are rich resources for mathematical activities in class-
rooms.
Counting is performed in all societies. However, not all of
these use our familiar base ten system. In the more than 700
language systems studied by Lancy (1983) in Papua New Guinea,
only one of the four types of counting system which he identified
uses a base of ten. The Zulu system, in common with those of
other Nguni languages, has remnants of a body-parts counting
system based on ten fingers, e.g., the Zulu word for six, isiThu-
pha, means "the right thumb". Also, some of the number words
change according to what one is counting (e.g., men, cattle,
stones, etc.). In some cattle-based African societies there is a
taboo on numbering one's cattle. It is considered unlucky to
count a man's cattle - which constitute his wealth. A man in
Botswana, when asked how he would know if his cattle were all
there, if he did not count them, replied, "If you walk into a
room filled with your relatives, would you have to count them to
know they were all there?" (Berry, 1985). Thus there is a concern
with individuals, and with small numbers. Children might not be
aware that the concern with large numbers and accuracy is a
product of industrial culture.
It is interesting for children to learn of the cultural
roots of what they often take for granted. Who invented the zero
in our Hindu-Arabic counting system? Why do we say four-teen,
but twenty-four, reversing the order? After all, in Dutch it is
viertien and vier-en-twintig; the order stays the same. Depending
which languages and macrocultures are represented in a particular
classroom, teachers have rich resources of cultural backgrounds
to draw upon. The teacher does not have to be expert in all of
these; children will take pride in sharing what they can find out
about their home culture, whether in relation to numeration,
measuring, or any other mathematical aspect.
In his research with the Oksapmin in Papua New Guinea, who
use 27 numerals in a body-parts counting system, Saxe (1991) has
illustrated clearly the interplay between home and school mathe-
matical cultures. Oksapmin children in a Western-style school
were using their body-parts system, unbeknown to the teacher, in
solving school mathematics problems. In this interplay of cul-
tures, a new culture was being constituted in the classroom.
Saxe (ibid.) also described the form-function shifts which took
place in the mathematical understanding of young Brazilian
candy-sellers as they attended school. The influence of the large
denominations of bills in the currency, on their mathematical
constructions, and the way in which their "school" mathematics
influenced their developing understandings of mathematics in
their selling practice, illustrate that in- and out-of-school
mathematics need not exist in separate compartments. Children's
out-of-school mathematical activities may also provide a rich
resource for teachers.
At Uthongathi, Kesiree (grade 7, Indian) commented, "You
just use mathematics everywhere, but you never realize that
you're using it; because it just comes". Kesiree was talking
about the mathematics involved in dressmaking. In designing and
in measuring there is scope for many activities with a cultural
background in mathematics classrooms. Tribal Ndebele homes in
Africa are decorated, outside and in, with colourful geometric
designs. Islamic mosaics in Mosque or home also contain intricate
geometric designs. As far as playing is concerned, the patterns
in many traditional African games (e.g., mancala and its varia-
tions) are the basis for many mathematical constructions (Pre-
smeg, 1992).
Given a wealth of possibilities for using ethnomathematics,
based on diverse cultural backgrounds of children within a given
classroom, what principles might be suggested whereby teachers
can use their own creativity to tap this rich resource? The
following are a few ideas.
A writing assignment might be used to encourage students'
awareness of the mathematics which is embedded in their cultural
backgrounds. If each student researches and writes about his or
her own culture, the teacher will also gain an enhanced under-
standing of where the students are coming from, i.e., the macro-
cultures of that classroom.
Following this exploration, classroom projects in which
students work in small groups on various mathematical cultural
activities could serve to deepen their understanding of elements
of mathematics in their own and other students' cultures. Sharing
of ideas within groups and later with the whole class would be
essential in the forging of new taken-as-shared understandings
which constitute the microculture of the classroom.
In conclusion, the benefits of such an approach, not only in
mathematics classrooms but in all multicultural classrooms every-
where, are expressed in the words of one Uthongathi student:
GRAGEN (grade 9, Indian): When I came to this school I
feel more relaxed, I feel
really happy, something inside
me. It's nice talking to all
the race groups because you
get all that racial feeling
out of you. So I feel free,
communicating with all other
racial groups, so you don't
get this barrier sort of
thing, like you're White don't
play with me. You know, once
you get to know them, they're
much more friendlier than you
think. It's just like to help
each other and be one big
family.
References
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some cross-cultural issues. For the Learning of Mathematics,
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Bishop, A.J. (1988). Mathematical Enculturation: A Cultural
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Lancy, D.F. (1983). Cross-Cultural Studies in Cognition and
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Leiter, K. (1980). A Primer on Ethnomethodology. Oxford: Oxford
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Presmeg, N.C. (1985). The Role of Visually Mediated Processes in
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Presmeg, N.C. (1992). Cultural Bases of Mathematical Prototypes.
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Quebec City, Aug. 16 - 23, 1992.
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