POLITICS OF MEANING IN MATHEMATICS EDUCATION

Ole Skovsmose

Aalborg University, Denmark and State University of São Paulo, BrazilOsk @ learning.aau.dk

AbstractBy a politics of meaning I refer to the social, economic, cultural and religious conditions for experiencing meaning. I refer as well to the layers of visons, assumptions, presumptions and preconceptions that might construct something as being meaningful. By addressing different politics of meaning in mathematics education, I want to show how meaning becomes formatted. In order to do this, I provide a foreground-interpretation of meaning. The basic idea is to relate meanings and foregrounds, acknowledging that foregrounds are formed bya range of factors, as well as by the person’s experiences of such factors. Politics of meaning can be analysed with reference to sexism, racism, instrumentalism, the school mathematics tradition, critical mathematics education, and the banality of expertise.

Key words: Politics of meaning, foreground-interpretation of meaning, sexism, racism, instrumentalism, school mathematics tradition, critical mathematics education, banality of expertise.

The first part of the film Life is Beautiful, directed by Roberto Benigni, takes place in the 1930s in Italy, where one could witness a fascination for German efficiency. In my book An Invitation to Critical Mathematics Education, I refer to a scene from this film in order to illustrate a grotesque political blindness that might be exercised through mathematics education. Here I want to return to this very same scene in order to illustrate what it could mean to address politics of meaning in mathematics education.

In the scene we meet an Italian educator who had visited Germany and was impressed by what she saw. There, children of 7 years of age were able to solve a problem like the following: “A lunatic costs the state 4 Marks per day, a cripple, 4.5 Marks per day, and an epileptic 3.5 Marks per day. The number of patients is 300,000. How much would we save if these individuals were eliminated?”

The Italian educator could not believe that seven year old German children had to solve a problem like this. It was after all a difficult calculation! A person listening to the exited educator’s explanation pointed out that it is just a question of multiplication (apparently assuming that the number of cripples and epileptics are the same): “300,000 times 4. Killing them all we will save 1.200.000 Marks a day. It is easy, right?” The educator agreed, but her point was that while 7 year old children in Germany could solve it, such a problem was far beyond the capacity of Italian children that age.

Difficult or not, are we dealing here with a meaningful exercise? Mathematical exercises, at least as they dominate the school mathematics tradition, demonstrate a grand variety of contextualisations. We meet workers A and B fully occupied in digging a ditch, apparently not needing any kind of break. We meet a happy family driving on holiday with an average speed of

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80 km per hour. We meet Peter on his way to the grocer’s shop to buy exactly four kilos of apples. One can think of these contextualisations as attempts to bring meaning to mathematical calculations. However, such contextualisations bring about a manifestation of a worldview, which in turn may represent a range of presumptions and priorities.

Presenting a certain contextualisation as being meaningful is a highly contested issue. By a politics of meaning I refer to the social, economic, cultural and religious conditions for experiencing meaning as well as to the layers of visons, assumptions, presumptions and preconceptions that might construct something as being meaningful. Thus the exercise from Life is Beautiful might make sense within a pathological Nazi-world-view. However, my general point is that any contextualisation in mathematics education represents a world-view within which meaning becomes articulated. This articulation is a highly political issue. (As I will return to at the end of the paper, I also let “politics of meaning” refer to the study of the formation of experiences of meaning. Thus “politics of meaning” come to include the same type of ambiguity as “mathematic education” referring to an educational practice as well as to the study of this practice.)

In the following, I will look at interpretations of meaning in mathematics education that to a large extend have provided a de-politicisation of the issue, and I will refer to some overall philosophic theories of meaning that resonates with this de-politicisation. Contrasting such interpretations, I will provide a foreground interpretation of meaning, which helps to reveal the complex formattings of meaning in mathematics education. This interpretation will establish some terrain for a politics of meaning in mathematics education, which I will address through some examples.

1. Meaning as a de-politicised issue

Conceptions of meaning have been profoundly elaborated in mathematics education, and one can identify different approaches. For instance, one addresses meaning in terms of possible references to mathematical notions; another relates meaning to participation in inquiry processes; and one associates meaning to the possible uses of mathematics.1

Acknowledging referential theories, it was assumed that manipulatives and concrete material in general, would help in bringing meaning into elementary mathematics education. Attempts have also been made to establish meaning through a careful decomposition of mathematical concepts.2 Thus the meaning of, say, the notion of a function has to be established through a carefully elaborated route that introduces the notions set, ordered couple, and set of ordered couples, before reaching the very notion of function. Some discussions about the use of computers in mathematics education have also emphasised the importance of providing mathematical notions with illustrative references. It is has been pointed out that computers open possibilities for particular forms of concretisations of mathematical concepts.3

The critique of the New Math Movement included a critique of the referential theory of meaning, and inspired by the work of Hans Freudenthal meaning became related to participation

1 For broader presentations of perspectives on meaning in mathematics education, see, for instance, Howson (2005); Kaiser (2008); Kilpatrick, Hoyles and Skovsmose (Eds.) (2005); Otte (2005); Thompson (2013); and Vollstedt (2011).2 See, for instance, Bieler (2005); Dreyfus and Hillel (1998); and Hillel and Dreyfus (2005).3 See, for instance, Hoyles (2005); and Laborde (2005).

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in mathematical activities. Thus instead of taking departure in meaning of a notion, one needs to relate meaning to the participation in mathematical activities. This way, students’ experiences of meaning become related to their involvement in inquiry processes. This idea makes integral part of Freudenthal’s outlook and of the project of realistic mathematics. However, it developed in many directions, and the whole inquiry approach in mathematics education elaborates further on meaning in terms of participation.4

The use-oriented interpretation of meaning has many proponents. The general idea is that students would come to experience meaning in mathematics education when they become familiar with applications of mathematical notions and ideas. This could be in terms of mathematical modelling, but it could as well be in terms of selected examples of uses of mathematics. Thus the idea is to relate meaning of mathematical notions and techniques to the use of these notions and techniques.

Certainly there are huge differences in interpreting meaning in mathematics education in terms of references, participation or uses. However, different as they are, these three patterns of interpretation also reveal a principal similarity: They assume the apolitical nature of meaning. This, however, is not particularly related to mathematics education; it is as well a characteristic of classic philosophical theories of meaning. And let me briefly summarise some of the philosophic positions that turn the discussion of meaning into a de-politicised issue.

Platonism represents a most elaborated suggestion for a referential theory of meaning. The meaning of a notion has to be identified as the entity to which the notion refers, and the world of proper references makes up the world of ideas. Gottlob Frege, assuming a version of Platonism, elaborated a referential theory drawing on a distinction between sense and reference.5 While “sense” has to do with experiences, “reference” has to do with logic. When addressing mathematics, his idea was to concentrate on the domain of “references” and to leave aside “senses” as a physiological issue irrelevant for grasping the essence of mathematics. For instance, while the sense of the concept function refers to the idea of relating two variables, the reference of the same concepts is a set of ordered couples. This idea brought Frege to emphasise that mathematics in general concerns sets, and that the whole semantics of mathematics could be expressed through a proper theory of sets. The world of ideas that constitutes all possible references to mathematical concepts is the world of sets. This referential theory provided the basis for logicism, and for the attempts to construct mathematics on a foundation of logic. It provided as well a basic for the whole New Math movement, establishing set theory not only as the logical but also are the educational foundation of mathematics. Frege’s referential theory of meaning is strictly located within the domain of logic, which turns the discussion of meaning into an analytical enterprise.

In Tractatus, Ludwig Wittgenstein (1922) elaborated a referential theory of meaning in terms of a picture theory. This theory led to the formulation of the principle of verification according to which a proposition has a meaning if and only if one can point out a possible procedure for its verification.6 With departure in this principle, meaning became related to conditions for providing empirical justifications. The principle of verification has not had a direct impact on the discussion of meaning in mathematics education, but it has been defining for forming the philosophic discussion of meaning as a strictly de-politicised analytical issue.

4 See, for instance Jaworski (2006).5 See Frege (1960) as well as Dummet (1981, 1991).6 See, for instance, Hempel (1959).

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In Philosophical Investigations, Wittgenstein (1958) abandoned any referential theory of meaning. He suggested that we give up looking for possible references to concepts. Instead, the meaning of a concept should be discussed in terms of its possible uses. A notion gets its meaning through the language games in which it may operate. By this use-oriented interpretation of meaning, Wittgenstein introduced a shift in the philosophy of meaning. According to Platonism, to Frege, and to the principle of verification, the social has no role to play in identifying the meaning of concepts, while Wittgenstein’s suggestion established the social as crucial for interpreting meaning. Important, however, is to recognise that although the opening of the social dimension in theories of meaning in principal makes space for addressing politics of meaning, this was not the step taken by Wittgenstein. His use-oriented interpretation of meaning became developed largely within an analytic approach to philosophical issues. This theory of meaning was elaborated further through John Austin’s ideas about “how to do things with words” and John Searle’s speech act theory.7 But also from their analytical approaches there is a long way to go before reaching a politics of meaning.

Thus in philosophy, the referential, verificational and use-oriented theories of meanings also provide a de-politicising of the issue. One finds an direct resonance between a referential theory of meaning in mathematics education and a referential theory in philosophy. One also witness some resonance between a use-oriented theory of meaning in mathematics education and a use-oriented theory in philosophy, although being aware that in mathematics education the use-oriented interpretation drawn on many different sources. However, the parallelism between theories of meaning in mathematics education and philosophy is not complete: while a verificational theory of meaning is largely a philosophy enterprise the participatory theory of meaning is largely an educational enterprise. Together, however, we have to do with a de-politicisation of discussions of meaning.

In order to provide a re-politicisation of this discussion, it becomes necessary to provide new interpretations of meaning, and in the following I will suggest a foreground-interpretation.

2. A Foreground-interpretation of meaning

The notion of “politics of meaning in mathematics education” emerged to me while I was formulating an intentionality-interpretation of meaning.8 I operate with a close connection between the notion of intentionality and the notion of foreground, so I do not try to make any proper distinction between an intentionality-interpretation of meaning and a foreground-interpretation of meaning. However, in the following I will formulate the ideas in terms of foregrounds.9

ForegroundsWhile the background of a person refers to his or her social heritages and the context in which he or she has grown up, the foreground of a person refers to the scope of his or her future possibilities. Like the background, the foreground is structured through a range of parameters

7 See Austin (1962, 1970), and Searle (1983).

8 See Skovsmose (2016). 9 In the previous I have drawn on formulations from Skovsmose (2015), and in the following I will do so as well.

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and circumstances. Thus possibilities might be limited due to a variety of obstructions. It was a surprise to me to find an expression of this idea (although not using the notion of foreground) in the opening pages of The World Bank’s Equity and Development: World Development Report 2006.

“Consider two South African children born on the same day in 2000. Nthabiseng is black, born in a poor family in a rural area in the Eastern Cape province, about 700 kilometres from Cape Town. Her mother had no formal schooling. Pieter is white, born in a wealthy family in Cape Town. His mother completed a college education at the nearby prestigious Stellenbosch University.

One the day of their birth, Nthabiseng and Pieter could hardly be held responsible for their family circumstances: their race, their parents’ income and education, their urban or rural location, or indeed their sex. Yet statistics suggests that those predetermined background variables will make a major difference for the lives they lead. Nthabiseng has 7.2 percent change 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 than 1 year. Nthabiseng is likely to be considerably poorer than Pieter throughout her life. Growing up, she is less likely to have access to clean water and sanitations, or to good schools. So the opportunities these two children face to reach their full human potentials are vastly different from the outset, through no fault of their own.” (Word Bank, 2006, p.1).10

The future of Nthabiseng and Pieter, their foregrounds, are conditioned by a range of parameters expressing their life expectancy, possibilities for education, likely economic conditions and so on. We are faced here with a profound economic, political, cultural and religious structuring of their future possibilities. It is a structuring that as well may include deep layers of sexism, racism and of presumptions in general. This observation does not only apply to Nthabiseng and Pieter, but to everybody.

A foreground, however, is not a simple expression of statistical parameters; it is as well formed through the persons’ experiences of possibilities and obstructions. Thus a foreground also reflects the person’s expectations, hopes, fears and frustrations.

I have been talking about a person’s foreground, but in many cases one can think of a foreground as referring to a group of people. Thus many people might be submitted to the same set of parameters: one can, for instance, think of all the girls from Nthabiseng’s neighbourhood. But although we might find some general patterns of foregrounds shared by larger groups of people, the foregrounds continue to reflect individual perspectives as well.

One can talk about foregrounds in the plural, and this plurality might be considered an intrinsic characteristic of foregrounds. Thus it does not make sense to talk about the foreground of a person as if it were a well-defined entity. The person makes interpretations, changes interpretations, and comes to think of new possibilities. As part of a project presented in my book Foregrounds (Skovsmose, 2014), we interviewed a group of young people, around 14 years old. One of them told us that he was ready to start working in a year or so. One possibility was to go to help his brother who was working as a bricklayer. He also mentioned that he might move to live with his sister in a big city, and that his sister was married to a famous professional football player. Being there, he might get opportunities to enter such a career. Then, after a little

10 This quotation was also presented in Skovsmose (2011).

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hesitation, he added that what he really wanted was to become a model. In a most direct way, he expressed the multiplicity of his foregrounds, and it might be relevant always to think of foregrounds in the plural, also when we have a particular person in mind.

MeaningsThe basic idea in a foreground-interpretation of meaning is to relate meanings and foregrounds. Let me illustrate by an example.11

In a school in Rio Claro, the city in Brazil where I live, the mathematics teacher was planning to do some project work. She proposed different topics, and so did the students. They suggested working on surfing. The teacher, however, had some doubts about this proposal. The school was situated in a poorer neighbourhood in Rio Claro, located in the interior of the São Paulo State. The teacher knew it was most unlikely that the students had been traveling, and only a few, if any, might have ever seen the ocean. How could a project about surfing make any sense to them?

If we think of meaning as established through references to situations familiar to the students, the teacher’s worry seems justified. It would be difficult for the students to connect features from their environment with surfing; the project seems devoid of meaning. The conclusion appears the same if we think of meaning in terms of use – far away from the sea, surfing is a useless activity.

However, if we consider meaning as having to do with foregrounds, the interpretation becomes different. We have, for instance, to consider the students’ hopes and aspirations. We have to consider their imaginations, and surfing could make up fascinating images. Maybe the children from Rio Claro had dreams of getting to the ocean. Maybe they had watched programmes about surfing on the TV. Maybe they imagined becoming famous – Brazil has many famous surfers. Their foregrounds might be structured in all possible ways, and also informed by a number of stereotypes. Anyway, it is with reference to such structured foregrounds that the students experience meaning of the activities in which they become involved. In fact the surfing project was concluded with much dedication from the students.

Naturally, one can consider relationships between the activities in which the students are involved and their background. This is the basic idea ofwaht can be referred to as a background-interpretation of meaning. However, the basic idea of a foreground-interpretation of meaning is that students’ experiences of meaning first of all emerge from relationships between the activities in which they are involved and their foregrounds. (When I talk about experiences of meaning, I always have in mind experiences of meaning as well as of lack of meaning. Thus a foreground-interpretation of meaningconcerns students’ experiences of an activity as being meaning or meaningless.)

HorizonsThe Portuguese translator of my book Travelling Through Education (Skovsmose, 2005) had to solve the problem of translating foreground. There is no word in Portuguese that matches “foreground”, so some more metaphoric translations had to be considered. Maria Bicudo, who supervised the translation, suggested horizonte futuro, and in fact the idea of future horizons provides a relevant elaboration of the concept of foreground.

One can imagine an open landscape with the horizon appearing out there as a fine line dividing land and sky. One can think of a horizon as an almost natural phenomenon. However,

11 For presentations of this example see also Skovsmose and Penteado (2015) and Skovsmose (2015).

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this is not the conception of horizon I find most adequate for exploring foregrounds. Instead of relating horizons to open landscapes, one can think of horizons as they emerge to us trying to find our way around in a huge metropolis. In this case, the horizon is not any straightforward natural entity; it is instead a complex fabrication. We can look in one direction and find our view blocked by huge buildings, while in another direction we might see mostly cars. We might notice a pedestrian crossing which could bring us to the other side of the road. From there we could make a new turn and see what comes. Horizons within a metropolis are fabricated entities, and it is such fabrications I have in mind when I think of foregrounds in terms of horizons.

Within the metropolis, the horizons structure the meaning to our walking around. We can experience something as interesting and want to take a closer look at it. We may need a cup of coffee and try to find the closest cafeteria. We may want to buy new shoes and start looking for shoe shops. We can look forward to meeting a friend and consider how to get to his or her place as fast as possible. There are so many different ways in which meaning becomes constructed when walking around in a metropolis. And certainly, given the horizons of the metropolis many things simply do not make sense: we cannot turn left until we get to a street turning left; we cannot get into a bus before we reach the bus stop; and we cannot get into the shoe shop if we arrive after the closing hour.

Like horizons so also foregrounds make part of fabricated topologies, and politics of meaning become addressed when we start paying attention to such fabrications.

Identities and foregroundsThe notion of identity has been explored profoundly in sociology, and this notion can be related to the notion of foreground. One could think of identity as something genuine for a person. This is, however, not the conception I want to refer to. Rather, I think of identity a highly elaborated feature. Such a conception has, for instance, been presented in the book Identity, where Zygmunt Bauman (2004) elaborates his ideas through a conversation with Benedetto Vecchi. I will quote a passage from the book, where I, however, substitute the word “identity” with “identity/foreground”. Then the quotation runs as follow:

“At one end of the pole of emergent global hierarchy are those who can compose and decompose their identities/foregrounds 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/foreground choice has been barred, people who are given no say in deciding they preferences and who in the end are burdened with identities/foregrounds enforces and imposed by others; identities/foregrounds which they themselves resents but are not allowed to shed and cannot manage to get rid of. Stereotyping, humiliating, dehumanizing, stigmatizing identities/foregrounds …” (Bauman, 2004, p. 38, italics in the original)

Here we find a clear expression of the idea that identities are constructed, and can be constructed in radically different ways for different groups of people. Exactly the same is the case with respect to foregrounds. Thus, even though I do not claim that identities and foregrounds are the same, I find that Bauman’s observations about identities can be applied to foregrounds. In particular we have to acknowledge that foregrounds need not be prosperous terrains spreading out in front of the person; instead they can be imposed by others and be stereotyping, humiliating, dehumanizing and stigmatizing. Such an observation brings us directly to the

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politics of meaning, as it is with reference to any such fabricated foregrounds that a person experiences meaning or lack of menaing.

3. Examples of politics of meaning

In order to explore politics of meaning it becomes important to consider how foregrounds become formed through a range of social, economic, cultural and religious factors including layers of visons, assumptions, presumptions and preconceptions. Through some examples, I am now going to illustrate features of politics of meaning in mathematics education.

Sexism and meaningThe conditions for a woman taking a walk in a city have changed profoundly during time. These conditions are set by social, economic, cultural and religious factors. It might be simply forbidden for a woman to walk around in public places without being accompanied by a man. It might also be considered inappropriate, if not suspicious, for a woman to walk around alone. In the modern metropolis the risks of being assaulted have to be considered; this applies to both women and men, although not necessarily in the same way. The topology of the metropolis refers to the physical structuring of space as well to social, economic, cultural and religious structurings, expressed through rules, costumes and priorities. Conditions for taking a walk are formed within these topologies, and so are the experienced of meaning of taking a walk.

This applies to all kind of directing oneself, also with respect to mathematics. During the 1960s, Denmark and the rest of the Western World witnessed a period of technological optimism. However, still during this perieod one met a profound exclusion of women from completing further studies in technical disciplines, including mathematics. At that time it was still not easy for women to walk in the direction of mathematics-dense studies. The dominant topologies of the time excluded mathematics from women’s horizons.

In order to interpret girls’ priorities and achievements with respect to mathematics, one can refer to their backgrounds. Thus one can refer to the ways girls are positioned in the family and the cultural traditions they become subjected to; and certainly one can obtain a range of interesting interpretations this way. However, I want to pay particular attention to the formation of girls’ foregrounds, and to the observations that these foregrounds can be imposed by others and be stereotyping, humiliating, dehumanizing and stigmatizing. Giving such impositions, girls come to experience meaning, of lack of meaning, in engaging with mathematics.

Thus in order to interpret girls’ achievements in mathematics, it becomes important to consider the political formatting of the topologies within which their foregrounds and their experiences of meaning become acted out.

Racism and meaningWith reference to South Africa, one can talk about the topology of apartheid. This topology concerns the structuring of the physical space; it concerns as well an ideological structuring provided by rules, traditions, preconception as well as by explicit racism. This topology was implemented during the apartheid period, although certainly anticipated by the whole colonial system.

The physical structuring concerns the segregation of different groups of people. Do we consider the city of Durban, which I have visited several times, one finds the black townships

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scattered around the city, while its centre was defined as a white area. The Indian areas were located as buffer zones between the black townships and the white areas. However, this structuring does not only concern the distribution of residential areas. It concerns the location of work places; it concerns the distance different groups have to travel to work; it concerns the organisation of the whole transport system; it concerns the location of hospitals and the distribution of schools. The topology concerns the distribution of any kind of possible services; and it concerns the distribution of wealth as well as of poverty. The ideological structuring concerns the priorities that constituted the overall racist world-view. It concerns the way one sees each other; the way one addresses each other; and what one expects from each other. The ideological structuring formed the explicit rules of the apartheid regime; it provided as well the legitimisation of the physical structuring of the country.

The apartheid regime has come to an end, but to a great extent the topology of apartheid remains. Thus the structuring of the physical space can only be changed during a longer period of time. One can only change location of neighbourhoods, the distribution of work places, and the infrastructure as part of long-lasting initiatives. The explicit rules of apartheid were changed when South Africa became a democracy. However, like the physical structuring the ideological structuring also demonstrates a considerable inertia. As a consequence, South Africa still suffers from a topology of apartheid.

Nthabiseng and Pieter are two children located within this topology, although at very different poles. This has a huge impact on the formation of their foregrounds and, as a consequence, of what they might experience as being meaningful or not. This also applies to mathematics education. What might they expect with respect to further education? What job opportunities could they imagine? What uses of mathematics might they come to experience in the future?

Acknowledging Bauman’s formulation we might find that Nthabiseng belongs to the group of students who are given no say in deciding about their possibilities. She might be located within a topolity which which makes space for only rather resericted foregrounds imposed by others. It can be foregrounds that she herself cannot manage to modify significantly. It could be humiliating and dehumanizing foregrounds. In this sense her foregrounds might be ruined due to the topologies within which they are located. Still her experiences of meaning are formed through such foregrounds. Pieter might be positioned at the other end of the pole. His parents’ economy might allow him to draw from a planet-wide pool of offers. This forms his foregrounds, which in turn represent his conditions for experiencing meaning, also when engaging with mathematics.

Students’ experiences of meaning have to do with the formation of the life-worlds in which they are situated; it has to do with the horizons that these life-worlds open in front of them.12

Instrumentalism and meaningStieg Mellin-Olsen (1981) describes instrumentalism as a rationale for learning mathematics not related to the content of the learning, but to the benefits that can be achieved through the learning. Thus one can try to master some mathematical techniques, not simply in order to understand mathematics better, but to be able to pass a coming test.

12 I do not use the notion of life-world as prescribed in phenomenology. I do not see life-worlds as existing a priori to social structurations. Instead I see them as a result of profound structurations (see Skovsmose, 2014).

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Experiences of meaning reflect the students’ foregrounds, and so do forms of instrumentalism. Foregrounds can include features of what one wants to achieve; they can include illusions; and they can be formed through layers of stereotypes. As foregrounds, so also forms of instrumentalism reflect the complexity of the topology in which the students are acting.

In the USA, the Algebra Project was organised by Bob Moses.13 The aim of the project was to improve the quality of mathematics education in poor communities and to provide better access to further education for black students. Mathematics exercises a powerful gatekeeping function, and Moses saw algebra as playing a particular role for students passing through the gate. He wanted to ensure that black students were not obstructed in their career opportunities by low scores in mathematics. In order to overcome such obstructions it became crucial to engaging back students in the existing curriculum; which forms the logic of the gatekeeping – and the aim of the Algebra Project was to help black students mastering this logic.

Black students participating in the project might experience meaning in what they were doing. This experience can hardly be interpreted in terms of traditional theories of meaning applied in mathematics education. We will get only a superficial understanding if we address the meaning associated to the Algebra Project in terms us references, participation or uses. Rather we have to pay attention to the opportunities that might emerge in front of the students. The meaning has to do with the students’ hopes, priorities and imaginations; it has as well to do with overcoming fears and aversions. In fact one can easily imagine that Nthabiseng would experience much meaning by participating in the Algebra Project.

I follow Mellin-Olsen in assuming that instrumentalism is an important phenomenon to consider with respect to mathematics education. However, I do not assume that instrumentalism as such represents a questionable attitude. Instrumentalism is a complex phenomenon; it has to do with experiences of meaning as related to the topology within which the students are operating. It has to be studied with reference to the complexity of the students’ life-worlds. The school mathematics tration and meaningThe school mathematics tradition can be characterised in the following way: (1) The activities in the classroom are first of all defined through the chosen textbook. The teacher provides an exposition on a particular topic, which defines the tasks for the students. (2) The pre-formulated exercises play a dominant role, as solving exercises is considered essential for the learning of mathematics. These exercises demonstrate three particular characteristics: all the information given is exact and should not be questioned; all the information given is necessary for solving the exercises, and also sufficient as no other information is needed; the exercises have one and only one correct answer. (3) One important feature of classroom practice is to eliminate errors, as “doing things correctly” is considered equivalent to “learning mathematics”. (4) The students’ performance has to be evaluated, for instance through the teacher’s questioning approach; through the teacher’s control of the students’ solutions of exercises; and through different forms of tests.14

A reasonable guess could be that when students, taught according to this tradition, have reached their 15th year, they have been presented for around 10.000 of such exercises. This sequence provides a deep formation of the students’ conception of mathematics. It becomes demonstrated again and again that mathematics provides a unique solution to a problem; thus it is possible to make a complete list of correct answers to all the 10.000 exercises. As a

13 See Moses and Cobb (2001).14 See Skovsmose and Penteado (2015).

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consequence the school mathematics tradition points towards an ideology of certainty, which claims that mathematics-based solutions to a problem demonstrate necessity, objectivity, and neutrality.

One could claim that the school mathematics tradition turns mathematical activities into meaningless exercises. However, one could also consider the possibility that the school mathematics tradition exercises a particular politics of meaning. Within this tradition, particular forms of instrumentalism can be stimulated. It can come to be experienced as meaningful, at least by some students, to complete the designated exercises, to try to provide the correct answers, to provide prompt answers to the teacher’s questions, and so on. Thus I do not see a particular activity as being meaningful or meaningless by itself. Rather, experiences of meaning are constructed, and also the school mathematics tradition exercises a partical politics of meaning.

Critical mathematics education and meaningHowever, there are radial different ways of exercising a politics of meaning, as for instance proposed by critical mathematics education. I do not differentiate between critical mathematics education and mathematics education for social justice, so it is a rather broad trend in mathematics education that I have in mind.15

In critical mathematics education one finds suggestions for project works addressing for instance: pollution, violence, social exclusion, distributions of income, sexism and racism.16 The introduction of such issues in mathematics education certainly represents concerns for establishing meaningful mathematics education. Thus it has been broadly assumed that socio-political significant examples become experienced as meaningful by the students. Without any hesitation I find that such examples are relevant. However, we cannot assume that political issues and topics concerning social injustice automatically become experienced as meaningful by the students. Nor in this case we can associate meaning as a property of the issues that become addressed.

When students perceive something as being meaningful, we have to do with a constructed perception. So when projects about elections, immigration, minimum wages, etc. may be experienced as meaningful by the students, it has much to do with the complex interactive processes that relate such issues to their foregrounds. Thus a foreground-interpretation of meaning brings us to acknowledge the importance of negotiating meaning. Such a negotiation makes part of a critical mathematics education. When issues of social justice and injustice come to make part of a mathematics education, we have to do with an example of a politics of meaning worked out through a critical mathematics education.

Banality of expertiseIn Life is Beautiful we listened to an application of mathematics, the horrible context of which was simply ignored. The only thing paid attention to was the pattern of the calculations. This way a politics of meaning becomes exercised not only through what is put into focus, but also through what is ignored. Foregrounds are defined through horizons, and a politics of meaning can be exercised through what is established within the horizon and what is put below the horizon.

15 See, for instance, Skovsmose (2011, 2014b).16 See, for instance Gutstein (2006, 2012). Bartell (2012). Peterson (2012) and Powell (2012).

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Putting beneath the horizon is an important process accompanying the students’ engagement with the 10.000 exercises. Students may have solved all these exercises without really addressing critically any of the 10.000 contextualisations. For solving any such exercise in accordance with the logic of the school mathematics tradition, one just has to remember that the information provided in terms of numbers is exact, necessary and sufficient; and that the exercise has one and only one correct answer. However, all this constitutes a politics of meaning.

Such a politics concerns school mathematics, but certainly also more advanced applications of mathematics. Sometimes I have characterised this politics in terms of a banality of expertise. This banality operates by lowering a lot of issues beneath the horizon. Thus when mathematics becomes applied as part of a particular expertise, one may concentrate on the very mathematic calculations, while the complexity of the situation with respect to which mathematics is applied is pushed below the horizon.

I have talked about the banality of expertise acknowledging Hannah Arendt’s observation with respect to the trail of Adolf Eichmann in Jerusalem in the beginning of the 1960s. Arendt wrote the book Eichmann in Jerusalem: A Report on the Banality of Evil about the whole process. Eichmann was responsible for organising the transport of Jewish people to Auschwitz: a most complex administrative task, which however Eichmann managed with an exceptional expertise. As Arendt pointed out, Eichmann did not appear as any evil person raging against the Jewish, nor did he in any other way demonstrate Nazi brutality. Rather, he appeared as a simple office assistant, who had been doing his job in a dedicated way obeying orders and regulations. To Arendt this is an example of a banality of evil. The evil is not exercises by persons dedicated to brutality, but as part of an expertise.

An expertise can be exercised through mathematics when, for instance: calculating the consequences of a company outsourcing certain parts of its production; estimating the possible impact of a certain promotion; providing the techniques for making drones operate; constructing algorithms for pattern recognition; and estimating the possible profit by not bringing a medicine at the market until the demand has increased sufficiently. To work with any such calculations mathematical expertise is necessary. The banality of mathematical expertise becomes manifest, when the mathematical issues become located within the horizon, and the context of the calculations becomes lowered beneath the horizon.

A banality of mathematical expertise is an example of a politics of. One can in fact see this banality as an advanced equivalent of the politics of meaning exercised by the school mathematics tradition.

4 Politics of meaning as a research area

We have seen examples of different politics of meaning in mathematics education. We have, for instance, related such politics to girls’ achievements in mathematics, and considered how the socio-political context establishes a selection of possibilities within girls’ horizons; and this selection might exclude mathematics.

One can associate a politics of meaning to the Algebra Project. Thus, a politics can resonate with an instrumentalism. However, we have to do with several versions of instrumentalism. One such version refers to a situation where students that have suffered social exclusion and injustices get the opportunity to come to master the logic of schooling. This opens

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new possibilities for them, it help to expand their foregrounds, and they might find such activities to be meaningful.

One can relate politics of meaning to the concerns for critical mathematics education. One cannot assume, however, that addressing forms of social suppression or economic inequalities automatically will be experienced by students as meaningful. The meaningfulness of such activities has to be constructed, and also this makes part of a politics of meaning. Another politics of meaning can be related to the school mathematics tradition, and one might find that a similar politics becomes exercised in advanced studies of mathematics, leading to a banality of expertise.

Thus there is a range of politics of meaning to be considered. To address examples of politics of meaning is an important research issue in mathematics education. I will refer to this research area as the Politics of Meaning in Mathematics Education. Thus I want to suggest this research area as crucial for the further development of mathematics education.

Acknowledgments

This paper is an extended verison of “Politics of Meaning in Mathematics Education: Short Version” that was presented at the Topic Study Group 53, Philosophy of Mathematics Education at the International Congress on Mathematics Education (ICME 13), July 24-31, 2016, Hamburg. A revised version of “Politics of Meaning in Mathematics Education: Short Version” is planned to appear as “Meaning in Mathematics Education: A Political Issue” in Revista Eletrônica de Educação Matemática edited by Maria Bicudo. I want to thank Ana Carolina Faustino, Denival Biotto Filho, Peter Gates, Renato Marcone, Raquel Milani, Amanda Queiroz Moura, João Luiz Muzinatti, Miriam Godoy Penteado, and Guilherme Henrique Gomes da Silva for their helpful comments and suggestions.

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