The problem-solving citizen

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S TUDIES IN EDUC A TION AL SCIEN CES: LICENTIA TE DISSERT A TIONS SERIES 20 1 4:33 AS D AHL MALMÖ UNIVERSIT

JONAS DAHL

THE PROBLEM-SOLVING

CITIZEN

isbn 978-91-9140-551-5 (

print

)

isbn 978-91-9140-552-2 (

pdf

)

issn 1653-6037

THE PR OBLEM-SOL VIN G CITIZEN

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Malmö högskola, 2014

Lärande och Samhälle

Developing Mathematics education

THE PROBLEM-SOLVING

CITIZEN

THE PROBLEM-SOLVING

CITIZEN

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Publikationen finns även elektroniskt, se www.mah.se/muep

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It’s alright, we told you what to dream

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FOREWORD ... 9

ABSTRACT ... 10

INTRODUCTION ... 11

SWEDEN ... 16

THE CITIZEN IN LIGHT OF THE MATHEMATICS CURRICULUM (ARTICLE 1) ... 22

THE PROBLEM-SOLVING CITIZEN ... 39

PROBLEMS AND PROBLEM SOLVING ... 41

Why problem solving? ... 41

Problems... 44

Difficulties in assessing problem-solving competence ... 45

INVESTIGATING THE CONSTRUCTION OF THE PROBLEM-SOLVING CITIZEN (ARTICLE 2) ... 47

METHODOLOGICAL CONSIDERATIONS ... 55

The pedagogic device and Social Activity Theory ... 55

Discursive saturation ... 56

Domains ... 58

THE NEED FOR PROBLEM SOLVING AS GLOBAL POLICYSPEAK: THE CASE OF SWEDEN (ARTICLE 3) ... 63

CONCLUSION ... 89

DISCUSSION ... 93

DEN PROBLEMLÖSANDE MEDBORGAREN - SVENSK SAMMANFATTNING ... 97

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FOREWORD

As a mathematics teacher in upper secondary school in Sweden, I

have always had an interest in issues that arise or could arise when

it comes to teaching students who are under-achieving in

mathe-matics. Accompanying this interest comes not only a question of

how to teach, but also a question of what and why. When I started

my PhD studies, I had an idea that maybe not all students needed

to understand mathematics. I saw a trend, at least in Sweden,

where mathematics education was changing from something based

on skills for solving problems or tasks to a more competence-based

curriculum (and education) where understanding was emphasised

more. To reach the goals and to pass the mathematics course, it is

no longer enough to show straightforward mathematical skills; one

must also show that one understands and is able to communicate

this understanding.

I had an idea that maybe not everyone needed to understand the

basis of mathematics in order to achieve a good life. As I see it

now, this was a rather naïve view of mathematics education.

First of all I would like to thank my head supervisor Tamsin

Meaney for all the time and effort. You are absolutely monsterful.

I would also like to thank my supervisor Leif Karlsson whom

without this thesis would have been bursting of strange metaphors

and be just a brabble. Thanks also to all my research colleagues at

Malmö Högskola, especially Maria Johansson for all her weird

thoughts and Richard Wester for all the fun trips. Last but not

least I would like to thank Cissi. Without you this would have been

impossible.

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ABSTRACT

The present thesis is made up by three articles and in all of these

the mathematics curriculum for upper secondary school in Sweden

is analysed. The main focus is the citizen and citizenship and the

point of departure is problem solving as a competence. Besides an

investigation of the connection between citizenship and the

curricu-lum or the role the citizen have in the curricucurricu-lum, questions about

what tensions appear when problem solving is recontextualised

into the curriculum are posed. Following an international trend in

(mathematics) education, the mathematics curriculum in Sweden

stresses demands made on the students and citizens instead of

rights that the students or citizens have. Demands that everyone

must become problem-solving citizens. By the use of Bernstein’s

theories about the pedagogic device and his division of different

knowledge forms into a vertical and a horizontal discourse, I

inves-tigate possible effects of these demands. Despite intentions that all

should be included, I show that there is a risk for exclusion instead.

Bernstein suggested that school reproduces social inequity. In this

thesis I discuss how this is done in the curriculum. My conclusion

points at a risk of segregation and exclusion of lower

socio-economic groups from influence, power and control. Furthermore,

the reproduction of social inequity is build more solidly into the

system with the new curriculum as although it is unclear whether

the purpose of the changes to the curriculum was really to divide

groups and exclude some from power.

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INTRODUCTION

In the curriculum for upper secondary school in Sweden, it is

stated:

The task of the school is to encourage all students to discover their own uniqueness as individuals and thereby actively

par-ticipate in the life of society by giving of their best in

responsi-ble freedom. (Skolverket, 2013, p. 4, my italics)

School should be for all, but what does that mean? Does it mean

that everyone should be given the opportunity to develop

accord-ing to their own background or foreground? Or does it just mean

“everyone should reach the goals”—goals set by someone else?

This leads to further questions about why: Why should everyone

reach the goals? and what: What should the goals be? Should they

be the same for everyone? In regard to the quote from Skolverket,

the National Agency for Education, one might ask what does it

mean to

actively participate? What does it take to be able to do

that? By combining both sets of considerations, I come to the

ques-tion: what is the connection between the goals and the ability to

actively participate?

I have, in three articles, discussed issues surrounding these

ques-tions in the context of mathematics education in upper secondary

school in Sweden. This thesis is hence built around the following

articles: “The citizen in light of the mathematics curriculum”

(re-ferred to as article 1), written together with Maria Johansson and

originally published in Educare in December 2013; “Investigating

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presented at the BCME8 conference in Nottingham, England, in

April 2014; and “The need for problem solving as global

poli-cyspeak: The case of Sweden” (article 3), submitted to Educational

Studies in Mathematics in June 2014.

The new mathematics curriculum for upper secondary school in

Sweden that was launched in 2011 could be viewed as part of an

international trend that is sometimes called “the cognitive

revolu-tion” (Schoenfeld, 2007) or “the competence reform” (Bergqvist &

Bergqvist, 2012). This reform includes a movement away from

skills-oriented curricula to competence-based curricula in

mathe-matics education which reflects a trend in mathemathe-matics education

over the past twenty to thirty years. Examples of this can be seen

in, for instance, Curriculum and Evaluation Standards for school

mathematics (NCTM, 1989), Principles and Standards for School

Mathematics (NCTM, 2000) and Adding It Up (Kilpatrick,

Swaf-ford, & Findell, 2001) in the USA and the KOM-project in

Den-mark (Niss, 2007). PISA (Programme for International Student

As-sessment) (OECD, 2012) and TIMSS (Trends in International

Mathematics and Science Study) (Mullis et al., 2007) also

empha-sise competences instead of ‘basic skills’ or ‘topic-knowledge’. The

PISA testing is part of, or even a frontrunner of, this trend and the

influence from PISA on the Swedish national curriculum is

signifi-cant. The recent change of the mathematics curriculum in Sweden

is explicitly stated to be due to the disappointing Swedish results in

PISA tests (Skolverket, 2012c) and the similarities between the

PISA framework and the Swedish mathematics curriculum is

high-lighted and discussed in article 1. In article 1, an analytical division

is made between mathematics demanded by and mathematics

de-veloped within a society. Our conclusion is that there is a stress on

the demanded in the curriculum. These are, of course, demands

made on the students but in the long run, on the citizens. This goes

back to the questions posed at the beginning of this chapter—

what? and why? What are the demands and why must it be these

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mathematical competences should be emphasised in these curricula

and frameworks, but one thing common to all of them, including

the Swedish curriculum (Skolverket, 2012a), is that problem

solv-ing is positioned as an important mathematical competence. For

instance, in Adding It Up (Kilpatrick et al., 2001), problem solving

is positioned as the core of mathematics and what other

compe-tences should support.

Hence, when it comes to the question above about what the

goals should be, part of the answer could be that students should

become problem solvers.

About why?—those who support the competence reform, such

as the European Union (EU) or OECD (Organisation for Economic

Co-operation and Development) do so by presenting it as necessary

for dealing with an increasingly changing, globalised world.

Coun-tries are situated as needing to be competitive in relation to other

countries and to achieve this, it is stated that citizens need to be

flexible and have the capability to engage in lifelong learning

(Nordin, 2012). According to Nordin (2012), this suggests that the

ideology of competition is recontextualised into the language of

education through official documents, such as curricula and

poli-cies. In this way, the language of economy and the market can be

seen to have contaminated the language of education (Nordin,

2012). Nordin (2012) also suggests that the key competences from

the EU are mostly oriented towards the labour market, meaning

the reasons behind these competences are so citizens can become

employable which Nordin views as an example of an international,

neo-liberal trend.

Seen in this light, the citizen who “actively participates” is a

citi-zen who is flexible, knowledge-seeking throughout life,

competi-tive, feels free and gives his or her best. To be(come) this kind of

citizen, it seems that mathematics competence, and especially

prob-lem-solving competence, is crucial. We have to be competent

mathematical problem solvers in order to ‘survive’ and contribute

in this rapidly-changing globalised world.

Therefore, one way of

seeing the goal of mathematics problem solving in schools, is to see

it as the development of the problem-solving citizen. This is viewed

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duties that citizens have placed upon them, and I use this as a point

of departure for articles 2 and 3.

Article 2 is a theoretical and methodological paper where I

dis-cuss if and how critical discourse analysis can be used to

opera-tionalise Bernstein’s pedagogic device.

In article 3, I use Bernstein’s (2000) view on citizens, citizenship

and what school should provide in contrast to the problem-solving

citizen. Bernstein identified three important conditions that need to

be realised in school for the creation of a democratic citizen:

en-hancement, inclusion and participation. Enhancement is “a

condi-tion for experiencing boundaries, be they social, intellectual or

per-sonal, not as prisons, or stereotypes, but as tension points

condens-ing the past and opencondens-ing possible futures” (Bernstein, 2000, p. xx,

italics in original). Inclusion is the right to be “included, socially,

intellectually, culturally and personally” (Bernstein, 2000, p. xx),

but also means the right to self-selected exclusion. Participation “is

the right to participate in the construction, maintenance and

trans-formation of order” (Bernstein, 2000, p. xxi). These three

condi-tions imply that citizenship is more than membership in a state and

a voter in elections; the rights of a citizen, as an included and

be-longing member of society is emphasised over economic duties to

be competitive and employable. Emphasising the need for people to

become problem-solving citizens seems to have very little in

com-mon with what is required to become a democratic citizen,

accord-ing to Bernstein. With this view on citizenship, what school should

provide students with is access to power, not only in one’s own

life, but power to transform the existing order. Bernstein’s three

conditions are interdependent; to be able to participate in this

Bernsteinian sense, one needs to be able to experience boundaries,

and one needs to be included. In article 3, I make use of Bernstein’s

ideas regarding the pedagogic device and his division of vertical

and horizontal discourses to investigate what tensions occur when

global policyspeak about competences and problem solving is

drawn upon in the Swedish curriculum.

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this thesis is the focus on the Swedish mathematics curriculum for

upper secondary school (years 10-12). In following the

interna-tional trend described above, the Swedish curriculum should,

firstly, be seen as an exemplar and the conclusions can, perhaps

with smaller variations, be transferred to other contexts. Secondly,

since “the structure of knowledge in the curriculum—its

bounda-ries, its exclusions and inclusions—could be seen as an expression

of the distribution of power in society” (Young, 2010), the purpose

is also to make a contribution to a more sociological or political

discussion. I am aware that the focus on class is a tricky concept

(Jobér, 2012). Instead of class, I use the term ‘socio-economic

status’ (SES), but this is also tricky, especially in the Swedish

con-text. There are no official statistics that connect SES and choice of

upper secondary education except for the education levels of the

students’ parents (discussed in article 2). In a report from 2005,

Broady and Börjesson (2005) divide social origin into 32 different

sub-groups, showing a correlation between social status and upper

secondary school preferences by using the parents’ educational

level as the main sectioning factor.

Since this project is situated in a Swedish context and in the new

mathematics curriculum, introduced in 2011, in the next chapter, I

explain the Swedish context and the Swedish curriculum.

Follow-ing that is article 1 about how the citizen is described in the

cur-riculum. It identifies the close connection between the PISA

frame-work and the Swedish mathematics curriculum, showing how

Sweden takes part in global policyspeak about education, which is

discussed in in more detail in article 3.

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SWEDEN

In article 2, I point out that the changes in the curriculum for

up-per secondary school in Sweden were made as a result of up-perceived

problems with upper secondary school such as the vocational

pro-grammes being considered too theoretical and too similar to the

academic preparatory programmes (Skolverket, 2012c). In this

sec-tion, I therefore begin with a brief look back before moving on to

explain how upper secondary school, and especially mathematics

education, is organised in Sweden.

In 1970, vocational programmes were incorporated in the

Swed-ish upper secondary school (Skolverket, 2012c) for the first time.

These were more practical than the more traditional academic

pre-paratory programmes and only two years in duration, compared to

three years for the other programmes. Compared to earlier

tional training that took place mainly in the workplace, these

voca-tional programmes (commonly referred to as ‘practical

pro-grammes’) offered subjects like mathematics, English and Swedish,

but were still not as theoretical as the other programmes,

(com-monly called ‘theoretical programmes’).

In 1994, all programmes in upper secondary school became

three years in duration and the common subjects for the

‘theoreti-cal’ and ‘practical programmes’ were increased to eight

(Skolverket, 2012c). There were common courses which meant

that all students took the same initial mathematics course. This

course A was the same for all programmes with the same syllabus,

grading criteria and final national test. Students in academic

pre-paratory programmes such as social science and economics also

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(E, ‘Mathematics – Widening’ and ‘Mathematics – Discrete’)

(Skolverket, 2012a). Course A gave basic eligibility to tertiary

edu-cation even if higher courses were demanded for some of the

terti-ary education programmes, especially those within economics or

natural science university faculties.

In 2011, non-compulsory upper secondary school was divided

into twelve vocational and six higher education preparatory

pro-grammes (Skolverket, 2012c). This can be seen in Table 1.

Table 1 The different programmes in upper secondary school.

Vocational programmes

Higher education preparatory

programmes

Child and Recreation

Programme

Business Management and

Economics Programme

Building and Construction

Programme

Arts Programme

Humanities Programme

Electricity and Energy

Programme

Natural Science Programme

Social Science Programme

Vehicle and Transport

Programme

Technology Programme

Business and Administration

Programme

Handicraft Programme

Hotel and Tourism Programme

Industrial Technology

Programme

Natural Resource Use

Programme

Restaurant Management and

Food Programme

HVAC and Property

Maintenance Programme

Health and Social Care

Programme

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stigated a national curriculum outlining values, tasks, goals and

guidelines. Each programme has its own diploma goals and, to

some extent, a unique setting of subjects. In all programmes,

Eng-lish, history, physical education and health, mathematics, science

studies, religion, social studies and Swedish are compulsory.

How-ever, to obtain a graduating diploma, the amount of courses that

must be completed differ (see Table 2).

Table 2 Requirements for obtaining a graduating diploma

Vocational programmes

Higher education preparatory

programmes

Students should have grades for

the education covering 2 500

credits, of which passing grades

provide 2 250 credits.

In the passing grades, the

fol-lowing courses are required:

Students should have grades for

the education covering 2 500

credits, of which passing grades

provide 2 250 credits. In the

passing grades, the following

courses are required:

• Swedish or Swedish as a

second language 1

• Swedish or Swedish as a

second language 1, 2 and 3

• English 5

• English 5 and 6

• Mathematics 1a

• Mathematics 1b or 1c

• Foundation courses of 400

credits

In addition, a pass in the

di-ploma project is required.

In addition, a pass in the

di-ploma project is required.

(Skolverket, 2012c, p. 21)

For some subjects, such as mathematics, the content differs

pending on the programme. Each subject has a syllabus that

de-scribes the courses that are to be studied within each programme.

Mathematics is divided into three paths for different programmes:

path A for vocational programmes; path C for the Natural Science

programme and Technology programme; and path B for the

Busi-ness Management and Economics programme, the Arts

gramme, the Humanities programme and the Social Science

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pro-receive a grade for each course, see Figure 1.

Figure 1 The different courses and paths for mathematics

(Skolverket, 2012c, p. 37)

One example is the Social Science programme. In mathematics, the

students who take path B, must complete two compulsory courses

(1b and 2b) and can take one or two non-compulsory courses (3b

and 4). As can be seen in Figure 1, course 4 is the same for all

pro-grammes. Each course consists of around 100 hours of teaching

and the grades for each course range from A (highest) to E (lowest

passing grade) and F (failing).

The subject syllabi contain: (a) Aims, describing the scope of the

subject, but may also include teaching methods; (b) Goals which

specify which parts of the aims that are to be graded; (c) Courses,

as mentioned above; (d) Core content for each course; and (e)

Knowledge requirements for the grades A, C and E (for example,

to achieve grade B, the student needs to fulfil everything under C

and most of the things under A). The aims and goals are the same

for all courses. This is elaborated further in articles 2 and 3.

The knowledge requirements are formulated as competences

(Swedish: förmågor) using a progression matrix and are the same

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tween grade E and grade A for problem-solving competence:

Grade E: Students can formulate, analyse and solve practical mathematical problems of a simple nature. These problems in-volve a few concepts and require simple interpretations. …

Grade A: Students can formulate, analyse and solve practical mathematical problems of a complex nature. These problems involve several concepts and require advanced interpretations.

In problem solving, students discover general relationships that are presented in symbolic algebra (Skolverket, 2012a, p. 6. Bold

in original).

When it comes to mathematics, the competences that the students

should develop are the same for all courses:

• Use and describe the meaning of mathematical concepts and

their inter-relationships

• Manage procedures and solve tasks of a standard nature with

and without tools

• Formulate, analyse and solve mathematical problems, and

as-sess selected strategies, methods and results

• Interpret a realistic situation and design a mathematical model,

as well as use and assess a model's properties and limitations

• Follow, apply and assess mathematical reasoning

• Communicate mathematical thinking orally, in writing, and in

action

• Relate mathematics to its importance and use in other subjects,

in a professional, social and historical context (Skolverket,

2012a, pp. 1-2)

In Sweden, there are national tests for all of the mathematics courses in

upper secondary school. The aim of the national tests is primarily to:

• support an equivalent and fair assessment and award of grades

•

provide a basis for an analysis of the extent to which knowledge

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lum:

The national tests also contribute to: • making the syllabi specific

• increasing student goal attainment (Skolverket, 2012c, p. 59)

Hence, the specific aim of the national tests is to make the syllabus

specific for teachers and students; therefore, in articles 2 and 3, I

make the case that the national tests should be considered part of

the curriculum.

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THE CITIZEN IN LIGHT OF THE

MATHEMATICS CURRICULUM

17 pages

The Citizen in Light of the Mathematics

Curriculum

Jonas Dahl & Maria C. Johansson

# " %  !# # !$   %  " % ( " # ! # ( !% ! % # " # ! # %

Keywords: citizen, curriculum, ethnomathematics, mathematical literacy, transition

Jonas Dahl, PhD student, Malmö University.

jonas.dahl@mah.se

Maria C Johansson, PhD student, Malmö University.

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Introduction

The aim of this article is to consider how school mathematics is connected to

citizenship in the Swedish curriculum. This is related to that of determining

the mathematics that every citizen needs, for instance to meet requirements for getting a job, to handle mathematical activities in order to cope with everyday issues or to understand and change society. The notion of citizen-ship in this article is connected to the mutual process between the society and the citizens in which the citizen is seen as having an active part in this dual interplay (EPACE, 2010).

Wedege (2010) suggests that determining the mathematics that every cit-izen needs has to do with two types of “knowing mathematics in the world”. The first type is knowledge developed in everyday life, for example, ethno-mathematics, folk ethno-mathematics, and adult numeracy. The second type is knowledge wanted in everyday life, such as mathematical literacy, math-ematical proficiency or mathmath-ematical competence. We draw on this distinc-tion using the labels ethnomathematics and mathematical literacy as analyti-cal constructs in order to interpret what is required of the citizen, as ex-pressed in the curriculum. This should be regarded as a way of approaching understanding, rather than creating or interpreting a dichotomy, as the prob-lem is far more complex. It can be understood as dividing school mathemat-ics and the curriculum along practical and theoretical considerations. How-ever, this can be counterproductive, or, as stated by Atweh and Brady (2009), when discussing real world applications versus the dominant dis-course of formalised academic mathematics:

Seen in this way, the intellectual quality of mathematics is measured primarily from within the discipline itself rather than the usefulness of that knowledge for the current and future everyday life of the student. (p. 271)

In focusing on the mathematics that is considered necessary for citizens, we do not discuss mathematics as an academic discipline per se, nor will we focus on the mathematical context. Instead, our intention is to focus on the relationship between the citizen, the mathematics s/he is supposed to need, and the mathematics s/he is supposed to develop in relation to what is writ-ten in the curriculum.

Ethnomathematics is connected to the mathematics people develop in everyday life. It focuses on cultural practices, which can be considered re-lated to mathematics and power (Knijnik, 1999; 2012). According to D’Ambrosio (2001; 2010), an ethnomathematical approach can contribute to peace, equity and human dignity. Research using an ethnomathematical per-spective reflects manyfoci, such as recognising uses of mathematics in

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dif-ferentcultures or on its contribution to a more global and just society (Ev-ans, Wedege, & Yasukawa, 2013).

Mathematical literacy also has various foci, aims, and implications. For the purposes of this article, this perspective is interpreted as a view on what people need. Jablonka (2003) writes on how mathematical literacy can be interpreted and states that mathematical literacy:

may be seen as the ability to use basic computational and geometrical skills in everyday contexts, as the knowledge and understanding of fundamental mathematical notions, as the ability to develop sophisti-cated mathematical models, or as the capacity for understanding and evaluating another’s use of numbers and mathematical models. (p. 76)

Therefore, it seems appropriate to label the mathematics that people need as mathematical literacy. Jablonka (2003) also notes that mathematical literacy is “about an individual’s capacity to use and apply [mathematical] know-ledge” (p. 78). The way mathematical literacy is understood by PISA (OECD, 2006)is similar, but with an emphasis on the individual’s responsi-bility to determine the mathematics they need as citizens

:

Mathematical literacy is an individual’s capacity to identify and

un-derstand the role that mathematics plays in the world, to make well-founded judgements and to use and engage with mathematics in ways that meet the needs of that individual’s life as a constructive, con-cerned and reflective citizen. (p. 72)

Explicitly, the PISA definition connects mathematical literacy to the zen’s ability to be reflective; although it is not fully clear upon what the citi-zen should reflect or even who the citiciti-zen is.

Using the analytical constructs of ethnomathematics and mathematical literacy, we outline the structure for the discussion in the following section. We then consider how the Swedish curriculum reflects the different con-structs. Using these labels provides insight into the relation between how the citizen is mentioned in the curriculum, both as an issue of demands from the society, in terms of mathematics and also whether there might be an interest in what the citizen could develop through that curriculum. We will also dis-cuss how citizenship is described in the curriculum, and suggest how such a discussion could contribute to a deeper understanding of the mathematically literate citizen and the understanding of her/him in the curriculum. However, we are aware that there are many different viewpoints and stakeholders within the field of mathematics education. Undertaking the challenge to

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in-vestigate a field of which we are a part has consequences. Bourdieu (2004) notes:

A scientist is a scientific field made flesh, an agent whose cognitive structures are homologues with the structure of the field and, as a con-sequence, constantly adjusted to the expectations inscribed in the field. (p. 41)

There is a need to be aware of this dilemma. We have been exposed to school mathematics, as well as having exposed others to it. This has given us insight that we could not possibly have gained from the outside. Trying to be reflexive, we aim to understand how different standpoints within the field of mathematics education may have influenced, or should influence, the curri-culum. For this purpose, we have chosen to examine a body of literature, which includes seminal articles written by leaders in their respective fields.

Our point of departure is social, critical, and inspired by the

sociomath-ematical perspective (Wedege, 2004; 2010), in which the citizens’

relation-ship to mathematics is in focus. Further, the interplay between general struc-tures and subjective matters is stressed in the sociomathematical perspective (Wedege, 2004; 2010). We suggest that understanding more about what the curriculum states regarding the citizen and mathematics could highlight the possible tension between what is relevant for the citizen and what is required from society.

The Swedish curriculum

In Sweden, school is compulsory until the ninth year, when students are sixteen years old. Upper secondary school is voluntary, although almost every student attends, as requirements from society make it, in reality, com-pulsory. Consequently, what is expressed in the curriculum is what, as inter-preted by Skolverket (The Swedish National Agency for Education), is seen as needed for students who become future citizens. Although Swedish stu-dents can attend different senior secondary programs (in natural science, social science, and pre-vocational), in the core area of mathematics, topics and competences are the same.

The Swedish mathematics curriculum is divided into core content and

competences. These are listed in tables 1 and 2. Core content refers to what

should be taught in mathematics classrooms, while competences refer to the skills the students are expected to develop and, therefore, provide the framework for assessment (Skolverket, 2011b). Table 2 outlines some fea-tures or ambitions that connect to the world outside of school. In the same table, we include PISA’s overarching ideas and competences. By doing this,

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we are not making any particular assumption; instead, we are showing how the Swedish curriculum is not an isolated national phenomenon.

In describing how the mathematics curriculum was designed, Skolverket (2011a) refers to drawing from “international experiences”. While no direct reference is made in this document to PISA, there are similarities between PISA’s definition of mathematical literacy and the Swedish mathematics curriculum. In later reports from Skolverket (see e.g. Skolverket, 2012), there are references to PISA, indicating that PISA was something that Skolverket was aware of when the new curriculum was being written. In Table 1, it can be seen that PISA uses the term “overarching ideas”, whilst in the Swedish curriculum, the correlating term is “core content”.

When it comes to the competences, there are further similarities and these can be seen in Table 2. However, the Swedish curriculum stresses that stu-dents should be able to “relate mathematics to its importance and use in oth-er subjects, in a professional, social and historical context”. This is one dif-ference between the Swedish curriculum and PISA, as these competences are mentioned by PISA only as the role mathematics plays in the world. Never-theless, there are similarities between this in the Swedish curriculum and PISA’s definition of mathematical literacy provided earlier.

Table 1 (OECD, 2006; Skolverket, 2011b)

PISA’s overarching ideas Core content in Swedish mathematics curriculum

Quantity Understanding of numbers, arithmetic and algebra

Space and shape Geometry

Change and relationships Relationships and change

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Table 2 (OECD, 2006; Skolverket, 2011b)

PISA’s competences Competences (förmågor) in Swedish

curricu-lum

Thinking and reasoning Follow, apply and assess mathematical

rea-soning.

Argumentation

-

Communication Communicate mathematical thinking orally,

in writing, and in action.

Modelling Interpret a realistic situation and design a

mathematical model, as well as use and as-sess a model's properties and limitations. Problem posing and solving Formulate, analyse and solve mathematical

problems, and assess selected strategies, methods and results.

Representation

-

Using symbolic, formal and technical language and oper-ations

Use and describe the meaning of mathemati-cal concepts and their inter-relationships.

Use of aids and tools Manage procedures and solve tasks of a

standard nature with and without tools.

-

Relate mathematics to its importance and use

in other subjects, in a professional, social and historical context.

The outcome of school mathematics is likely to be related to the curriculum and may influence the citizen and his/her life trajectories in many possible ways. Therefore, what needs to be problematised is how mathematics educa-tion could provide the students with a foundaeduca-tion for active citizenship, or question if this is even possible. What does it mean to be an active citizen, and how is it connected to the specifics requirements in the mathematics curriculum?

In order to problematise and discuss these issues, we examine in the fol-lowing section what the two approaches, mathematical literacy and ethno-mathematics, contribute to the role that mathematics plays in producing cer-tain kinds of citizens, according to the curriculum.

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Mathematical literacy

Wedege (2010) stated: “it is obvious that any definition [of mathematical literacy] is value based and related to a specific cultural and societal con-text.”(p. 31) Understanding these values is connected to who makes the defi-nition (Jablonka, 2003). The defidefi-nition from PISA on mathematical literacy is based on the assumption that mathematical knowledge has a functional value (Wedege, 2010). It “deals with the extent to which 15-year-old stu-dents can be regarded as informed, reflective citizens and intelligent con-sumers” (OECD, 2006, p. 72).

PISA’s definition of mathematical literary seems to stress the importance of the citizen to be “concerned and reflective.” Although the inclusion of “intelligent consumers,” partly contradicts this intention, as the meaning of being a consumer is not questioned. We find another contradiction between the statement: “…in ways that meet the needs of that individual’s life as a constructive, concerned and reflective citizen” and “every country needs mathematically literate citizens to deal with a very complex and rapidly changing society” (OECD, 2006, p. 76, our italics). These statements could form a contradiction if the citizens’ wellbeing and self-fulfilment conflicts with providing society with a sufficient work force.

PISA’s definition of “mathematical literacy” is one of many, as the ter-minology varies among researchers. These standpoints are value based and one aim seems to be to avoid what is “too basic”. For instance, Hoyles, Noss, Kent, and Bakker (2010) suggest that numeracy and mathematical literacy are not sufficient for describing, for example, advanced workplace mathematics. They use the term technomathematical literacy to describe workers’ competence to communicate by means of mathematical and spe-cific technological tools.

Kilpatrick’s (2001) definition is closer to that of PISA’s, but he also sug-gests the word literacy to have a value-based connotation; therefore, he uses the notion mathematical proficiency instead. Within this, he defines five strands:

(a) conceptual understanding, which refers to the student’s compre-hension of mathematical concepts, operations, and relations; (b)

pro-cedural fluency, or the student’s skill in carrying out mathematical

procedures flexibly, accurately, efficiently, and appropriately; (c)

stra-tegic competence, the student’s ability to formulate, represent, and

solve mathematical problems; (d) adaptive reasoning, the capacity for logical thought and for reflection on, explanation of, and justification of mathematical arguments; and (e) productive disposition, which in-cludes the student’s habitual inclination to see mathematics as a sen-sible, useful, and worthwhile subject to be learned, coupled with a

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be-lief in the value of diligent work and in one’s own efficacy as a doer of mathematics. (p. 107)

Kilpatrick et al. (2001) further suggest that these strands are interwoven and interdependent, and should equip students with the ability “to cope with mathematical challenges of daily life and enable them to continue their stud-ies of mathematics in high school and beyond” (p. 116).

The various definitions of mathematical literacy seem to have a common-ality in the focus on different strategies, strands and procedures. In this ap-proach, it seems as if these procedures are considered important for the citi-zen’s wellbeing and ability to cope with issues of everyday life.

The term mathematical literacy, as well as other related terms, is not used in the Swedish curriculum. However, there are statements which seem to suggest that there is in the curriculum the sort of mathematics that students need for active citizenship:

The upper secondary school should provide a good foundation for work and further studies and also for personal development and active participation in the life of society.

The subjects, which mainly contribute to providing a good foundation for personal development and active participation in the life of society, are the upper secondary foundation subjects [e.g. mathematics] (Skolverket, 2011a, p. 8)

The role mathematics could play in contexts such as the social and the professional is taken into account in the ethnomathematical approach, which instead focuses on cultural aspects of mathematics and, hence, diversity and equity due to the connection between power and culture (Knijnik, 2012).

Ethnomathematics

Following Wedege (2010), we interpret the ethnomathematical approach as mathematics developed outside of school contexts, also with the potential to be of interest for school and for informing curricula (Meaney & Lange, 2013).

Ethnomathematics is not only about practical mathematics, but offers an approach focusing on interpreting mathematical activities exercised in dif-ferent groups. Ethnomathematics is described by D’Ambrosio (2010) as relating to the mathematics practiced by different cultural groups, such as workers, indigenous societies, and other groups with shared traditions and objectives.

Ethnomathematics aims to link tradition and modernity, and it provides an alternative to the Western influence in mathematics education (D'Ambro-sio, 2001; 2010). This focus on the cultural practices of different groups

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groups questions the domination of Western culture within school mathemat-ics because of its exclusion of large sectors of the world’s population (D'Ambrosio, 2010).

Another path within the ethnomathematical approach aims to recognise cross-cultural mathematical activities practiced by groups of people within, for example, different workplaces (Evans, Wedege, & Yasukawa, 2013; Wedege, 2010).

In ethnomathematics, a central element is the notion of culture (Knijnik, 1999; 2012) and different cultural uses of mathematics, which also reflect power relations and hierarchies in the respective society. Nevertheless, the question still remains if all humans need or want the same mathematics. Jablonka (2003, p. 79) states: “research shows that there is a diversity of functional forms of numeracy that individuals and groups possess, which are well suited for their particular purposes.” This implies that different forms of mathematics should be seen as genuine in their own right (Knijnik, 2012; Radford, 2008b).

One difficulty that surrounds the ethnomathematical approach is its im-plementation into school contexts, and some critics of ethnomathematics point to this. For example, Rowlands and Carson (2002) critique ethnomath-ematicians for seeing mathematics as a weapon of cultural imperialism and state: “There seems to be a consensus in mathematics education that formal, academic mathematics is somehow intrinsically oppressive and oppressive

because of its rationalism” (p. 81).

In contrast, Meaney and Lange (2013) investigated the benefits gained from recognising ethnomathematics in the classroom. They suggest that the students’ backgrounds and foregrounds have to be considered when adopting an ethnomathematical perspective (Meaney & Lange, 2013). Meaney and Lange also remark on the need to reflect upon the current context in relation to the students’ background, and also what they are aspiring toward, so that neither the potential in the foreground, nor the background, are cut off. Therefore, it is necessary to be sensitive to the possible gains and losses for students when ethnomathematics is brought into the classroom. The critical questions are whether the experiences students bring to the classroom are considered, and whether school mathematics is likely to be valued in their future mathematical endeavours. Bringing students’ cultures into school mathematical activities is also accompanied by the risk of exoticising, ro-manticising and trivialising certain cultures. On the other hand, if these are excluded from the classroom, there is a risk of negligence and acceptance of injustice. This is mentioned by Pais (2011), who used a socio-political basis for his critique of bringing ethnomathematics into the mathematics class-room:

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There is no easy way out of this paradox: whether school should be reserved for learning of the globalized knowledge allowing everyone to participate in our high-tech world, or a school that incorporates di-versity but runs the risk of domestication of the Other. (p. 221)

Even the discourse surrounding globalised knowledge can be seen as prob-lematic. Dowling (2005) questions viewing mathematics from this perspec-tive due to the irony of revealing mathematical content in different cultures’ practices by referring to European mathematics.

This is the myth of emancipation. […] European mathematics consti-tutes recognition principles which are projected onto the other, so that mathematics can be ‘discovered’ under its gaze. The myth then an-nounces that the mathematics was there already. (p. 15)

Atweh (2012) brings up the discussion from another point of view, where the Other, with a capital O, is important in the responsible mathematics educa-tion he argues for. In his approach, ethics is the foundaeduca-tion of mathematics education. From this point of view, the encounter of the authentic Other is at the very heart of engaging with each other and in learning. Therefore, not focusing on gender, ethnicity and other inequalities could be just as negative as the stereotype of the traditional mathematics education.

In the context of the Swedish curriculum, ethnomathematics is not men-tioned, as is the case of mathematical literacy. However, the absence of eth-nomathematics in the curriculum differs in that it states that education should provide students with a foundation for work. Seeing education as the provid-er implies seeing the student as a receivprovid-er and not a contributor. From the ethnomathematical viewpoint, as we have discussed, this idea is challenged.

The two approaches

In both approaches—ethnomathematics and mathematical literacy—there are worthwhile aims which strive to equip citizens with the mathematics seen as crucial to coping with their lives. The question is what role the edu-cational curriculum of a society may have in accomplishing these worth-while aims for its citizens, presupposing that school mathematics is useful in the world outside of school. This is often referred to as an issue of transfer, which means that knowledge from one context is used in another (Evans, 1999). Evans (1999) further argues that having a simplistic view of the con-tinuity of knowledge between different contexts may be as insufficient as being stuck in the narrow interpretation of learning as being situated.

Certain assumptions about the world follow both ethnomathematics and mathematical literacy. In the mathematical literacy approach, the technical

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development of the world is in focus and must be maintained, supported or increased, along with the empowerment of individuals. However, this is also the case with the ethnomathematical approach, wherein globalisation and cultural diversity are in focus, but with the added challenge of the influence of the Western world.

It is not our aim to create a polarised discussion. Instead, our intention is to provide an understanding of the different viewpoints in mathematical education. However, this may also be a reflection of a tension existing with-in the field of mathematics education itself. Instead of bewith-ing productive for the field and its influence on the curriculum, this dichotomy runs the risk of getting stuck in debate. The need for citizens’ empowerment and technical development does not necessarily counter the need for taking cultural di-versity and equity into account. Instead, we suggest that it is necessary to consider the social world, and how the citizen can be viewed in this world from a critical perspective.

The citizen seen from a social and critical perspective

The citizen, who is also a student whilst at school, needs to be considered in relationship to society. As mentioned earlier, the notion of citizenship con-cerns the mutual interplay between the citizen and society. This gives the rather complex view of a citizen as a political being. From a socio-political approach, it does not make sense to view mathematics as separate from the world or the social practice in which the activity is embedded (e.g. Jablonka, 2003; Knijnik, 2012; Meaney & Lange, 2013; Radford, 2008a; Radford, 2008b; Radford, 2012; Wedege, 2010). From this socio-political point of view, it becomes necessary to consider how these activities are re-lated to power, and how and why the subordination occurs (Knijnik, 2012). For example, Knijnik (2012) describes Western mathematics as being the ruler that mathematics is measured by. Thus, mathematical activities are given value in relation to their similarities with school mathematics. Fur-thermore, mathematics used in cultural practices, such as in workplaces, can be difficult to describe as it is interwoven in technical and social routines (see, e.g., FitzSimons, 2013; Williams & Wake, 2007). From this point of departure, it does not make sense to see schools as producers of mathemati-cally literate citizens equipped with a pre-packed solution kit. The individual student and citizen are likely to be influenced by others, both in school and out of school.

Atweh (2012) and Radford (2008a) share a similar vision of human to-getherness, as a way of learning. This joint endeavour is also valid for math-ematics education, in which learning is about being (Meaney & Lange, 2013; Radford, 2008a). With this extended view of learning, the aspect of ethics is highlighted as a foundation of life and mathematics education (Atweh &

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Brady, 2008). In their view on ethics, Atweh and Brady (2008) emphasised a moral obligation towards others. They also stress the importance of access to high quality, equitable mathematics education, which is more than math-ematics being seen as a set of procedures whose quality is measured with its own tools. Instead, their interpretation of quality is as follows:

Quality in mathematics education is measured not as, or not only as formal abstraction and generalization, but by its capacity to transform aspects of the life of the students both as current and future citizens. (p. 272)

Education should not only engage students in interpreting the world, but also in the endeavour of changing the world, which Atweh and Brady (2008) consider to be the basis for socially responsible mathematics education. Skovsmose (1994) has a similar view, describing this as critical mathematics

education. From this, follows the assumption that education has a role to

play in either maintaining or changing the social conditions of society (Skovsmose, 1994).

Skovsmose (2005) also highlights the opportunities that a society pro-vides for different groups, depending on gender, age, class, race, economic resources, and culture. He suggests that one way of doing this is to not only take students’ backgrounds into consideration, but also their foregrounds. The word foreground could be interpreted as horizons of action, which in-cludes the motives or possibilities to move between contexts. Horizons of action, in relation to learners in transition, are described by Meaney and Lange (2013). They see contexts as different systems of knowledge and con-clude that the transition, which is reflected upon, can be a source of learning. Given this, the transition between context, and what is valued in different contexts, may affect both the students and the citizens’ learning and becom-ing. The learning that can occur from reflecting on the transition is highly dependent on whether it is, for the citizen, a minor or a major issue (Meaney & Lange, 2013).

We interpret the demands made on citizens through the concept of math-ematical literacy, and we see what citizens do and create in different cultures and situations, as ethnomathematics. As previously mentioned, this division between ethnomathematics and mathematical literacy is an analytical con-struction, rather than a straightforward matter of distinct approaches. We suggest that the moving between needs and demands, as the notions are used by Wedege (2010), can be captured in the concept of transition as it is used by Meaney and Lange (2013). Citizens are under ongoing transitions be-tween different contexts that are made available during the trajectories of life. Context, as defined above by Meaney and Lange, as different systems of knowledge, may include different views on mathematics. For example,

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school mathematics can be a requirement to gain a job or entry into higher education, even if that school mathematics may not be needed once the citi-zen obtains a certain type of job or has been accepted for higher education. It is also important to be aware of, and to be able to reflect on, the transitions that we as citizens undergo. Otherwise, it is likely that mathematics in out-of-school contexts takes school mathematics as the measure of what it means to be good at mathematics. If school mathematics is taken as the measure, engagement in other activities or ways of seeing mathematics may not be promoted.

Connections between school mathematics and

out-of-school contexts

School mathematics exists and is generally claimed to be useful. This is something that we as citizens are, of necessity, exposed to. Consequently, we assume that the transitions, either to or from school, have an impact on the citizen, even though the issue of transfer of mathematics from school to the world outside could be questioned. Nevertheless, mathematics is often un-derstood as having only one direction, according to Lundin (2012):

Mathematical knowledge is taken to be a useful tool for action and understanding in the world “out there”, outside the school. On the oth-er hand though, engagement with reality as it is, is not believed to lead to the formation of mathematical knowledge. (p. 3)

Some researchers argue that it does not make sense to talk about school mathematics as being useful at all. Lundin (2008) writes about mathematics and mathematical knowledge as a sublime object in that it is assumed to be invisibly present, powerful and emerging inside people in the game of school mathematics, which has been played so many times and so seriously that we gain faith in its connection to reality. He suggests that the real function of school mathematics is one of sorting students. This is in line with Dowling (2005) who writes about different myths produced about school mathemat-ics. The myths give us a false idea about the benefit we gain from school mathematics whilst, at the same time, differentiating students and reproduc-ing inequity (Dowlreproduc-ing, 2005).

While Dowling (2005) and Lundin (2008, 2012) question the value of school mathematics in the world outside school, Skolverket (2011b; 2011c) seems to see the curriculum as a straightforward matter or, at the very least, leaving the connection unquestioned. According to the curriculum, the idea of providing students with good foundational studies, work and life in soci-ety seem to be central and accomplished through the core content and the competences outlined in the mathematics curriculum.

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Mathematics, curriculum, and the citizen:

Concluding discussion

In this article, we have suggested that mathematical literacy is the dominant approach adopted in the curriculum. The assumptions in mathematical liter-acy and, hence, in the curriculum, tend to consider the empowerment of the individual, and also the need for society to have mathematically literate citi-zens. However, the conclusion that can be drawn from this is that the state of the world and the mathematical procedures and strategies that the citizen will need to adapt to this world, are taken for granted in the curriculum. This can be considered its major weakness.

The ethnomathematical approach challenges this view by considering dif-ferent cultural practices. There is also, within this approach, dissatisfaction with the dominance of Western culture in mathematics education. It thus highlights the state of the world as being. Yet, what is not clearly considered is how these different cultural practices could inform the mathematics curri-culum without, at the same time, emphasising otherness or contributing to the stereotyping of different cultures considered as Other.

Although there are valid points made for citizenship development in both approaches, neither of these is sufficient, not even if taken together. We con-sider that it would be beneficial to concon-sider ethics as the foundation of math-ematics education. Our reason for this is inspired by socially responsible mathematics education (Atweh, 2012) and its view on being for the Other, which moves mathematics education from an individualistic to a collective endeavour. Further, we suggest that this should be made explicit in the curri-culum, and that the citizen be addressed before the mathematical procedures. In the curriculum, there are few assumptions made explicitly about the citi-zen and about the world, and these are expressed in terms of the technical world rather than the social. This is not in alignment with seeing mathemat-ics as a social activity in which humans can engage together. Consequently, there is a need to have a different view of citizens and their potential for changing society for the better.

In other words, being a citizen and a human is more than just being math-ematically literate. After all, “learning is both a process of knowing and a process of becoming” (Radford, 2008, p. 225). First, changing the view of the citizen implies a different understanding of the mathematics in the curri-culum. Being an “intelligent consumer” as mentioned by PISA, is different from gaining something collectively; for example, a more sustainable world or a more equal social order. Being a consumer is in the interest of only a relatively few citizens globally, while the issue of a more equal society is the concern of a much wider range of the world’s population.

In order to get closer to the citizen and what type of mathematics educa-tion the citizen may need, we suggest that the expression “citizen in the

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world” is made explicit in the curriculum, and is addressed before the math-ematics. We see mathematical activities as shared experiences in which it is necessary to consider the transitions that students or citizens make during and after leaving school. This could be a way of achieving permeability be-tween school and the surrounding world. Even if there is no transfer of mathematics possible, there are still learners in transition between contexts.

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