By the same time she read the Diary of Anne Frank and she was moved by the fate of the young Jewish girl. Only later it came to her knowledge that the executioners of Bergen-Belsen and other extermination camps tried to justify themselves with the excuse that they had strictly obeyed orders. Probably then it occurred to her that obeying is also to give up the responsibility on oneself, it is becoming an object that can be manipulated.

The country – Portugal – was then living under the dictatorship of the man who created the above precept. Obviously it was intended to generate docile, submissive, dependent individuals. School was a place of non-thinking, of non-debating, of non-criticising; school was a place where teachers prescribed and students followed their prescriptions.

Illiteracy reached very high rates among the population but the dictator undervalued schooling, claiming that knowing how to read only facilitated the contact with subversive literature. We can understand what he meant by that, knowing as we do that illiteracy is connected to forms of political or ideological ignorance that work as a matrix of the way we perceive the world (Giroux, 1989) and as a refusal in believing that our acts can change society.

The Portuguese people was then living under the anaesthesia of the trilogy: Fado, Fátima and Football. Fado was a convenient a way of making poverty, inequality and injustice look as inevitable. Religion worked as a consolation on such inevitability and football was a form of expelling the dissatisfaction and resentment, in directing them to the referees and the adepts of the opposing team.

But the individuals'
development is also influenced by structures which are previous and external
to school and shape their interests, their achievements, their believes
and their aims (Lobrot, 1992). Some people had the chance of growing up
in an environment that favoured the ideals of liberty, equity and social
justice. There were those who realised that could be more than just observers
of reality; that they could play a role in the transformation of society.
This made possible the revolution of April 1974 and the establishment of
a representative democracy.

People are divorced from politicians. The majority is a lethargic mass, resigned to the inevitability of a distant and hermetic power. They could not find ways of extending and deepening democracy both in local communities and in production companies. Therefore they do not participate in the creation of a common destiny. Citizens are confining themselves to vote. But unlike what Schumpeter (1987) argues there is much more to democracy than the regular election of the people's representatives.

We live in a time where family has lost importance in the education of the new generations. Young people spend much of the day on their own, partly due to the fact that women left home to embrace professional careers and partly as a result of parents having a more demanding and time consuming rhythm of life.

The room left empty by the family is being occupied by streets or television or computer games or... While people are more and more uncommitted to forms of social intervention they are showing a growing need to take action, to become protagonists. And this happens to be fulfilled in a number of ways: when one is a member of a gang, one is a leading person in a reality show, one is the hero of some exciting virtual adventure.

This thirst of action
finds no echo in school. It still refuses students' central role in their
learning process. It is not yet a place where students may choose the subjects
and issues of their study and where they can be co-enquirers with the teacher,
free to question both the curriculum and the pedagogy (Freire, 1972).

Paul Ernest (1991) has assembled the main features of this perspective under the label of the Public Educators' ideology. For our present concerns we will underline the epistemological and ethical grounds of this perspective. It assumes that knowledge, ethics, social, political and economical issues are strongly inter-related. Therefore, knowledge is connected with forms of empowerment and is tied to possibilities of intervention in reality. In a word, knowledge and real life are not to be viewed as separate but must be integrated. The principles of egalitarianism and social justice also underpin the educational aims of this ideology.

Raising critical awareness or achieving critical consciousness is probably the outmost realisation of school under this kind of philosophy. This means to provide students the tools to become protagonists "here and today", it means no longer postponing the possibility of listening to their voices and valuing their knowledge.

We are thus suggesting the kind of critical pedagogy that, in the words of Giroux and Simon (1989), "takes into consideration how the symbolic and material transactions of the everyday provide the basis for rethinking how people give meaning and ethical substance to their experiences and voices" (p. 237).

In mathematics education, many have argued for such a change in proposing new aims for mathematics education. Keitel (1993) has discussed the need to unveil the implicit mathematics of our everyday routines. Skovsmose (1992; 1994) has pointed out the importance of recognising the formatting power of mathematics in our present society. Both have elected school mathematics as the place for developing reflective knowledge. Once again, we can see the influence of Freire's ideas concerning the way "problem posing" education, of a genuine reflective nature, entails a continuous act of the world's unveiling (1972, p. 99).

Our view of the role of mathematics education includes both the development of a democratic competence and the actual practice of democratic behaviours. School mathematics is foreseen as a place of thinking, debating and criticising. But we are also expecting school mathematics to be the medium through which people can discover what it is they have become and what it is they no longer want to be.

Along with Keitel (1993), we believe that mathematics education should evolve around situations where students and teachers are confronted with open-ended questions that favour the emergence of divergent perceptions of reality and give room to conflicting interests and values.

Sharing with Skovsmose
(1992; 1994) the conviction that knowing is a factor of power and simultaneously
a possibility of reaction to power, we consider mathematical education
as a ground to awake people to social justice. Moreover, as Niss (1994)
recalls us, speaking of equity is also speaking of mathematics and mathematical
knowledge. There is not a precise allocation for mathematics where we can
go and find it. Because it is hidden, mathematics looks as if it is absent.
But it is "more like an all-permeating ether" (Niss, 1994, p. 372).

Many of the academic subjects included in the curriculum of a Business degree are oriented to the learning of maximising profits while neglecting other concerns of social nature. The development of a democratic competence was not a purpose of the Calculus course. However, some of the problem situations used in teaching mathematical topics were suitable to promote the act of acquiring a new consciousness.

The data upon which
we want to reflect were collected during a normal class where students
worked in groups on a problem situation that concerned the income distribution
of a certain population. A mathematical model for the cumulative income
distribution was presented. It was the function f(x)=x^{2},
where x represents the lowest x% of population ranked by income and f(x)
is the cumulative percentage of income earned by that part of the population.

One of the tasks suggested to students was to find out the income of the wealthier half of the population and the income of the poorer half. Another request was the interpretation of the outputs of f(0) and f(1).

The resulting dialogue of a group of five students is now transcribed. In this first piece of dialogue some of the students already reveal an awareness of reality in terms of social justice. But others, particularly Cristina, who turns out to be the best of the group in strictly mathematical issues, shows a propensity to keep mathematics and reality apart. At start she refuses to make connections between the mathematical model, with its implications, and the real world.

[1] Miguel: Now we just need to compute f(0.5).

[2] Paulo: Which gives 25%.

[3] Miguel: So 25% is the income of the poorest half of the population. This means that the other half is receiving 75%. It’s a striking difference!

[4] Cristina: What are you saying? I don’t get it.

[5] Paulo: He’s saying that the first half, which is the worse paid, gets 25% of the total income. Therefore, the second half is receiving all the remaining, that is 75%. These are the better paid, they’re the richer people.

[6] Cristina: Sure, according to this model...

[7] Eduardo: Yeah, I doubt that the income can ever be so unfairly distributed... (Speaking with an ironic tone in his voice).

[8] Isabel: Well, you’d better not!

[9] Eduardo: OK. Let’s move on to this one: how do you interpret the fact of having f(0)=0 and f(1)=1?

[10] Paulo: Well, if there’s no population there can’t be any distribution of incomes.

[11] Eduardo: No people, no income.

[12] Paulo: The f(1)=1 means that 100% of the people receive the whole income.

[13] Eduardo: Exactly.

[14] Isabel: Which means that there is no embezzlement of money.

[15] Cristina: There’s no sense here to speak of an embezzlement of money...

[16] Isabel: I mean that there are no false donations, no frauds, no fake payments, no funds deviations and no tax evasions.

[17] Eduardo: You’re making a good point there...

Miguel finds the difference of incomes between the two halves striking [3]. To Paulo it means the drawing of a line separating the poor people from the rich people [5].

On the other hand, Cristina seems to insinuate that it’s only a result drawn from a theoretical model. She tends to detach the mathematical context, which may reflect a common belief that mathematics classes are supposed to deal with mathematical ideas only. At the same time she can be expressing her naiveness in admitting the fact that reality cannot be like the model is suggesting [6].

Isabel and Eduardo who reveal a more accurate and critical perception of the social and economical reality sense much the contrary. Using an ironic tone in his speech, in response to Cristina’s ingenuity, Eduardo is perfectly convinced that incomes can be very unfairly distributed in the real world [7].

We may notice that students did not insist in stretching their interpretations further on. This is probably a consequence of the problem formulation, which does not make a reference to a specific situation or to a concrete population. At the moment we are producing this paper we are expecting to see how other students of a similar class will react to the same problem, having this time the income distribution for Portugal, which was in 1993 the most unequal of the European Community (Eurostat, 1997).

Anyway it is still obvious that students brought their views of the world and particular life experiences to the interpretation of the model. It is feasible to admit that such views and experiences are not shaped by the same economical, social and political backgrounds.

The question about the images of the Lorenz function at 0 and 1 would seem to be quite straightforward. Unexpectedly, however, it stimulated another critical look at the meaning of the mathematical model. Again, Isabel manifests her critical consciousness and sharp perception of reality. She integrates in her reasoning what we would call an administrative organisation of incomes distribution: the distribution of taxable incomes. Therefore, as she points out, what seems to be a simple fact, namely that 100% of the population would receive 100% of the income deserves some caution [14].

There are well known
phenomena of tax evasion and illicit deviation of funds [16]. Those situations
justify Isabel’s doubting that the declared income to the fiscal bureau
coincides with the true income of the population. Assuming that the model
of income distribution is based on the declared income – which, by the
way, is realistically the case – the fact that f(1)=1 must be interpreted
as Isabel does. If there are cases of corruption among the population,
then the income declared will not match the real income of the population.
That is why the equality f(1)=1 really says that the income received by
the whole population is the total *declared* income. Although she
does not say it explicitly, she may be guessing that the distribution would
be much more unequal if the model took into account the real incomes of
all the members of the population.

There is strong evidence in Isabel’s thinking of her knowledge of reality, and namely of the recurrent stories of financial corruption taking place in Portugal at various levels. For instance there are some political disputes about the introduction of a minimal tax for liberal professionals who are declaring incomes rather below their true value. A few years ago there was also a case of a major fraud involving fake receipts, which is still waiting for a court’s decision. Isabel seems to be an informed person who recognises and critically evaluates important current social problems. She is also able to bring them to the foreground as the opportunity comes.

Once more, the mathematical model is put in question as to its reliability; this time what is criticised is its way of hiding part of the truth about the real world.

Having explored the Lorenz function given at the beginning of the task proposal, students were then asked to find the Lorenz curve that would match complete equity. They seemed to get it mostly by intuition, but they looked for a way of supporting their answer. They also discussed what that model could represent in the real world.

*[19]
Cristina: Equally?*

*[20]
Paulo: Yes, to be equally distributed among all it should be the function
f(x)=x. Each share of the population gets the same share of the income.
For instance 20% of the population gets 20% of the income: f(0.2) is equal
to 0.2.*

*[21]
Miguel: If we were to draw a graph, it would be a straight-line and not
a curve.*

*[22]
Cristina: It must be that. Each point of the X-axis has that very image
on the Y-axis. It can only be the bisectrix of the first quadrant: f(x)=x.*

*[23]
Isabel: Now, explain me that, will you!*

*[24]
Paulo: Mathematically, everyone gets the same, there are no differences.
Naturally if we attend to the actual situation of some real country, knowing
about its social and economical conditions, we won’t believe it can be
attained. For instance, in Brazil, 10% of the population gets 90% of the
total national income.*

*[25]
Cristina: I see. The distribution has to be of one to one. Everyone gets
the same amount of money.*

*[26]
Isabel: That’s what happens in left government countries.*

Isabel however does not share this pessimistic opinion. She thinks that some countries – those with left political governments – are examples of economical policies where distribution of incomes fit the one to one pattern of distribution [26]. In her perspective, left ideologies sustain the ideals of social justice. But in choosing those as paradigms of equality, she also seems to forget or to disregard the failure of left policies in what concerns equity.

In due course, we shall see how Isabel will continue to argue in support of her conviction on the possibility of having social and economic justice.

In a later phase of students’ activity the issue under discussion is the concept of Gini index as a measure of income concentration. The model to compute the Gini index is given by the formula where f(x) is the Lorenz function. Students start to work on the calculations to evaluate the Gini Index for a sequence of income distributions: x2, x3, x4,..., xn. They use their knowledge on the properties of integration and transform the integral of a difference into a difference of integrals. The most relevant part of their discussions, however, stands on the commentaries they produce as they get their mathematical answers.

*[28]
Paulo: We get 1/3. The index is 1/3.*

*[29]
Miguel: So, in the case of this population the Gini index is 1/3. This
tells us already something...*

*[30]
Paulo: Indeed, it tells us that there is a deviation. We’re founding a
deviation from that extreme case where the distribution is equitable.*

*[31]
Cristina: Can we say that the Gini index shows the deviation? I mean, when
there isn’t equity the injustice can be exposed with this.*

*[32]
Isabel: It could be used for that purpose. Unions could take advantage
of it to claim for raises. They could say that profits are being unfairly
distributed...*

*[33]
Miguel: Ideally we wish there wouldn’t be any deviation. To get that, the
distribution had to fit the straight line, but that’s very unlikely. In
every country this must escape from the straight line.*

*[34]
Isabel: But one can try to put salaries a little more balanced... Of course,
there’s also the unemployment, which doesn’t help either. But there should
be an effort to make this closer to the straight line...*

*[35]
Paulo: That’s where the problem is. The deviation looks too large in this
case. If the curve is a little smoother, maybe it will approach the straight
line a bit more and the deviation can go more unnoticed...*

The idea of deviation turns out to be a powerful image in students’ understanding of the meaning of the Gini index [30-31]. Having the straight-line y=x as the reference to which they compare other cases of income distribution, they see the parabola as a detour. But there is another sense for the word deviation that may have contributed to their analysis. Deviation has an ethical sense of moving away from the good track, the

At last Cristina changes her attitude and starts to relate the model to the real world and she is now taking a stand when looking at that reality. We can see how she considers the index as an instrument to expose injustice.

As we would expect, Isabel approves this idea and extends it by suggesting that the Gini index could be an indicator of the low value of salaries and of the unfairness of profit distribution [32]. Moreover, she pleads for citizens’ claiming mechanisms like the unions, proving once more to be aware of such means of intervention.

Miguel describes the
straight-line distribution as the ideal situation. But like Paulo thought
before he also believes that it fails to be the real case all over the
world [33]. Only Isabel refuses to resign herself to the inevitability
of the Gini deviations. She recalls the unions’ role in demanding better
salaries and even though knowing about the negative effects of unemployment
she still finds possible to reduce injustice and poverty [34]. To her convincing
words she obtains the agreement of Paulo who thinks that differences in
people’s incomes could be made less flagrant [35]. We can not assure what
Paulo is really intending to say. He would prefer to see the gap between
the actual line and the straight-line being reduced. But he may just be
trying to disguise what he sees as an uncomfortable situation of inequality.
Although he is agreeing with Isabel he is probably less committed than
she is.

In contrast, there is also one student – Cristina – who tends to separate the mathematical procedures and results from the real world. She acts as if the model had an abstract nature and she prefers to look at it as a formal entity, which is meant to motivate or illustrate a mathematical exploration.

What happens in the course of the activity reveals a progressive change of attitude towards the acquiring of a critical consciousness. This is a result of an opportunity for students to share information and knowledge, to express their personal points of view on controversial issues, namely on the character, utopian or not, of an equitable distribution of income.

The continuous swinging between mathematics and reality allows for a better understanding of each of them. As a matter of fact, mathematics is working as an instrument to uncover reality in its social, economical and political aspects.

As the activity evolves, more and more elements of a reflective nature are introduced coming down to a critical approach to reality. At a certain point we can see how the entire group is awake to social justice.

The type of task proposal
here presented is only a weak example; we must recall that our purpose
was just the investigation of students’ construction of meanings. Nevertheless
it is strong enough to make us understand how mathematics education can
contribute to the development of a democratic competence without assuming
the role of indoctrination.

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