mathematics learning with ardinas at Cabo Verde
and João Filipe Matos
Centro de Investigação
Faculdade de Ciências
da Universidade de Lisboa
Among several mathematics
education researchers learning is starting to be seen as a social practice
(Lave, 1988) and this idea is now being used to look into school mathematics
learning (Adler, 1996; Santos, 1997). However, several questions emerge
when we intend to think about school learning from this point of view.
Lave's results come from studies on adults in situations with relevant
differences from schooling. For instance, practices in which adults were
involved in Lave's studies were deeply connected to a (chosen) process
of becoming. However, it is growing among us a strong belief that, for
most of young people (12 to 15 years old) schooling is not explicitly associated
to a process of becoming but it is a transitory life-space. Becoming is
not the intentionality and purpose of pupils' school practice. Therefore
we feel the need to clarify the meaning of learning as social practice,
particularly in these aspects that we see as fundamental.
Our view of mathematics
learning at school draws from the idea that school life is an everyday
practice (Lave, 1988). But life at school has its own culture, its history
and we must remember that this practice is lived at an institution. One
of the goals of our study is to understand which are the elements of this
practice and how do they relate to the learning that takes place there.
In oder to understand what are constitutive elements of that practice,
we believe that it is useful to look at an out of school practice. Why
do we feel that this is useful? First, because as educators we live the
school from the inside and this turns difficult to be sensitive to important
aspects of the practice. Looking at and trying to describe a practice which
is not familiar is a way to put ourselves as learners about that practice.
Second, we searched for a practice out of school that incorporates some
common elements to the school practice. These elements are: this practice
is lived essentially by youngsters in school age, it has a non professional
character, it is part of an institution, it is a transitory situation in
the lives of the participants, and we could identify a visible relation
to mathematics use.
To describe and analyse
this practice we are using an analytical tool proposed by Lave (1996) which
we share and believe that is coherent to an approach to learning as a social
practice. This analytical tool is "a set of questions for interrogating
anything claiming to be an example (...) of learning" (Lave, 1996, p.156):
that is, a direction of movement or change of learning (not the
same as goal directed activity),
relation: a general specification of relations between subjects and
the social world (not necessarily to be constructed as learners and things
3. Learning mechanisms:
ways by which learning comes about" (Lave, 1996, p.156).
However we felt that we
need some powerful ideas in order to go deeper in the understanding of
this practice. For instance, we found in the work of Wittgenstein (1953)
some interesting ideas that seem to be consistent to Lave's approach. We
can say that Wittgenstein is concerned with "us as being able to 'go on'
with each other" (1953, nos.146-155) "reacting and responding in ways that
makes it possible for us to continue our relationships" (Shotter, 1997,p.1).
One of the implications of this view is that an important aspect of people
life is a concern with how to sustain participation. In the case of the
ardinas' practice, to learn that practice implies to learn certain
things (as mathematics use) that help them to maintain their participation
in that practice. Following this line we are working for example on the
idea of rules and following rules from Wittgenstein (1967)
in order to make more visible the subject-world relation.
At the moment we are
focusing on some examples illustrating how we are using the concepts of
telos and learning mechanisms in the analysis of this practice.
The practice of ardinas
at Praia, Cabo Verde
The newspaper A Semana
is sold only on the street once a week at Praia in Cabo Verde. Every Friday
(hopefully at morning), at the newspaper office, the newspapers are devilered
to an adult (Egídio) who is responsible for the selling and for
paying back to the administration. Egídio gives a number of newspapers
to each one of the 19 ardinas (from 50 to 150 exemplars each). Immediately
after that the ardinas rush to their selling places in the city
and try to sell the newspaper to the costumers as fast as possible during
that day. The ardinas sell the newspaper at places chosen by themselves
according to the rhythm of selling. The newspaper cost 100 escudos (fixed
by the newspaper office) and the ardinas should pay (at the end
of the day) 87.5 escudos to Egídio for each newspaper sold. The
ardinas are young boys aged from 12 to 20 years (mainly about 15-16-17).
Some of them begun to sell one month ago and others are selling for four
years. Most of them sell newspapers in order to help their families ("to
help my mother"). The level of schooling of these boys goes from 2nd grade
to 9th grade. Right now only three of the 19 ardinas are studying
at the local school as five of them dropped out the school last year and
the others left it several years ago.
come from two places. One group is from Praia, the capital of Cabo Verde
(from a poor and problematic borough) and the other comes from S. Martinho
(a rural small village near Praia). Egídio is the one who invites
or accepts boys for the job of ardina. Certainly there is an history
of these groups. Five months ago the ardinas were all from Praia.
However problems arrived with some of them and Egídio brought 10
boys from S. Martinho (the place where he also lives) to substitute the
others. Egídio gave some hints to this group from S. Martinho about
the selling process (the price of the newspaper, some good places for selling).
When a newcomer from Praia joints the group, Egídio defines an oldtimer
who will be in charge of the newcomer (passing him a small number of his
newspapers to sell, protecting him from piratas and receiving from
him the money to pay to Egídio).
Telos and learning
At the beginning the ardinas
almost only know (i) where to receive the newspapers, (ii) the price of
the newspaper, and (iii) where and to whom pay for the newspapers sold.
To all of them (newcomers and oldtimers) to be a good ardina is
to respect and follow the rules (payment rules) and to sell quickly. At
the beginning some of their feelings about the selling process are: to
be afraid of being roubred or lose money, not having the ability to know
how and which people to address in the street and where to go, the big
weight they have to carry during the day, the difficulties of getting enough
coins and bills to facilitate exchange. The direction of movement of learning
(telos) to be an ardina draws from these elements. During the process
of trying to be a good ardina they use some strategies. From these strategies
emerge the use and learning of mathematics. For instance, (i) during the
day they check the money they earned according to the number of newspapers
already sold — involving estimation or mapping money onto number of newspapers
sold; (ii) when giving exchange to the costumers they vary the way they
do it in order to keep certain kind of coins and bills — involving linear
combinations of numbers; (iii) the preview of the amount they have to give
back to Egídio at the end of the day (never through a direct calculation
using the value 87.5 escudos) — involving complex processes of calculation
using different strategies from additional reasoning to proportional reasoning.
A promising approach?
We are not yet in a position
to make conclusions about the usefulness of the approach we are using in
the study of learning. However, we feel that what is emerging from the
analysis of data (which we tried to give a taste of) show that it is fruitful
in order to help us to have a better understanding of Lave's approach of
learning as a social practice. This would help us to identify how this
approach could be useful to study youngsters' mathematics learning.
Adler, J. (1996)
Lave and Wenger's social practice theory and teaching and learning school
mathematics. In L. Puig & A. Gutierrez (Eds.) Proceedings of the
20th Meeting of the International Group for the Psychology of Mathematics
Education, vol. II, 3-10.
Lave, J. (1988). Cognition
in practice: Mind, mathematics and culture in everyday life. Cambridge:
Cambridge University Press.
Lave, J. (1996). Teaching,
as learning, in practice. Mind, Culture and Activity 3(3), pps 149-164.
Santos, M. & Matos,
J. (1997) Students appropriation of mathematical artifacts during their
participation in a practice: à propos de A. Sfard…" In E. Pekkhonen
(Ed.) Proceedings of the 21h Meeting of the International Group for
the Psychology of Mathematics Education, vol. 4, pp.128-135.
Shotter, J. (1997).
Wittgenstein in Practice: from 'the way of theory' to a 'social poetics'.
Wittgenstein, L. (1953).
Philosophical Investigations. Oxford: Blackwell.