Learning about mathematics learning with ardinas at Cabo Verde
Madalena Santos and  João Filipe Matos
Centro de Investigação em Educação
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):


"1.Telos: that is, a direction of movement or change of learning (not the same as goal directed activity),

2. Subject-world relation: a general specification of relations between subjects and the social world (not necessarily to be constructed as learners and things to-be-learned),

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.

The ardinas 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 mechanisms

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.


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Lave, J. (1996). Teaching, as learning, in practice. Mind, Culture and Activity 3(3), pps 149-164.

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Shotter, J. (1997). Wittgenstein in Practice: from 'the way of theory' to a 'social poetics'. http://www.focusing.org/Shotter.html.

Wittgenstein, L. (1953). Philosophical Investigations. Oxford: Blackwell.