This is a text I wrote almost six years ago, for the summer session in 2010 of the Nordic Summer University – held in Majvik outside of Helsinki. It builds in my PhD thesis The Mathematics of Schooling (Lundin 2008). I agree with everything I wrote in 2010, but would not put it this way today. In fact, at this very conference I was recommended to read Robert Pfaller‘s Die Illusionen der anderen (2002). That book turned out to be fantastic and it transformed my conception of the issues I describe here. The result of that transformation can be seen all over this blog (that I write together with Ditte, whom I met at this same summer session). What has happened, put shortly, is that I have shifted focus from the object of mathematics (viz. the title of my dissertation) to the practices that bring this object into existence and at the same time are, in a way, put in motion by this object. I am very glad to be able to say that Robert Pfaller will be one of the key-note speakers at the NSU summer session of 2016.
I wrote this text as a response to difficulties of getting my central point across. I still very much experience such difficulties and perhaps the text can do some good, here, on my and Ditte’s blog.
The sketch above is made by my brilliant friend Marie Wenger. Pupils and a teacher, perhaps, stand there before the dark woods of schooling, frightened but longing to reach the magical ”matematikens dimhöljda toppar” from where you can see everywhere and where everything is possible.
The context of discovery
As many others, I became critically inclined towards mathematics education as a pupil. I found the exercises of the maths-classroom excruciatingly boring. As I moved along the more or less pre-established path for the kind of student I turned out to be, I also became increasingly aware of the relative uselessness of all mathematics learned in school: in primary and secondary school as well as at Chalmers, where I studied mathematics and computer science. It was common knowledge that the main reason for being there was not to learn mathematics (or to learn anything useful at all in particular), but to “learn how to learn”, to get useful “contacts” and so on. This annoyed me.
After Chalmers, because I had also studied some social science, I got the opportunity to work as a PhD student. I was to explore, as I put it then, “why so many had to study so much mathematics which they will never use outside school” – surprising as it may sound that this project was accepted at a department for the mathematical sciences. (I want to point out that I, at this point, understood the relation between mathematics and schooling in a way which I later found out was quite typical: I asked myself how it could be that mathematics was so misrepresented in school; how the mathematics of schooling could be so useless and boring despite its intrinsic usefulness and beauty – or however it is put.)
I started reading public reports on mathematics education, was introduced to the field of matematikdidaktik (i.e. research in mathematics education) and interviewed a couple of mathematicians and teachers – everything in a methodology and theoretical framework inspired by the French sociologist Pierre Bourdieu (see e.g. Bourdieu 1996a; Bourdieu 1996b). I was at that time a member of the research group for the Sociology of Education and Culture (SEC) at Uppsala University, lead by professor Donald Broady, an expert of Bourdieu and French epistemology. I was to draw the map of a Swedish field of mathematics education.
I became increasingly fascinated by what I read in public reports about mathematics education. I found many claims regarding mathematics very strange, e.g. that it is a “precondition for large parts of the development of society and it permeates the whole society […]” (Skolverket, 2000), that you need mathematical knowledge to ”live and act in a democratic society”, that it leads to ”general education, economic growth and citizenship” (NCM, 2001, p. 81), and that it is a precondition for “self confidence” (NCM 2001, p. 21). I thought I knew pretty much about mathematics, and however useful it was, this was not it!
My readings in matematikdidaktik also surprised me, to the point of alienating me from my fellow PhD students. At one point we were having a lesson about “mathematical problem solving”. To me it was obvious that no sane person would ever try to solve these kind of problems outside the school context. I thus found the investigations of how to solve them, not to mention the idea that they were somehow relevant to thinking-in-general and everyday life, totally misguided. Needless to say, I failed to convince the auditorium that this was the true state of the matter. To the contrary I was to learn about how mathematics is often ”invisible for the inexperienced viewer” (Skolverket, 2000; Skovsmose 1994) which is reason for a certain “paradox of relevance” (Niss 1980, p. 10) for mathematics education, i.e. that unlearned people tend to view mathematical knowledge as much less useful than it in fact is. (Thus, the fact that mathematics does not seem to be “out there”, and that mathematical knowledge does not seem to be relevant, is used as an argument for even more mathematics education: it is important, Skovsmose and Niss claim, that people learn to recognize the presence and importance of mathematics.)
The turning point for me came through the study of history. This began as an innocent attemt to give my then thoroughly sociological study some historical grounding. I was curious regarding a certain figure of speech in which it is claimed that the practices of mathematics education are “traditional” – in the sense of promoting memorization and route learning – in contrast with “modern” ideas of individualized creative practices leading to the formation of mathematical concepts. I was curious as to when the transition took place, from traditional to modern ideas.
Taking what I then saw as a giant leap backwards in time, I looked into the discussion on mathematics education of the 1970s. This, it turned out, was an era of great emphasis of “modern” ideas. Proponents of memorization were nowhere to be found. I thus went further back, to the 1950s. Still modernization. To make a long story short, this particular way of talking about “traditional methods” can first be found in Sweden in the 1850s. The “traditional” kind of memorization and route learning seems, in Sweden, to have gone out of fashion at the beginning of that century.
I was fascinated by this finding. Previously, not knowing anything about history, I thought the field of mathematics education to be in rapid change – more rapid perhaps even than other parts of society. Now the opposite seem more likely, that mathematics education, including the way of talking about it, is and has been very constant. No less surprising to me was the fact that the history supporting this conclusion seemed to be almost completely unknown.
To this was, at approximately this point, added the somewhat painful, but at the same time very interesting fact, that no one cared very much about my – as it seemed to me – totally fascinating discoveries. Because, reflecting on the matter, who could possibly care? It seemed that the structure of academic disciplines – with its strict divisions between mathematics, education, history and sociology – made this history intrinsically un-researchable. And what was the news really? That it is hard to teach mathematics and has been so for a long time? The way of stating the problem, the self understanding of mathematics education, made the matter seem intrinsically un-interesting. I had a strong feeling I was on to something.
Through Bourdieu’s then recently published Science de la science et réflexivité (Bourdieu 2001), I was introduced to history-sociology- and-theory of science. There I found much of interest regarding mathematics (Hacking 2000; Hacking 1975; Porter 1995; Cartwright 1983; Alder 1997; Alder 1999; Funkenstein 1986; Gaukroger 2006; Richards 1988; Dear 1995; Toulmin 1992). Suddenly, mathematics was no longer – as it always is from the perspective of mathematics education – given beforehand, but instead a changing set of images, conceptions, ideas, words, practices, institutions, coming into use under particular circumstances at particular times and places. I learned that a certain overrating of the instrumental efficiency of mathematical knowledge is an integral part of modernity. And, for that matter, what does it mean that mathematics is “used”? I read about nominalism (Gillespie 2008), the nature of concepts (Rorty 1980) and reflected on how technology somehow seems to “bear witness” (Latour 1987) to mathematical agency, as a techno-theology similar to the natural theology of earlier times (Gascoigne 1988; Clark 1999).
This accorded very well with some present day studies of mathematics education taking antropology (Lave 1997), Foucault (Walkerdine 1988) and Basil Bernstein (Dowling 1998) as their respective points of departure. Mathematics education, I could conclude, did not generally lead to useful knowledge. The idea of “transfer” was mistaken (Lave). The idea of a “child” to which the practices of mathematics education were adjusted could more appropriately be seen as the constitution of this very object (Walkerdine). Mathematical knowledge did not describe reality, and was not used in everyday life, but rather propagated myths about this being the case (Dowling).
Evidence had piled up, but evidence of what?
I had to abandon Bourdieu. As is especially apparent in his book about science, he is quite scientistic (for a critique of Bourdieus view on science from a fellow Frenchman, see Latour 1993; Latour 2005). Bourdieu’s is a method for studying the social conditions for different appropriations of the universal – not a method for studing the constitution of the universal as such, which was what I felt was needed. That I found in the works of the Slovenian philosopher Slavoj Žižek (starting with The sublime object of ideology of 1989 I then followed references to Laclau and Mouffe 1985; Butler, Žižek, and Laclau 2000; Sloterdijk 1988; Sohn-Rethel 1978; Žižek 1999).
Four functions of mathematics
So what was going on? Let me start by saying a few things about how mathematics is talked about. Taking this as a point of departure, I will argue that mathematics plays four interrelated roles.
First, outside school in public discourse, “mathematics” (together with “knowledge”) is used as what Laclau and Mouffe (1985), and later Laclau (2001) calls a nodal point. It binds together series of words such as “self confidence”, “economic growth”, “technology”, “science”, “democracy” and “citizenship” by being placed as a single agent that can bring them all about. When used in this way, the meaning of “mathematics” is never made more precise by referring to what kind of mathematics is meant (e.g. refering to “algebra”, “arithmetics” or “geometry”). It is used as self evidently meaningful.
The implicit reference of “mathematics” does however go to the context of schooling. Mathematics is to strengthen democraty and create economic growth through education. It is education that is to bring mathematical knowledge about. Mathematical knowledge thus, secondly, becomes a link between society and the institutionalized practices of the education system.
Thirdly, the situation in school, in the discourse of teachers and in mathematics education research, is different. There the idea of sublime objects structuring desire (Žižek 1989) is more to the point than the interconnecting function of a word devoid of content. In the discourse of mathematics education, mathematics functions as a kind of attribute, injecting mening into otherwise quite bland words such as “thinking”, “knowledge”, “problem solving”, “concepts” and “creativity”. Interlinked with mathematics, to “mathematical thinking”, “mathematical knowledge”, “mathematical problem solving”, “mathematical concepts” and “mathematical creativity” they entice fascination and can be the focus of long term research projects. Importantly, these objects infuse meaning into the specific practices of mathematics education. The practices are justified by reference to these objects.
Finally, the purpose of schooling is to build, create, form these objects (concepts, knowledge) in its subjects. Mathematics thus also constitutes the link from education to society, as the thing that students bring – in many different forms – from school to their everyday and professional life.
I talk about the first two of these functions as pertaining to the (clean, empty, shiny) “outside” of mathematics, and the last two as its (rich, intriguing) “inside”. They corrrespond to two different ways for mathematics to exist: on the one hand as a kind of essence of the natural and social world and on the other as properties of people, as their “knowledge” and “competence”. I think of them as the two sides of a soap bubble containing mathematics education: its outside reflecting society, bringing its imaginary together as if that was the essence of education, its inside reflecting the practices of education, as if they caught the essence of life in modern society. Itself vanishingly thin, mathematics determine how school and society see each other.
It is interesting to note that a characteristic feature of mathematics, as talked about in the context of education, is that it is always seen from a certain distance. No one is expected to master it. It is brought in from somewhere else (i.e. from science). One thus knows that it is many different things, but not exactly what. There is always some part of it that is beyond the horizon. This makes it fit very well with what Žižek talks about as a sublime object of an ideology.
Mathematics contributes to the constitution of school and society as two interdependent units. Importantly, they are constituted as interdependent in a particular way. The society bound together by mathematics is not just any society, and it is not just any practices that are to bring mathematical knowledge about. In the next section, I will talk about some specific properties of mathematics education.
Three doctrines of mathematics education
I have singled out three doctrines which form the core of how mathematics education understands itself. It is important to note that they are seemingly derived from the properties of mathematics or, more specifically, from the combined properties of man, mathematics and nature.
The first doctrine of mathematics education is that it is important to make a distinction between on the one hand digits and words (spoken and written) which are signs and symbols, and on the other the concepts and numbers which stand “behind” them. The task of mathematics education is, accordingly, to form, to make present, in its subjects, mathematical concepts. “Having” these concepts is synonymous with understanding mathematics. This conceptual understanding is the essential prerequisite for attaining the good of mathematics.
Historically this doctrine comes from the study of Euclid’s Elements, with its empirical constructions signifying objects residing in another realm.
The teaching method most clearly reflecting this doctrine was called “socratic” or “heuristic” in the 19th century, with references going back to Plato. In this method, the teacher uses cleverly constructed questions to “bring out” the mathematical truths from the pupil. The point of this is that – according to the first doctrine of mathematics education – mathematical concepts can only take form through something which one could perhaps call constructive discovery. Concepts cannot be “transferred” through discursive explanation. They must be built by the student himself.
The second doctrine of mathematics education is that mathematical concepts (at least the most fundamental ones) only take form as the result of the immediate experience of material reality. If the central term of the first doctrine is “självverksamhet”, the central term of the second is “åskådlighet”, in the original german: anschaulichkeit. Important here is to distinguish this doctrine from the use of objects as a means for explanation, i.e. as an illustration. What is at issue here is the gradual formation, the building, bildung, of concepts, as the result of a sustained and – as far as possible – non discursive sensual experience of material reality. The argument, as presented at the beginning of the 20th century, runs as follows: numbers are abstract. Children lack the ability for abstract thinking. They thus need to think the numbers as connected to collections of things. When number concepts has taken form in this connected form, they can gradually be disconnected from objects. At this point, and not before, can digits be introduced as “abbreviations” or “representations” of the concepts. If digits are introduced earlier, the children will learn how to manipulate the digits, without ever forming the concepts that should stand “behind them”, ending up hopelessly trapped in non-conceptual thinking.
Historically, this doctrine seems to have taken form at the end of the 18th in a context of Christian romanticism – I think here primarily of Pestalozzi and Fröbel.
The third doctrine of mathematics education is that mathematical concepts become practically useful in a certain context only by being formed in that very context. There is a twist, though (not emically acknowledged): context here means written representations of context. That concepts are to be formed “in context” thus translates to the applied tasks we are all familiar with from our own time in school. This doctrine implies the identification of textual representations of practice with the practice itself.
The effects of the teaching methods corresponding to this doctrine has been explored by the English sociologist Paul Dowling (1998). He suggests that it leads to the propagation of two myths: that reality is inherently mathematical, making knowledge of mathematics a privileged kind of knowledge of the world and second that mathematics “participates” in everyday practice, making knowledge of mathematics a prerequisite for proper participation in these practices.
Dowling’s analysis, which builds on a close reading of a teaching material for English secondary education, sheds light on the current standing of the ontological transformation, most of which took place in the 17th century, which moved mathematics from being a human construction – a model of the world essentially different from, and certainly less real than, the world in itself – to being the essence of the world, more real than whatever can be seen and measured (Gringas 2001; Dear 1995).
These three doctrines have, when translated into teaching methods, the effect of connecting mathematics to:
- rational thinking and our inner, “glassy”, thruth-reflecting essence
- material reality, making it inherently mathematical and
- everyday and professional life.
Mathematics education thus contributes to the constitution of what may be called a cosmos. It does not seem far fetched to call this a, or perhaps the, modern cosmos, as described from different perspectives in (Taylor 2007) and (Toulmin 1992).
Socially constructed reality
Now I want to focus on how the seeming constance over time and place of mathematics is the result of quite concious efforts of standardization (Cf. Porter 1986; Porter 1995; Desrosières 1998). That the activity of a mathematics classroom is probably one of the most easily recognized in modern society is no coincidence. It is the result of centralized teachers education, centralized production of teaching material, standardized tests of student performance and centrally coordinated inspections of teaching practice. All this is of course done in the name of mathematics (and in the name of the child). We are to believe that the homogeniety is the result of successful adaption to objective reality. My point is that it is the other way around: mathematics and the properties of the child seems to be parts of a “reality” which everybody takes to be “the same” because of the homogeniety of schooling.
One should also note how the reality of mathematics is socially supported. We learn that we meet mathematics every day – and indeed, we meet it every day in school. We learn that we need to master it to get by in our everyday life – and this is certainly true of the everyday of schooling. We learn that knowledge of mathematics is a prerequesite for self confidence and full participation in society, and the regulations of the education system certainly makes this come true for many. It does not seem far fetched in this case to speak of a socially constructed reality.
This reality probably gains power from the fact that mathematics education is a long series of tests, small ones in the form of applied tasks and bigger ones leading to scores and grades, in which everybody are forced to invest time and energy. Those who succeed derive pleasure from this experience, those who fail lose self confidence. Importantly, the tests are interpreted in terms of mathematics; what one succeeds or fails in is always mathematics. When a failure is talked about as a failure in “mathematical problem solving” and explained in terms of a lack of “conceptual understanding of mathematics”, mathematics is filled with meaning in parallell with the practice it explains.
In the cosmos of mathematics, the child, children and child-hood (cf. Ariès 1962) play an important role. How school practices which purportedly take “the child” as their major point of departure contribute to the constitution of this child-object is well known (Walkerdine 1988; Walkerdine 2002). Children lack the ability for “abstract “ or “conceptual” thinking and “develop”, in stages, from bodies without minds to fully human beings, with characteristically “glassy” conceptual thinking. What I want to point to is how this development goes hand in hand with the objects to be known and thought, i.e. mathematics. Mathematics is used as an interpretative framework for talking about how children change as they grow older. Conceptual mathematical thinking is a sign of adulthood.
This story of development, often described in terms of movement along a path, corresponds on a sociological level (reconnecting with Bourdieu) to the hierarchical distribution of people. The system of education contributes to the reproduction of society by distributing children between different social positions more or less according to the positions of their parents (Bourdieu 1996a; Bourdieu and Passeron 2008; Bourdieu and Passeron 1964). In this, mathematics education is very helpful by producing seemingly objective scores according to which children can be allowed or denied access to different paths .
It is mathematics that make these scores meaningful. Here it is perhaps again right to speak of mathematics as a node, through which the scores spread out to encompass the world. By “being mathematics” the scores are equivalent to rational thinking, instrumental effectivity and knowledge of the world.
It may seem obvious that mathematical tests, arranged according to strict formal rules and filled as they always are with representations of reality, are a kind of ritualized enactments of competence. But I do not think the mechanisms of this ritual are at all easy to understand. Especially not the mechanisms through which the performance of the formalized school-setting is made to represent competence relevant outside school. In the case of mathematics education, I think it is mathematics that makes this magic possible by its two modes of existence: as part of the world and as knowledge, by being formable and measurable in school, and, as formed knowledge, usable outside school.
Scores are talked about as amounts of knowledge, implying knowledge as an abstract quantity somehow residing in each individual. I think one should take this very seriously. We learn that all humans are born equal. Mathematics education simultaneously creates and discovers this abstract quantity which makes humans, as adults, hierarchically different. Mathematics functions as a kind of reference object – again the idea of a node is probably useful – to which everybody are related, a powerful, ever present object, of which we have more or less “in us”.
One can compare this quantitative knowledge-essence of mathematics education with Marx’ idea of money as a fetish. Marx said that the value of money is thoroughly social, but that we act as if it was an inherent property of money. This, Žižek adds, must be understood as a symptom of the fact that we do not, in modern society, see people as essentially different. We say that everybody have the same value and explain their different positions in society by the contingent and non essential fact that they have different amounts of money.
What is mathematical knowledge to this? The value of grades (i.e. test scores) are of course thoroughly social, in approximately the same way as money. And the interpretation of the scores as an individual quantity of a universal substance can be seen as hiding this social nature (Cf. here also the distinction made between “canonical” and “self referential” messages in Rappaport 1999).
But the constitution of the universal substance is, it seems to me, very different in the cases of money and knowledge. We acknowledge, as Marx and Žižek says, the properties of money in our acts of using them, i.e. this is how they get these properties. Mathematics is, I believe, constituted in the formalized setting of the classroom. And the distribution of money is a matter quite different in nature from the distribution of knowledge.
Importantly, knowledge is distributed according to centrally constructed plans: you, as a human or citizen, are not to decide what to learn, when to learn it, how to learn it, and who you are to learn it from. This state of affairs is purportedly derived from the properties of the substance thus distributed. As regards money, prices and wages are determined without central planning, what to buy and when to buy it is supposedly a matter of free choice. I suspect that these two matters are somehow related.
Distribution of abscence
Talking about the distribution of mathematical knowledge, it is actually more accurate to talk about distribution of lack of knowledge. It is typically claimed that mathematics education should bring about mathematical knowledge, but that it fails to do this, instead producing anxiety, boredom and segregation by use of traditional, mechanical, formalistic, non dialogic, textbook-guided methods devoid of mathematical content. Mathematics is essentially not there, in school, in the socially extended body of the education system. It is present, however, in its “head”, in the centralized agencies for research, planning, inspection and development. From there flows a never ending stream of purportedly reforming directives.
This typical stance towards mathematics and mathematics education can be described in terms of a distinction between potential and actuality. Mathematics is thence not actual, not actually there, in mathematics education. On the other hand mathematics has an inherent potential to lead to the good of society. Thus, there is a constant drive for reform of educational practice, to make mathematics present, to actualize its potential. These reforms are guided by the particular properties of the potential of mathematics.
Ivan Illich was right in pointing out that people today usually do not believe in education (Illich 1972; Illich and Cayley 1992). The normal stance is cynicism. Interestingly, this explains why the normal consequence of education is not the good connected to its subject matter (e.g. mathematics). Mathematics is not there, and this means that what is distributed to the individuals as they pass through the education system is not so much the substance of mathematical knowledge as lack of such knowledge, a kind of guilt-laden emptyness, representing their relative lack of worth in comparison with the impersonal and indifferent universality of mathematics (Cf. Boltanski and Thévenot 2006).
Maybe surprisingly, the cynicism regarding mathematics education often seems to extend even to mathematics itself. I have found that central proponents of mathematics education, i.e. authors of public reports, doubt that mathematics really is a precondition for handling everyday life, a pillar of democracy etc. This complicates matters considerably, not the least when arguing “against” mathematics education. It is very hard not to seem to kick a dead horse.
What has to be acknowledged here is that there always is a “part” of mathematics that escapes the cynical stance. This is a part beyond personal knowledge, seemingly completely unrelated to mathematics education, seemingly idiotic to doubt. This is, I would say, the sublime object of mathematics education.
Mathematics in this sense fits very well with Žižek’s description of a fetish: mathematics is still largely unaffected by the falling apart of the great narratives of modernity, it is still “out there” keeping things together – from the distance created by the divide between the two cultures. Without it, mathematics education would be a total mess, and then, what would be left of the education system at large, and then, of society?
Thus, it is exactly the creative, beautiful, sublime usefulness of mathematics, most distant from boring mathematics education, that is its politically most powerful result. It is this object that must be put into question.
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