Democratic Access to PowerfulMathematical Ideas*

Ole Skovsmose and Paola Valero**

Abstract. The emergence of the informational society creates theparadoxes of inclusion and citizenship, which call into question anysimple interpretation of the meaning of “democratic access topowerful mathematical ideas”. In exploring this thesis we put forwardways of understanding what “powerful mathematical ideas”represent logically, psychologically, culturally and sociologically. Asa way of tackling the issues of democratic access to these ideas, weelaborate on three arenas of mathematics education practices whereit is possible to build a meaningful participation to committedpolitical action, namely the classroom, school organization, andsociety both locally and globally. To conclude we explore thepotentialities of the space of investigation into democratic access topowerful mathematical ideas defined by the four interpretations of“powerful” and by the three arenas of democratic access. We pointto the necessity of covering this whole space of research in order togive a full picture of the complexity of mathematics education in ourcurrent informational society.

Carlos had to move out of his home. His mother seems to be worried.She lost her job and the money she made through great effort to payfor the small house is in the hands of the bank. Carlos, a tenth gradestudent, is one of the many Colombian youngsters who will finishhigh school at the beginning of the 21st Century. Many of thesestudents seem to be confused about their future. Teachers insist on theimportance of schooling and learning, especially mathematics. Yethow could that help in their actual situation? On the other side of theworld, in Denmark, Nicolai got seriously sick after eating a home-

* This paper will be published in L. D. English (Ed.), Handbook of internationalresearch in mathematics education: Directions for the 21st century. Mahwah, NJ:Lawrence Erlbaum Associates.** Although our names appear in alphabetic order, we want to acknowledge ourequal contributions to the writing of this paper.

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made ice cream. Contracting salmonella via eggs, chicken or variousmeat products is not an unknown situation in Denmark. People blamequality control, but do they, in fact, know what it is about? Does theeducation that Nicolai gains in school help him understand thedangers of his apparently “safe” society?

These are cases of real students in two different countries, and,as we shall argue, their life experiences are significant to mathematicseducation. In the task of advancing the field of research on thephenomena connected to the learning and teaching of mathematics,we start from the consideration of the global informational society asthe complex social, political, cultural and economic context. Withinthis context both world and local trends intermesh, and newchallenges to mathematics education practices and research emerge.Based on the contradictions of this current social order, we proposethe paradox of inclusion and the paradox of citizenship as two centralproblems that mathematics education has to face. With this purpose inmind, we proceed to give meaning to the term “powerfulmathematical ideas” in four different ways. Then we discuss thenotion of “democratic access” and question the simple identificationof democracy with universal access. Finally, we argue that facing theparadoxes of inclusion and citizenship represents a struggle for theprovision of “democratic access to powerful mathematical ideas” inmathematics education, in both practice and research.

Paradoxes of the informational society

After the breakdown of the wall between East and West, Fukuyama(1989, 1992) declared “the end of history”. This statement resonateswith what theories of post-industrial society had been claiming sincethe 1970’s, namely that the world has reached a state in which thesources of value –and therefore of power– can be described not onlyin terms of labor and capital, but also and primarily in terms ofknowledge and information. This state in the transformation ofcapitalism has also been called the information society (Bell, 1980).Together with the consideration of value and power, there has been achange in the kind of citizens that this new type of social orderrequires. People need to be able to deal with knowledge andinformation in continuous processes of learning. This particular shift

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is what has been called the learning society (Ranson, 1998). In whatwe will refer to as the “informational society” (following Castells,1999), subsuming both the information society and the learningsociety, the impact of technology goes beyond industrial production,and, in fact, affects political, economic, social and cultural structures.

A discussion of the informational society cannot be separatedfrom a consideration of globalization as the process responsible forestablishing the “world village”. Globalization refers to the fact thatevents in one part of the world may be caused by, and at the sametime influence, events in others parts. Our environment –described inpolitical, sociological, economic, or ecological terms– is continuouslyreconstructed in a process that receives inputs from all corners of theworld. Simultaneously, our actions have implications for even themost remote corners of the planet. However, globalization also relatesto the apparently shared belief that a given kind of environment isdesirable, and that there is some kind of universal commitment to theachievement of certain ideals like democracy, market freedom andindividual competitiveness. The myth of the “end of history” can beinterpreted as the legitimization of a false universalism (Eagleton,1996). Together with the discourse of globalization comes a newdiscourse of colonization. In a similar way that the first Europeanwaves of colonization, from the 14th to the 18th centuries, broughtnew languages, religions and social orders that trampled downindigenous cultures, the new global colonization also imposes newways of living, producing, and thinking. D’Ambrosio (1996) seesscience, including mathematics, as also playing a role in this culturalinvasion; and, of course, mathematics education is not an innocentonlooker of the situation.1

1 As an example of what globalization means, we can analyze the ThirdInternational Mathematics and Science Survey (TIMSS) as a representativeinternational study that produced knowledge and information about the state ofmathematical and science education in the world. Despite all the discussions aboutthe problems of TIMSS as a legitimate ranking system and means of comparison,one of the conclusions that at several levels seemed to be drawn is the necessity offollowing the model of high-scoring countries like Singapore and Japan. Thus, atthe International Round Table at the start of ICME 9, a director from theSingapore Ministry of Education explained their conception of mathematics andscience education and how they have been able to achieve success. The setting ofthe whole round table can be interpreted as an attempt to put forward in theinternational community of mathematics educators a model that is desirable to

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Castells (1999) criticizes some of the dominant descriptions ofthe post-industrial society based exclusively on the North American-European context. He emphasizes that a theory of the informationalsociety should refer not only to the fact that certain countries andcertain regions are becoming closely interrelated, but also to the factthat people, countries and regions are excluded, apparently for notbeing of any relevance to the construction of the informationaleconomy. Since access to knowledge is clearly important in theinformational society, “[t]he ability to generate new knowledge andto gather strategic information depends on access to the flows of suchknowledge and information [...]. It follows that the power oforganizations and the future of individuals depend on theirpositioning vis-à-vis such sources of knowledge and on their capacityto understand and process such knowledge” (p. 60). The access to theflow of knowledge and information constitutes a major divisionbetween those in the core of the informational society and thoseoutside it. According to Castells, exclusion is devastating since “[t]hestructural logic of the information age bears the seeds of a new,fundamental barbarism” (p. 60). All the outsiders belong tostructurally irrelevant areas in the informational society, andconstitute what Castells calls the “Fourth World”.

This observation draws attention to the complex dynamics ofglobalization. At the same time that we are becoming similar, we arealso moving apart. The interplay between the global and the local is agame that connects many parts of the world in a network of flows,and simultaneously excludes regions and people from specificcommunities and countries in the world. The Fourth World includesnot only large regions of Africa, Latin America and Asia, butcertainly also carves out large chunks of Europe, USA, Japan andAustralia. Many people who either live in poverty or are isolatedfrom the centers of informational and technological production andexchange in these countries such as political refugees and illegalimmigrants in the USA, old people in rural areas in Japan, aboriginalcommunities in Australia and young junkie and punk communities inGermany are apparently superfluous in this world order.

follow. On the other hand, the systematically scant mention of countries that didvery poorly –like Colombia and South Africa– shows the dispensability of thesecases in what is accepted as relevant internationally.

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Nevertheless, the Fourth World has some relevant roles to playfor the informational economy. First of all, it supplies spaces fordumping ecological problems and other side-effects of industrialproduction. It also provides a market area to be flooded and a cheapsource to supply the material flow of goods needed in theinformational economy. The globalization linked with theinformational economy seems to continue a provocative exploitationof certain parts of the world as a given. A concern for equity seemsnot to be part of this type of globalization.

Globalization is also responsible for determining what counts asfunctional people in the free-flowing, informational economy. Thissocial order is characterized by a strong capacity for renewal andflexibility in individuals and social organizations, which manifestsitself through an enterprise capacity. Individuals and groups becomeorganized under the principle of continuous learning as a mechanismof adaptation to rapid and constant environmental changes. This ideahas implications for dominant current educational conceptions.Learning is conceived as a continuous “learning to learn” in order tofulfill societal requirements. Notions like “constructivist teaching andlearning”, “active students and teachers”, “rich educationalenvironments”, “technology inclusive experiences” and, morerecently, “accountable and efficient educational services”, togetherwith “satisfied parent and student clientele” dominate the learningsociety discourse (Masschelein, 2000; Apple, 2000).

Despite the apparent suitability of some of these “learning tolearn” notions, this whole discourse should be carefully questioned.There is a risk of reducing learning to a mechanism of individualsurvival, which opposes a conception of learning as a human activitywhereby unique beings search for meaning in an attempt to initiateevents that contribute to securing a sustainable, durable, commonworld. In other words, the possibilities of education as a questioningof the self, a judgement of the meaning of life, a construction of acommon world, and a criticism of the given order of things, arehighly at stake (Masschelein, 2000). Furthermore, Flecha (1999, p.67) notes that “the knowledge prioritized by the new forms of life isdistributed unevenly among individuals, according to social group,gender, ethnic group, and age. At the same time, the knowledgepossessed by marginalized groups is dismissed, even if it is richer andmore complex than prioritized knowledge. More is therefore given to

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those who have more and less to those who have less, forming aclosed circle of cultural inequality.”2

In mathematics education the lifelong “learning to learn” ideashave been taken as a desirable goal to be reached in the 21st Century.Mathematicians in the 1960’s took seriously the duty of setting amathematical education upon which could rest “the ever heavierburden of the scientific and technological superstructure” (OEEC,1961, p. 18). Nowadays, a large portion of our mathematicseducation community is apparently committed to the edification ofcompetent citizens of the emerging and rapidly changinginformational society. As the National Council of Teachers ofMathematics (NCTM) Standards 2000 (NCTM, 2000, p. 3-4) state,the capacity to understand and do mathematics is more relevant thanever, since it allows one to “have significantly enhanced opportunitiesand options” for shaping one’s future. This formulation implies thatacquiring mathematical competencies is a condition for being able toadapt, and therefore, both survive and help sustain this type of socialdevelopment. The need and desire for more mathematically ablepeople, as expressed in the discourse of “more mathematics for all”,may contribute to spreading a utilitarian value of mathematicseducation that in the long run serves as a tool for the survival of thesmartest. The contradiction between the social expectations emergingfrom this kind of discourse and actual practices where mathematics isused as a social filter determining who has access to further success(Smith, 2000; Volmink, 1994; Zevenbergen, 2000b), in fact getsresolved in favor of those who pass the gates of mathematics.Therefore, without anticipating it, mathematics education maysupport the dangers of the learning society.

We find that the “informational society” is a contested concept.3It contains contradictions, and it can develop in different directions.We shall try to summarize this fact by formulating two paradoxes ofparticular importance for mathematics education. The paradox ofinclusion refers to the fact that the current globalization model of

2 We could go further in this argument by asking who actually benefits from theexpansion of the learning society discourse. Plausible explanations about the forcesassociated with recent reform trends in several countries can be found in, forinstance, Apple (1996, 2000).3 Here we are inspired by Young (1998) who presents the idea of “learningsociety” as a contested concept.

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social organization, which embraces universal access and inclusion asa stated principle, is also conducive to a deep exclusion of certainsocial sectors. The paradox of citizenship alludes to the fact that thelearning society, claiming the need of relevant, meaningful educationfor current social challenges, at the same time reduces learning to amatter of necessity for adapting the individual to social demands. Theparadox of citizenship concerns in particular the notion of Bildung4

which refers to the development of general competencies forcitizenship, especially the capacity to act critically in society, and inthis way have an impact on it. This paradox refers to the fact that, onthe one hand, education seems ready to prepare for active citizenship,but, on the other hand, it seems to ensure adaptation of the individualto the given social order.

Although from our field of research and practice, mathematicseducation, we cannot solve the paradoxes, we find it necessary to facethem. If not, mathematics education could act blindly in the furtherdevelopment of current society. We engage in the task of exploringthe significance of these two paradoxes from the particularperspective of mathematics education by examining the notions of“powerful mathematical ideas” and “democratic access”. We will doso by reference to two examples.

Two examples

Terrible small numbers

Salmonella pollution is an everyday danger in Denmark. In one wayor another, Nicolai knew that an “innocent” home-made ice cream,prepared with infected eggs, could be enough to make him sick.Students in school hear and can read about salmonella infection. Anewspaper article under the headline “We have to live withsalmonella” reads as follows:

4 For a discussion of the notion of Bildung see Klafki (1986) and Biesta (2000).There is no adequate English translation of the German word Bildung, although“liberal education” has been suggested.

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Experts estimate that a steady number of more than1.000 Danes will be sick with salmonella each year.The Minister of Food, Henrik Dam Kristensen (SocialDemocracy) says that we won’t succeed in wiping itout. Danes have to live with the permanent risk ofgetting sick from salmonella via Danish meat and eggproducts […] This was one of the conclusions fromthe report given to the Minister by the DanishZoonosis Center, the advisory institution in thesematters. According to the Minister, the paper does notlead to any changes in the strategy against salmonella,but Danes must learn to live with the infection risk.We will still prepare tests and investigations so that wecan come as close as we can to zero risk. However,that is not the same as ensuring that salmonellainfected eggs, chicken and pork will not pass thecontrol. Today it is impossible to make people believethat a 100 percent secure control is in place (Politiken,2000, our translation).

If risks, as stated by Beck (1992), are an essential constituent of ourcurrent world, how could school and specially mathematics teachingand learning provide tools for analyzing those risks in a meaningfulway? The project, “Terrible small numbers” tries to address thisquestion.5 Together with their students, the teachers participating inthe project collected 500 black photo film cases to simulate eggs–after all, they resemble eggs in size, lack of transparency andpossibility to be “opened” for examination. Inside each egg there wasa yellow centicube, except in some of them in which a blue centicubewas placed. The blue “yolk” represented a salmonella infected egg.

During the first sequence of activities the mix of healthy andsalmonella infected eggs was made in front of the whole class.Everybody knew that out of the 500 eggs, 50 were infected. Thestudents then had to take samples consisting of 10 eggs, and to countthe number of infected eggs. Intuitively the students expected to get

5 This project is described in Alrø et al., (2000a, 2000b), in Danish. It is acollaboration between two Danish teachers, Henning Bødtkjer and MikaelSkånstrøm, and three researchers, Helle Alrø, Morten Blomhøj and OleSkovsmose. “Terrible small numbers” has been tried out in different classrooms,but here we primarily provide a general overview of its main ideas.

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one blue egg in each sample, but after some experiments they foundthat in some cases they could get 3 blue eggs –or even more– out of10. How could that be? Was it because the mix was not done in aproper way? Was it bad luck, somehow? The basic question to beaddressed by this experiment has to do with the reliability ofinformation provided by samples. How can it be that a sample doesnot always tell the “truth” about the whole population? And howshould we operate in a situation where we do not know anythingabout the whole population, except from what a sample might tell?How can we, in this case, evaluate the reliability of numericalinformation?

In a second sequence of experiments, students were presentedwith two types of eggs, Spanish and Greek, to buy for retail sale inshops. In both types there were some infected eggs, but this time thestudents did not know how many. In order to make a decision aboutwhich type of eggs to buy for retailing, they needed to run a qualitycontrol test. It was impossible to test all eggs, as eggs opened in thequality control cannot be used for sale. Furthermore, it was expensiveto get eggs checked for salmonella, so the students’ (the retailers’)budget was affected by control costs. They had to consider carefullyhow many Spanish and Greek eggs needed to be sampled in order tomake a decision about which type to buy. The concern for making aresponsible decision was confronted with the interest of making ahealthy business.

A third sequence of activities dealt with the evaluation of therisks of getting salmonella from egg-made food products. The startingpoint for the preparation of the products was a mixture of 500 eggs, 5of which were infected. A preliminary question was to calculate theprobability of finding a blue egg –it was not difficult to come to5/500 = 0.01. Now, if we want to make an ice-cream portion out of 6eggs –and of course we would like them all to be healthy– theprobability of getting a salmonella-free portion is (1-0.01)6 andtherefore the risk of infection is 1-(1-0.01)6. To get to this formulawas not simple. The students started by suggesting that if theprobability of getting an infected egg is 0.01, then, when picking 6eggs, the probability must be 0.06. However, by finding the properformula, the students had an opportunity to contrast mathematicalcalculations with empirical experimentation.

The project tried to provide ground for a discussion of thedifference between ideal mathematical calculations and empirically

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obtained figures, as well as a debate about the possibility ofcalculating risks in general. The notion of risk can be summarized inmathematical terms by the equation R(A) = P(A)C (A). Here Arepresents an event. The risk, R(A), is the product of the probabilitythat A happens, P(A), and the consequences of A happening, C(A). Inother words, the risk of eating an ice cream dessert equals theprobability of being infected times the “cost” of being infected, andnaturally the “costs” increase with the size of the dessert since manymore people may taste it.

Macro-figures becoming macro-dangers

A country in an unstable economic and political situation is a perfectscenario for witnessing the macro-dangers of macro-figures.Colombia, in the last decade of the 20th Century, represented a deeplytroubled society, in conflict between democratic consolidation andinternational globalization demands. In this scenario, where almostpre-modern, modern and post-modern living conditions coexist,students struggle to find good reasons for going through schooling–if, of course, they have a chance of doing so. Carlos certainly findsit difficult to see the role of so much studying in his future. It is evenharder now that his family had to leave the house that his motherstarted paying for some years ago. Recently the monthly mortgagepayments became so high that she had to give up. When she tried tosell the house, she could not recover a single cent of what she hadinvested, and her best solution was to give it back to the bank as partof the debt payment.

Carlos was not the only student who, between 1998 and 1999,lost his home. Such was the story that many people lived inColombia. For the first time people started being concerned tounderstand what the UPAC (Unidad de Poder Adquisitivo Constante[Unit of Constant Buying Power]), introduced in 1971, could mean intheir lives. Certainly a mathematical investigation in the classroomcould be of help. In what follows, we imagine the general guidelinesof a project, “Macro-figures becoming Macro-dangers” with 10th or11th grade students.6 The project may allow students to reflect about

6 This example builds on discussions with Colombian teachers during the seminar“Cómo desarrollar una educación matemática crítica en el salón de clase” led byOle Skovsmose in Bogotá, Colombia (October 8-9, 1999), on Paola Valero’s

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the use of mathematics as a power resource through economic andsocial models.

From where to start? We could ask students to ask their familiesand friends about the UPAC and its predicaments. We want theproject to be of relevance for the students’ actual situation. As one ofthe essential inquiry sources, we all can collect receipts for mortgagepayments during the last one or two years. We can ask for help fromthe social science teachers to get information about the UPAC systemand the reasons why back in the 1970’s the government adopted it. Isit possible to discover the assumptions of the system? Are they stillvalid? The UPAC system, which was intended to promote privatesavings and housing acquisition, was designed under the assumptionthat, on the one hand, devaluation, inflation and interest rates couldbe controlled by the government (Currie, 1984; Perry, 1989), and onthe other hand, that the country would have a steady economicgrowth.

In the case of mortgage payment, the UPAC system operates inthe following way. In order to calculate nominal interest (n) on amortgage, the system considers inflation (i), the effective interest rate(e), which is estimated at 6% annually, and a risk factor (r). Thenominal interest, n, is then determined by the formula (Vélez, 1997):

n = (1+i)(1+e)(1+r) – 1

For instance, in normal conditions, if i = 0.06, e = 0.06, and r = 0.01,then n = 0.13, which would be a reasonable case. At a time of deepeconomic crisis the nominal interest gets out of control due tovariations in inflation, the effective interest rate and the risk factor, asactually happened in Colombia in the period between 1997 and 2000.For instance, in a crisis situation, if i = 0.18, e = 0.20, and r = 0.10,then n = 0.55, generating an aberrant situation. Besides, in thespecific Colombian case at the end of the 1990’s, when people couldnot afford to cover the payments –e.g., due to unemployment– andwere forced to sell their properties, they lost all their savings since thevalue of real estate decreased as part of the crisis itself.

presentation “Desenmascarar las matemáticas: Un reto para los profesores delpróximo milenio” in Portimão (Portugal) during ProfMat 99 (November 10-14,1999), and specially on follow-up discussions with Jaqueline Cruz and VerónicaTocasuche, secondary school teachers in Colombia, and with Pedro Gómez. Theseideas have not been implemented yet.

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After a first exploration we could start looking at specific cases–that of Carlos’, maybe, if he and his family agree– and make groupsbased on students who have similar cases. The main purpose of thework could be to advise specific families in the process of negotiatingnew payment systems with the bank. Based on the payment receiptsgathered, students can study the connections between the differentfigures in a given period of time –the total amount of the mortgage,the interest rate, the proportion of the debt that has been actuallyrepaid, the payments for interest, etc. We could go as deep as we needinto the mathematical exploration of the situation.

Then we could enter into a discussion about the consequences ofthe model. We could prepare a report for the families, explainingwhat happened during the time they paid their mortgages andproposing suitable alternatives to deal with bank proposals about there-negotiation of their mortgage and the adoption of the new modelproposed by the government7. As one of the aims of the project wewould like to grasp the potentiality that a mathematical classroominvestigation could have for initiating changes in the students’ lives.Is what we all gain during the development of the project enough toact politically around the families in trouble?

Does this thought experiment illustrate essential aspects of whatto consider in an enquiry in the mathematics classroom? Is itimportant to realize this project? Likewise, what can we say about“Terrible small numbers”? If mathematics education should face theparadoxes of inclusion and citizenship of the informational society,such questions become important. However, in order to discuss inmore detail the possibilities of tackling the paradoxes, we have toexplore what could be the meaning of “powerful mathematical ideas”.Subsequently, we will discuss the different aspects of providing“democratic access”, and then return to the paradoxes.

7 From January 2000 the UVR system (Unidad de Valor Real [Unit of RealValue]) replaced the UPAC. The UVR established a simpler index based on thenational basic cost of living.

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Powerful mathematical ideas…

To say that something is powerful is tantamount to affirming that itcan exercise power. If we state that mathematical ideas can exercisepower, we should try to clarify the following questions: What are weunderstanding by “power”? What is the source of the power ofmathematical ideas? What are the consequences of that power? Inwhat follows, we put forward different possible interpretations.

… logically speaking

Mathematical ideas can be seen as powerful from a logical point ofview. In this sense, power refers to the characteristic of some keyideas that enable us to establish new links among theories and providenew meaning to previously defined concepts. In this sense, one cancertainly assert that plenty of powerful mathematical ideas haveemerged throughout the history of the discipline.

In particular we can associate the notion of powerfulmathematical ideas with abstraction. Thus, a concept may beinterpreted as powerful to the extent that it provides new insight intoa different set of concepts. The notion of group illustrates the logicalpower of making abstractions. A group can be defined as a set, M,consisting of certain elements, and an operation, *, which to any pairof elements from M associates an element from M, and which fulfilscertain properties. Exemplars of groups are then recognized all overmathematics, a basic one being the set of integers together with theoperation “addition”. A wide range of other mathematical structures,besides group, are recognized, like ring, vector space, metric space,topological space, all defined solely by their formal properties andnot by any qualities of their elements. Such formal structures make itpossible to bring an understanding obtained in one area ofmathematics to apply in an apparently completely different area. Inthis way, abstractions have led to a class of powerful mathematicalideas, logically speaking. The power or strength of those ideas, then,can be defined as an intrinsic and essential characteristic of theirposition in the hierarchy of mathematics, which allowed them toinfluence other ideas as to re-accommodate and re-define them. Oncea more abstract mathematical idea provides a new conceptualizationfor previously existing notions, the whole building of mathematics is

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restructured given the legitimacy of the new ruling and organizingprinciple. In this sense, powerful mathematical ideas, logicallyspeaking, have an intrinsic power exercised within the realm ofmathematics.

Such powerful mathematical ideas can be expressed in a logicalarchitecture, as exemplified by the work of the Bourbaki group. Acloser look at this strictly modernist edifice reveals also what“powerful”, in a logical sense, could mean for mathematics education.If mathematics education is conceived as having the role ofenculturating students into established mathematical knowledge andits ways of working, then it is easy –as the proponents of the NewMath Movement in the 1960’s thought– to generate a list of powerfulmathematical ideas around which to organize the curriculum. Bymeans of such logically basic ideas, all other ideas could be defined.

Although the particular approach and aims of the modernmathematics education wave have almost disappeared from schoolcurricula, there is still a dominance in practice of the idea thatmathematics curricula consist of a list of essential, powerfulmathematical ideas and topics to be learned. The amazing similarityand stability in the structure of national mathematics curricula acrossthe world (Kilpatrick, 1996) show the strength of the shared belief inthe logical power of given mathematical ideas. Independently fromthe orientation of the approach to school mathematics, such as theBack-to-Basics in the USA and the National Numeracy Strategy inthe UK –which stress the traditional priorities of mathematical topics–or the NCTM Standards –which represent a more progressivecurricular proposal– all these views try to grasp the essence ofpowerful mathematical ideas from this logical point of view. Muchmathematics education simply assumes that mathematical ideas arepowerful primarily in a logical sense. This justifies that mathematicsteaching can concentrate on providing students access to “real”mathematics, either by following the school mathematics tradition oreven by a progressive establishment of a scaffolding which makes itpossible for the students to construct mathematics by and forthemselves.

From this perspective, what can we make of the projects“Macro-figures becoming macro-dangers” and “Terrible smallnumbers”? Could they lead to powerful mathematical ideas, logicallyspeaking? Certainly we could imagine possible ways of strengtheninga mathematical focus. In the case of “Terrible small numbers” one

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could have gone deeper into the mathematical significance ofexpressions like 1 – (1– p)x, and into other probability notions andthe connections among them. This could have brought the studentsinto a whole exploration of probability theory. In the planning of theproject “Macro-figures becoming macro-dangers” one could startconsidering the equations:

n = (1 + i)(1 + e)(1 + r) – 1 = (i + e + r) + (ie + ir + er) + ier

In particular, by making this algebraic reduction it becomes clear thatn > i + e + r. Furthermore, the project could provide a nice entranceto algebra, and once more it can be illustrated that abstraction is anessential element of powerful mathematical ideas. The calculationscould also open a route directly into the exploration of exponentialfunctions, as the project makes it relevant to consider how a functionlike f(t) = (1 + n)t , with t referring to time, operates.

The logically based interpretation of powerful mathematicalideas legitimates doing mathematics for the sake of the internalcharacteristics of mathematics. It supports the desirability of allowingstudents to experiment and play with ideas and ways of working thatin themselves appear powerful. Nevertheless, this perspectiveembraces some risks. It could accentuate the paradox of inclusion,since it will justify the provision of an abstract curriculum that, asmuch research has documented, systematically closes the possibilityfor the majority of students of participating in a meaningfulmathematics education experience (Boaler, 1997). This perspectivecould also contribute to exacerbating the paradox of citizenship sincemathematics education could end up offering knowledge whichappears relevant for students to their further career opportunities, butwhose relevance beyond this is limited.

… psychologically speaking

We could also associate power with the individual’s experience inlearning mathematical ideas. In this sense, power is determined inrelation to learning potentialities. From this perspective, what countsas significant ideas is what students can come to grasp and makemeaning of in their process of developing mathematical thinking. Infact, the majority of research in mathematics education in the decades

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of the 1980’s and 1990’s is an important source for the identificationof this kind of powerful ideas.

Influenced by the work of Piaget and, more recently, Vygotskyon the development of human cognition, mathematics educators haveformulated different theoretical frameworks to describe what thelearning of mathematics is about8. These theories have also served thepurpose of describing basic principles for what should be achievedthrough the mathematical schooling experience. Verschaffel & DeCorte (1996) offer an example in the case of arithmetic. First theystate the leading principles for an arithmetic learning-teaching inschool in terms of learning mathematics as a social and cooperativeconstructive activity, the role of meaningful contexts, and theprogression towards higher levels of abstraction and formalization(pp. 102-103). Then they formulate some major aspects that need tobe given more attention in connection with, for example, theacquisition of number concept and number sense (pp. 105-111).These aspects include counting at the expense of logical operationalskills in the early grades, allowing an awareness of multiple uses ofnumbers, promoting number sense and estimation, and going beyondwhole numbers. In contrast to a logical interpretation of powerfulmathematical ideas, such items do not emphasize the mathematicalcontent involved in the learning process, but focus instead on themental operations that go together with the acquisition of themathematical notions. In the case of algebra, Kieran (1992) providesa list of similarly powerful mathematical ideas.

One important notion emphasized in the learning of algebra–and of more complex mathematics– is that of the duality betweenconceptions of mathematics as processes and as objects (Sfard, 1991),which in the French didactique des mathématiques version isformulated as the dialectic between mathematics as tools and asobjects (Douady, 1987). This whole discussion, which has certainlyinfluenced the understanding of mathematics learning and teaching9,combines a mathematical analysis about the nature of mathematicalobjects with an analysis of learning processes. In this way, the point

8 For a discussion of the influence of Piagetian and Vygotskian ideas onmathematics education see Skott (2000, pp. 24-39) and Lerman (2000).9 One of the signs of the influence of this work is the extent, in quantity andquality, to which Sfard’s paper has been quoted in research in mathematicseducation since it appeared in 1991.

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of the power of mathematical ideas is connected to the degree inwhich they can be integrated into the students’ understanding throughprocesses of interiorization, condensation and reification. The wholeissue of understanding (Sierpinska, 1994) is, therefore, the key todefining the potential that mathematical ideas can have once locatedin the domain of human learning.

In addition, an emphasis on affective, motivational andidiosyncratic aspects of both students’ and teachers’ understanding ofmathematics –and of its learning and teaching– is also considered acentral part of the generation of powerful mathematical ideas,psychologically speaking. The realization that meaningfulmathematical ideas are only acquired –or constructed– if theindividual has a favorable mental disposition to engage in the processof learning generated a complementary set of ideas such as theimportance of students’ and teachers’ attitudes and beliefs towardsmathematics and its teaching and learning. In this sense, some meta-mathematical thinking notions –like competencies in problem solving,metacognition and sense making (Schoenfeld, 1992) came to go handin hand with mathematical ideas. This combination constitutespowerful clusters in a psychological sense.

For mathematics education all these principles implied theadvance in ideas of reform along the lines of, for example, theNCTM Standards, whose progressive proposals represent acombination of powerful mathematical ideas in both a logical andpsychological sense. To illustrate this combination, we can see howthe description of the Standards (NCTM, 2000) plays with theidentification and integration of mathematical topics (e.g., numberand operations, algebra, data analysis and probability), mathematicalrelated activities (e.g., problem solving and communication) andcompetencies in those topics and activities (i.e., understand numbers,ways of representing numbers, relationships among numbers, andnumber systems, use mathematical models to represent andunderstand quantitative relationships, monitor and reflect on theprocess of mathematical problem solving, and communicate theirmathematical thinking coherently and clearly to peers, teachers, andothers).

Considering our two projects, following the psychologicalinterpretation, we could discuss the role of the contextualization onwhich the projects are based. The projects bring into the classroomconcrete situations that the students can use as a basis for

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understanding. In this sense, each project provides a frame for thestudents to become familiar with mathematical notions that intervenein the situation. Its main role is, as a facilitator and as a motivationaldevice, to bring students into mathematics. In the case of “Terriblesmall numbers”, the students are familiar with the whole issue ofsalmonella. The proximity of the topic to their lives can providepossibilities to make connections between already internalizedconcepts and new ideas to come. The experimentation with thesamples of eggs opens further links to which the ideas of probabilityand risk can be connected. In “Macro-figures becoming macro-dangers” the extraction of basic data for the mathematical analysisfrom real sources can be viewed as an especially engaging activity,which can motivate students to learn the mathematical aspects behindthe real cases. In particular, the students could observe a newsignificance of making algebraic reductions. They can revealconnections that are not so easy to identify if only numericalcalculations are used.

In most cases, the psychological interpretation of powerfulmathematical ideas rests on the assumption that human learningprocesses are universal, even though strong cultural and socialdifferences may affect meaning construction. It also assumes that,therefore, those ideas are transferable into diverse situations and that,given this transferability, they constitute useful knowledge. We findthis interpretation problematic in the light of recent studies that haveevidenced and developed a radically different view of knowledge andhuman cognition. First of all, recent studies have shown (Lerman,2000) that the individual’s social and cultural situatedness –inparticular ethnic, social or gender groups at a given historicalmoment– has an impact on her cognitive development. Secondly, ithas been suggested that knowledge is not a mental possession butparticipation in communities of practice (Lave, 1988; Lave &Wenger, 1991). From this perspective, there is no possible knowledgetransfer but different types of participation and action in differentcontextualized situations (Boaler, 1997; Wedege, 1999). A view ofmathematical ideas from a broader perspective is, then, necessary.

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... culturally speaking

If students should experience the relevance and meaningfulness oftheir learning in relation to their socio-cultural experience, it isnecessary to consider what counts as powerful from the situatedlearners’ perspective. We could then try to relate powerfulmathematical ideas to the opportunities for students to participate inthe practices of a smaller community or of the society at large. Thesepossibilities have to do with the students’ foreground, which refers tothe way students interpret and conceptualize –explicitly or implicitly,consciously or unconsciously– their future, feasible, life conditionsgiven the social, cultural, economic and political environment inwhich they live.10 It also refers to the students’ interpretations andconceptualizations of their possibilities to engage in meaningfulaction. Naturally, the foreground is modulated by the background ofthe students, that is, their “socially constructed network ofrelationships and meanings” (Skovsmose, 1994, p. 179) whichbelongs to their personal history. However, the foreground providesresources and reasons for the students to get involved –or not– intheir learning as acting persons. In other words, the foregroundallows students to focus their intentions on the activities connected tolearning. We see intentions as primarily constructed from the aperson’s foreground. So, mathematical ideas can become powerful tostudents in as much as they provide opportunities to envision adesirable range of future possibilities.

Many studies have tried to identify what “powerful mathematicalideas” could mean from a cultural perspective. In this context,“cultural perspective” refers to radical and political interpretations, asfor instance described in Frankenstein (1995).11 She has tried toidentify issues that specifically concern the political situation of herworking-class, urban adults involved in remedial mathematicsprograms. She shows how questions about unemployment, military

10 For a discussion of the notion of foreground, see Skovsmose (1994).11 A much more narrow interpretation of “cultural” is found in, for instance,Seeger, Voigt & Waschescio (Eds.) (1998) where the culture of the mathematicsclassroom is interpreted as first of all referring to interaction and communication inthe classroom. Other interpretations of culture are present in the work of Cobband colleagues for whom the mathematics classroom is the micro-community ofpractice where sociomathematical and more general social norms are built (Cobb,2000).

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expenditure, taxation, economic policy, etc. can be dealt with as acentral part of mathematics education. Being able to handle suchquestions means developing a relevant competence for actingpolitically as critical citizens. In this way, powerful mathematicalideas become defined first of all with reference to the situation of thelearner in a given socio-cultural situation. This radical perspective isalso present in many of the papers in Powell & Frankenstein (1997).

Knijnik (1996, 1997) provides another example of culturally andpolitically powerful mathematical ideas in the case of the Brazilianlandless movement. From an ethnomathematical approach, she,together with teachers from the community, has found ways ofbridging the gap between academic mathematics and people’s popularmathematical knowledge as a way of enhancing possibilities of socialchange. The emergence of a “synthesis-knowledge” that rescues andvalues popular understandings but also raises awareness about itslimitations, is one of the results of relevant pedagogical work inmathematics for the community.

Mukhopadhyay (1998) also presents an interpretation ofmathematics education as a tool for adopting a critical stance towardscurrent popular culture. She exemplifies her point of view with amathematical investigation in the classroom about Barbie dolls. Thisinvestigation, starting from making a model of Barbie of “normal”height, can promote the adoption of a critical attitude towards thestereotypes with which we are confronted and which have aninfluence on youth behavior, such as women wanting to have a bodylike Barbie, and having serious eating disorders. Generally speaking,mathematics education becomes powerful in a cultural sense, when itsupports people’s empowerment in relation to their life conditions.

Both “Terrible small numbers” and “Macro-figures becomingmacro-dangers” illustrate what it could mean to consider the politicaldimension of the students’ culture. Danish students know aboutsalmonella poisoning, and many Colombian students may haveexperienced the consequences of the disturbance in the logic of theUPAC-system. Therefore, we know that it is possible to relate thecontent of mathematics education to the students’ background.Nevertheless, it might be easy to miss the relation with theirforeground. How could our two projects touch students’ learningintentions by touching their foreground? We imagine that for someDanish students experimenting about the meaning of quality controlin food products could generate a learning intention related to their

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capacity for making decisions about types of aliments appropriate forconsumption. For Colombian students, especially for those whoactually lived through the negative consequences of the break downof the UPAC system, we could imagine at least two strong ways inwhich their foreground is touched. For some, talking about the issueitself can be so painful that a resistance to get engaged in the topicwill dominate. In this case learning intentions could emerge inopposition to the proposed learning environment.12 On the otherhand, for some students the project could generate learning intentionsin relation to their capacity for helping their families make a criticaldecision about housing and real state acquisition.

The issue of touching students’ foreground is a delicate point insome of the ethnomathematical approaches. Some studies identifymathematical competencies built into the students’ culture, forinstance competencies related to basket or fabric weaving andornamentation in some Mozambican communities (Gerdes, 1996,1997), as a starting point for mathematics education. However, thereis no guarantee that, although belonging to the cultural background ofa particular group of students, these geometrical competencies wouldbe considered as relevant, engaging or motivating.13 Students’intentions for learning might be related, first of all, to theirforeground. So, can ethnomathematics be accused for providing arestricted access for some children to mathematical ideas, or just anaccess to mathematical ideas without sufficient potential for touchingthe students’ foreground? We need to point in the direction of thepotentials of mathematical ideas for developing critical citizenshipand mathemacy as efficient tools for a critical reading of mathematicsand of how mathematics may operate in the social environment.

…sociologically speaking

Powerful mathematical ideas can be investigated from a sociologicalperspective as well. Such ideas can be defined in relation to the extentto which they are used as a resource for action in society.

12 It is important to note that teachers, working from a culturally empoweringperspective, may face cases of students resisting the teachers’ “empowering”game, since the students can envision traditional teaching as a valuable contributionto their foreground.13 For a critique of ethnomathematics, see Vithal & Skovsmose (1997).

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Mathematics does not exist as an independent knowledge insociety. Social actors, and not only mathematicians, use mathematicsas a descriptive and a prescriptive tool. Mathematics, including itsapplied forms such as engineering mathematics and mathematicaleconomics, is part of the available resources for technological action,including planning and decision making. We use the term“technological action” (Skovsmose, 1994) in the broadest possibleway, including making decisions about how to manage the economyof the family, establishing a new security system for electroniccommunications, investigating traffic regulations, organizinginsurance policies, making quality control of mechanicalconstructions, providing a booking system for airlines, testing ofalgorithms and computer programs, and very many other activitiesthat nowadays are present in most working places. In various waysmathematics constitutes a resource for such actions.

In order to express this wildly growing multitude of “actionsthrough mathematics”, we highlight three characteristics of the way inwhich social actors operate with and through mathematics in atechnological enterprise. Firstly, by means of mathematics, it ispossible to establish a space of (technological) alternatives to apresent situation. However, mathematics also provides a limitation onthis space of alternatives. In this sense, mathematics serves as a sourcefor technological imagination, which is also an imagination with lotsof blind spots. Secondly, we seem able to investigate particular detailsof a not-yet-realized plan. However, hypothetical reasoning aboutdetails of imagined constructions, supported by mathematics, also laysa trap, since mathematics imposes a limitation of the perspectivesfrom which hypothetical situations are investigated. In particular,risks can emerge from the gaps in hypothetical reasoning, whichmight overlook whole sets of consequences of certain technologicalimplementations. Finally, being a resource for technological actionand decision making, mathematics becomes an inseparable part of ourpresent reality and of other aspects of society. We come to live in anenvironment created and supported by means of mathematics. Inparticular, the development of the informational society is closelylinked to the spread of mathematical based technologies.14

14 For a discussion of mathematics in action and the notion of the formattingpower of mathematics, see Skovsmose (1999), and Skovsmose & Yasukawa(2000).

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Talking about powerful mathematical ideas, sociologicallyspeaking, we do not simply have in mind a long list of mathematicalmodeling examples or applications of mathematics that have beenpresented in many textbooks serving the purpose of motivatingstudents and illustrating that mathematics can be an useful subject.15

We want to draw attention to the fact that mathematics operates as anintegrated part of many technological actions, and that such actions,as any other, may have unpredictably good or bad consequences. Thequality of the consequences of a technological action is notguaranteed by the quality of the mathematical base behind it. Thethree aforementioned characteristics of how we operate with andthrough mathematics may help one to grasp the complexity ofmathematics in action, and to draw attention to the basic uncertaintywhich is associated with any mathematical idea put into operation inany technological context.16 Therefore, a critique of mathematics inaction is necessary.

We can illustrate some of the aspects of powerful mathematicalideas, sociologically speaking, by considering “Macro-figuresbecoming macro-dangers”. This project builds on an actual situationin which a mathematical model has been the basis for an economicpolicy with great social impact. A main task for the students could beto reveal the connections between the blind spots of the hypotheticalreasoning of the model, and the emergence of certain socialuncertainties. A particular issue is to consider the relationshipbetween the growth of, on the one hand, the functions fi(t) = (1 + i)t, fe(t) = (1 + e) t and fr(t) = (1 + r) t , and, on the other hand, thegrowth of the function fn(t) = (1 + n ) t, where n , e , i, and r areconnected by the formula: n = (1+i)(1+e)(1+r) – 1. By studying thesefunctions, we witness some elements of the logic of the UPACsystem. We experience the drama of “actions through mathematics”.The system of mortgage payment is determined by this logic. Inparticular, it becomes relevant to clarify to what extend the growth offn(t) gets out of control (economically speaking) even though thegrowth of fi(t), fe(t) and fr(t) seems “reasonable”. Thus, the project

15 De Lange (1996) presents a discussion of applied mathematics in education. Wefind that his view of applied, realistic, mathematics is in many aspects differentfrom what we see as sociologically relevant and powerful.16 For a discussion of uncertainty about mathematics, see Skovsmose (1998,2000a).

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can illustrate how mathematical calculations used for social decisionmaking can provoke new risk structures for certain groups of people.

“Terrible small numbers” also shows the relationship betweenrisks and mathematics. In this project, students were brought into asituation where economic and epistemic interests were confronted.This contradiction is exemplary in many technological designprocesses. How much further investigation is needed in order to makea well-justified decision? How big a sample of eggs has to beinvestigated in order to decide whether to put the eggs in the marketor not? In many cases mathematical modeling makes it tempting andpossible to jump into conclusions about what to do; such conclusionsmay bring new risk structures to our future.

So far in our analysis we have tried to show different interpretationsof “powerful mathematical ideas”. Each one can be related to centralnotions like the level of abstraction in the mathematical architecture,the meaningfulness of acquired mathematical notions, the waylearners can experience an empowerment as citizens, and the criticalconcern about how mathematics operates as a resource for action in atechnological environment. Each one of these interpretations cansuggest a response to the paradoxes of inclusion and citizenship.Thus, if the logical and the psychological interpretation dominate, theparadoxes seem to disappear. What better can be done than bringingstudents to master the highest level of abstraction in a meaningfulway? Considering the sociological and the cultural interpretation of“powerful”, both paradoxes reappear in a strong version. Mathematicseducation cannot ignore these. However, the discussion of providingdemocratic access to powerful mathematical ideas becomes morecomplex when we also consider the notion of democracy. We shalldiscuss this issue in the following section.

Democratic Access…

All students, everywhere in the world, have the right to education.We can go further and say that all students in the world should havethe possibility of learning mathematics. Democratic access, in thissense, refers to the actual possibility of providing “mathematics forall”. We find that the idea of “democratic access”, understood as the

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right of participating in mathematics education, is very complex, andwe would like to discuss in more detail what the expression couldrepresent.

“Democratic access” designates the possibility of entering akind of mathematics education that contributes to the consolidation ofdemocratic social relations. As we have argued elsewhere (Skovsmose& Valero, 2000), a critical view on the connection betweenmathematics education and democracy situates the notion ofdemocracy in the sphere of everyday social interactions, and redefinesit as purposeful, open political actions undertaken by a group ofpeople. This action is collective, has the purpose of transforming theliving conditions of those involved, allows people to engage in adeliberative communication process for problem-solving, andpromotes coflection, that is, the thinking process by means of whichpeople, together, bend back on each other’s thoughts and actions in aconscious way (Valero, 1999).

Democratic access in mathematics education, in the sense wehave indicated, is played out in very many different arenas where thepractices of mathematics education take place. We will comment onthree such arenas that we find fundamental: the classroom, the schoolorganization, and the local and global society.

... in the classroom

The mathematics classroom is a micro-society where democraticrelationships between students and teacher and among students mustbe present if education is to provide any form of democratic access.Democratic relationships that allow collaboration, transformation,deliberation and coflection are central in opening possibilities for acritique of mathematical contents in the class and of their significancein social actions based on them.

Communication in the classroom can follow many patterns, butin order to establish a spirit of democracy, dialogue and critique areindispensable components. Thus, a mathematics classroom governedby bureaucratic absolutism or by a communication that does notincorporate possibilities for politicizing the whole mathematicseducation experience does not represent democratic life. Alrø &Skovsmose (1996) provide an example of a communicative modelwith a democratic concern in mathematics education. The notion of

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Inquiry Cooperation Model refers to a variety of communicative actssupporting an inquiry process. Such a process cannot be simplyguided by the teacher; rather the students act in the process ofinvestigation in cooperation with the teacher. The elements of theInquiry Cooperation Model are: getting in contact, discovering,identifying, thinking aloud, challenging, reformulating, negotiatingand evaluating.

The nature of some of these acts can be clarified with referenceto the project “Terrible small numbers”. When the students carry outthe experiment related to the quality control of eggs, the process isnot reduced to a set of exercises organized in a certain sequence. Theopenness allows students to “own” the learning process and toexperience what it could mean to be responsible for making decisions.When the students work on their own and the teacher wants tointervene, students should not feel threatened in their ownership ofthe process of investigation. The teacher has to get in contact andthen she can challenge them: How could it be that in some of thesamples there are more that one egg with salmonella? The studentscan try to identify sources for explanation: Maybe the teacher did notmix the eggs sufficiently? Discoveries can be made: Could it be thatsamples do not always “tell the truth” about the whole population?During the process of negotiat ion , where different possibleexplanations are considered, thinking aloud is possible. Thinkingaloud is a way of providing public access to a line of thought, and itcan be open for negotiations and reformulations. Any result of such aprocess can be evaluated.

In the section “...sociologically speaking” we have outlined threeaspects of actions through mathematics: Mathematics helps to openpossibilities by providing the basis for a technological imagination;mathematics supports investigation of particular aspects of not-yet-realized constructions, and, when realized, mathematics operates as anintegrated part of the technological device. If such aspects ofmathematics in action should be addressed critically in themathematics classroom, then mathematical content needs to becontextualized, not only in terms of the provision of a “task-context”but mainly in terms of a “situation-context” (Wedege, 1999). In“Terrible small numbers” it would not be possible to introduce adiscussion about reliability and make students experience what itwould mean to make decisions, if the figures were not stronglyrelated to actual situations happening in the social, political, economic

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and cultural environment in which learning takes place. In general,we find that contextualization is a precondition for problematizing“trust in numbers”. Such a problematizing is essential for establishinga critical citizenship.

The contextualization is not simply a motivational device–although it might be motivating. It is a condition for establishing adiscussion of how mathematics can operate as a source of power in asociological sense since it invites a critical examination of howmathematics in fact is put into operation. A rich contextualizationcould help an Inquiry Cooperation Model enter the classroom, and, inthis way, influence the structure and content of the discussion.

We are aware of the fact that the project “Terrible smallnumbers” took place in a school situation, and that this particularcontext provides a frame for interpreting the activities. Studentsworked with eggs that were not real, and their calculations had noactual consequences. Although they calculated the risk of producingan ice cream dessert out of 6 eggs, there was no real ice creamproduction in the classroom. Still, there is an important difference toa traditional exercise context where a problem can refer to prices,goods and amounts to be bought, but where these prices, goods andamounts operate in a complete different way than real prices, goodsand amounts. The traditional school mathematics exercise isaccompanied by a set of metaphysical assumptions, notably that thedescription provided by the text is exactly true and it cannot bechallenged. If a problem makes us buy apples and their price is set at$3.10 per kilo, it does not make sense if one student knows that thesame kind of apples can be bought around the corner for $2.30. If weare asked to buy 3 kg of apples, it does not make sense to questionwhether we can expect the scale to show exactly 3 kg –although weall know that apples are big units and it is very difficult to have aweight of exactly 3 kg. The information provided by the text of theexercise is what we need for solving the problem, and the problemhas one and only one correct answer (Mukhopadhyay, 1998;Skovsmose, 2000b).

An essential task of the contextualization is to crack themetaphysical assumptions of the exercise paradigm. This metaphysicswas challenged by the project “Terrible small numbers”, and this isessential for a critique to make sense. Opening the classroom for in-depth reflections to take place is a condition for mathematicseducation to be part of a democratic endeavor.

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... in the school organization

Although many opportunities for establishing democratic access topowerful mathematical ideas are present in the classroom, this is notthe only or the main arena for establishing such an access. In fact,recent research work has acknowledged the importance and necessityof understanding classroom practices in connection with the wholecontext of the school organization and, more generally, theeducational institution (Krainer, 1999; Perry et al., 1998; Stein &Brown, 1997; Valero, 2000). Teachers and students in the classroomare not isolated from the way mathematics teachers and school leaderswork in order to shape mathematics education through bothcurriculum planning and teachers’ professional development. So,when we discuss democratic access, we also have to consider howmathematics education practices in the school as a whole operate andcreate opportunities for –or obstacles to– this endeavor.

In the context of the school organization, we want to drawattention to the importance of who organizes the curriculum and howit gets organized. Let us assume that we, as well-intended policymakers, want to provide a curriculum that ensures that students getdemocratic access to powerful mathematical ideas –independently ofwhat interpretation of “powerful” we have in mind– and that we, as aresult, offer a very detailed plan including topics and ways ofworking in the classroom. Let us assume, furthermore, that thiscurriculum is in fact put into operation. Then, this very detailedplanning itself will obstruct the realization of our democraticintentions since the top-down model closes possibilities for the peopleinvolved in the actual curriculum development to own the process.

This conflict points to the basic complementarity17 incurriculum thinking. The very process of planning carefully and indetail an access to any kind of ideas obstructs in itself the possibilityof making this access democratic. The latter presupposes thatteachers, students and school leaders are acting subjects in identifying,planning and implementing the curriculum. (Naturally other groupssuch as parents, could be considered as well.) This view implies that

17 Vithal (2000a) elaborates on the notion of complementarity in mathematicseducation as the possibility of “bringing together irreconcilably conflicting butnecessary positions or theories” (p. 307), in order to provide better and fullerunderstandings of what we study in mathematics education.

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certain curricular decisions need to be taken in the community ofparticipants in the school mathematics practices. The way acurriculum is organized, then, depends on the relations inside anetwork of school mathematics practices (Valero, 2000), whereteachers as individuals, the group of mathematics teachers, thestudents, and the school leaders can share their expectations about themeaningfulness of a mathematics educational experience. Whoparticipates in formulating a curriculum and how such a formulationis implemented are constantly implicated in an unsolvable andnecessary tension between specificity and freedom.

The planning of “Macro-figures becoming macro-dangers” is amicro-curriculum design process in which basic components areidentified and developed in close connection to the classroom.Practicing teachers, as a response to their own and their students’experiences, identified the idea of the project. It was not a suggestionfrom a textbook or an external authority, but it emerged from asituation that needed understanding since it was affecting the life ofthe school community. The potentialities for collaboration amongteachers and students in the development of the project, as well as fortransforming their understandings –and eventually their situationconcerning mortgages– are key elements in the project. Were itpredetermined that the projects should serve as an introduction to,say, algebraic calculations, then the significance of the projects couldeasily be lost. Then it would be impossible for the students, as well asfor the teachers, to maintain ownership of the project. Theexperimental character of the curriculum design process exemplifiedby “Macro-figures becoming macro-dangers” highlights theimportance of creating a “laboratory for curriculum development”(Vithal, 2000a). This notion refers to cooperation between differentparticipants in the network of school mathematics practices, in orderto build an open curriculum planning process that can acknowledgedemocratic concerns.

... in the local and global society

There is a contradiction between establishing mathematics educationin terms of democratic access to powerful ideas and, at the same time,letting that education serve differentiation functions in society by, forinstance, ranking students in a way that significantly influences their

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future career possibilities. The emphasis on high-stakes assessment orclassroom assessment in most countries can clearly contribute to theparadoxes of inclusion and citizenship (Morgan, 2000). Furthermore,mathematics education –at least in certain forms– generates differentsituations of exclusion according to gender, race, language, class andabledness (e.g., Grevholm & Hanna, 1995; Keitel, 1998; Khuzwayo,1998; Rogers & Kaiser, 1995; Secada, Fennema & Adajian, 1996;Zevenbergen, 2000a). If we want to end this exclusion, then weshould allow entry to all into mathematical learning.

The difficulty of establishing mathematics education as ademocratic resource can be clearly illustrated by the followingdilemma. Ethnomathematics (D’Ambrosio, 1996; Powell &Frankenstein, 1997) has represented a challenge to Eurocentrismfirstly, by demonstrating that all cultures, not least traditional ones,demonstrate a deep mathematical insight, and secondly, by showingthat building on this knowledge it is possible to reconstruct amathematics education which does not recapitulate the priorities ofcolonisation. This has led to the formulation of ethnomathematicalcurricula for disadvantaged populations like, for example, the landlesspeople in Brazil (Knijknik, 1997) or Mozambican peasants (Gerdes,1997). Could it, be however, be that offering ethnomathematicaleducation to certain disadvantaged groups prevents them from beingactive members of the informational society, and therefore doomsthem to a life in the Fourth World?

In a similar way, critical mathematics education (Skovsmose,1994; Vithal, 2000a) has been proposed as an educational philosophyto address the risk of a mathematics education that contributes to thecreation of citizens uncritical towards the devastating effects ofmathematics in society. Nevertheless, in a research and developmentproject intending to open possibilities for critical mathematicseducation with immigrant students in Catalonia (Gorgorió & Planas,2000), the researchers perceived certain interpretations of that type ofeducation as a “soft” program that could be suitable for this particular kind of students. This view contrasts with the position –non-explicit,but nonetheless easy to elicit in actions and proposals– of theeducational authorities defending the need for “hard core”mathematics education programs for those students expected tosucceed within the educational system, in particular, the localstudents. We see that this interpretation could lead to a situation inwhich so-called “critical mathematics” programs serve as a second-

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rate curriculum for immigrants and political refugees since, after all,it may not provide them with the “hard” mathematical knowledge thatis needed for climbing the ladder of social prestige in thatcommunity. Here we directly face the paradox of inclusion. In thisway, an attempt for inclusion could result in growing exclusion, and aconcern for citizenship could come to establish citizenship among theexcluded.

In the creation of the Fourth World in the present informationalsociety, as described by Castells (1999), the barbarism of the paradoxof inclusion is associated with mathematics education. Mathematicseducation could help to secure access to the informational society aswell as to establish and legitimize exclusion from it. For a teacher inan under-resourced educational system, it is difficult to provide newpossibilities in life for the students beyond what is already wellknown to them as their background. Thus, a fundamental discussionabout mathematics education in many developing countries concernsthe relocation of resources as a way of distributing possibilities inradically new ways. If this does not happen, then, for instance,historical black schools in South Africa are doomed to belong to theFourth World. The situation in South Africa is exemplary for theproblem of how an unequal distribution of resources obstructsdemocratic ideals.

One particular aspect concerning mathematics education and theinformational society is the use of technology in teaching andlearning. We have to consider how mathematics becomes installed inmore and more technological devices (Wedege, 2000) and how itoperates “behind the screen”, making it possible to use mathematizedtools without presupposing a deep understanding of their underlyingmathematical structure, maybe even without being aware of the factthat a complexity of mathematics is in operation. The implication ofthis is that the necessary competencies to operate with thesetechnologies split people into two main categories: those who canoperate on the surface of the technology, and those who can constructand reconstruct it. Both competencies are essential, and therefore it isimportant to discuss how mathematics education operates with regardto this splitting. Furthermore, it has to be discussed how mathematicseducation manages its global function, in a world where access tocomputers is still for the few and selected. Given the nature of theinformational society, mathematics education occupies a sensitiveposition where possibilities in the informational age are distributed

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among students, regions and nations. In mathematics education thebarbarism of this distribution becomes visible.

Globalization also concerns the particular content of what islearnt. The project “Macro-figures becoming macro-dangers” isaddressing issues which represent general aspects of how risks anduncertainties are distributed. The project tries to illustrate how largescale economic figures and decision making can have particulareffects, and contributes to the creation of macro-dangers. Thistransformation from figures to dangers is a basic aspect ofglobalization, where large-scale decision making distributes risk anduncertainties in a formidable way. In this sense, “Macro-figuresbecoming macro-dangers” gets a particular exemplary value.

Facing the paradoxes

Let us recapitulate the paradoxes of the informational society as weoriginally described them. The paradox of inclusion refers to the factthat the current globalization, which proclaims universal access andinclusion as a stated principle, can also be associated with processes ofexclusion. As part of the development of a universal framework forglobal connections, strong processes of exclusion and isolation aresimultaneously taking place. Among other things, this brings about a“Fourth World” whose many new citizens are already to be found inthe mathematics classroom. The paradox of citizenship refers to thecelebration of a learning society that emphasizes the need for relevantand meaningful education for the further development of social,political and cultural structures, while in reality that education mayhave only a functional relevance for the system.

Does mathematics education in fact face the paradoxes? Up tonow we have been mainly referring to mathematics education as afield of practice, but now we will concentrate on mathematicseducation as a research field. In Figure 1, we present the space forinvestigating democratic access to powerful mathematical ideas.

Reviewing research literature in mathematics education, thereare unfortunately different ways in which the field ignores theparadoxes of inclusion and citizenship. One way of doing so is byconcentrating on particular interpretations of powerful mathematicalideas, mainly the logical and the psychological ones, where the

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emphasis is put on the development of mathematical thinkingindependently of any context wherein it takes place. The secondpossibility is through ignoring that mathematics education is part of ademocratic endeavor, or simply (and rhetorically) assuming thatmathematics education, due to the very nature of mathematicalthinking, constitutes a profound democratic enterprise. In this view,powerful mathematical ideas have an intrinsic democratic value(Skovsmose & Valero, 2000). A third and more moderate escape iscarried out through only considering selected aspects of whatdemocracy could involve. In particular, much discussion has beenfocused on democracy in the classroom, but ignored the other arenaswhere meaningful participation in political action through differentkinds of powerful mathematical ideas is built.

Gómez (2000) carried out a classification of the paperspublished in 1997 in three main international journals in mathematicseducation –the Journal for Research in Mathematics Education(JRME), Educational Studies in Mathematics (ESM), Recherches enDidactique des Mathématiques (RDM)– and the proceedings of the21st meeting of the International Group for the Psychology ofMathematics Education (PME). Although he acknowledges that hissample is not representative of all international research published inthe area, he considers it as an indicator of the type of research carriedout. He concludes that “mathematics education research production iscentered mainly on cognitive problems and phenomena; that it hasother minor areas of interest; and that it shows very little productionon those themes related to the practices that influence somehow theteaching and learning of mathematics from the institutional ornational point of view.” (p. 95). Translated to our space of research(Figure 1) Gómez’ results indicate that there is a high concentrationin the lower, left area of our space. In order to check these results, weclassified the papers published in JRME, RDM, ESM, For theLearning of Mathematics (FLM), Suma, and the International Journalof Mathematics Education in Science and Technology (IJMEST)published between January 1999 and October 2000, in the differentareas of our space of investigation. The results are indicated in Figure1.

This distribution shows that the majority of these papers have aconcern for interpretations of powerful mathematical ideas in alogical and psychological sense and in the arena of classroominteractions. There is, however, also a considerable amount of papers

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highlighting cultural interpretations. However, there are areas such asthe society and school arenas for democratic access, and thesociological interpretation of powerful mathematical ideas, that eitherhave not been very much explored or have a low priority in what isactually published in research journals. Naturally, we are not claimingthat each and every research project in mathematics education shouldaddress the full range of issues mentioned. However, it is highlyproblematic that dominant research trends in mathematics educationoperates within a limited scope of the space of investigatingdemocratic access to powerful mathematical ideas. Such aparadigmatic limitation effectively obstructs the possibilities formathematics education to face the paradoxes of the informationalsociety.

Society

School

Are

nas

for

dem

ocra

tic a

cces

s

Classroom

Logical Psychological Cultural Sociological

Interpretations of powerful mathematical ideas

Figure 1: Indication of distribution of research papers in the space of investigationon democratic access to powerful mathematical ideas

What could it mean, then, for mathematics education research to facethe paradoxes of inclusion and citizenship? We shall try to indicatepossible answers to this question by raising clusters of questions thatpoint in the direction of under- or non-researched issues.18

(1) Democracy understood as a collective, political action for thepurpose of transformation is lived-through everyday experience,

18 At this point we could also enter into the discussion of what it means toresearch a situation that “does not exist” since there is also a connection betweenthe lacks in the space of investigation and lacks in actual practices in schoolmathematics. Since this whole discussion is so broad and it is not our intention totackle it here, we merely mention the work of Skovsmose & Borba (2000) andVithal (2000a, 2000b) who have tried to approach this question.

35

including the mathematics classroom. How do particular forms ofmathematics education, including the interaction and communicationin the classroom that they generate, acknowledge democratic values?Which are the forms of interaction in the classroom that openpossibilities for politicization and critique of both the mathematicalcontent and the interaction itself? How does mathematics educationacknowledge that the micro-society of the classroom is related to thelocal and global society wherein students live? Are the forms oflearning in school related to forms of learning in workplaces andorganizations and in everyday situations?

(2) The contextualization of school mathematics is an importantdoor to enter into cultural and sociological interpretations of powerfulmathematical ideas. Do we deal with a contextualization which,primarily, observes the metaphysics of the exercise paradigm, or dowith have to do with deeper, real-life references? Does thecontextualization of school mathematics touch on both the students’background and foreground in significant ways? Do we try toilluminate issues where the content of mathematics education preparesthe students to operate as critical citizens in a context wheremathematics and mathematically based decision making are inoperation?

(3) The dynamics of school mathematics practices, understood asthe complex interaction among teachers, school leaders and studentsin the school organization, needs exploration. A particular issueconcerns who participates in the curricular decisions and where dothey take place. Does curriculum planning and implementation openpossibilities to bring into the classroom different interpretations ofwhat powerful mathematical ideas mean? What do teachers as a groupand the school leaders value as an appropriate mathematics educationgiven their students’ backgrounds and foregrounds? In particular, it isimportant to consider how local curricula can come to operate insociety. Could a particular curricular design and implementationcome to constitute “second-rate” mathematics education, that doomsstudents to exclusion or to uncritical acceptance of society? How doesthe process of exclusion of certain social groups –defined in terms ofgender, race, class and ability– operate in the school organization as awhole?

(4) It is relevant to consider how Information andCommunication Technologies (ICT) open and reorganize newlearning possibilities (Balacheff & Kaput, 1996; Borba, 1997, 1999).

36

What do ICT mean in boosting culturally and sociologically powerfulmathematical ideas? Part of this understanding has to do withunderstanding how the actual global distribution of ICT learning-possibilities stands. Obviously, we have to do with the most unequaldistribution of the ICT-facilities around the world. What does thismeans for the role of mathematics education in under-resourcedclassrooms and schools? In particular, what does this imply for theformation of the “Fourth World”? Does the reorganization of learningpossibilities also include a reorganization of inclusion as well asexclusion from the informational society? Is research in mathematicseducation too often set up in such a way that certain social andeconomic resources are taken for granted, although they can be takenas such only in certain (privileged) parts of the world?

(5) As we have indicated previously, mathematics operates as aresource of power in a variety of actions and decision making in allareas of life. Does mathematics education open possibilities forstudents to see this resource in operation? How can “actions throughmathematics” be illustarted in mathematics education? How far do wego in making mathematics education a critical activity, addressingboth the wonders and the horrors of actions through mathematics?What does it mean to offer a mathematics education that tries toillustrate such contrasting aspects of powerful mathematical ideas?

(6) Finally, through mathematics education in all its arenas –theclassroom, the school and society– we are contributing to theconstruction of public images of mathematics and mathematicseducation. How does this process of building social images andideologies of mathematics and mathematics education happen in thedifferent practices of mathematics education? Which are thecharacteristics of the discourse that we construct in order that it canactually attribute so much power and democratic relevance to oursubject? What are we doing in the classroom, in schools, in society, tostrengthen mathematics as a power-knowledge? What are the broadestsocial consequences of our practices? Could it be reproducing a kindof world where the paradoxes of equality and citizenship can easilysurvive?

If as mathematics educators in research and practice we areconcerned about the lives and experiences of students like Nicolai andCarlos, we should consider even more seriously the importance ofbroadening our interpretations of what democratic access to powerfulmathematical ideas means. Furthermore, we should clearly keep in

37

mind the necessity of tackling the inclusion and citizenship paradoxesin our endeavor in the coming century.

Acknowledgements

We would like to thank Pedro Gómez, Brian Greer, Núria Gorgorio,Lena Lindenskov, Swapna Mukhopadhyay and Tine Wedege for theircomments to previous versions of this chapter. This paper is part ofthe research initiated by the Center for Learning Mathematics, aninter-institutional center of the Danish University of Education,Roskilde University and Aalborg University in Denmark.

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Authors’ Affiliation

Ole SkovsmoseCentre for Educational Development in University Science,Aalborg UniversityFredrik Bajers Vej 7B, DK-9220 Aalborg ØstDenmarkTel. +45 96358080, Ext. 9781Fax +45E-mail: osk@dcn.auc.dk

Paola ValeroInstitute for Curriculum StudiesDanish University of EducationEmdrupvej 101, DK-2400 Copenhagen NVDenmarkTel. +45 39696633Fax +45 39696626E-mail: paola@dpu.dk