Travelling Through Education
Uncertainty, Mathematics, Responsibility
Travelling Through Education
Uncertainty, Mathematics, Responsibility
Ole Skovsmose
Department of Education and Learning
Aalborg University
Denmark
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Contents
Acknowledgements
vii
Introduction
ix
Part 1: Mathematics education is everywhere
1. ‘Cinema Paradiso’
2. Mathematics education makes wonders
3. In the classroom
4. A cultural perspective
5. The politics of learning obstacles
6. Mathematics education is everywhere
7. Globalisation
8. Knowledge processing
9. Ghettoising
10. Mathematics education is critical
11. Critical mathematics education
1
4
8
13
17
21
25
28
33
38
43
Part 2: Mathematics in action
12. Ideology of certainty and virtual reality
13. Epistemic transparency provides certainty
14. Transparency and progress
15. The assumption of progress
16. Gentle and clean?
17. Modelling as picturing
18. The picture theory
19. Mathematics everywhere!
20. Mathematics in action
21. More mathematics in action
22. Three aspects of mathematics in action
23. Mathematics and power
24. Speechless
48
50
56
65
68
71
74
77
79
82
86
91
96
Part 3: Aporia
25. The paradox of reason
26. Technology
27. One- and two-dimensional thinking
28. The apparatus of reason
29. Critique seems impossible
v
100
103
107
112
116
TRAVELLING THROUGH EDUCATION
30.
31.
32.
33.
34.
35.
36.
Aporia
Bad faith
Critique: A soluble concept?
Challenges to social theorising
Challenges to the philosophy of mathematics
Challenges to mathematics education
Modernity and the Holocaust
120
123
128
133
137
140
143
Part 4: Mathemacy can mean hope
37. Headache
38. Radical constructivism
39. Rational constructivism
40. Mathematics can mean many things
41. Facing an aporia
42. Mathematics can be real
43. Knowledge can mean action
44. Reflections can be public
45. Learning can mean dialogue
46. Learners can be noisy
47. Conflicts can set the scene
48. Mathemacy can mean hope
49. Ghettoising can never be ignored
50. Globalisation is all over
51. Explosive concepts
52. ‘Cinema Paradiso’, long version
148
151
155
159
163
166
170
173
176
179
183
186
188
192
195
197
Recycling
202
Some summing up
209
References
217
Name index
237
Subject index
241
vi
Acknowledgements
Many people have made comments and suggestions for the improvement of the manuscript. My main debt of gratitude is to Helle Alrø for
her support in developing the notion of dialogue as part of a critical
epistemology; Mathume Bopape for making me ‘see’ things I have
never seen before; Herbert Khuzwayo for specifying what racism in
educational research could imply; Paola Valero for helping in clarifying
the critical relationship between mathematics education and democracy;
Renuka Vithal for pointing out the complexity of the notion of culture;
Keiko Yasukawa for clarifying the notion of ‘mathematics in a package’;
and Miriam Godoy Penteado, my wife, for her overall concern for the
manuscript as well as for the author of this book.
I also wish to express my gratitude to Sikunder Ali Baber, Irineu
Bicudo, Morten Blomhøj, Marcelo Borba, Jessica Carter, Ole Ravn
Christensen, Kathrine Krageskov Eriksen, Gail FitzSimons, Núria
Gorgorió, Tom Børsen Hansen, Arne Astrup Juul, Lena Lindenskov,
Rasmus Hedegaard Nielsen, Núria Planas, Diana Stentoft, John
Volmink and Tine Wedege for their critical and constructive comments;
and Leone Burton for, as well, so carefully bringing my English into
shape.
In fact now considering this extensive support I have received, I
cannot but wonder how I could have spent such a long time writing
this book. I started in 1999, and since then I have added, changed,
deleted and added again to the manuscript.
Over that period bits and pieces of the manuscript have been
elaborated into individual papers and presentations. ‘Mathematics in
Action: A Challenge for Social Theorising’ was presented at the 2001
Annual Meeting of the Canadian Mathematics Education Study Group.
This paper has also been published in a Portuguese translation in M. A.
V. Bicudo and M. C. Borba (Eds.), Educação Matemática: Pesquisa em
Movimento, São Paulo, Cortez Editoria, 2004; in a Greek translation in
Themata stin Ekpaídefsi, 4(2-3), 2004; and in a revised form in the
Philosophy of Mathematics Education Journal, (18), 2004. ‘Students’ Foreground and the Politics of Learning Obstacles’ has been presented at
the Second International Congress on Ethnomathematics in 2002. A
Portuguese version of the paper is published in J. P. M. Ribeiro, M. do
Carmo S. Domite and R. Ferreira (Eds.), Etnomatemática: Papel, valor e
significado, São Paulo, Zouk, 2004; and a revised version appeared in For
the Learning of Mathematics, 25(1), 2005. ‘Ghettoising and Globalisation:
vii
TRAVELLING THROUGH EDUCATION
A Challenge for Mathematics Education’ was presented at the XI InterAmerican Conference on Mathematics Education, 2003.
The whole study has been carried out as part of the research initiated
by the Centre for Research in Learning Mathematics, and it is part of
the project ‘Learning from Diversity’, which is supported financially by
the Danish Research Council for the Humanities and Aalborg
University.
Aalborg, February 2005
Ole Skovsmose
viii
Introduction
In 1993 I was on my way to South Africa, to attend the conference
‘Political Dimensions of Mathematics Education’.1 The academic
boycott of South Africa had just been lifted. It was my first visit to the
country. At that time the process of democratisation seemed to have
been secured. Nelson Mandela was free, and free elections were
pending. But fear was also present: Could the apartheid regime still
somehow strike back?
Since the mid-1970s I had been working on the development of
critical mathematics education, and in 1993 I was completing the draft
of Towards a Philosophy of Critical Mathematics Education (published in
1994). The presentation in this book is first of all based on interpretations of experiences from a Danish school context. Going to
South Africa was the first time I was going to talk about mathematics
education in a political and cultural situation so different from the one I
knew. I had read quite a bit about the historical and political development in South Africa and about the terrors of the apartheid regime, so I
felt somewhat prepared.
I was due to arrive in Durban a few days before the ‘Political
Dimensions of Mathematics Education’ was scheduled. A small oneday pre-conference in Durban had been arranged, and I was also
invited to give lectures at other places as well during my first days in
Durban. I was well-prepared, I had all my transparencies in my hand
luggage. Nevertheless, I felt uncertain. For what was I in fact prepared?
I was met at the airport and taken to a hotel at the beach front in
Durban, where a row of good hotels was facing the Indian Ocean with
the waves rolling in towards a sandy beach. I stepped out of the car. I
have no idea how I managed to do it, but I slammed the door of the car
on one of my fingers. It did hurt, and the finger was bleeding. People
were so concerned. I went to the bathroom, while people got me
checked in at the hotel. Later, when we were sitting in the lobby talking,
I greatly appreciated the glass of iced water in which I could cool my
finger. People insisted on taking me to a doctor.
– No. No, that is not necessary, I said. But people and the pain in my
finger soon convinced me that it might be a good idea anyway.
The surgery was positioned in a township far away from the
beachfront and the row of hotels. We went away from the ‘white’
neighbourhoods of the city. Probably we passed some Indian neigh1
See Julie, Angelis and Davis (Eds.) (1993).
ix
TRAVELLING THROUGH EDUCATION
bourhoods that, as I later came to know, were organised as bufferzones between the ‘white’ areas and the ‘black’ townships. We turned
into neighbourhoods that I would not have experienced, were it not for
my bleeding finger. I could see the red dust from the sandy road being
whirled into the air behind us as we drove deeper into the townships.
We saw many groups of black people, apparently waiting for something. They looked at the car passing by, indifferent to the red dust. I
saw how houses seemed to grow smaller and smaller, before finally
taking the shape of huts made up of black plastic bags and pieces of
wood scraped together. People walked along the roads, some of the
women carried heavy bunches on top of their heads.
– Shall I pull off the nail? the doctor asked. – It will fall off anyway,
but if I pull it off now the new nail will grow out better, he continued.
I did not think it was really necessary for him to take the trouble to
pull off anything, so finger, nail and pain were enveloped in a carefully
elaborated white bandage, and I returned to the hotel. It was positioned
in front of the previously ‘whites only’ part of the beach.
That evening in my hotel room, I changed my well-prepared talks
and my overheads. What I had seen through the window of the car on
my way to and from the surgery, made it clear to me that I had to
present things in a quite different way. The examples and references I
had selected from the Danish context and which I found could be
interesting did not appear so interesting and so relevant any longer. My
perspective of critical mathematics education started to change.
In one lecture the following day, I referred to the article ‘Education
after Auschwitz’ (‘Erziehung nach Auschwitz’) by Theodor W. Adorno.
In the opening sentence of the article, Adorno states that the very first
demand of education is that an Auschwitz shall never happen again.
This statement can be taken quite literally, and with his Jewish
background, Adorno could associate strong and personal meanings to
formulating this demand. The statement definitely includes a critique of
German education, which did not establish any educational hindrance
to the sweeping success of the Nazi ideology. The formulation of the
demand can also be taken as a metaphor, claiming that education must
play an active role in social development. This brings us to critical
education: education cannot just represent an adaptation to the political
and economic priorities (whatever they might be); education must also
engage in political processes including a concern for democracy.
The meaning attached to this claim of critical education naturally
depends on the notion of democracy held by the claimant. I do not see
x
INTRODUCTION
democracy as only referring to procedures for election – although this
is an essential element of democracy. How essential becomes obvious
as soon as democratic elections are obstructed. Thus, the slogan ‘One
man, one vote’ had a strong political meaning in apartheid South Africa.
To me, democracy also refers to a ‘way of life’: to ways of negotiating
and making changes. Democracy refers to political procedures as well
as to forms of action in groups and communities.2
By means of the expression ‘Education after Auschwitz’, Adorno
signified a critical challenge: any education must prevent a new
Auschwitz from happening. The expression ‘Education after Apartheid’
also offers a challenge: Any education must prevent a new apartheid
from happening. The student uprising in Soweto 1976 demonstrated
the relevance of considering what education after apartheid could
mean. To me the expression ‘education after apartheid’ brings a new
dimension to critical education.
Critical education emerged during the 1960s, with much inspiration
from Critical Theory. Critical mathematics education originated during
the 1970s in a European environment, and during the 1980s a version
emerged in the USA. The notion of ethnomathematics developed in
Brazil, and after Ubiratan D’Ambrosio in 1984 presented the ideas at
the International Congress on Mathematical Education in Adelaide, the
ideas got wider attention and initiated a strong trend towards critical
mathematics education. But none of these different trends in critical
mathematics education applies directly to the South African situation.
Instead, I see this situation as a challenge to previously established
interpretations of critical mathematics education.
In Towards a Philosophy of Critical Mathematics Education, I included
several classroom experiences, which I used as resources for presenting
a critical perspective on mathematics education. In Dialogue and Learning
in Mathematics Education: Intention, Reflection, Critique, written together with
Helle Alrø, we also investigated different examples of classroom
practices. We do this with an intention of getting closer to a theory of
learning, which could resonate with the concerns of critical mathematics education. The present book, Travelling Through Education, cannot
be read as a direct continuation of my previous work. Here I concentrate on philosophic considerations; I want to establish a conceptual
‘sensitivity’ for a new critical mathematics education. This work was
2
For a presentation of a broader concept of democracy, see Valero (1999); and
Skovsmose and Valero (2001).
xi
TRAVELLING THROUGH EDUCATION
initiated by the broader perspective on educational issues that I started
to glimpse that day in Durban.
This is a notebook of bits and pieces from a conceptual travel. But,
in a different sense, it also represents a report on travel. While the main
part of Towards Philosophy of Critical Mathematics Education was drafted in
the protected environment of Cambridge, this present book has been
drafted in different places around the world. The main part of the
manuscript was written in Brazil, Denmark and England, whilst notes
have also been inspired by visits to other countries. So, the book not
only represents conceptual travel, it also reflects seasons of real travelling. And as real travelling means meeting people, so, during my travel,
have I benefited from the contributions of very many people (I am
afraid that I am not able to acknowledge them adequately).
In Part 1, I comment on the critical position of mathematics education, and also indicate some concerns of critical mathematics education.
In Part 2, I make comments on mathematics in action, and consider the
discussion of mathematics as an applied discipline in the contexts of
technology, management, engineering, economics, etc. In Part 3, I
comment on mathematics and science in general. I generalise these
comments into a discussion of ‘reason’ and of the ‘apparatus of reason’.
In Part 4, I return to the discussion of mathematics education, and
comment on notions that could become ‘sensitive’ to the critical
position of mathematics education.
It is possible to travel around the world so much that one may not
feel at home anywhere. One may simply lose sight of one’s roots. This
can be a problem, but hopefully it also has some advantages. I am
travelling between different academic fields. I touch upon mathematics
and mathematics education, but I do not deal with these areas as is
usually done in mathematics education. I will touch upon the philosophy of mathematics, technology and science, but not as is usual in the
philosophy of mathematic, technology and science. I address sociological issues, but I do not pretend that I am carrying out a sociological
study: I am only just glancing over issues such as globalisation,
ghettoising, the learning society, risk society. I seem to be travelling if
not simply jumping from here to there. But this is, anyway, what I am
doing. I find it important to become aware of connections between
many very different issues. In short, I want to illustrate that travelling,
in the academic fields as well, makes sense even when you lose sight of
your (academic) roots. Travelling also includes up-rooting.
xii
Part 1
Mathematics education is everywhere
1
‘CINEMA PARADISO’ is not only the title of a magnificent film, but
also the name of the glorious centre of a small Italian village: the
cinema where children, teenagers and adults experienced the full scale
of human emotions in a unifying and protective darkness. Outside
‘Cinema Paradiso’ reality was waiting. Part of that reality, for the
children, was the school and the mathematics teacher, who emphasised
the importance of understanding basic principles of arithmetic by
hammering a boy’s forehead against the blackboard. The boy had a
hideous discoloured mark on his forehead, which into his adult life
served as a reminder of his weak performances in mathematics. I see
the scene as if the instructor of ‘Cinema Paradiso’, Gieseppe Tornatore,
indicated that the mark was caused by the aggressive mathematics
teacher’s habits of hammering, against the blackboard, the heads of
those who showed a lack of mathematical understanding.
In The State Nobility, Pierre Bourdieu remarks that elite schools
“always attach a great deal of importance to subjects and activities that
are formal, gratuitous, and not very gratifying because they have been
reduced to mere intellectual and physical discipline: dead languages, for
example, treated as pretexts for purely formal grammar exercises rather
than as instruments that could afford access to works and civilization
… or today’s modern mathematics that, despite its apparent efficacy, is
no less derealizing and gratuitous than the former gymnastics of the
classics” (Bourdieu, 1996: 110-111).
For a mathematics educator, this is most depressing reading. It
appears that mathematics education serves a social function by
providing a stratification that may include the marking of students. This
stratification separates those who will get access to power and prestige
from those who will not. It is also remarkable that mathematics
education seems to provide a legitimisation of this stratification. Not
only is the stratification acted upon by authority, it is also accepted by
its victims as being, somehow, objective. To Bourdieu this justification
is part of ‘state magic’. The stratification is public as is it’s labelling. And
this labelling is incorporated into the life conditions of the students.
(However, when trying to draw conclusions from Bourdieu’s remark,
1
TRAVELLING THROUGH EDUCATION
we must remember that he refers to a French context in the past.) Prior
to this development in the institutionalisation of schooling, social
stratification was ensured through a system for establishing and
preserving hierarchies such as a class-system. However, the educational
system has now subsumed this function, and many studies have documented the role of mathematics education in this process. Not only
girls are ‘counted out’ but also other groups. There is no lack of studies
demonstrating the horrors mathematics education can support, if not
create.
Naturally, mathematics education can be organised in many different
ways within a particular society, and therefore it is not straightforward
to make any general conclusions about mathematics education. The
notion of ‘school mathematics tradition’ has been suggested as referring
to mathematics education as it takes place in ‘normal’ and ‘regular’
cases. This tradition is dominated by the exercise paradigm, and a whole
terminology has developed around it, drawing heavily on the metaphor
of travel (travelling along the apparently endless sequence of exercises):
– We are on time. – We are a bit behind, but we will catch up in the
end. The roles played by mathematics education can be understood
from the perspective of the many problematic social functions
exercised by the school mathematics tradition. Thus, most studies,
documenting problematic functions of mathematics education refer,
explicitly or implicitly, to this tradition in mathematics education.
Assuming that mathematics education (as structured according to the
school mathematics tradition) might produce hideous marks, it
becomes relevant to ask: Who is responsible for this? In The State
Nobility, Bourdieu refers to an investigation, which identifies ‘categories
of perception’ and ‘forms of expression’ used by mathematics teachers
to label differences in students’ performances. These categories and
forms of expression enable the teachers “to suppress or repress the
social dimension of both recorded and expected performances and to
dismiss any questioning of the causes, both those causes that are
beyond their [the teachers’] control, and are thus independent of them,
and those that are entirely dependent upon them” (Bourdieu, 1996: 1011). In other words, Bourdieu suggests that mathematics teachers
operate with a terminology that makes it possible for them to ignore
social aspects of students’ performances in school. Michael Apple
makes a related observation: “In the process of individualizing its view
of students, it [mathematics education] has lost any serious sense of the
social structures and race, gender and class relations that form these
2
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
individuals. Furthermore, it is then unable to situate areas such as
mathematics education in a wider, social context that includes larger
programs for democratic education and a more democratic society.”
(Apple, 1995: 331)3 Apple is not addressing the teachers in particular
but mathematics education in general. His point is, however, similar to
Bourdieu’s: The community of mathematics education manifests an
ignorance of social, political and cultural aspects in the lives of students.
I do not sympathise with Bourdieu’s formulations, if they point to
mathematics teachers representing the problematic link in mathematics
education. Although that is not to deny that teachers may participate in
and contribute to a discourse that suppresses the social dimension of
mathematics education. Nor do I agree with the generality of Apple’s
statement. There are some discussions in mathematics education that are
highly sensitive to issues of race, gender and class.4 Nonetheless much
research in mathematics education does ignore questions about the
socio-political functions of mathematics education. In fact, one
representative of socio-political blind research is found in the French
tradition in mathematics education. By concentrating on constructs
such as ‘didactical transposition’ and ‘learning obstacles’ (presented as
an epistemological entity), mathematics education easily turns context
blind, and the research does not support mathematics teachers in
interpreting, say, the politics of public labelling.
In Counting Girls Out, Valerie Walkerdine provides a bleak picture of
what mathematics and mathematics education might be doing: “We
have argued that modern government works through apparatuses like
schools, hospitals, law courts, social work offices, which depend upon
what Michel Foucault has described as technologies of the social: scientific knowledges encoded in practices which define the population to be
managed – not through simple and overt coercion, but by techniques
which naturalise the desired state in the bourgeois order: a rational
citizen who rationally and freely accepts that order and obeys through
‘his own free will’, as it were. Those knowledges, apparatuses, practices,
seek constantly to define and map processes which will naturally
produce this subject. They constantly define girls and women as
3
4
Lerman (2001b) also draws attention to this observation made by Apple. See also
Apple (2000). For discussion of broader socio-political issues related to mathematics
education, see, for instance, Skovsmose and Valero (2002b); Valero (2004); and
Valero and Zevenbergen (Eds.) (2004).
The notion of race is problematic. However, Apple does use it.
3
TRAVELLING THROUGH EDUCATION
pathological, deviating from the norm and lacking, but they also define
them as necessary to the procreation and rearing of democratic
citizens.” (Walkerdine, 1989: 205) In Discipline and Punish, Foucault describes both prisons and schools as technologies of the social, making
sure that the population can be managed. According to Walkerdine,
mathematics education is one such technique, which helps to ensure the
functioning of the social order, not by overt coercion, but by making
certain that rational citizens, using their ‘free will’, accept the imposed
order. One result of exercising the rational thought in mathematics and
mathematics education is that girls are ‘counted out’.
2
MATHEMATICS EDUCATION MAKES WONDERS. What happened in France in 1959 at the Royaumont seminar, organised and
financed by the Organisation for European Economic Co-operation
(OEEC), later to become Organisation for Economic Co-operation and
Development (OECD), is well documented.5 The initiative was provoked by the ‘Sputnik shock’, and the assumption was simply that the
links between mathematics and the technological upgrading of society
demanded that something radical must be done to improve
mathematics education. The mathematician Marshall H. Stone gave the
introductory lecture to that seminar. He said: “In fact, it is no longer
possible to treat adequately the place of mathematics in our schools
without going into its relations with modern science and technology.
Indeed, if there is a crisis in education at this time – and there are many
of us who believe so – it has arisen largely because no technological
society of the kind we are in the process of creating can develop freely
and soundly until education has adjusted itself to the vastly increased
role played by modern science in human affairs.” (OEEC, 1961: 17) He
continued later: “Thus the teaching of mathematics is coming to be
more and more clearly recognized as the true foundation of the
technological society which it is the destiny of our time to create. We
are literally compelled by this destiny to reform our mathematical
instruction as to adapt and strengthen it for its utilitarian role carrying
5
See OEEC (1961).
4
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
the ever heavier burden of the scientific and technological superstructure which rests upon it.” (1961: 18)
Could we imagine any more impressive picture: Mathematics and
mathematics education are carrying the scientific and technological
superstructure of our present society! A tremendous responsibility is
assumed and a heroic picture is painted of mathematics education as a
main vehicle for technological development. And when development is
interpreted in optimistic terms, which could easily be done in the late
1950s and during the 1960s, then the picture gets an impressive framing
as well. Stone emphasised that we must “adapt and strengthen”
mathematics education because of its “utilitarian role”. If we understand technological development as a simple and ultimate good, such an
adaptation and strengthening could be unproblematic, which was the
way it appeared to Stone and to many philosophers of technology at
that time. This was a period illuminated by technological optimism
claiming that adequate responses to social, political and economic problems could be found in a proper technological development. (Ecological catastrophes and the risk society were still waiting for the sociologists to discover and for everybody to experience.)
Stone indicated that a utilitarian argument for mathematics education
could be improved by an essentialist claim: intrinsic values of mathematics ensure that mathematics and mathematics education (carried out
in the proper way) provide cultural and technological progress. By its
very nature, mathematics education is a praiseworthy task. As a consequence, the best thing to do is to get down to business and identify a
curriculum that can bring students into mathematics. This idea was
clearly manifested at the Royaumont seminar, and Stone’s introductory
lecture was followed by a lecture by Jean Dieudonné, who, inspired by
the Bourbaki-terminology, outlined what he found to be an adequate
mathematics curriculum.6 This marked the start of the New Math
movement, emphasising the importance of introducing students to the
logical architecture of pure mathematics.
That mathematics can be seen as an ultimate good brings about
‘ambassadors’ for mathematics, Dieudonné being one of them. Such
‘ambassadors’ regard mathematics as an essential aspect of our culture,
a unique form of thinking and analysing, an indispensable conceptual
tool for our understanding of nature. Consequently, they believe that it
can be nothing but an ultimate good to bring students into this frame
6
Dieudonné’s lecture is published in OEEC (1961).
5
TRAVELLING THROUGH EDUCATION
of knowledge and thinking. According to this perspective a particular
pragmatic argument for educational reform in mathematics can easily
appear superficial (although it could be politically useful).
I see Elementary Mathematics from an Advanced Standpoint by Christian
Felix Klein as the work of such an ambassador. The original German
version was published in 1908. It contains two volumes: the first deals
with arithmetic, algebra and analysis, although not in any systematic
way; the second volume deals with geometry. In these two volumes
Klein provides a wide-ranging view of elementary (although not that
elementary) mathematics that, at the same time, demonstrates its
complexity. This presentation of mathematical topics from the “viewpoint of modern science”, as stated by Klein in the preface, is aimed at
teachers of mathematics in secondary schools; but certainly it turned
into a much broader invitation to engage with the mathematical way of
thinking. Klein provides an exposition of mathematical themes like the
fundamental theorem of algebra, equations with complex parameters,
trigonometric series, geometric transformations, projective transformations, etc. Furthermore, he presents and discusses different perspectives on mathematics, such as the axiomatic and formal approach
proposed by Peano; he discusses Dedekind’s cut and ways of introducing irrational numbers; as well as Hilbert’s presentation of the
foundation of geometry. Klein’s many historical observations also
include a careful exposition of the Euclidean organisation of geometry.
Klein’s masterpiece symbolises the claim that learning mathematics has
a value in itself. In this way Klein’s work can be related to essentialism
in mathematics education. This essentialism is, however, classic. It was
the grand topics in mathematics, developed during a historical process,
which define the knowledge that is an ultimate good with which mathematics education engages.
Dieudonné suggestions at the Royaumont seminar provide another
example of essentialism. As already mentioned, his lecture introduced
the so-called modern approach in mathematics education. It was a
direct reflection of Dieudonné’s active participation in the Bourbakigroup, and the structuralist outlook of Bourbaki.7 Basic to mathematics
are three mother-structures that could be described in the language of
set theory. The structural approach of Bourbaki provided a new
organisation of mathematics, and in particular ‘Euclid must go’, as
Dieudonné announced. The implication of this approach is that classic
7
See, for instance, Bourbaki (1950) and Dieudonné (1970).
6
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
organisations of mathematics in a variety of topics, as, for instance,
arithmetic, algebra, analysis and geometry, turn out to be old-fashioned.
The idea of basing the organisation of mathematical curricula on the
logical architecture of mathematics became appreciated the world over.
This idea came to represent a new form of structural essentialism.
Modern mathematics exemplifies that mathematics contains structures
which mathematics education can bring to students. And there appears
to be no doubt that bringing students and the essence of mathematics
together is simply a splendid thing to do.
Hans Freudenthal can also be seen as an ambassador of mathematics. However, he steadily claimed that the structuralism, advocated
by Dieudonné and many others, is basically problematic. The essence
of mathematics, according to Freudenthal, is not to be found in
mathematical structures, or in any piece of mathematical architecture,
but in the processes that brings about such structures. Freudenthal
might have been inspired by intuitionism introduced by L. E. J.
Brouwer, who saw mathematics as a mental activity. Formal structures
could only be dead plaster casts of living mathematical thoughts. To
Brouwer and to intuitionism in general, the essence of mathematics was
to be found in the processes of doing mathematics, i.e. in mathematical
thinking.8 This caused Freudenthal to initiate an important trend in
mathematics education stressing that mathematics is a human activity.
His idea has survived longer than structuralism. It is now taken, in
many countries, as establishing students’ thinking and mathematical
activities as the core of mathematics education. Freudenthal’s monumental work, Mathematics as an Educational Task from 1973 became a
trend-setter. However, maybe his idea that mathematics as an
educational task is valuable in itself is most clearly presented in his
Didactical Phenomenology of Mathematical Structures from 1983. The main
point of this book is to clarify the ways in which, within mathematics
education, students can grasp the essence of mathematics; and this
clarification constitutes, at the same time, reasons for doing so.
Pragmatic arguments for mathematics education always appear superfluous to an essentialist.
Nowadays many ambassadors of mathematics present the idea that
mathematics and mathematical thinking are worthwhile of themselves.
This is the main assumption of what I refer to as essentialism in
8
See Brouwer (1975a, 1975b).
7
TRAVELLING THROUGH EDUCATION
mathematics education.9 Naturally, there might be something problematic associated with certain forms of mathematics education – thus
Dieudonné could criticise the classical approach, and Freudenthal the
structuralist approach. In general, it is claimed that educational
problems emerge because of poor organisation of curricula or of
classroom practices that make it difficult to expose the essence of
mathematics. According to essentialist arguments, the way forward is to
identify what is essential in mathematics (whether structures or forms
of thinking) and make this apparent in mathematics education. Then we
can experience wonders in mathematics education. In recent days
mathematics education the essentialist line of thought is found in much
enthusiasm for mathematical details. It is difficult not to feel inspired by
this enthusiasm. And I appreciate this enthusiasm.
However, I have a worry. Essentialist thinking might tempt educators not to consider the broader socio-political context of mathematics
education. Certainly problems can be found, but according to an
essentialist line of thought, there are solutions to be found by
scrutinising the nature of mathematics more carefully, by digging into
the mathematics. Perhaps the essence of mathematics has not been
located properly, so a better grasp of what mathematics is about might
be necessary. Essentialism means that we can go to mathematics in
order to look for solutions. This brings about an enthusiasm, but also
an internalism, which makes a socio-political perspective superfluous.
3
IN THE CLASSROOM. There are many different mathematics
classrooms around the world. If we do not consider the experimental
classrooms, nor classrooms doing project work, and if we do not
consider unpleasant classrooms with a dominating teacher (like the one
we met in ‘Cinema Paradiso’), we are still left with the majority of
9
Here it is important to observe that, logically speaking, it is also possible to define a
‘negative’ essentialism, namely that what is done by mathematics education is
problematic because of the very nature of mathematics; and some may interpret the
remarks by Walkerdine, referred to previously, as pointing in this direction.
However, in my terminology I reserve the use of ‘essentialism’ for ‘positive
essentialism’.
8
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
mathematics classrooms. They represent that school mathematics
tradition to which I referred to previously.
This tradition has been characterised in different ways, and I am
making my own attempt. The school mathematics tradition is
dominated by the use of a textbook, which is followed more or less
page-by-page. Other kinds of material are used only as supplements.
The textbook sets the scene. All lessons are structured more or less in
the same way.10 One element of the lesson is that the teacher makes an
exposition of some theoretical ideas. This exposition is done as a
classroom plenary, where students often have the possibility to interrupt and to raise questions. A second element of the lesson is that the
students solve exercises, either individually or in groups. Normally these
exercises are formulated in the textbook. The number of exercises to be
solved, given by the teacher, is adjusted in such a way that not all of the
exercises can be solved in school, some of them have to be worked on
as homework. Some time is spent by the teacher correcting the
students’ solutions to the exercises. This can be done in the classroom
plenary, where students might present solutions at the blackboard; in
this case, the teacher has the possibility to offer other solutions or to
make more systematic expositions of solutions to difficult exercises.
Solutions of selected exercises can be handed over to the teacher in
written form (say once per week), and the teacher, then, has to hand
back the corrected exercises. The exercises are formulated in such a way
that each of them has one and only one correct answer. It should be
easy to correct solutions from a checklist. The school mathematics
tradition also involves the checking of students’ understanding of some
theoretical bits and pieces, which the students must explain, sometimes
being called to the blackboard to do so. Different forms of tests also
are part of the tradition; teachers may believe that tests may help them
to evaluate the students’ understanding of parts of the curriculum. The
school mathematics tradition has often been associated with an
unpleasant teacher, like the mathematics teacher in ‘Cinema Paradiso’,
but I do not want to make this association. I have knowledge of many
classroom settings with a comfortable, warm atmosphere, where the
schedule outlined above is followed. Naturally, minor variations can be
observed in the schedule, but the point is that the school mathematics
10
Contrary to this, when mathematics education is organised as project work, students
are, sometimes, working in groups, sometimes they are collecting data, sometimes
the teacher gives a lecture. The lessons might be very different from each other.
9
TRAVELLING THROUGH EDUCATION
tradition is represented by variations of the same organisational
structure.11
Let us look at some students from such a classroom. I do not want
to focus on students who are called good or excellent by their teacher;
nor on the students whom the teacher finds problematic, either because
they have special difficulties in mathematics or because they are noisy. I
want to look at students who do reasonably well in mathematics, who
do their homework regularly, although not always, who solve the
exercises as best they can. They might need some help, but with the
help of parents and friends they get through. In other words, I am
considering the ‘normal’, ‘regular’ or ‘average’ students, who often tend
to become invisible in the classroom. Groups of ‘silent girls’ are to be
found in the literature, but there are also ‘silent boys’.
These groups of ‘normal’ students may leave school and continue
into further education where not too much mathematics, in any
traditional sense of mathematics, will be expected. (That their later jobs
may contain quite a substantial part of implicit mathematics is a
different point.) They might become shop assistants, tax accountants,
salesmen, bus drivers, firemen, laboratory assistants; some will be
employed in insurance companies, some will work in industry; some
will become teachers, even mathematics teachers. How to describe the
mathematics education that these people received? What purpose did it
serve? Could we say that their mathematics education prepared them
for their particular job function? The answer to this is both yes and no.
The exercises they might have done could have taken the form: ‘Solve
the equation …’, ‘Construct a triangle with the sides …’, ‘Calculate the
difference between …’ Many times the order is not stated explicitly, but
an exercise like ‘324 + 2555 + 4556’ can be read as ‘calculate the sum
324 + 2555 + 4556’. The long sequence of exercises, characteristic of
the school mathematics tradition, can be seen as a long sequence of
orders that the students must follow.
If we take a look at overall descriptions of aims and intentions for
particular mathematics education programmes, we frequently find
statements about developing capacities in creativity, systematic thinking,
problem solving and communication. However, in reality, estimating
the total number of exercises that a student is supposed to solve during
primary and secondary school, we would probably arrive at a number in
11
A similar description of the ‘school mathematics tradition’ is presented in Alrø and
Skovsmose (2002); see also Richard (1991).
10
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
the region of 10,000. A student completing tertiary education and
proceeding with further studies including mathematics will complete a
considerably higher number of exercises. Anyway, let us take a look at
these 10,000 excises as a whole. Let us imagine that we read the whole
text aloud. It would sound like a long sequence of commands. It would
be difficult to hear an invitation to creativity in these 10,000
commandments. In what ways does this help students to grasp some of
the essence of mathematics?
According to many officially stated aims of mathematics education,
the idea of creativity and the importance of developing mathematical
competencies that can be used in everyday life situations are stressed.
Consequently, the school mathematics tradition, including its commands, appears to be a failure, certainly for the great number of
‘normal’ students. This tradition appears to represent a huge dysfunctionality in the educational systems. How could it be, then, that this
tradition has developed as a ‘tradition’? It seems to be a very expensive
social experiment, which goes wrong, year after year after year. How
could that be? Could it be that although the school mathematics
tradition appears a great mistake for the majority of students, the
tradition, nevertheless, can be deemed to be successful for the minority
of students who continue their studies and become engineers,
economists, dentists, computer scientists, mathematicians, etc.? Could it
be that mathematics education in fact acts as one of the pillars of the
technological society by preparing well that minority of students who
are to become ‘technicians’, quite independent of the fact that a
majority of students are left behind? Could it be that mathematics
education operates as an efficient social apparatus for selection, precisely by leaving behind a large group of students as not being ‘suitable’
for any further and expensive technological education?
Another possibility is that mathematics education, and in particular
the school mathematics tradition, might have other functions than
those of which we are normally aware. Could it be that ‘normal’
students in fact learn ‘something’, although not strictly speaking mathematics (and certainly not mathematical creativity), and that this
‘something’ serves an important social function? If we look back again
at the 10,000 commandments, what do they look like? Certainly, not
like any of those tasks with which applied mathematics occupies itself,
tasks in which creativity is needed to construct a model of a selected
piece of reality. Nor do they look like anything a working
mathematician is doing. However, they might have some similarities
11
TRAVELLING THROUGH EDUCATION
with those routine tasks, which are found everywhere in production and
administration. An accountant has to do sums day after day. A
laboratory assistant has to do a series of routine tasks in a careful way.
Numbers have to be read from measuring instruments and put into
schemes, and this must be done correctly.12 All such jobs do not invite
creative ways of using numbers and figures. Instead things have to be
handled carefully and correctly in a pre-described way. Could it be that
the school mathematics tradition is a well functioning preparation for a
majority of students who come to serve in such job-functions?
We have to be aware of the possibility, strongly indicated by
Bourdieu, that the actual social and political functions of a particular
mathematics education do not directly depend on the official part of
the curriculum but also on the social, political context in which the
schooling takes place. Although the mathematics curriculum may be
described in certain attractive terms, the actual socio-political function
of bringing students through this curriculum could still be to produce
and to legitimate a state nobility. However, we need not consider simply
the group of students who are successful in mathematics. We could
equally consider the group of ‘normal’ students, who are not going to
become celebrated as any ‘state nobility’. Mathematics education might
not only designate the ‘state nobility’, it might as well help to identify
‘state functionaries’. And doing so in an efficient way, could be the
grand (and hidden) success of the school mathematics tradition.
Nonetheless, a large group of students might be left, and they will have
learned a substantial lesson: that mathematics is not for them. To
silence a group of people in this way might also serve a socio-political
and economic function. So, the school mathematics tradition might also
squeeze out a group of ‘disposable people’, that must be satisfied with
whatever kind of job comes their way.
12
Lindenskov (2003) has argued that the carefulness and precision, which is a
precondition for solving exercises within the school mathematics tradition, represent
important competencies, when we consider the scarcity of resources during, say, the
1950s.
12
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
4
A CULTURAL PERSPECTIVE. Until 1993 I did not carefully
consider the notion of culture, but becoming involved in the South
African project changed this. What I, in brief, refer to as the South
African project is a collaboration between institutions in South Africa
and Denmark established shortly after 1994. A main aim of the project
was to create an environment for researching mathematics education
from a democratic perspective.13 Coming to know a country just leaving
the apartheid era made it possible to observe the extreme complexity of
the notion of culture. One’s doubts about what mathematics education,
and in particular the school mathematics tradition, might be doing were
deepened.
Previously, I had not been aware that culture could be used in a
negative way. Naturally, culture can refer to tradition and folklore. But
if people think I should represent Danish culture at an international
conference by dressing up in old Danish style, then I would feel
ridiculous. When expressed by the apartheid system, a notion like Zulu
culture could come to operate in an oppressive way.14 Zulu culture not
only refers to traditions, but may also constitute a trap. An ‘appreciation’ of Zulu culture was associated with the assumption that people
belonging to this culture stood outside Western development. The Zulu
culture could be picturesque and include, for instance, dances with
traditional weapons. The notion of culture could get a negative
connotation, referring to people ‘out there’ and ‘down there’. This
could support a justification that people with such a ‘different’ culture
had better stay in their homelands.15 To me, if cultural concerns result
13
14
15
The project has now come to a successful conclusion as Mathume Bopape, Nomsa
Dlamini, Herbert Khuzwayo, Maga Moodley, Anandhavelli Naidoo, and Renuka
Vithal have all obtained their Doctoral degrees. It has been a pleasure for me to
participate in the project.
See also the discussion of culture in Adler (2001a); Bopape (2002); Cotton and
Hardy (2004); and Vithal (2003).
The complexity of the notion of culture can be illustrated when we consider the
notion of ubuntu (in Zulu, in Sotho the word is botha). Bopape drew my attention to
this concept, and Dlamini has also helped me to clarify its different connotations.
Ubuntu refers to solidarity. Thus, it refers to the feeling of shared concern and
responsibility. Ubuntu is part of democracy as developed in the African traditions of
negotiation, which Mandela refers to in his Autobiography: Long Walk to Freedom.
Ubuntu is a word with attractive connotations, but it can also refer to a concern for
traditions and authority. It can include what many would consider an exaggerated
13
TRAVELLING THROUGH EDUCATION
in an unreserved cheering of traditions, this appears problematic. Thus,
I would want to distance myself from many trends that are said to
belong to the Danish culture. In paying attention to culture I want to be
wary of celebrating the traditional. If we think of the growing
reservation towards foreigners, inspired by the rhetoric of the right, this
has a tendency to become part of the general concepts of being Danish
and of being concerned with the preservation of Danish values.
‘Culture’ is a contested concept. Culture is changing and developing, it
includes a complex mix of new and old elements, both attractive and
problematic.
Mathematics education is part of changes in culture, and considering
the possible roles of mathematics education from a cultural perspective
raises uncertainty about how mathematics is part of social and
technological development. In particular, the ethnomathematical
tradition has opened the discussion of the relationship between mathematics education and cultural changes. Such studies indicate that
mathematics education serves a global function, and it could easily
become an instrument for cultural imperialism and come to represent
Western Culture. Ubiratan D’Ambrosio has emphasised the fact that
education, and mathematics education in particular, can be discussed in
terms of colonialism.16 The content and form of mathematics education
may express ideas, principles and ways of thinking which are highly
culturally bound, but at the same time may be foreign to the situation in
which actual mathematics education takes place. However, before I go
any further I have to present a reservation about the very notion of
‘ethnomathematics’. To connect ‘mathematics’ with the prefix ‘ethno’,
which in many languages and in many situations connotes ‘ethnicity’ is
to me problematic. Naturally, in the literature of ethnomathematics, it is
clearly stated that ‘ethno’ does not mean what ‘ethno’ often connotes.
‘Ethno’ simply refers to ‘culture’. I am not suggesting that there is a
problem with the ethnomathematical programme, rather I have a
problem with the word.17 What makes sense, and what is emphasised in
16
17
concern for authorities in a community. Ubuntu, thus, comes to represent a tension
between attractive and non-attractive features.
See, for instance, D’ Ambrosio (2001). Bauchspies (in print) discusses to what
degree learning and the learning of mathematics can mean colonisation. For more
general discussions of ethnomathematics, see, for instance, Gerdes (1996); Powell
and Frankenstein (Eds.) (1997); Ribeiro, Domite and Ferreira. (Eds.) (2004); and
Knijnik, Wanderer and Oliveira (Eds.) (2004).
See also Vithal and Skovsmose (1997).
14
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
the ethnomathematical literature, are the connections between culture
and mathematics.18
Authors of ethnomathematical studies have pointed out that what is
done by mathematics education from a global perspective can be
related to processes of colonisation. We can think of different steps in
the process of colonisation. One was initiated by the discoveries made
by the Portuguese, the Spanish, the Dutch, and the English. This
process was one of invasion and direct military suppression, accompanied by a cultural suppression in terms of a forceful introduction of a
new language and a new religion. With Brazil in mind, the suppression
included the physical movement of the Indians. New language and new
religion were accompanied by new schemes of production. Production
for whom? It is clear that the introduction of the coffee crop in Brazil
was not for the local market. The colonies represented a resource and a
supply for the colonising countries. The scientific way of thinking can
also be linked to processes of colonialism. And education in mathematics can equally be seen as an element of a cultural invasion. Thus,
Western mathematics has been described as representing Western
values and a dominant way of thinking. This brings a clear perspective
to the ethnomathematical approach, which includes an awareness of the
ways of thinking that are part of indigenous cultures.19
In ‘Western Mathematics: The Secret Weapon of Cultural Imperialism’, Alan Bishop refers to a textbook which contains the following
problems: “If the cricketer scores altogether r runs in x innings, n times
not out, his average is r/(x-n). Find his average if he scores 204 runs in
15 innings, 3 times not out.” And: “The escalator at the Holborn tube
station is 156 feet long and makes the ascent in 65 seconds. Find the
18
19
This implies that ‘engineering mathematics’, ‘mathematics for economy’,
‘mathematics in physics’, ‘mathematics in cryptography’ all represent different
branches of ethnomathematics, as does ‘Chinese mathematics’, the ‘mathematics of
the Incas’, the ‘mathematics of street children’, etc. However, recent ethnomathematical studies do not include many investigations of engineering mathematics, etc.
The ethnomathematical perspective easily can become problematic, if it includes a
simple appreciation of picturesque ways of thinking. Culture is a contested concept,
and acknowledging this Knijnik, in her studies of the mathematics of the
Movimento Sem Terra (The Movement of the Landless People), clarifies that the
culturally embedded way of thinking should be criticised and developed and acted
upon. This brings a very important aspect to the notion of ethnomathematics. See
Knijnik (1998, 2002, 2004).
15
TRAVELLING THROUGH EDUCATION
speed in miles per hour.” Working with such exercises has a very
different meaning to children in London than to children in Tanzania,
who in fact, during the British colonial times, were faced with such
problems. The use of a textbook that included such exercises was
recommended by the English colonial education officer. Thus, in
“India and Africa, schools and colleges were established which, in their
education, mirrored once again their comparable institutions in the
‘home’ country” (Bishop, 1990: 55). It was simply a deliberate strategy
to instruct ‘in the best from the West’, which certainly was assumed to
be superior to any other options.20
When an English textbook is used in Tanzania, the exercises and
their contextualisation could serve imperialism. In the most literal way,
students who are successful in this educational programme might
become well-adjusted state functionaries. In colonial times it was
particularly relevant for an empire to select and nominate such
functionaries carefully. So, from the point of view of running an
English empire, the colonial officer in Tanzania might have come up
with good recommendations. Stated in general terms: “So, it is clear
that through the three media of trade, administration and education, the
symbolisations and structures of Western mathematics would have
been imposed on the indigenous cultures just as significantly as were
those linguistic symbolisations and structures of English, French,
Dutch or whichever the European language of the particular dominant
colonial power in the country.” (Bishop, 1990: 56) Certainly, the
domination of language means domination of culture. Bishop’s point is
that the domination of other systems, like education and mathematics
education in particular, represents a similar domination. As part of the
processes of colonialism, designating ‘state nobility’, ‘state functionaries’
and ‘disposable people’ get a despicable significance. Through these
mechanisms, mathematics education can be characterised as a weapon
of Western imperialism.
20
To reconsider further Bishop’s claim that mathematics can be considered the secret
weapon of cultural imperialism, it is interesting to study Reynolds and Cutcliffs
(Eds.) (1997) where case studies related to technology and colonisation are
presented.
16
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
5
THE POLITICS OF LEARNING OBSTACLES. In 1954 Hendrik
Verwoerd made the following statement in his address to the South
African Senate: “When I have control over Native education I will
reform it so that the Natives will be taught from childhood to realise
that the equality with Europeans is not for them … People who believe
in equality are not desirable teachers for Natives … What is the use of
teaching the Bantu mathematics when he cannot use it in practice? This
idea is quite absurd.” (Quoted from Khuzwayo, 2000: 4) To Verwoerd,
the paramount task was to make sure that blacks did not get any access
to power. Exclusion from mathematics was part of this strategy. In
other words, learning obstacles can be explicitly established in the most
direct way. Here we are far away from the theoretical notion of learning
obstacles, analysed in terms of students’ preconceptions, if not misconceptions, of some mathematical notions and ideas. This epistemic
interpretation of learning obstacles is not the only one possible, and
Verwoerd’s statements emphasise that learning obstacles can take the
most direct form.
Verwoerd expressed strong ideas about education. In the case where
a political system exercises (illegitimate) power, it is essential to control
the educational system. Fundamental Pedagogy is an expression of the
necessity of controlling the educational system in order to control the
mind of people. Fundamental Pedagogy was the subject of obligatory
study for all teacher students (black, Indian, coloured, white) in
apartheid South Africa. Fundamental Pedagogy was the way in which
the state dictated the need to control the educational system in order to
control the minds of its population. The message of Fundamental
Pedagogy was that equality is not for blacks (including Indians and
coloured), and that believing in equality is not for teachers.21 In more
general terms: By means of education it is possible to ensure a
‘boundary’, an ‘apartheid’, not just in terms of ‘race’ but in terms of
‘achievement’ as well. What Bourdieu observed as a sociological fact,
Verwoerd expressed as a political strategy. In some baroque way we see
a clear statement of the social impact of mathematics education:
Excluding people from mathematics education upheld a social
exclusion.
21
For critical analyses of Fundamental Pedagogy, see Khuzwayo (2000); Vithal (2003);
and Bopape (2002).
17
TRAVELLING THROUGH EDUCATION
Let me illustrate what I mean by ‘the politics of learning obstacles’ by
summarising one aspect of white research in black education carried
out during the apartheid past of South Africa. Here the interpretation
of learning obstacles was a big issue, as some interpretations could help
to explain away the brutality of the apartheid regime. Racism was a
basic category, and we can easily identify the basic assumption of
‘classic racism’. In schools, the weak performances of black children
were accounted for by referring to certain ‘facts’. That black children
did not perform as well as white children had to be understood in terms
of biological structures, established thousands and thousands of years
over time. Certainly, such an explanation established a solid distance
between the apartheid regime and the causes for what was observed in
the classroom. In particular, it was proposed that children’s learning
obstacles have nothing to do with the school structure, and certainly
nothing to do with apartheid politics. These obstacles were to be found
in the black children themselves. These children brought their own
defeat along with them, right into the classroom. There was nothing
that could, or should, be done about it. Black children were inevitably
linked to their own bad performances, which were just a different
expression of their skin colour. The political dimension of school
performances was efficiently explained away by ‘classic racism’.
However, ‘progressive racism’ has also found its voice. The idea that
social aspects, rather than biological framing, play a fundamental part in
a person’s intellectual and emotional development led to new priorities
within white research into black education. Instead of searching for a
biological explanation for weak performances of black children, social
factors could be identified. In his study of research carried out at the
Orange Free State University, Herbert Khuzwayo (2000) investigated a
study from 1981 made by A. C. Wilkinson: ‘An analysis of the problems
experienced by pupils in mathematics at standard 5 level in the
developing states in the South African context’. The Wilkinson-study,
which represents ‘white research in black education’ explores problems
experienced by black students in mathematics. In this study the reason
for black children’s weak performances in school are sought in their
social background (and not in their ‘biological background’ as suggested
by classic racism). The study indicates an explanation in terms of family
traditions and, in particular, in terms of the dominant role of the father
in the black family. According to Wilkinson, this aspect of the family
helps to explain that the creativity, and also the mathematical creativity
of the black child, is ‘eliminated’. Thus, the structure of the black family
18
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
becomes a main factor in ‘explaining’ the black child’s weak performances in school.
As in the case of classic racism, it is not the school, nor any apartheid
politics, which has to be blamed for the weak performances of the
black child. The child, again, brings the cause for his or her weak
performances into school. Now the explanations are not in terms of
biological structures, but in terms of socially constituted psychological
patterns, which cause obstacles to creativity and mathematical thinking.
The problem is, thus, to be found in the cultural background of the
child. In other words, black culture produces the learning obstacles of
black children (and consequently not a suppressing white culture).
Black children’s problems in school are established in advance and
should not be located in the school structure. As the black children
themselves bring their own learning obstacles to school, the best the
school can do is to compensate for such cultural deficiencies.22
Bourdieu’s observations, as referred to previously, certainly make sense
in this context also. Racism establishes categories of perception and
forms of expression that “suppress or repress the social dimension of
both recorded and expected performances” and “dismiss any
questioning of the causes”.
In 1996, together with Mathume Bopape, I visited a school in a
South African township on the outskirts of Pietersburg. Bopape has
made a study of mathematics education in the most desolate parts of
South Africa, and he showed me what a school might look like. Broken
windows. Doors were missing. All electrical installations were missing
as well. There was a hole in the roof. Maybe the tiles had been removed
by somebody who found that his house needed the tiles a bit more than
the school building. When it was raining the children had to move away
from this part of the classroom. The classroom was either too hot, too
cold, or too wet. It looked like a place where teachers and students
would meet with a shared intention of leaving this ugly place as soon as
possible. What appeared to be the most obvious learning obstacle to
the children in this school: their skin colour, their dominant fathers, or
the hole in the roof? It was all too obvious: entering the classroom, the
first thing one noticed were the physical learning obstacles. And one
was there right on top of our heads.
22
For a critique of such deficient theories see, for instance, Ginsburg (1997); and
Gorgorió and Planas (2000, 2001).
19
TRAVELLING THROUGH EDUCATION
How is it that the research in mathematics education has not noticed
this hole in the roof? It is not mentioned in the research of children
with learning difficulties, as far as white research in black education is
concerned. In fact, much research seems able to explain away the
obvious: Black children are simply treated completely differently, and
their future has been spoiled by the apartheid regime. To ignore this
fact is a political act. Learning obstacles need not be sought in the social
background of the child. They can also be researched in the actual
situation of the children. The distribution of wealth and poverty also
includes a distribution of learning possibilities and learning obstacles.
This distribution is a basic political act. Paying attention to this means
re-establishing a politics of learning obstacles.
Unfortunately a dominant discourse in mathematics education
exemplifies Apple’s reference to the loss in mathematics education of
“any serious sense” of social structures that form the individual. Certain
patterns of research ‘help’ us to ignore simply collected data – as, say, a
hole in the roof – because we are used to interpret the performances in
school in terms of, primarily, the background of the children. I find it
problematic to understand the performance of somebody by referring,
first of all, to his or her background. This is a strategy by means of
which the political nature of learning obstacles can be eliminated. If we
instead try to understand performances in terms of both the
background, the here-and-now situation, as well as by the foreground of
the child, then the political nature of the learning obstacles becomes
more obvious. By the foreground of a person I understand those
opportunities that the social, political and cultural situation provides for
the person. However, not the opportunities as they might exist in any
‘objective’ form, but the opportunities actually perceived by the person
in question. I see the foreground as an important element in understanding a person’s intentions and actions. The intentions of a person
refer not only to his or her background, but also to the way he or she
experiences possibilities. Intentions express expectations, aspirations
and hopes. Because I see learning as action (not all sorts of learning but
some), it is not surprising that I relate students’ performances in school
not only to their background, but also to their foreground.23
The learning obstacles have not only to be sought in the historical
past of the person, but in the actual social constitution of that person as
23
For a discussion of ‘foreground’, see Alrø and Skovsmose (2002); and Skovsmose
(1994, 2005).
20
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
well as in the opportunities which the actual social and political system
make available to him or her. In particular, the apartheid system did
ruin the future of black children, and this could destroy black students’
incitement to learn. When a society has ‘ruined the future’ of some
group of children, then it has established learning obstacles. The
apartheid regime has come to an end, but the ghost of apartheid is still
in operation, and new ways of establishing differences have been set in
operation.
6
MATHEMATICS EDUCATION IS EVERYWHERE. This statement
may begin to make more sense, if we consider to what we can refer by
‘mathematics education’. There are classrooms all over the world where
mathematics is taught and learnt. Students listen to teachers’
expositions; they try to solve exercises. Students are given homework,
and parents easily become involved. Teachers participate in in-service
education in order to encounter new ideas and new material. When
students enter universities or technical colleges to study subjects such as
economy, engineering, land surveying, chemistry, pharmacy, physical
education, biology, astronomy, computer science, statistics, geology,
meteorology, natural science and, not to forget, mathematics, they meet
mathematics again.
Furthermore, rich sites for learning mathematic are found outside
the regular educational system. Mathematics is in operation in many
workplaces, in banking, in carpeting, in every shop. We are not
accustomed to think of the cashier at a supermarket as using mathematics. However, the automatic reading of the bar code and payment
by credit card presuppose that a huge mathematical apparatus is
operating. The reading of the bar code relies on a complicated
mathematics-based technical device that can be linked to automatic
steering of the stock. Using the credit card includes a good deal of
electronic communication, and mathematics security policy is applied.
Mathematics is condensed in programs and ready-to-use packages
installed in computers. The person using the system need not be, and is
rarely aware of the details of the mathematics operating behind the
screen. However, other types of competencies, also mathematical, are
important: – Could the price of these three items in fact be that much?
21
TRAVELLING THROUGH EDUCATION
Learning how to operate with systems that include mathematics
presupposes the learning of some mathematics, although it can be of a
very different nature from the mathematics in the packages. The
persons who have taught the cashier how to operate the cash register
may never have thought of this as an example of mathematics
education.
It has become widely recognised that mathematics is in operation in
many different workplaces. The discussion is raised by the very title of
the book What Counts as Mathematics? written by Gail FitzSimons. She
studies the technologies of power in adult and vocational education.24
In fact it is impossible to consider a workplace, where computers are in
use, without acknowledging that mathematics is in operation. In all
such situations teaching and learning of mathematics is taking place.
Ethnomathematical studies have helped to clarify that mathematic is
included in all cultures and that processes of the teaching and learning
of mathematics are part of any enculturation. When some techniques,
say of house building, are passed on to the next generation, we also
witness a process of mathematics education.
Mathematics education is part of everyday interaction and communication. There is mathematics included in the process of buying bread
and a newspaper on a Sunday morning. Then, when reading the
newspaper while having breakfast, more mathematics is introduced or
used. We read about inflation, sport results, lotto systems, the
likelihood that one football team is going to win against another on
their home ground, that the stock market is going down, that the prices
of petrol are going up, that there is a likelihood of the results of the
next election bringing a certain party to power. Special offers are
announced on almost every page of the newspaper. The business
section contains information about companies that must close, others
that are likely to be bought by foreign companies. All such
considerations are based on mathematical calculations. Reading
critically through such information might presuppose some
understanding of numbers, of calculations as well as of the scope of
certainties and uncertainties linked to applications of mathematics.
24
Other interesting studies of mathematics at the work place, which at the same time
broaden the conception of mathematics, are found in: Bessot and Ridgway (Eds.)
(2000); Coben, O’Donoghue and FitzSimons (Eds.) (2000); Harris (Ed.) (1991); and
Wedege (2000, 2002a, 2002b).
22
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
Reading a morning newspaper can be a process of engaging with
mathematics.
Thus, by ‘education’ I do not refer to any particular school context.
Mathematics education can take place in any situation. I use the word
‘mathematics education’, when I want to refer to situations where
processes of teaching or learning of mathematics are taking place. Thus
‘mathematics education’ becomes a covering label, and I want to ignore
the connotations indicating only the teaching-learning processes taking
place in schools. Mathematics education takes place everywhere.
As already indicated, many different groups of people might be
involved in mathematics education. We have students from all over the
world: from rich neighbourhoods, from favelas, students with many
different cultural backgrounds, students who have had very different
opportunities, aspirations and foregrounds. We also have students who
have never gone to school – but streets are also sites for mathematics
education.25 We not only have contact with students, but also with their
parents and teachers. Furthermore, many different discourses contribute to mathematics education. Students comment on the teaching, and
when asked about mathematics, they often talk about their mathematics
teachers. Teachers talk about students, and when listening to teachers
from one level of the educational system we may get the impression
that students coming from the previous level are becoming weaker and
weaker, year by year. Researchers might cooperate with teachers, and
teachers might find the discourse of research too remote from a
discourse that could deal with daily problems. The politicians might
suggest new initiatives for improving mathematics education; they
might ignore research results and develop a terminology about
efficiency, and competitiveness that is more businesslike than the one
used by teachers and mathematics educators.
We could try to talk about systems of mathematics education or,
maybe, about socio-political systems of mathematics education,
referring to both the groups of people and the discourses, which are
part of mathematics education.26 We can, however, not think of
mathematics education as a simple system, where we use ‘system’ as
25
26
See, for instance, Nunes, Schliemann and Carraher (1993).
A careful study of the ‘Institutional System of Mathematics Education’ has been
carried out by Valero (2002a). The different actors operating in this system have
been located, and that mathematics education described in its complex sociopolitical reality.
23
TRAVELLING THROUGH EDUCATION
presented in system theory. Mathematics education appears to be far
from any system, where the particular elements fit into an overall
picture and serve a purpose. Mathematics education is a most heterogeneous entity. It might be better to think of mathematics education as
a set of systems, without any straightforward uniform functionality. The
concept of structuration is basic to Anthony Giddens’ interpretation of
what sociology is about – not social ‘objects’, not ‘facts’, nor ‘systems’,
but ‘processes’.27 I do not pretend to grasp the full significance of
Gidden’s interpretation of ‘structuration’, but I find it illuminating to
think of mathematics education as a ‘family of structurations’. When I
refer to mathematics education as a system, it is only for simplicity. I
always have in mind that mathematics education includes many
different processes, many different groups of people, and many
different discourses. Mathematics education is a very loose concept.
And let it be like this.
In society we find many different families of structurations (which I
also refer to as ‘systems’ in order to simplify the terminology). We can
think of transport systems, hospital systems, postal systems,
communication systems, economic systems, educational systems, as
well as of the systems of mathematics education. Some of these systems
(families of structurations) can be considered to be of particular
significance for social theorising. Thus, social theorising, inspired by
Karl Marx, has paid a special attention to the economic structures as
providing a basic dynamic to social development, meaning that the
‘laws’ of economic progress define the main parameters in social
theorising. The development of other socio-political systems can then
be explained by referring to how they are related to the economic
systems. After the explosive development of information and
communication technologies, it has been indicated that the main steps
in social development reflect the emergence of new forms of
information and knowledge processing. I have not been aware,
however, of any social theorising that has paid special attention to, say,
the postal system. (Naturally, studies can be made of the postal system,
but that is a different issue.) This system is considered ‘insignificant’. By
a ‘significant’ social system I understand a system that seems to
influence other social systems to such a degree that social theorising
must pay special attention to it in order to formulate adequate
interpretations of social phenomena. The educational system has
27
See, for instance, Giddens (1984).
24
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
sometimes caught the attention of social theorising, although it might
not have any prominent place among the leading families of
structuration. Nor has mathematics education been considered as being
of particular significance for understanding the more basic processes in
society. (From the perspective of much social theorising, mathematics
education appears as insignificant as the postal system.)
The observations previously referred to by Stone, Bourdieu, Bishop
and D’Ambrosio indicate, however, that mathematics education might
have some significance. In order to indicate further the possible
significance of mathematics education in today’s socio-political processes, I shall, in the following three sections, consider (a) globalisation,
(b) knowledge processing and (c) ghettoising. Globalisation represents
one aspect of the informational society, namely that we become linked
together in many new ways, economically, culturally, and ecologically.
Knowledge processing plays a crucial role in the whole networking,
thus we can refer to the informational society as a learning society.
Finally, the growths of ghettoes signify that the linking-together
processes of globalisation do not mean a linking together in solidarity.
Globalisation can also mean further exploitation. I do not see
globalisation and ghettoising as two different processes, but as different
perspectives on the networking of the informational society.
Mathematics and mathematics education may have special significance
in knowledge processing, and in this way they will operate together
within processes of globalisation and ghettoising.
7
GLOBALISATION can concern all aspects of life. Globalisation refers
to economic issues, meaning that economic enterprises in one part of
the world affect economies in quite different parts. The hectic activity
on the international stock market can be seen as an expression of globalisation, strongly supported by electronic networking. For instance,
the ranking of ‘risk countries’ can vary from day to day. Globalisation
refers also to ecological phenomena. What is taking place in one part of
the world in terms of pollution, cutting down of the rainforest, etc. has
effects on quit different ecological environments. Globalisation refers
to growing political awareness of what is happening in different parts of
the world, thus the Gulf War may signify the USA interest in the supply
25
TRAVELLING THROUGH EDUCATION
of oil, and the USA’s war against terrorism affects (if not destroys) a
wide range of communities, families and persons, with no relationship
to terrorism whatsoever. The tribunal in The Hague represents the
political dimension of globalisation, as the international community
takes actions against war criminals (of some wars at least). Globalisation
refers to cultural trends, and at present, this often means
Americanisation. Traditions, values and fashions, which represent the
USA, have a growing impact on youth cultures, which become less and
less rooted in national traditions. Thus, when a McDonald’s sign is in
sight, in cities in, say, the Brazilian provinces, it is cheered by the youth!
Globalisation refers to the stream of information and news and to
communication around the world. Thus, the Internet represents a
strong underlying currency for globalisation. Globalisation refers to a
mesh of economic, ecological, political, cultural and communicational
trends.
Globalisation has some attractive connotations. It can include a
sense of being together and of sharing concerns for each other – as if
the globe were turning into a grand community. Maybe a better
understanding of globalisation is reached if we strip the notion of the
positive associations, and simply let globalisation refer to the fact that
new connections are established between previously unconnected social
entities. Globalisation can refer to the fact that what is happening and
done by one group of people may affect, for good or bad, a completely
different group of people, even those unaware of the nature of the
effect. Thus, globalisation can refer to interrelations and to loss of
transparency. Globalisation can mean a destruction of communities.
The concept of globalisation contains both positive and negative
connotations: “For some, ‘globalization’ is what we are bound to do if
we wish to be happy; for others ‘globalization’ is the cause of our
unhappiness. For everybody, though, ‘globalization’ is the intractable
fate of the world, an irreversible process …” (Bauman, 1998: 1).
Economy has been globalised28, and let me quote Immanuel
Wallerstein: “The modern world-system is a capitalist world-economy,
which means that it is governed by the drive for the endless
accumulation of capital, sometimes called the law of value.”
(Wallerstein, 1999: 35) The economic processes within this system can
be geared up by means of information and communication
28
See Archibugi and Lundvall (Eds.) (2001: 2).
26
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
technologies.29 As a consequence, it also makes sense to talk about an
informational economy. One aspect of this world system is the
extension and movements of supply chains, i.e. those chains that lead
from raw material to the final commodity. Along these lines we find
refinements of the raw material into the final product, and we find spin
off of profit at each step, basically characterised by the fact that the
closer we come to the final product, the higher is the profit spin off.
The direction of the supply chain can, nowadays, be changed according
to new priorities. Thus Albert J. Dunlap, one of the world’s leading
business managers expresses himself in the following way: “The
company belongs to the people who invest in it – not to its employees,
suppliers, not the locality in which it is situated” (quoted after Bauman,
1998: 6). The meaning of this statement is clear: the company is a freely
moving entity, and certainly the big companies can move their supply
chains as they want. The company is not ‘located’ in any particular
physical environment, and it has no obligations towards any particular
community. Where the company is placed for the time being, and what
the company is producing for the time being, are determined by the
people to whom the company belongs, and these are the people (or
some of the people) who invest in the company. Sticking to this
economic principle has been highly facilitated by the information and
communication technologies. The permanent (possibility of) moving
the company in order to ‘capitalise’ is a defining element of
globalisation, and it signifies power.
According to a neo-classic economy, the informational economy
might appear attractive, as it makes it easier for the individual
companies to pursue their own interests, for instance by moving supply
chains. The assumption is that individual entrepreneurship will provide
an overall consistency and make sure that ‘the world shall go from glory
to glory, from wealth to wealth, and therefore from satisfaction to
satisfaction’. Naturally, others have questioned that such entrepreneurship sums up a common-wealth, and if we follow a socialist analysis,
29
See also Harvey (1990: 180) who emphasises the following three features of any
capitalist mode of production: (a) “Capitalism is growth-oriented. A steady rate of
growth is essential for the health of a capitalist economic system, since it is only
through growth that profits can be assured and the accumulation of capital be
sustained. … (b) Growth in real values rests on the exploitation of living labour in
production. This is not to say that labour gets little, but that growth is always
predicated on a gap between what labour gets and what it creates… (c) Capitalism is
necessarily technologically and organizationally dynamic…”
27
TRAVELLING THROUGH EDUCATION
the endless accumulation of capital ends in disaster when no redistribution is ensured. From a classic Marxist perspective, no processes of
redistribution will even be able to compensate for the disasters of
capitalism. In this study, I shall not comment further on these overall
aspects of the informational economy. I just want to emphasise that I
do not believe in any neo-classical assumption about the overall
consistency of entrepreneurship. I find that a common redistribution of
wealth is a minimum claim to be made, but fundamentally I feel
extremely uncertain about where the globalised informational economy
might bring us and what it might mean in terms of concentration of
capital, power and wealth.
8
KNOWLEDGE PROCESSING. In ‘The Social Framework of the
Information Society’ from 1980, Daniel Bell emphasises that a new
‘axial principle’ for social development can be located. While capital and
labour have been conceived as the main resources for value, a new
important resource for value has emerged. In what Bell describes as a
post-industrial society theoretical knowledge when codified will be ‘the
director’ of social change. Thus, Bell suggests a new important idea for
the understanding of social development. It becomes vital to see
knowledge production as playing an important role. And when we
consider the codification of knowledge, then computer technology
becomes crucial. Knowledge has a new economic significance as
computer technology offers a new way of codifying and processing
knowledge.
The importance of knowledge can be condensed into a knowledge
theory of value. If we consider industrial society, Bell highlights that
machine technology, natural resources and energy supply are the
‘transforming agencies’. However, within post-industrial society the
crucial variables become information and knowledge. Classical
economics takes for granted that capital and labour play the central
roles in the production of value. According to this assumption, the
function of production, Q, is defined as a function of two variables Q =
Q(C, L), where Q denotes output, C capital input, and L labour input.
Bell emphasises, however, that “when knowledge becomes involved in
some systematic form in the applied transformation of resources …
28
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
then one can say that knowledge, not labour, is the source of value”
(Bell, 1980: 506). In this sense knowledge becomes an axial principle for
productivity.30 Knowledge and information are becoming “the strategic
resource and transforming agent of the post-industrial society” (1980:
531). This will imply transformation of the formation of the whole of
society: “In the coming century, the emergence of a new social
framework based on telecommunications may be decisive for the way
in which economic and social exchanges are conducted, the way
knowledge is created and retrieved, and the character of the
occupations and the work in which men engage. The revolution in the
organization and processing of information and knowledge, in which
the computer plays a central role, has as its context the development of
what I have called the post-industrial society.” (1980: 500)
Many studies are in line with Bell’s presentation of the ‘information
society’. However, it has also been criticised for providing, not a global
scenario, but a North American and European perspective, if not
simply a USA perspective on economic development. It has been
claimed that Third World countries, as well as other social forces, have
been eliminated from the picture. The result has been a most peculiar
world view, useful for underlining USA-priorities in politics and
business. This world view includes a systematic blindness to economic,
environmental, political and cultural horrors which are caused by
processes of globalisation in some provinces of our global village.
Acknowledging this blindness, Manuel Castells talks about the
‘informational society’. I share Castells’ concern about the narrow
perspective that might be associated with the expression ‘information
society’, and therefore I shall use ‘informational society’ as well as
‘informational economy’. 31
Although Bell suggests a knowledge theory of value, the notions of
knowledge and information are not further analysed. In fact
‘knowledge’ and ‘information’ operate as ‘dummies’ in his theory of
value. To me, it appears surprising that it is not necessary to make any
further specification about the nature of knowledge and information
that may serve as strategic resources in this new social order we seem to
30
31
See, for instance, Tomlinson (2001) for remarks about the function of production.
Could it have the format Q = Q(C, L, S), where S refers to communication and/or
business services?
See Castells (1999). Castells has used ‘network society’ as the title of one of his
books, and this notion is also useful.
29
TRAVELLING THROUGH EDUCATION
be entering. Does any kind of knowledge support processes of
globalisation? Hardly. Could there be a particular area of knowledge
that is significant to the informational society? If we study Bell (1980)
and Castells (1996, 1997, 1998), the answer seems to be ‘no’. At least
the nature and the content of knowledge and information they have in
mind, are only being addressed in the most general terms. We are left
without any specification of what kind of knowledge and information
might function as productive forces.
A strong tradition in philosophy pays special attention to the
definition of knowledge. Knowledge and the development of knowledge are essential elements in epistemology. How is it possible for
knowledge to emerge? What could be the source of knowledge? How
can we justify beliefs in such a way that we can claim beliefs to
constitute knowledge? In short: What is knowledge, and how do we get
it? To Plato, a theory of knowledge is connected to a theory of the
state. Knowledge is essential for the governing of the state, and, in
principle, that put the philosophers into power. John Locke also
considered an analysis of knowledge and human understanding as being
essential for identifying adequate principles of governing. We can
expand on this idea and suggest that social theorising must include
considerations about the specific nature of knowledge and information.
I find that it is important to clarify the following: Does every kind of
knowledge have a productive force in the informational society? Or is it
so that certain types of knowledge provide particular productive
resources?32
Let us see how Castells operates with the notion of knowledge. He
states: “In the new, informational mode of development the source of
productivity lies in the technology of knowledge generation, information processing, and symbolic communication. To be sure, knowledge
32
Another line of argument is, however, possible. Information has been associated
with information and communication technology (ICT). The thesis could be that the
ICT, and not information and knowledge itself, is the source (if not the cause) of
social and economic development. In this way we have updated a classic thesis in
sociology that principal changes in social development are linked to certain
technological developments. As ICT becomes a new technological infrastructure, we
can expect a new social superstructure. Therefore, the thesis of the new axial
principle can be reinterpreted as first of all having to do with, not knowledge in any
specific form, but with the emerging of a new technology, ICT. This might make
some sense, but still I do not think that we can escape the discussion of what kind
of knowledge may play an essential role. The construction, development and
operation of ICT are also based on knowledge, but not any kind of knowledge.
30
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
and information are critical elements in all modes of development, since
the process of production is always based on some level of knowledge
and in the processing of information.” (Castells, 1996: 17) It is clear
that the content and the clarity of this statement rest upon a
specification of what can be understood as information and as
knowledge. Castells adds a footnote to this section: “For the sake of
clarity … I find it necessary to provide a definition of knowledge and
information, even if such an intellectual satisfying gesture introduces a
dose of the arbitrary in the discourse, as social scientists who have
struggled with the issue know well.” Then Castells remarks that he has
no compelling reason to improve the definition of knowledge which
was provided by Bell, who defines knowledge as “a set or organized
statements of facts or ideas, presenting a reasoned judgement or an
experimental result, which is transmitted to others through some
communication medium in some systematic form” (Bell, 1973: 175;
quoted after Castells, 1996: 17).
It is clear that Castells does not take this part of his work very
seriously. He puts the definition of knowledge as a footnote, and he
also makes an excuse by referring to this footnoting as being an
‘intellectual gesture’. Furthermore, Castells does not apply this definition in any serious way later in his work, he lets the notion of
‘knowledge’ and ‘information’ stay as cloudy concepts throughout his
study of the informational society. The rough clarification, however,
that is conveyed is that knowledge is a particular kind of information, as
knowledge is information associated with a kind of justification. My
problem is that this simplification does not take us anywhere. (I am
sure that Castells has realised this.) But I find that it is essential to make
a much stronger specification of the notion of knowledge in order to
obtain a deeper understanding of some of the basic social processes of
the informational society (and I am afraid that Castells has not realised
this).
I simply do not think that we can talk about information and
knowledge in such overall terms as has been common in sociology.
Different areas of knowledge and information may play completely
different roles in the informational society. The different types of
knowledge may relate in quite different ways to the axial principle for
production. Just to make an illustration of my point. We can collect
information about people, their names and their relatives. We continue
to receive new information about the war crimes done all over the
world. We receive information about the football results, and we know
31
TRAVELLING THROUGH EDUCATION
that Manchester United won the triple in 1999. Some of us know that
this turned Peter Schmeichel into a legend in Manchester United. We
have some knowledge about mathematics and mathematical formulas,
for instance we know something about the distribution of prime
numbers. It is possible to collect information about peoples’ uses of
credit cards and about the professional cyclist’s use (or not use) of
helmets. We know about films stars, chemical formulas, cooking
traditions in Greece, the face of Marilyn Monroe. We get an awful lot
of extra information from the Internet. My thesis is that different forms
of information and knowledge play very different roles in the
development of the informational society. If we want to arrive at
deeper understanding of the processes in which the informational
society is created, then we have to be specific about the notions of
information and knowledge. Sociological understanding becomes
dependent on epistemic case studies.
The idea of knowledge being an axial principle of production has led
to considerations of the production of knowledge. Many recent studies
suggest that the production of knowledge becomes of particular
importance for economic development, so the question is not simply
having access to knowledge or not. Consideration of the production of
knowledge brings us directly to the concept of learning, and not
surprisingly we now find the notion of learning society being used in
different contexts.33 Daniele Archibugi and Bengt-Åke Lundvall (Eds.)
(2001) find it relevant to talk about a learning economy (and not simply
about a knowledge-based economy), as they see the economic
significance of the production as well as of the destruction of
knowledge. It is not knowledge as such, but the processes leading to
and from knowledge, which have economic impact. Certain forms of
knowledge can become obsolete and turn into economic obstacles.
Learning can be associated with schools, being an important site for
learning. But much learning takes place outside school. Learning is part
33
For critical comments on the notion of learning society, see, for instance, Young
(1998) who writes: “The idea of a learning society, as well as the associated ideas of
an information society and skill revolution reflects real economic changes and at least
a partial recognition that the mode of production and the conditions of profitability
of European companies have changed.” (1998: 141) Young suggests considering
‘learning society’ to be a contested concept “in which the different meanings given
to it not only reflect different interests but simply different versions of the future
and the different policies for getting there” (1998: 141).
32
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
of daily life. Learning takes place in companies and at work places. And
it is recognised that an organisation can become a ‘learner’.
I find that a further understanding of the informational society
(learning society and learning economy) and processes of globalisation
could be better clarified if we pay special attention to the production
and codification of mathematical knowledge, i.e. to the learning of
mathematics. This brings us to consider the role of mathematics education as taking place in formal and informal settings. I find that the
emergence of the informational society, including the processes of
globalisation, makes it important that the technological and sociopolitical roles of mathematics and of mathematics education are
carefully discussed.34 To me it is not surprising that it is possible to find
mathematics education everywhere (and later I will try to illustrate that
it is possible to find mathematics everywhere). This raises many
questions: Does the development of the informational society provide a
new meaning to the statement made by Stone: that mathematics
education can be seen as the basis on which the technological superstructure of society is resting? Does mathematics education provide a
knowledge production, essential for the ‘new axial principle’? Could the
functions of mathematics education, simultaneously, be expressed in
even more drastic formulations than those indicated by Bourdieu?
Could mathematics education also influence the informational society
by providing ‘operational’ forms of stratification, selection, demarcation, and legitimising? Could this education, as indicated by Walkerdine,
represent a forceful ‘technology of the social’, bringing about ‘rational
citizens’, ready for accepting the given order of the globalised informational economy?
9
GHETTOISING. In the End of Millennium, Castells defines social
exclusion as “the process by which certain individuals and groups are
systematically barred from access to positions that would enable them
to an autonomous livelihood within the social standards framed by
institutions and values in a given context” (Castells, 1998: 73). Castells
34
See also Skovsmose and Valero (2002a) for mentioning the possibility of
establishing a political economy of mathematics education.
33
TRAVELLING THROUGH EDUCATION
emphasises that “the ascent of informational, global capitalism is indeed
characterised by simultaneous economic development and underdevelopment, social inclusion and social exclusion” (1998: 82).
Zygmunt Bauman makes the following observation: “Globalization
divides as much as it unites; it divides as it unites – the causes of
division being identical with those which promote the uniformity of the
globe.” (Bauman, 1998: 2) To me, globalisation and ghettoising
represent different aspects of (or different perspectives on) the
informational society.
Castells makes the following observation which brings him to
consider the notion of the Fourth World: “This widespread, multiform
process of social exclusion leads to the constitution of what I call,
taking the liberty of a cosmic metaphor, the black holes of informational capitalism. These are the regions of society from which,
statistically speaking, there is no escape from the pain and destruction
inflicted on the human condition of those who, in one way or another,
enter these social landscapes.” (Castells, 1998: 162) The Fourth World
makes up the black holes of informational capitalism: “The Fourth
World comprises large areas of the globe, such as much of Sub-Saharan
Africa, and impoverished rural areas of Latin America and Asia. But it
is also present in literally every country, and every city, in this new
geography of social exclusion. It is formed of American inner-city
ghettos, Spanish enclaves of mass youth unemployment, French
banlieues warehousing North Africans, Japanese Yoseba quarters, and
Asian mega-cities’ shanty towns. And it is populated by millions of
homeless, incarcerated, prostituted, criminalized, brutalized, stigmatized, sick and illiterate persons. They are the majority in some areas, the
minority in others, and a tiny minority in a few privileged contexts. But,
everywhere, they are growing in number, and increasingly in visibility,
as the selective triage of informational capitalism, and the political
breakdown of the welfare state, intensify social exclusion. In the current
historical context, the rise of the Fourth World is inseparable from the
rise of informational, global capitalism.” (1998: 164-165)
It might be possible to think of a ghetto as a small community. A
ghetto might be reserved for a certain group of people, who stand out
from the society in which they live. Although they might not be
welcome in society, in the ghetto they can support each other and live
according to their own culture and traditions. The Jewish ghettoes
established over centuries within European cities might serve as an
example. Isaac B. Singer’s description In My Father’s Court, at the time
34
TRAVELLING THROUGH EDUCATION
hyperghetto has lost its positive role of collective buffer, making it a
deadly machinery for naked social relegation” (see Bauman, 2001: 122).
Apparently, the need for flexibility of the labour force only
encompasses some groups of people. Mobility is not necessary to other
groups, and, therefore, they have to be confined and made immobile.
Exclusion and ghettoising that accompanies globalisation demonstrate a
new brutality.
Being disposable and being together cannot be expected to fertilise
any feeling of solidarity: “No ‘collective buffer’ can be forged in the
contemporary ghettos for the single reason that ghetto experience
dissolves solidarity and destroys mutual trust before they have been
given a chance to take root. A ghetto is not a greenhouse of community
feelings. It is on the contrary a laboratory for social disintegration,
atomization and anomie.” (Bauman, 2001: 122) It is difficult to imagine
that solidarity can emerge in a modern ghetto: “Ghetto life does not
sediment community. Sharing stigma and public humiliation does not
make the sufferers into brothers; it feeds mutual derision, contempt and
hatred. A stigmatized person may like or dislike another bearer of
stigma, stigmatized individuals may live in peace or be at war with each
other – but one thing they are unlikely to do is to develop mutual
respect.” (2001: 121-122)
In the most direct way, ghettoising could mean building a wall
between ‘them’ and ‘us’.35 What does the wall look like that separates
the Fourth World from other worlds? In some cases we literately find a
wall separating rich and poor. For instance, in some cities in Brazil
condominios are constructed, meaning that a whole neighbourhood is
surrounded by a wall. The border between Mexico and the USA also
appears like a wall separating the Fourth World from the informational
society. The wall being raised between Israel and Palestine, placed well
inside the land of Palestine, seems to turn a whole country into a
35
In September in 1989, I spent one day walking along the Berlin wall on the West
side. I started in Kreuzberg, and I literally let my fingers touch the wall hundreds of
times until I came to the Brandenburger Tor. A very interesting tour. The wall was
decorated, and so is the small piece of the wall which I bought some months later
when the wall had gone down. In short intervals one could climb a small wooden
tower that made it possible to look into East Germany, observing the zone which
was so difficult to pass. I also saw the famous Checkpoint Charlie. At night the noman zone looked like a well illuminated highway running through the whole city, but
instead of the noise of traffic, the permanent barking of dogs was heard. This was
the separation between East and West.
36
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
ghetto. In other cases the wall is less tangible. In all cases, however, it
contributes to the new geography of the social order.
Globalisation means inclusion. Connections are made between
groups, regions, businesses. Globalisation also means exclusion, as the
connections made are only partial and serve only particular interests.36
While some groups, regions and businesses are connected, other
groups, regions and business are excluded. In this sense globalisation
and ghettoising become aspects of the same process. Globalisation and
ghettoising – inclusion and exclusion – have to do with schooling,
education and learning in general. Schools are sites for inclusion and
exclusion. Schools may provide access to the informational society.
Schooling signifies new social possibilities for many students. Certainly
schools could also signify the opposite. In this sense, many schools are
positioned on the borderline between the Fourth World and the
informational society. Schooling can be seen as a support for entering
the informational society, but it also becomes a gatekeeper and an
‘excluder’ from this ‘networking’. Schooling (or exclusion from schooling) can mean a preparation for the dumping of people into the Fourth
World. The remarkable statement of Verwoerd thus represents a clear
indication of what it means to put people into a ghetto, in this case
represented by ‘homelands’. In this ironic way he was very much ahead
of his time. The grand apartheid was built upon the idea that people
should be separated, and that black people have no role to play in white
society, beyond the level of unskilled labour. The existence of ghettos
in the informational society seems to indicate that this society has no
need for everybody. Only a part of the global population fits into the
networking, the rest are better left in their homelands.
In the time of apartheid, learning obstacles had their obscure
significance. But do learning obstacles reach a new significance when
we consider processes of globalisation? Castells refers to the millions of
homeless, incarcerated, prostituted, criminalised, brutalised, stigmatised,
sick and illiterate persons, who literally are expelled from the
informational society. They can simply be disposed of from the whole
economic enterprise. They are worth nothing as consumers and they
have no value as possible human resources for production. The most
‘rational’ thing to do is to prevent the Fourth World from interfering
36
See, for instance, the discussion of social exclusion in learning economy in
Schienstock (2001).
37
TRAVELLING THROUGH EDUCATION
with the well-functioning processes of the informational society.37 In
this way, the Fourth World turns into an over-dimensioned learning
obstacle. And, according to Verwoerd: mathematics is not for them.
10
MATHEMATICS EDUCATION IS CRITICAL. If we see mathematics education as part of global processes preparing the ground for
the informational society (and I see mathematics education this way),
we must also be aware that the process of globalisation “divides as
much as it unites”. In fact, according to Bauman, “it divides as it
unites”, and further, “the causes of division being identical with those
which promote the uniformity of the globe”. This is an essential
reminder to mathematics education. We should not be surprised if this
education divides as much as it unites. If we see mathematics education
as part of universal processes of globalisation, then we should also see it
as part of the universal processes of making exclusions. Designating
‘state nobility’, ‘state functionaries’ and ‘disposable people’ may be part
of the same educational processes.
Perhaps one function of mathematics education, within the new
geography of social exclusion, is to operate at the checkpoints denoting
inclusion or exclusion along the border: “Mathematics is not only an
impenetrable mystery to many, but has also, more than any other
subject, been cast in the role as an ‘objective’ judge, in order to decide
who in the society ‘can’ and who ‘cannot’. It therefore serves as the gate
keeper to participation in the decision making processes of society. To
deny some access to participation in mathematics is then also to
determine, a priori, who will move ahead and who will stay behind.”
(Volmink, 1994: 51-52) This statement by John Volmink can be read as
a dramatic description of the role of mathematics education in marking
a division between those who become included in and those who
become excluded from the informational society. (I shall not propose
that mathematics education, or education in general, provides the main
cause for social inclusion and exclusion. At a global level, many causes
interconnect. But the mathematics classroom might be an important
site to consider.)
37
Hardt and Negri (2004) talk about a new phenomenon of global apartheid.
38
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
I find the role of mathematics education in the further formation of
the informational society to be critical. To me, this means two things.
First, that mathematics education plays a significant role in socio-political
processes. This is indicated by some of the observations referred to
previously: that mathematics education can be seen as the foundation
of the technological society; that it can be seen as a cultural invasion;
that it provides forms of knowledge and techniques of particular
relevance for the informational society; that learning mathematics is
closely related to the developing of competencies for handling
information and communication technologies (ICT). I see the development of the informational society, including the processes of globalisation and ghettoising, as connected to mathematics education and to
the learning of mathematics. This, however, does not mean that I claim
that mathematics education is a socially determining factor. The
organisation of mathematics education is influenced by numerous and
very different factors. My claim is simply that mathematics education
can play a significant role in interaction with many other socio-political
actors and factors.
Second, I find that mathematics education is critical as, in many of its
forms, it plays an indeterminate role (or a possible double role).38 How
mathematics education in fact is operating in different contexts is not
well-defined. It could be that mathematics education ensures an
adjustment and functionality of a future labour force, say, by
regimenting students with the long sequence of exercises formulated in
a short and clear language of orders and commands. It could be that
mathematics education provides a competence basic for any citizenship,
critical or not. It could be that mathematics education provides an
entrance to a magnificent world of ideas and theories with both
aesthetic values and technological relevance by resourcing technological
imagination. It could be that such an imagination is a prerequisite for
the identification of new techniques and technological constructions
and for the further formation of the informational society. But this
could also mean that mathematics education takes part in processes of
exclusion. My point with these remarks is to indicate that the socio-
38
Maybe the notion of being ‘indeterminate’ could be developed further by
reconsidering the notion of being ‘undecidable’, as discussed by Torfing (1999) with
reference to discourse theory. Being indeterminate also signifies not being predetermined.
39
TRAVELLING THROUGH EDUCATION
political roles of mathematics are indeterminate. The roles of being
hero or scoundrel can all be acted out by mathematics education.
By claiming that the role of mathematics education is critical, I mean
that the socio-political roles of mathematics education are both
significant and indeterminate. Mathematics education could operate in
very different ways, and this could really make a difference! In other
words, I use ‘critical’ here in the same sense as when we talk about a
patient’s condition as being critical. This means that he or she could
survive, but also that nothing can be taken for granted. His or her
condition is simply critical.39
Several considerations about mathematics education suggest that the
role of mathematics education cannot be critical as this part of the
educational system contains intrinsic qualities that will ensure that
mathematics education will serve attractive aims. This line of argument
might include some form of essentialism, as mentioned previously
when I referred to some ‘ambassadors’ of mathematics. Thus, some
studies have claimed the existence of an intrinsic connection between
mathematics education and democratic values. Such an interpretation
has been presented by Colin Hannaford (1998). His idea is clearly stated
in the title of his paper ‘Mathematics Teaching is Democratic Education’ with ‘is’ italicised. With reference to the Ancient Greeks,
Hannaford claims that the rational and open debate (which he refers to
as techno logos), essential for democracy, can as well be related to an
inherent component in mathematics. Thus, mathematical thinking
represents not only formal thinking, useful to follow in order to
produce deductive chains and to recognise the necessity linked to
39
The present discussion of the critical position of mathematics education is based on
Skovsmose and Valero (2001), where the relationship between mathematics
education and democracy is presented as being critical. For further discussion of the
critical relationship between mathematics education and democracy see, for instance,
Skovsmose (1990, 1994, 1998b), Vithal (1999); and Valero (1999). Valero (1999)
presents an important notion of democracy as action characterised as being
collective, transformative, deliberative and coflective. The notion of coflection refers
to a collective process of reflection. Valero observes that ‘reflection’ comes from
Latin, that ‘re’ means ‘back’ or ‘again’, and that ‘flexio’ means bending. Then she
brings the idea together in the following way: “Coflection – ‘co’-‘flection’ – is the
word that refers to the meta-thinking process by means of which people, together,
bend on each other’s thoughts and actions in a conscious way. That is, people
together think about the actions they undertook, but adopt a critical position
towards them.” (Valero, 1999: 22) See also the presentation of deliberative
democracy in Bohman and Rehg (Eds.) (1997).
40
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
mathematical proofs, it also expresses a way of thinking that is useful
for the development of the city state, the polis. A new standard of
rationality was introduced in public, and mathematics became an
exemplary representative for this way of thinking. That mathematics
became exercised in a deductive form and that democracy became a
ruling principle in Ancient Greece can, thus, be seen as a clear
indication of the intrinsic connection between a mathematical way of
thinking and a democratic way of life. In Hannaford’s formulation:
“Democracy … depends on trust and respect between people extending beyond the limits of family and tribe, to include all the members of
an entire society. To be productive, what this mutual respect then needs
is some kind of systematic, clear and open argument by which people
can communicate and co-operate with each other intelligently. In other
words, it needs the sort of argument used in mathematics.” (1998: 182)
Mathematics comes to represent a mastery of a rationality, which is
intrinsically connected to a democratic way of being. This idea has
implications for how to see mathematics education: “If children are
taught mathematics well, it will teach them much of the freedom, skills,
and of course the disciplines of expression, dissent and tolerance that
democracy needs to succeed.” (1998: 186)40
It has also been suggested that the relationship between mathematics
education and democratic values are almost contradictory. This line of
argumentation can be linked to what could be called ‘negative
essentialism’: by the very nature of mathematics, mathematics education
would include problematic effects. Such a negative interpretation can be
read into the comments made by Bourdieu, but many other studies
have more directly exposed anti-democratic functions of mathematics
education. For instance, mathematics education can provide an
‘occupation of the mind’, as Herbert Khuzwayo (1998) summarises in
his account of the history of mathematics education during the
40
Hannaford follows up with a reservation: If mathematics is not taught well then we
can end up with a form of mathematics education, which could destroy democracy
(see, Hannaford, 1998: 186). Thus, Hannaford’s point is not that any teaching of
mathematics, but only mathematics “taught well” will ensure democratic values.
Hannaford also extends the conception of the intrinsic values of mathematics to be
applicable to the community of mathematicians: “More Mathematicians are working
and teaching in the world than ever before. They produce more new mathematics
than history has ever known. They communicate, they criticize, they co-operate,
better. that ever before. And they work democratically.” (1998: 186) Interesting to
compare with the impression presented by Burton (1999, 2001, 2004).
41
TRAVELLING THROUGH EDUCATION
apartheid period in South Africa. Mathematics education could support
all kinds of social inequalities, already established in society and further
developed by mathematics education itself. The gate keeping, exercised
by mathematics education, is a clear anti-democratic device. As referred
to previously, Walkerdine presents a bleak interpretation of the actual
function of mathematics education (in England). Following her study,
we can extrapolate the conclusion that mathematics education and
democracy are foreign to each other, as mathematics education
supports the development of a rationality that cannot be related to any
form of democratic thinking. Thus, in The Mastery of Reason, Walkerdine
notices that political arithmetic was “an attempt to use arithmetic
techniques to calculate those aspects of the population which could
then be amenable to scientific forms of government” (Walkerdine,
1988: 214). With reference to the emergence of technologies for
administration based on formal sciences, she adds that practices of
schooling now begin “to produce a new professional class – an
educated bourgeoisie who could calculate and reason scientifically – and
a proletariat who would be reasonable in order to be governed” (1988:
214). These remarks square with remarks by Bourdieu, referred to
previously, about the appointment of a state nobility. 41 I can clearly
follow Walkerdine in her worry about how mathematics education
might help (somebody) to master a reason (so different from the reason
referred to by, say, Hannaford), which seems highly functional and
applicable in exercising administrative and technological power. It
becomes tempting to generalise into the claim that the school mathe41
In Dowling (1998) we might also get close to what I consider a ‘negative
essentialism’. However, Walkerdine (1989) also indicates the possibility of
alternatives as she talks about transforming the mathematical discourse in order “to
produce a discursive practice which would not separate rationalisation from affect
and from the social” (1989: 27), and she adds: “This would not be a feminist or
female Mathematics, precisely because it would not be a Mathematics as we
understand it today.” (1989: 27) It might be possible to imagine an alternative, but
the alternative has not much to do with the mathematics we know today.
A different form of negative essentialism can be found in some forms of general
critical education, which in fact are characterised by not paying attention to
mathematics education. The claim seems to be that mathematics education cannot
be part of any critical endeavour because of the very nature of mathematics. This
negative essentialism has played some role in the educational discourse during the
1970s and 1980s. It might have been caused by some interpretations of Critical
Theory, for instance as formulated by Marcuse, which tend to identify ‘mathematical
reasoning’ with ‘instrumental reason’.
42
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
matics tradition represents a counter-democratic element in education,
providing some with skills for the mastery of reason, and others with a
docile attitude of being ‘reasonable’. Still, it has to be kept in mind that
Walkerdine has made her investigations at a particular time in a
particular country where education is developed in a particular way. I
could agree with what Walkerdine claims, if she refers only to the
school mathematics tradition. But I find that there are alternatives
outside this tradition.
I would not question that it is possible to observe many examples of
mathematics education, which demonstrate a bleak picture of the role
of mathematics education. By claiming that the socio-political role of
mathematics education is critical, my point is, however, to emphasise
that such bleak pictures need not be true. The bleakness might be well
documented by observations, but these are observations of educational
practices in some situations and in some contexts. It need not be so in
all situations. It is possible to consider alternative forms of mathematics
education, as well as very different social roles of mathematics education. To claim that the socio-political role of mathematics education is
critical means to claim that alternatives are possible and that finding
alternatives could make a difference. I do not relate mathematics
education to any optimistic position claiming the existence of an
intrinsic connection between mathematics education and, say, democratic values. Nor do I claim that mathematics education per se will serve
anti-democratic interests. Instead I simply claim that the socio-political role
of mathematics education is critical as it is significant and indeterminate. No actual
functions of mathematics education represent the essence of this
education. There is no such essence.
11
CRITICAL MATHEMATICS EDUCATION. I do not see mathematics education containing any strong ‘spine’. Mathematics education
could collapse into dictatorial forms and support the most problematic
features of any social development, exemplified by the collapse during
the 1930s of mathematics education in Germany into a Naziaccommodative form. Mathematics education may also have a potential
to develop strong support for democratic ideals, although this potential
is not realised by any strength intrinsic to mathematics education. How
43
TRAVELLING THROUGH EDUCATION
mathematics education may operate in relation to democratic ideals will
depend on the context, on the way the curriculum is organised, on the
way the students’ perspectives are recognised, etc. Essentialism has
been suggested in different forms. For instance, technological optimism
claims that technological development is progressive and attractive, as
this development ensures not only technological progress, but also
progress in related areas, like economy and welfare. In the philosophy
of technology, however, essentialism has been doubted, and I also
doubt any essentialism with respect to mathematics education.
The critical nature of mathematics education represents a great
uncertainty. Naturally, it is possible to try to ignore this uncertainty.
This can, for instance, be done by assuming that mathematics education
somehow can become ‘determined’ to serve some attractive social
functions when organised in, say, a national curriculum crowned by
some well-chosen aims and objectives. But I find this an illusion. The
functions of mathematics education cannot be determined (or redetermined) by introducing some overall guiding principles put at the
top of the curriculum. To change the ‘indeterminism’ of mathematics
education is not a simple task. There are no straightforward procedures
for ‘determining’. The functions of mathematics education might
depend on many different particulars of the context in which the
curriculum is acted out. To acknowledge the critical nature of
mathematics education, including all the uncertainties related to this
subject, is a characteristic of critical mathematics education.42
Critical mathematics education is not to be understood as a special
branch of mathematics education. It cannot be identified with a certain
classroom methodology. It cannot be constituted by a specific
curriculum. Instead, I see critical mathematics education as defined in
terms of some concerns emerging from realising the critical nature of
42
Both ‘crisis’ and ‘critique’ are derived from the Greek word krinein, which refers to
‘separating’, ‘judging’ and ‘deciding’. As pointed out by Connerton (1980: 17), in
antiquity, the notion krisis, could refer to legal issues, thus Aristotle used the term to
denote a juridical decision. Later, a medical use of krisis was developed, and as
already mentioned, it refers to a decisive turning point in an illness, changing for the
better or for turning fatal. Finally, kritikos came to refer to the study of texts. To me
these observations bring together nicely the different connotations of crisis and
critique. A ‘critical situation’ or a ‘crisis’ brings about a need for action and
involvement, i.e. a need for critique.
44
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
mathematics education.43 If no intrinsic relationship exists between
mathematics education and some attractive socio-political development,
then the relationship has to be acted out with reference to the particular
context.
Critical mathematics education is a response to the critical position
of mathematics education. Later, in Part 4, I clarify what a conceptual
sensitivity to some concerns of critical mathematics education could
mean.44 I address issues about: (a) what realism with respect to
mathematics could mean; (b) how knowledge can mean action; (c) how
reflections can be public; (d) how learning can be dialogic; (e) how
learners can be ‘realised’; (f) how conflicts can set the scene; (g) how
mathemacy can mean hope; (h) how ghettoising may operate; and (i)
how globalisation could mean both inclusion and exclusion. Such
enumeration of issues might appear unsystematic, but I do not hope for
systematisation. I only look for a conceptual sensitivity that can be
fruitful for critical mathematics education. Here I briefly refer to three
issues, which could give some idea about the concerns I have in mind. I
comment on the socio-political setting of mathematics education; on a
competence that could be associated with mathematics education; and
on the students. In other words, I say a few things about context,
content, and learners.
First, as mathematics education is not assumed to possess any
essence, critical mathematics education is concerned about the different
possible roles which mathematics education could play in a particular
socio-political setting. Critical mathematics education is concerned
about how mathematics education might be stratifying, selecting,
determining, and legitimising inclusions and exclusions. It is also
concerned about the possible different routes the processes of
globalisation might take. Globalisation is a contested concept, and,
therefore, it can be developed in different ways. It is open to sociopolitical inputs, and mathematics education can be seen as one element
of the processes of globalisation. By a borderline school, I understand a
school from where students can find access to the informational society
as well as experience a route to the Fourth World.45 Critical mathematics education must be concerned about what is happening in such
43
44
45
See also Frankenstein (1987, 1989, 1995); Gutstein (2003); Mellin-Olsen (1987); and
Nickson (2002).
See Skovsmose and Nielsen (1996).
For a discussion of borderline schools, see Penteado and Skovsmose (2002).
45
TRAVELLING THROUGH EDUCATION
schools. What kind of opportunities do they provide for the students?
Critical mathematics education must consider educational issues from
the ‘top’ as well as from the ‘bottom’. It is important to consider
mathematics education from the perspective of globalisation including
all the attractive features that globalisation might include. But it is just
as important to consider what mathematics education could mean for
the potentially excluded. To indicate with an example: Many studies
seem to reveal that introducing computers in the classroom provides
significant new learning possibilities. What does that observation mean
when we consider well-researched schools? What does the same
observation mean when we consider schools in poor areas of the world
where there might be holes in the roof and no possibility to gain access
to any computer, nor to electricity? Both questions have the same
significance for critical mathematics education.
Second, critical mathematics education is concerned about the nature
of those competencies which mathematics education might support.
Knowledge and power are connected, not least with respect to mathematics. Learning, and learning mathematics in particular, could mean
empowerment. But it could easily come to mean empowerment for
some, as the educational process produces both inclusion and
exclusion. The content of mathematics education concerns some forms
of knowledge that play a significant role in the further formation of the
informational society. The notion of mathemacy signifies competencies
related to mathematics, similar to the notion of literacy, as developed by
Paolo Freire. The task of Freire was not simply to teach illiterate people
to read and write, as reading could also mean reading a socio-political
situation, and not just a text, as open to interpretations and to critique
(still one should keep in mind that Freire’s programme was extremely
efficient in making people literate, in the traditional sense of ‘literate’).46
In this sense Freire expanded the programme of literacy into a support
for the development of critical citizenship, implying that people not
only need to see themselves as affected by political processes, but also
as possible participants in such processes. Like literacy, so also
mathemacy refers to different competencies. One of these is to deal
with mathematical notions, a second one is to apply such notions in
different contexts, a third is to reflect on such applications. This
reflective component is crucial for the competence of mathemacy.
More generally, critical mathematics education is concerned with the
46
See, for instance, Gadotti, (Ed.) (1996).
46
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
up to the First World War, tells about the small life of Krochmalna
Street within a Jewish neighbourhood of Warsaw. Singer’s father, the
rabbi, had to solve the different problems emerging in the community.
Such a neighbourhood might serve well as one illustration of
‘community’, as made by Bauman in Community. It is, however, important to observe that those ghettos that are growing in the process of
globalisation might be very different from ‘communities’. In fact, it can
be claimed that globalisation means a destruction of communities.
In Community, Bauman makes different observations about the new
forms of ghettos, which could also be called hyperghettos: “We may say
that the prisons are ghettos with walls, while ghettoes are prisons
without walls.” (Bauman, 2001: 121) As part of the positive and
attractive characteristics of globalisation, moving and travelling have
been celebrated. Thus, globalisation means to be aware of different
cultures, traditions, and places. Globalisation means a celebration of
being inter-national. Ghettoising means exactly the opposite. It means
to be prevented from moving. Ghettoised people are immobilised
people. There is no need for these people, and certainly not for them
moving around. Considering the celebration of globalisation, to be
hyperghettoised is a much harder experience than being enclosed in a
‘classic’ ghetto, which at least represents a community and not only a
prison. As emphasised by Bauman: “Ghettoes and prisons are two
varieties of the strategies for ‘tying the undesirable to the ground’ of
confinement and immobilization.” (2001: 120)
Does it make sense to confine people? In the informational society,
the flexibility of the labour force is celebrated, although what is flexible
is the supply chain. It is important that the qualifications of the labour
force can be developed, as one characteristic of the informational
society is the rapid change in demands for labour. Thus, one aspect of
globalisation is that any scheme of production can spread globally. So,
if we consider the ghetto being a reservoir for extra labour force, the
construction of the modern ghetto seems irrational. The point,
however, is that the modern ghetto does not serve as any reservoir.
And certainly not as a reservoir for possible consumers who could help
to speed up the informational economy. The modern ghetto can be
considered a dumping ground for people who have no role to play in
the informational economy. There is no need for their labour or for
their demands. They are disposable people. Bauman refers to Loïc
Wacquant who observes that “whereas the ghetto in its classic form
acted partly as a protective shield against brutal racial exclusion, the
35
PART 1: MATHEMATICS EDUCATION IS EVERYWHERE
development of the competence of mathemacy in such a way that it
provides empowerment similar to the empowerment expressed by
literacy. One direct meaning of empowerment refers to the possibility
for the individual to go beyond the limitations that a socio-political
situation has imposed on a group of people. More generally,
mathemacy means a support for critical citizenship, whatever group of
people we might have in mind. With this background it becomes
important to consider the question: In what way is it possible to
establish mathematical learning that might support the development of
mathemacy? It is no way a given that this question can be answered in
any clear and satisfactory way, but it stays as a concern of critical
mathematics education.
Third, critical mathematics education must be aware of the situation
of the students. It must consider what background the students have,
but also be aware of what possibilities for the future a particular society
might provide to different groups of students. A way of establishing
this awareness is to consider not only the background of the students
but also their foreground. This will open the routes for a more direct
consideration of how different societies provide opportunities (or the
opposite) for different groups, depending on gender, age, class, ‘race’,
economic resources and culture. Critical mathematics education must
always be concerned about the issues of equality, and therefore it must
try to consider the nature of learning obstacles that different groups of
students might face. By also considering the foreground of the students,
critical mathematics education becomes a pedagogy of hope.
There is a big issue that has not been addressed directly in what has
been said so far. Mathematics itself. Mathematics represents a concern
of critical mathematics education. Mathematics itself must be considered, not only from an educational but also from a philosophic and
sociological perspective. Mathematics represents an important aspect of
the development of rationality or ‘reason’. It represents a huge variety
of cultural techniques integrated in handcrafts, daily life routines,
science, technologies, economy, business, industry, military all over the
world. Furthermore, mathematics itself appears to represent a particular
aspect of globalisation and therefore of ghettoising. Mathematics is in
action in a variety of techniques and technologies, which defines both
the informational society and the global networking.
47