The students had the textbooks in front of them, opened on page 35, and they started doing the exercises. I was sitting next to John, and I, trying to operate as a supportive observer, asked him: ‘So what are you going to do?’ John did not even look at me. He was already deeply concentrating on doing the exercises on page thirty-five. I asked a second time, but he completely ignored me. He was progressing rapidly through the exercises. His handwriting was not very clear but as far as I could see, correct answers were produced with efficiency. Whilst he was writing, he half-covered what he wrote with his left hand so that the boy sitting on the other side of him and I had difficulties seeing and following his calculations. I looked around in the classroom: several other students seemed as focussed as John.

When the first students had finished the exercises, they immediately got up and went to stand in a small row next to the teacher’s desk. Whilst the students were doing the calculations, the teacher had walked around in the classroom giving supportive comments. He smiled at the students and they smiled back. But when the first student had finished the exercises, the teacher had already taken his seat at his desk. The first student showed what she had done. It was not John; he was number three in the row. The teacher looked at the solutions and nodded: ‘Very well done!’ and he made a huge tick in the student’s notebook. The second in the row got at huge tick as well. Everybody in the row received a huge tick and positive comments. This procedure turned out to be the same in the following lessons, and so did the students’ competition for becoming the first in the row. John was one of the candidates, but there were also several others.

During my visits to that classroom I paid particular attention to a small group of girls who were sitting in the back of the classroom. They were not really participating in the competition to be among the first in the row. They worked on the designated exercises, but at their own rhythm. They wrote things in their notebooks. Sometimes it was wrong, and they had to erase it. But that was not to be done immediately. They had erasers of different sizes and colours and smells. Maybe the eraser at hand had the smell of a banana. But it could be that the mistake had to be erased with an eraser with the smell of strawberry. So one had to ask around for an eraser with the appropriate smell. Certainly such an eraser was available, as the group of girls were well equipped with erasers. They also had different pencils and different pencil sharpeners. It might well happen that a number like ‘2’ needed to be written with a newly sharpened pencil. So if an exercise seemed to result in such a number, better do the sharpening of the pencil in advance. There were in fact so many things to take care of!

The group of girls had created a small pleasant world of their own, not disturbed too much by the general rush hour sweeping through the classroom. They had shifted their intentions away from participating in the performance game. But the consequence was not that they did nothing. They had created their own priorities, and they were engaged in a productive form of underground construction of meaning. They had established their own vision about what was important to do. My observation of this group of girls made me pay particular attention to students’ intentions for learning, and the possible redirection of such intentions. It made me formulate the notion of foreground.¹

Foregrounds are not panoramic and coherent pictures of possibilities. They are fragmented, partial, inconsistent, dizzy; they are criss-crossed by ruptures. Foregrounds may be frightening, as they contain uncertainty, nightmares, danger, threats. Foregrounds may include dreams which, from the beginning, might be recognised as being completely out of reach. A foreground is a perplexing constellation of aspirations and frustrations. A foreground is an ongoing rebuilding of such constellations. There is no stability with respect to foregrounds, and maybe it is most adequate to see a foreground as a process. This observation makes it relevant to talk about ‘foregrounding’ and to see ‘to foreground’ as a verb.²

In its introduction The World Bank’s World Development Report 2006 refers to two South African children born on the same day in 2000:³

Nthabiseng is black, born to a poor family in a rural area in the Eastern Cape province, about 700 kilometers from Cape Town. Her mother had no formal schooling. Pieter is white, born to a wealthy family in Cape Town. His mother completed a college education at the nearby prestigious Stellenbosch University. (World Bank, 2006, p. 1)

The report went on to point out that:

… Nthabiseng has 7.2 percent chance of dying in the first year of her life, more than twice Pieter’s 3 percent. Pieter can look forward to 68 years of life, Nthabiseng to 50. Pieter can expect to complete 12 years of formal schooling, Nthabiseng less that 1 year. […] Nthabiseng is likely to be considerable poorer than Pieter throughout her life. […] Growing up, she is less likely to have access to clean water and sanitation, or to good schools. (World Bank, 2006, p. 1)

Nthabiseng and Pieter’s foregrounds are structured through different values of the parameters designating propensities in life. Naturally, they can interpret their situations and possibilities in different ways. Their futures are not determined by statistics (not in any strict sense of determining), but the statistical framing of their aspirations and hopes are radically different. The difficulties and obstructions they are going to encounter in life will be radically different. As their foregrounds are different, their ways of acting might be very different, in particular with respect to schooling. If one wants to understand their priorities and their ways of acting in school, one has to understand how they might experience going to school. What kind of meaning would they associate to schooling? What kind of possibilities does schooling provide for them? What new elements might schooling bring to their foregrounds?

Foregrounds and backgrounds are related. Naturally, one can claim that the foreground of a person is shaped by the background of the person. This is clearly illustrated by the case of Nthabiseng and Pieter. Principal statistical parameters seem to become extended from the past and into the future. However, we have to be careful if we are going to stipulate any causal transaction from background to foreground. Pieter is, for instance, expected to live much longer than Nthabiseng, but in what sense can we see this as caused by their personal backgrounds? The difference of the foreground-parameters with respect to Nthabiseng and Pieter is produced through an extensive exploitation of one group of people with respect to another. The foreground-parameters have to be interpreted in terms of complex socio-political relationships. They can be formed through exploitation and domination. In case one tries to simplify the explanation by relating the background and foreground of, say, Nthabiseng, one could be trapped by the assumption that the explanation of Nthabiseng’s difficulties, for instance with respect to education, have to be searched for in her personal background. This would invite a deficit-interpretation of her achievements in school. Instead I suggest that foreground-parameters should be analysed as expressions of complex relational patterns of domination, exploitation and exclusion. This, in turn, would show the inadequacy of a deficit-interpretation.

These investigations revealed many things, but here I will emphasise first of all what can be called the multiplicity of a foreground. In fact, it is an open question whether we should use the singular or plural when we refer to the foreground(s) of a particular person. A foreground refers to opportunities which the social, political, cultural and economic contexts make available for the person, and to the way in which they are interpreted and experienced by the person. However, it is possible to change perspectives, to see new possibilities, to create new possibilities; it is possible to fall into a gloomy mood that annihilates any form of hope. The foreground is not any well-defined territory of perceived opportunities. A person might, simultaneously, envisage different foregrounds that might contradict one another. It is important to acknowledge the multiplicity of foregrounds, also when we have to do with a specific person at a specific moment. Foregrounds are packed with contingencies, they represent uncertainties, they contain risks, they are unpredictable. They represent hope and aspirations as well as defeat and misery. They seem ready to split up and multiply.

In particular, this multiplicity can be dominant when we consider a person in a borderland position⁵. Such a position is experienced by a person who can envisage radically different opportunities, as for instance these young Indian students to whom I have just referred. They are familiar with the life of young people in the city, and they know what it means to prepare for further studies. They know what the Brazilian society may offer by way of living conditions. They know the Indian community and the Indian language. They know about working in the fields. They can easily envisage very different scenarios. They can see themselves leaving the Indian community and moving to the city, getting a job and getting married. But they can also remain within the Indian community. In this sense they are positioned in a borderland which opens to radically different foregrounds.

One of the Indian students expressed clearly that remaining in the Indian community was a high priority. He liked to work in the fields and to be part of the solidarity associated with working for everybody. Another student expressed his aspirations to study medicine. The reason he gave was that health was one of the principal problems in Indian communities in Brazil. He wanted to return as a qualified doctor. Foreground can be expressed, it can be changed, it can be reworked. There is not much continuity in this process of foregrounding. The student who preferred to work in the fields might come to see other possibilities. The student who wanted to study medicine might become interested in working in the city or becoming a researcher at the university. Foregrounds are multiple, not least for students in a borderland position.

Sikunder Ali Baber, who is from Pakistan, completed his PhD in Denmark. He studied the conditions of immigrant students in Denmark, particularly students from Pakistan (Baber, 2007). He paid special attention to the students’ foregrounds, and to how they saw their possibilities. One of the principal points of his study was that immigrant students’ foregrounds were somehow polarised. They felt that they, as immigrant students, had to perform much better in school than the average in order to have any chance in Denmark. If your school performances were average or below, you had no possibilities, except helping in your father’s small shop. You were a lost case. For a ‘regular’ Danish child, however, being average provided a comfortable position. The point of Baber’s study was not to document whether or not the immigrant students’ interpretation was correct or not, statistically speaking. The point was to clarify features of their foregrounds; and these features had implications for the meaning they might associate to going to school, for their experience of being excluded, for their choice of stepping out of the race in school or not.

Foregrounds contain strong collective features. Above I talked about the foreground of a person; we said that the foreground of Nthabiseng was radically different from the foreground of Pieter. One could, however, also talk about the foreground of a group of persons. Thus we could pay attention to the statistical parameters which form part of the constitution of Nthabiseng’s foreground. These values outlining expectations of length of life, income, and schooling she shares with many other people from her neighbourhood. In this sense, Nthabiseng shares foreground with many others, and it seems possible to talk about the foreground of a group of people. Furthermore, experiences can be shared. Interpretations also have strong collective features. Thus, Nthabiseng’s aspirations with respect to education are emerging as a result of a collective process. Foregrounds are constructed through shared life-experiences, shared visions and aspirations, shared interpretations of threats and possibilities, and shared frustrations. They are constructed through communication. Naturally, foregrounds can also be imposed on a group of people, through exploitation and stereotyping. Foregrounds are elaborated though complex social processes.

This means that one can talk about the student’s foreground (for instance, when one wants to emphasise the individual features of a foreground); or about the students’ foreground (when one wants to emphasise the collective aspects of a foreground). One can talk about the student’s foregrounds, when one wants to address the multiplicity of foregrounds. Lastly, and in general, one can talk about students’ foregrounds. Whatever formulation one chooses, there are always some important features that tend to be left out.

He said that he was a mathematics teacher and that in his school there was a boy who had given up. He showed no interest in doing anything. He was well known in the school, and his doing nothing was clearly demonstrated to everybody in the classroom independent of which teacher happened to give the lesson. The boy was recognised amongst his friends as a trendsetter. He did not spoil the class with loud comments, but the way he put down his head on his folded arms when the teacher was about to say the first words was a manifest public announcement of his lack of interest. The teachers had tried everything to get him interested, but his head remained down on his folded arms.

One day the principal of the school asked the boy to come to his office. Here he asked the boy, ‘What do you want? What do you want to become?’ The boy smiled and said, ‘I want to get into the military.’ The principal nodded and said that there was a military school at a military camp not so far away. He asked the boy to get into his car: ‘Let us go there.’ And off they went to see the military camp, and to get an idea of what the school demanded and what it meant to be trained for the military. The boy and the principal walked around, saw something, talked with somebody. It was not a long excursion; they were back in school a few hours later.

From that day on, the boy did not rest his head on his arms any longer. After the excursion he changed completely and became a perfectly attentive student. It appeared that just showing an interest in his foreground established a new relationship between the student and what was taking place in the school. The point of the story is not that the boy had experienced some particular and well-defined relevance of the topics dealt with in the school whilst he visited the military school, for instance that reading a map is crucial in any military education. It appears that the very fact that the principal showed an interest in the student’s aspirations turned the situation around.

Meaning is constructed, and this also applies to the meaning that students may construct with respect to activities in the classroom. I interpret meaningfulness in terms of relationships, and in particular in terms of relationships between, on the one hand, what is taking place in the classroom and the activities the students are invited to participate in, and, on the other hand, the foreground of the students. However, such relationships need not be elaborated in complete detail in order for students to experience meaningfulness. Some approaches in mathematics education have been searching for meaning by relating the activities in the classroom to a particular aspect of the background of the students. I do not find this to be irrelevant, but I find that meaning construction with respect to learning has a more intimate relationship to the foreground of the person. What can be experienced as meaningful education by the students is an open question. However, showing an interest in the student’s foreground means showing interest in what is crucial for establishing meaningfulness.

As part of the project Learning from diversity we also interviewed young people from a Brazilian favela.⁶ We asked the young people what they wanted to do in the future, and how they saw mathematics with respect to this. For instance, one point of this foreground investigation was to understand how the students might see the relationship between the mathematics they experienced in school and their aspirations for the future. It was generally recognised that mathematics was relevant for many different kinds of further studies and that mathematical techniques were essential for engineers, doctors, dentists, et cetera. What was completely obscure, however, was the particular nature of this relevance. It was not possible for them to point out any particular relationships between issues in the curriculum and out-of-school practises assumed to be applying mathematics. The relevance of mathematics could only be expressed in general terms; it could not be exemplified. Or, as pointed out by one student, when we consider the solutions of quadratic equations of the form ax² + bx + c = 0, and we are asked to calculated the discriminant, Delta = b² – 4ac, we are lost. It is not possible to imagine any real-life situation – in engineering, medicine, economy, computing, whatever – where one needs to calculate Delta.

One can make different interpretation of this Delta syndrome. One can, for instance, claim that the syndrome emerges from a general misunderstanding, namely that the relevance of mathematics can be broken down into the relevance of particular elements of mathematics. However, this can bring about misleading questions such as: What is the practical relevance of calculating Delta? Instead one may claim that the relevance of mathematics has to be acknowledged through the relevance of the overall features of mathematics, and not via its particular elements. This might sound reasonable, but still there remains the question of how the particular activities in the classroom then might be related to the overall features of mathematics. And this, again, makes us return to the point that meaningfulness, as experienced by the students, emerges from relationships between what is taking place in the classroom and their foregrounds. The Delta syndrome reminds us that this relationship is of a most complex nature and often obscure.

Students in different situations have different options for dealing with the Delta syndrome. Naturally, students can operate within the mathematics classroom as if one has to do with a competition, as was symbolised by the competition in getting first in the row next to the teacher’s desk. Thus, students can ascribe meaning to activities through their instrumental value. Thus instrumentalism can provide much energy to some students’ activities.⁷ In general instrumentalism with respect to learning mathematics refers to an activity, not motivated by aspirations of understanding mathematics, but by aspirations of obtaining something such as passing a test, entering further education, or being the winner of the page 35 race.

Through an instrumental approach one might become able, in a determined way, to concentrate on some classroom activities which otherwise might appear meaningless. Some students will get strong support from home in dealing with the Delta syndrome in an instrumental way. Parents could help with homework, whether it makes sense or not, always making clear the importance of good performance in school for future career opportunities. But for students with less support from home, the Delta syndrome might be devastating. Thus Pieter and Nthabiseng have very different conditions for experiencing meaning in mathematics education, as well as for coping with the Delta syndrome.

The students were fifteen or sixteen years old, and I asked them to do a little experimentation (my English was translated into Spanish by a research colleague). I wanted them to pay attention to their imagination, to their hopes and dreams with respect to the future. In other words, I wanted to conduct a foreground investigation, although on a very small scale.

I asked the students: ‘Just close your eyes, and start to imagine. Imagine yourself in the future. Imagine yourself in ten years. Imagine that you can become exactly what you dream of becoming.’ There was a little laughing, but everybody was sitting with closed eyes. I continued: ‘Now, imagine something you would really like to be and do in ten years …. We have good time, just close your eyes and do some dreaming. And imagine that you succeed in doing what you dream of doing.’ I could see their mouths moving a little bit. Some were smiling, some looked grave and focused. Many changed expression now and then. ‘No, no, don’t look’, I said, ‘just continue imagining.’ So they were sitting there and imagining, with their eyes closed.

After a while I asked them to open their eyes. They looked around with shy smiles. I asked if somebody wanted to tell about their imaginings. My idea was the following: Firstly, I wanted to listen to what they would dream of when asked to dream freely. Secondly, I wanted to ask them to dream with open eyes, and imagine more realistically how they saw themselves in ten years’ time. In this way, I would get an impression of the more realistic features of their foreground. Thirdly, I wanted to ask them how they saw the relationship between what they were doing in mathematics and what they had imagined, whatever it might be – either wild or realistic dreaming.

After a while a girl said, ‘I was dreaming about becoming a hairdresser.’ And one of the boys said he was dreaming about working as an electrician. He would like to put up wires in houses. I was rather surprised. I had expected to listen to something about becoming famous, becoming a singer, becoming a professional football player. Had the students misunderstood? I had after all asked them to do some ‘wild dreaming’. I was about to say something, and to explain that later we would come back to a more realistic perspective of the future. But then I understood: I had, in fact, been listening to their wild dreaming.

Ghettoising means tying people to the ground. This can be done in a concrete way, as in case of Palestinians who cannot leave the ghettoes as they do not have human rights or passports. People can also be tied to the ground by not having resources for moving anywhere. Ghettoising makes dreaming hurtful. In a ghetto, dreams are kept in cages and this brings us to consider what a ruined foreground can mean.

However, let me just add one more thing with respect to the Barcelona example: After I had listened to the students’ wild dreaming, I asked if they saw relationships between what they were doing in the mathematics classroom and what they wanted to do in the future. They said that they could see many relationships. I asked them to exemplify. The boy who wanted to become an electrician explained how an electrician could figure out if there was enough wire in the roll he had to take with him to complete a certain task. The students explained that one need not unroll the whole wire and measure its length. The wire could remain in the roll. One could, instead, measure the diameter of the roll, count the number of rounds of wire in the role, multiply these two numbers and multiply the result with 3. Then one knew how much wire one had available in the roll. This was in fact the common practice among electricians, and this practice had been explained by the mathematics teacher. Maybe the boy wanted to become an electrician only after the teacher had explained how to measure the length of a wire without unrolling the whole wire. Meanings are constructed, and foregrounds are constructed.

I find that one key to understanding students’ achievements at school is their foregrounds, including their interpretation of possibilities. In particular it is difficult to fight for something that appears unattainable. This applies to students who find it impossible to complete a first-in-the-row race, and it applies to students in a ghetto. In this sense a ruined foreground may form a most profound learning obstacle. A ruined foreground can be the most direct cause of failure in school. Let us look at some examples.

In his doctoral thesis Herbert Khuzwayo (2000) studied the history of mathematics education in South Africa during the apartheid period, from 1948 to 1994. In particular he investigated what could be called ‘white research on black education’. One element in this ‘research’ was to find out why, according to some statistics, Black students could not figure out mathematics. Naturally such statistics had been both produced and interpreted within a racist framework and formulated through a deficit-model, and the conclusion was that the cause of Black students’ failure was associated with the Black students themselves.

One can, however, get a different reading of achievement figures if one considers the students’ foregrounds. What did Black students’ foregrounds look like during the apartheid era? Could they consider studying engineering? Or any other kind of technical studies? Many possibilities were simply eliminated from their foregrounds – maybe due to the explicit apartheid rules; maybe due to engraved social and economic suppression. Black students’ foregrounds were ruined. What sense could it make to Black students to struggle with mathematical issues in order to qualify for further studies that were in any event inaccessible? For them there were no possibility to pursue any form of further technical studies requiring mathematical skills. One reasonable approach would be to do as the girls did, keep quiet and stay out of the school race, which could bring only defeat; another would be to become an activist. Ruined foregrounds of Black students during the apartheid era established the most brutal form of learning obstacle. Looking around the world today, however, one finds many examples of ruined foregrounds.

Let us consider a different situation. If we go back to, say, the 1930s in Denmark, we find that women, it seemed, could not do well in mathematics. How are we to explain this phenomenon? One could try to relate this to the background of the girls. Their failure in mathematics could have been related to their upbringing. However, we could also try to consider their foregrounds. How did girls during the 1930s in Denmark see their future opportunities? They could easily observe that mathematics was for men, as all further studies that drew heavily on mathematics, like engineering for instance, were extremely male-dominated. It might have been very difficult for girls at that time to envision themselves as dealing with technical issues. Mathematics did not play any part in any practices that could be associated with their foregrounds. A most reasonable thing, then, was not to struggle with this subject. Better simply to pull out of the race. However, Denmark did not remain the same. New opportunities arose, and girls’ performances with respect to mathematics changed.

Once, Mathuma Bopape invited me to visit different townships schools in the Pietersburg area in South Africa.⁸ One school looked more broken-down than many others. We were standing in a classroom that looked as if it had suffered a light bombing, and up there, right above our heads, there was a hole in the ceiling. It may be that some houses in the neighbourhood had needed the tiles a bit more than the school building. When it was raining the chairs in the classroom had to be moved away from this part of the classroom. If we should try to explain the observation that Black children have lower achievement rates in school than White children, I think it would be wise to remember the hole in the ceiling. For me it is the most obvious learning obstacle one could think of. This hole, however, has been overlooked my much educational research, in particular by ‘white research on black education’. By providing different patterns of explanation this research has been engaged in explaining away the obvious: that Black children’s foregrounds have been ruined. But not only this: their actual learning conditions have been ruined as well.

He continues:

It is exclusion, rather than the exploitation suggested a century and half ago by Marx, that today underlies the most conspicuous cases of social polarization, of deepening inequality, and of rising volumes of human poverty, misery and humiliation. (Bauman, 2004, p. 41)

This is a strong reminder of the fact that the exclusion of economically ‘irrelevant’ groups is part of the global order today.

Bauman describes the drama of inclusion–exclusion in terms of extreme conditions for constructing identities in the following way:

At one pole of the emergent global hierarchy are those who can compose and decompose their identities more or less at will, drawing from the uncommonly large, planet-wide pool of offers. At the other pole are crowded those whose access to identity choice has been barred, people who are given no say in the deciding their preferences and who in the end are burdened with identities enforced and imposed by others; identities which they themselves resent but are not allowed to shed and cannot manage to get rid of. Stereotyping, humiliating, dehumanizing, stigmatization identities … (Bauman, 2004, p. 38)

Conditions for constructing identities are polarised. Globalisation, representing the capitalist order of today, establishes some groups of people as being in a position of making an apparently free build-up of identity (although certainly subjected to the capitalist logic of consumption), whilst others have to cope with imposed and stigmatised identities. This constitutes that part of the new global order where exploitation has turned into exclusion.

Let us now repeat Bauman’s formulation, elaborating them a little further and talking about students’ foregrounds instead of their identity.⁹ In this way I want to emphasise that the discussion of inclusion–exclusion is closely related to the discussion of foregrounds.

At one pole of the global hierarchy that is emerging through the processes of globalisation we find those groups of students who have the opportunity of composing and decomposing their foregrounds more or less at will – although we cannot ignore the fact that these compositions are taking place within the capitalist layout of globalisation. There are plenty of resources from which they can bring together a variety of possibilities in life. They can draw on the uncommonly large, planet-wide pool of offers to which they, maybe due to the affluence of their families, have direct access. They can get the support needed to make the best of their time in school. They can go to private schools if this turns out to be most beneficial. They can choose the education that prepares them for the opportunities in life that to them seem most attractive.

At the other pole are crowded those many students whose access to the free formation of foregrounds has been barred. Here are students who have no (or very little) say in establishing their future possibilities in life, and who in the end are troubled with foregrounds imposed, as well as ruined, by others. These are the students who – tied to the ground in a ghetto or located in a borderland position – find their dreams to be put in cages. These are the students whose foregrounds might represent real learning obstacles. There is no lack of stereotyping that accompanies ghettoising and exclusion. Students from poor conditions might easily be classified as being troublemakers and obstructive, at best they might be classified as suffering a range of deficiencies due to their personal backgrounds. If we consider, for instance, immigrant students in Denmark, then stereotyping, humiliation, dehumanisation, and stigmatisation form part of the public formation of their foregrounds.

Dramatic processes of inclusion–exclusion are established though schooling. And if we follow Bauman in claiming that the malfunction of capitalist economy is now operating through exclusion, it becomes crucial to consider the role of schooling with respect to this malfunction. Processes of inclusion and exclusion operate to a great degree through schooling. These processes can be experienced by every student: John, the silent girls, Pieter, Nthabiseng, anybody.

It becomes important to consider how foregrounds might be ruined; how they might be reconstructed; how it might be possible to add new elements to them; and how schooling might provide students with new possibilities. It becomes important to consider how foregrounds might represent hopes, despairs, and uncertainties.

Baber, S.A. (2007). Interplay of citizenship, education and mathematics: Formation of foregrounds of Pakistani immigrants in Denmark. Unpublished doctoral dissertation. Aalborg University, Denmark.

Bopape, M. (2002). Mathematics school based in-service training (SBINSET): A study of factors contributing towards success or failure of SBINSET in the South African School context. Unpublished doctoral dissertation. Aalborg University, Denmark.

Bauman, Z. (2004). Identity: Conversations with Benedetto Vecchi. Cambridge: Polity Press.

Khuzwayo, H. (2000). Selected views and critical perspectives: An account of mathematics education in South Africa from 1948 to 1994. Unpublished doctoral dissertation. Aalborg University, Denmark.

Laursen, I.L. (2008). At fremrette – et studie af etniske minoritetskvinders subjektivt fortolkede livsverden og arbejdslivsforventninger i komplekse samfund [Forward-directing: A study of ethnic minority women’s subjectively interpreted life-world and work expectations in a complex society]. Unpublished master’s thesis. Department of Learning, Education and Philosophy, Aalborg University, Denmark.

Mellin-Olsen, S. (1981). Instrumentalism as an educational concept. Educational Studies in Mathematics, 12, 351−367. http://dx.doi.org/10.1007/BF00311065

Penteado, M.G., & Skovsmose, O. (2009). How to draw with a worn-out mouse? Searching for social justice through collaboration. Journal of Mathematics Teacher Education, 12(3), 217−230. http://dx.doi.org/10.1007/s10857-009-9103-6

Skovsmose, O. (1994). Towards a philosophy of critical mathematics education. Dordrecht: Kluwer Academic Publishers.

Skovsmose, O. (2005a). Travelling through education: Uncertainty, mathematics, responsibility. Rotterdam: Sense Publishers.

Skovsmose, O. (2005b). Foregrounds and politics of learning obstacles. For the Learning of Mathematics, 25(1), 4−10. Available from http://www.jstor.org/stable/40248476. Reprinted in U. Gellert, & E. Jablonka (Eds.), Mathematisation – Demathematisation: Social, philosophical, sociological and educational ramifications (pp. 81−94). Rotterdam: Sense Publishers.

Skovsmose, O. (2011). An invitation to critical mathematics education. Rotterdam: Sense Publishers.

Skovsmose, O., Alrø, H., & Valero, P. in collaboration with Silvério, A.P., & Scandiuzzi, P.P. (2007). “Before you divide you have to add”: Inter-viewing Indian students’ foregrounds. In B. Sriraman (Ed.), International perspectives on social justice in mathematics education. The Montana Mathematics Enthusiast, Monograph 1, 151−167. Available from http://www.math.umt.edu/tmme/Monograph1/Skovsmose_etal_FINAL_pp151_168.pdf

Skovsmose, O., & Greer, B. (Eds.) (2012). Opening the cage: Critique and politics of mathematics education. Rotterdam: Sense Publishers.

Skovsmose, O., & Penteado, M.G. (2011). Ghettoes in the classroom and the construction of possibilities. In B. Atweh, M. Graven, W. Secada, & P. Valero. (Eds.), Mapping equity and quality in mathematics education (pp. 77−90). New York, NY: Springer.

Skovsmose, O., Scandiuzzi, P.P., Valero, P., & Alrø, H. (2008). Learning mathematics in a borderland position: Students’ foregrounds and intentionality in a Brazilian favela. Journal of Urban Mathematics Education, 1(1), 35−59. Available from http://ed osprey.gsu.edu/ojs/index.php/JUME/article/view/4/4

Stentoft, D. (2009). Challenging research conceptions in (mathematics) education: Telling stories of multiplicity and complexity. Unpublished master’s thesis. Department of Learning, Education and Philosophy, Aalborg University, Denmark.

World Bank (2006). World development report 2006: Equity and development. Washington and New York: A copublication of The World Bank and Oxford University Press. Available from http://wwwwds.worldbank.org/external/default/WDSContentServer/IW3P/IB/2005/09/20/000112742_0050920110826/Rendered/PDF/322040World0Development0Report02006.pdf

2.In her Master’s thesis, Iben Lindgaard Laursen studied the foreground of immigrant women in Denmark. She emphasised the importance of seeing ‘to foreground’ as a verb. She used the Danish word fremrette, which means ‘forward-directing’ when directly translated into English. In the English abstract of her thesis, she translates fremrette as ‘envisage’ (Laursen, 2008).

3.Renuka Vithal referred to these paragraphs in the World Bank Report in her lecture at the symposium Mathematics education, democracy and development: Challenges for the 21st century, Faculty of Education, University of KwaZulu-Natal, Durban, on 04 April 2008.

4.See, for instance, Alrø, Skovsmose and Valero (2009). Important contributions to the project ‘Learning from diversity’ were also provided through the doctoral studies by Sikunder Ali Baber (2007) and Diana Stentoft (2009).

5.For an introduction of the notion of borderland position see Skovsmose, Scandiuzzi Valero and Alrø (2008). See also Penteado and Skovsmose (2009), where the notion is further explored.

6.See Skovsmose, Scandiuzzi, Valero and Alrø (2008).

7.Instrumentalism has been carefully addressed in Mellin-Olsen (1981).

8.I refer to this event in Skovsmose (2005a). See also Bopape (2002).