Mathematical Literacy Grade 12 Textbook Pdf

This chapter investigates different perspectives on mathematical literacy that vary with the values and rationales of the stakeholders who promote it. The central argument is that it is not possible to promote a conception of mathematical literacy without at the same time — implicitly or explicitly — promoting a particular social practice. It is argued that mathematical literacy focussing on citizenship also refers to the possibility of critically evaluating aspects of the surrounding culture a culture that is more or less colonised by practices that involve mathematics. Thus the ability to understand and to evaluate these practices should form a component of mathematical literacy.

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3

Mathematical

Literacy

EVA

JABLONKA

Freie Universitiit Berlin, Germany

ABSTRACT

This chapter investigates different perspectives

on

mathematical literacy that vary

with the values and rationales

of

the stakeholders

who

promote

it.

The central

argument

is that it is not possible to promote a conception

of

mathematical literacy

without at the same time - implicitly or explicitly - promoting a particular social

practice.

It

is

argued that mathematical literacy focussing

on

citizenship also refers

to the possibility

of

critically evaluating aspects

of

the surrounding culture -

a culture that

is

more or less colonised by practices that involve mathematics. Thus

the ability to understand and to evaluate these practices should form a component

of

mathematical literacy.

1.

INTRODUCTION

There

is

an expanding body of literature referring to the terms 'numeracy' and

'mathematical literacy', although sometimes these terms are used only as a

synonym for mathematical knowledge.

On

the other hand, much of the literature

does not refer specifically to 'mathematical literacy', but

is

relevant because of

its concern with issues such as the goals of mathematics education, mathematics

for all, the public image of mathematics,

or

with the role of mathematical

knowledge for scientific and technological literacy. Accordingly, the references

given in this chapter are neither comprehensive nor restricted to the discussion

of numeracy

or

mathematical literacy.

Section 2 of this chapter briefly deals with the development

of

the terms

'numeracy' and 'mathematical literacy'. One main part of this chapter

is

divided

into

five

subsections.

It

provides a critical account of different perspectives on

mathematical literacy. The central argument

is

that it

is

not possible to promote

a conception of mathematical literacy without at the same time - implicitly

or

explicitly - promoting a particular social practice. Section

3.1,

Mathematical

Literacy for Developing Human Capital

considers attempts

of

developing a cross-

cultural definition of mathematical literacy for the purpose of generating measur-

able standards. Section

3.2,

Mathematical Literacy for Cultural Identity reviews

literature on ethnomathematics with respect to its implications for conceptualis-

ing mathematical literacy. Section

3.3,

Mathematical Literacy for Social Change

75

Second International Handbook

of

Mathematics Education, 75-102

A.J. Bishop, M.A. Clements.

C.

Keitel,

J.

Kilpatrick and F.K.S. Leung (eds.)

© 2003 Dordrecht: Kluwer Academic Publishers. Printed

in

Great Britain.

76

Jablonka

deals with a conception of mathematical literacy

that

promotes the use of

mathematical knowledge for analysing critical features

of

societal reality within

a process

of

pursuing a different reality. Section

3.4,

Mathematical Literacy for

Environmental Awareness

considers the possibilities of linking mathematical

literacy

not

only

to

an individual's capacity

to

solve personal

and

local problems,

but

also to global environmental concerns. Section

3.5,

Mathematical Literacy

for Evaluating Mathematics

schematically develops a conception

of

mathematical

literacy

that

reflects the view of the

author

more

than

the previous sections.

It

is

argued

that

mathematical literacy focussing

on

citizenship should refer

to

the

aim

of

critically evaluating aspects

of

the surrounding culture - a culture

that

is more

or

less colonised by practices

that

involve mathematics. Thus the ability

to

understand

and

to evaluate these practices should form a

component

of

mathematical literacy.

2.

FROM

NUMERACY

TO

MATHEMATICAL

LITERACY

It

is

indisputable

that

in today's society the ability to deal with numbers and to

interpret quantitative information

is

an

important

component

of

literacy in

addition to speaking, writing

and

reading. At the same time, however, it

is

difficult

to

say what the distinct meanings

of

'numeracy' and 'mathematical

literacy' are.

There are a

number

of

perspectives

on

numeracy

or

mathematical literacy

that

vary with respect to the culture

and

the context of the stakeholders who

promote

it.

It

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 sophisticated mathe-

matical models,

or

as the capacity for understanding

and

evaluating another's

use

of

numbers

and

mathematical models. These different interpretations reflect

different rationales and values

of

proponents, such as the desire

to

standardise

and

measure the

output

of formal mathematics education, popularise academic

mathematics, vocationalise general mathematics education,

or

educate critical

citizens.

According to the 1959 edition

of

the Webster's Collegiate Dictionary, in which

the term appears for the first time, 'numerate' means

"marked

by the capacity

for quantitative thought and expression". This definition reflects the meaning

of

numeracy from the Crowther Report (DES, 1959); this report was concerned

with the education of students in the 15-18 age group. Being 'numerate', meant

to

have a rather sophisticated understanding

of

mathematics

and

science (see

Brown et

aI.,

1998). Noss (1997) observes a narrowing

of

meaning in the

Cockcroft Report

(DESjWO,

1982) towards the ability to perform basic arithme-

tic operations and

to

decode information given in the form

of

graphical represen-

tations. He sees this narrow, number-based interpretation which excludes

important

ideas of mathematics (e.g., geometry, algebra

and

proof)

as linked to

the culture of utility.

Mathematical Literacy

77

'Innumeracy'

and

'mathematical illiteracy' became more familiar terms in 1988

when

John

A.

Paulos published his book 'Innumeracy. Mathematical Illiteracy

and

Its Consequences', a national best-seller in the USA. He shows many

authentic examples

of

innumeracy

and

develops a conception of 'numeracy' as

being able

to

understand better the quantitative aspects

of

one's environment.

Relatively simple number facts

and

some elementary ideas from probability

and

statistics are used for estimating, for understanding large numbers by linking

them to concrete examples of things, for building relationships between quanti-

ties,

and

for evaluating chances. However, looking

at

the quantitative aspects

of

our

environment does

not

automatically make

us

see something

of

interest.

The volume

'On

the Shoulders

of

Giants: New Approaches to Numeracy'

(Steen, 1990), developed under the auspices of the 1989 Mathematical Sciences

Education Board Curriculum Committee, provides

an

example of a different

interpretation of numeracy

that

is

informed by the practice

of

mathematics in

research

and

in science. It starts with deep mathematical ideas, such as dimension,

quantity, uncertainty, shape

and

change

and

shows a vision

of

the richness

of

mathematics as the language of patterns. This interpretation of numeracy does

not

primarily stress the idea of critically evaluating another's use

and

misuse

of

data

and

numbers,

an

ability that should be seen as a central ingredient of a

'liberating literacy' (Cremin, 1988).

Definitions

of

numeracy commonly include 'number sense'

and

'symbol sense',

which are asserted a mediating role between symbolic (numeric

or

algebraic)

representations

and

their interpretations.

Number

sense refers to informal aspects

of quantitative reasoning, such as the knowledge

of

situation-specific quantities,

common sense in employing numbers as measures,

and

the ability to make

order-of-magnitude approximations (McIntosh, Reys, & Reys, 1992). Symbol

sense includes being comfortable in using

and

interpreting algebraic expressions,

an

ability

that

relies upon generating numeric, graphic

or

computer representa-

tions of algebraic expresions (Fey, 1990; Arcavi, 1994).

The National Council

on

Education

and

the Disciplines (Steen, 2001) prefers

to speak of 'quantitative literacy' in stressing the importance

of

inquiring into

the meaning

of

numeracy in a society

that

keeps increasing the use

of

numbers

and

quantitative information. Nevertheless, the term 'numeracy'

is

still widely

used in adult mathematics education programs, even if the interpretation goes

far beyond the mere functional use

of

numerical

and

technical skills to process,

communicate,

and

interpret numerical information (Benn, 1997; Gal, 2000;

FitzSimons et

aI.,

1996).

'Mathematical literacy'

and

'numeracy'

cannot

be literally translated into

many languages so their meanings have to be paraphrased.

In

German, for

example, there

is

not

even a common word for 'literacy',

but

only for 'illiteracy'.

'Numeracy', however, connotes numbers

and

calculations with numbers. Hence

in this chapter the term 'mathematical literacy'

is

used intentionally to focus

attention on its connection to mathematics and to being literate. Thus it refers

metaphorically to a mathematically educated and well-informed individual.

78

Jablonka

3.

DEFINING

MATHEMATICAL LITERACIES

Any attempt at defining 'mathematical literacy' faces the problem that it cannot

be conceptualised exclusively in terms of mathematical knowledge, because it

is

about an individual's capacity to use and apply this knowledge. Thus it has to

be

conceived of in functional terms as applicable to the situations in which this

knowledge

is

to

be

used.

Knoblauch (1990), referring to reading and writing literacy, states that it

is

"Always literacy for something - for professional competence in a technological

world, for civic responsibility and the preservation of heritage, for personal

growth and self-fulfilment, for social and political change" (p. 76). This applies

in essence also to mathematical literacy. The contexts introduced, for example,

in the test items of the Third International Science and Mathematics Study

(TIMSS) show that the conception of mathematical literacy tends to

be

biased

towards the application of mathematics in the natural sciences (mostly physics

and biology) as

well

as in business and industry (Jablonka, 2000). But these are

'real-life situations' only for a small minority of the students who are going to

have a profession in science and industry.

Many attempts at conceptualising mathematical literacy start with labour

force demands, arguing that the introduction of microcomputers, and high

technology in general, implies a need for a higher level of skills. Analytical

thinking, evaluation, representation and searching for information are seen as

increasingly important for the workplace. This shift also expresses the need for

skills that are transferable from

job

to

job

and that keep up with changing

requirements. Definitions of mathematical literacy reflect this fact with respect

to technologies based upon mathematics, such as micro-electronics, genetic

engineering and 'biological technologies'. However, this

is

challenged by the

argument that only a

few

specialists are needed in 'high-tech' industries. But in

occupations which employ the products of those high-tech industries - products

that operate as black-boxes - a decline in skills can be witnessed, as

well

as a

drop in the decision-making functions of the workers. Even in skilled jobs there

is

a separation of conception from execution, as, for example, in computer

programming (Apple, 1997).

Dench et al. (1998) carried

out

a study in the

UK

showing the importance of

numeracy from the employers' point of

view.

Some argued that Information

Technology increases the need to work with numbers because it provides more

data for analysis.

On

the other hand, many employers saw a reduced need in

the application of number skills because Information Technology converts an

increasing number of tasks into routine ones. The argument that a growing

technological sophistication calls for higher levels of mathematical knowledge

is

only valid for a limited number of professions.

A conception of mathematical literacy linked to socio-economic needs in terms

of the marketability of skills might be called 'functional' in that it refers only to

the individual's ability to respond according to the given needs and constraints

of society.

On

the other hand, emphasising individual needs not only in terms

Mathematical Literacy

79

of survival, but also in terms of skills needed for effective participation in a

democratic society

is

quite a different rationale for determining the scope of

mathematical literacy. Consequently, communication

by

means of mathematical

language, interpretation of statements that contain quantitative arguments, and

critical evaluation of mathematical models are all essential to an emancipatory

mathematical literacy in technologically advanced societies.

The results of international comparative studies are contributing to a growing

public interest in mathematical literacy. The wish to generate measurable stan-

dards of mathematical literacy (as, for example, in TIMSS) motivates the search

for a canon of mathematical knowledge, methodological skills and mathematical

attitudes that can be introduced into various social contexts. Clarke (this volume)

discusses the issue of cultural authorship of international comparative studies.

He argues that the design should implement collaborative processes through

which the educational, philosophical and cultural positions are given voice in

the interpretation of

data

and the reporting of the research. In the mathematics

literacy test of TIMSS a general criterion in selecting the items was that they

should involve questions that could arise in 'real-life' situations. (lEA, 1997,

p.

iv).

But in the end, the standards of knowledge and skills to be tested are

conceptualised in the insiders' mathematically oriented terms, and the results of

quantitative studies such as TIMSS commonly show how little mathematics

is

understood in these terms.

On

the other hand, research shows that there are a diversity of functional

forms of 'numeracy' that individuals and groups possess, which are well suited

to their particular purposes. This fact points to the problem that whilst it

is

certainly true that work practices, everyday activities

and

arguments in the

media do embody some kind of mathematics this does not necessarily mean that

these practices have an intrinsical mathematical essence,

at

least not in terms of

traditional school

and

academic mathematics.

The assumption that it makes sense to search for a universally applicable

canon of mathematical skills that can be separated from the context of their use

is

doubtful from the perspective of a socio-cultural view of mathematics.

It

is

questionable whether mathematical skills can be separated from the social dimen-

sion of action and from the purposes and goals of the activity in which they are

embedded. Thus, the description of transferable methodological

or

process skills

(as, for example, skills of mathematical problem solving, modelling, generalising,

reasoning, and communicating)

is

by

no means unproblematic, especially when

referring to the application of mathematics. Such a description ignores the

interests and values involved in posing and solving particular problems

by

means

of mathematics.

Another difference in the conception of mathematical literacy

is

the extent to

which valuing mathematics

is

seen as a precondition. Or, even whether precondi-

tions should include a critical stance (FitzSimons et al., 1996). A case in point

is

the definition of 'mathematical proficiency' in the report from the National

Research Council of the National Academies by the Mathematics Learning

Study Committee (Kilpatrick, Swafford, & Findell, 2001). This definition includes

80

JabLonka

conceptual understanding of mathematical concepts, operations, and relations,

procedural fluency, strategic competence, and adaptive reasoning as well as a

'productive disposition' - that

is,

the habitual inclination to see mathematics as

sensible, useful, and worthwhile, coupled with a belief in diligence and one's

own efficacy.

Different conceptions of mathematical literacy are related to how the relation-

ship between mathematics, the surrounding culture, and the curriculum

is

con-

ceived. The different perceptions of this relationship are the leitmotif for the

following account. However, the following does not provide a detailed exposition

of perspectives, but an attempt to categorise different and, in some cases, conflict-

ing ingredients of mathematical literacy.

3.1. Mathematical Literacy for Developing Human Capital

"Our

mathematical concepts, structures and ideas have been invented as tools

to organise the phenomena of the physical, social and mental world"

(Freudenthal 1983, quoted in

OECD

1999,

p.

41). This statement prompts an

optimistic interpretation of the power of mathematical thinking for solving

individual and social problems, while conceiving of the mathematical tools

themselves as culture-free. It gives rise to a conception of mathematical literacy

in terms of the ability to analyse, reason and communicate ideas and results by

posing and solving mathematical problems. This comprises a mathematisation

and

modelling perspective.

One example of this very broad and demanding definition of mathematical

literacy

is

used in the

OECD's

International Programme for Student Assessment

(PISA): "Mathematical literacy

is

the capacity to identify, to understand and to

engage in mathematics and make well-founded judgements about the role that

mathematics plays, as needed for an individual's current and future

life,

occupa-

tionallife, social

life

with peers and relatives, and

life

as a constructive, concerned

and reflective citizen." (OECD

1999,

p.

50).

PISA aims

at

assessing mathematical literacy standards for the purpose of

comparative international analysis in

28

OECD

member countries as well as in

Brazil, China, Latvia, and the Russian Federation. Thus, it

is

claimed that the

given definition

is

a cross-cultural definition of mathematical literacy. But the

concept refers

to

'the world', and the reader learns that this means "the natural,

social and cultural setting

in

which the individual lives" (OECD, 1999, p.41).

This setting may be colonised by mathematical objects and concepts to a greater

or

lesser extent, depending on the technological and economic development of

a country and on the life-styles of its inhabitants.

The cultural setting in PISA

is

introduced through the situations and contexts

used in the problem statements. Thus, these are crucial ingredients

in

presenting

an operational definition of this theoretical concept of mathematical literacy.

Referring to the individual as an informed citizen, the problems are, for example,

drawn from the context of pollution, traffic safety

or

population growth.

It

is

hard to imagine that these are 'real problems' for all the students in the

32

Mathematical Literacy

81

countries. But this

is

apparently no problem with respect to the underlying

conception of mathematical literacy, because the 'authentic' situations are only

a means for re-contextualising mathematical concepts. Standardisation and

authenticity

do

not go together. In the end it

is

not the situations themselves,

which are of interest, but only their mathematical descriptions

(see

also Jablonka

& Gellert, 2002).

This conception of mathematical literacy aims to look at the world through

mathematical eyes.

It

emphasises higher-order thinking (developing and applying

general problem solving skills) rather than basic mathematical skills. To engage

in mathematical problem solving also implies a positive attitude towards and

an appreciation of mathematics and its benefits. However, such an engagement

in solving 'real life' problems

by

means of mathematics

is

always situated in a

social practice,

be

it in the workplace,

or

elsewhere when operating a personal

computer, when reading a train schedule, listening to a fortune teller, doing

handicraft, shopping

or

getting advice from a bank employee. Thus a conception

of mathematical literacy as behaving mathematically - a definition

not intrinsi-

cally related to the social community in which this behaviour

is

to be performed

- may equally be underpinned by educational arguments advocating critical

citizenship for participation in the public

life

of

an

economically advanced society

as

well

as

by

work force demands in underdeveloped countries:

"Mathematics

is

an efficient tool to assist in resolving complex problems

such as population growth, flood, storms, epidemics and so on, which affect

the day to day

life

of inhabitants of a country

....

They

[the

underdeveloped

countries] do not have properly trained teachers

or

mathematicians who

can accept the challenge of the problems of science, technology and society

waiting for mathematical modelling and mathematical solutions."

(Banu, 1991, pp.

117,

118)

Mathematical literacy

is

defined as a bundle of knowledge, skills and values

that transcend the difficulties arising from cultural differences and economic

inequalities because mathematics and mathematics education themselves are not

seen as culture-bound and value-driven. Mathematical literacy

is

connected to

learning how to think, but not to learning what to think about.

It

is

almost naive to believe that mathematical theorising as such would alter

ideological, political and economic conditions

(see

De Castell, 2000, for a similar

argument concerning literacy). Depending on these conditions, being mathemati-

cally literate

mayor

may not increase an individual's success in the workforce,

and raising the mathematical literacy standards of a population (and thereby

enlarging the mathematically skilled workforce) need not raise the material and

economic (let alone the democratic) development of a country.

Even so, the

GECD

suggests that there

is

indeed a simple relationship.

It

maintains that the PISA

is

intended to estimate and compare the stock of

'human capital'; that

is,

"the knowledge, skills, competencies and other attributes

82

JabLonka

embodied in individuals that are relevant to personal, social and economic well-

being" (OECD,

1999,

p.

11). A similar claim

is

made in the report from the

National Research Council by the Mathematics Learning Study Committee,

demanding that "all young Americans must learn to think mathematically, and

they must think mathematically to learn." and "For the United States to continue

its technological leadership as a nation requires that more students pursue

educational paths that enable them to become scientists, mathematicians, and

engineers" (Kilpatrick, Swafford,

& Findell, 2001,

p.

2).

Conceptualising mathematical literacy as thinking mathematically does not

bridge the gap between informal and formal mathematics. The tension

is

not

easily resolved by progressive mathematisation of situations. Becoming literate

is

always,

by

definition, a means of being introduced into a system of symbols

and reasoning invented to represent something outside of this system; the symbol

system of mathematics

is

one that has highly decontextualised 'meanings' (at

least when negative number, fractions and algebra are involved). So the construc-

tion of meaning

is

particularly difficult in mathematics when contrasted with

reading and writing literacy.

3.2.

Mathematical Literacy for Cultural Identity

Research has shown that there are a lot of out-of-school practices explicitly

involving mathematical concepts - for example, street vending, doing woodwork,

tailoring, home management

or

gambling

(see,

for example, Nunes et ai, 1993).

These can be considered as informal numeracy practices that are embedded in

different social activities. They differ in the kinds of mathematics that are

employed, in the purposes for employing that kind of mathematics, as

well

as

in the associated beliefs about the nature of mathematics and in the values about

the appropriateness of the (mathematical) problem solution (Baker, 1996).

Analysing these practices in order to uncover the students' values and beliefs in

comparison to those informing their formal school-numeracy practice shows

how far apart these practices mostly are. Thus, linking out-of-school practices

with school mathematics

is

seen to facilitate the transition from these practices

to the practice of school mathematics, even though the functioning of this

transition

is

not yet quite clear.

On

the other hand, in the workplace, graduates of school do not automatically

apply the mathematical techniques they have been taught. They invent

or

use

techniques that meet the purpose of the tasks (AAMT,

1997;

Noss, Hoyles, &

Pozzi, 1998). According to a broad definition of 'ethnomathematics' this concept

also refers to the distinct practices of labour groups (D'Ambrosio, 1985). The

official mathematics curriculum does not usually reflect the ethnomathematical

techniques used in the workplace. This means that these competencies do not

develop from learning mathematics

at

school. The local study of the practices

used in workplaces can form the starting point for developing teaching activities

that are meaningful to workers and at the same time value the diversity of their

workplace culture (FitzSimons, 2000). However, this does not guarantee that

Mathematical Literacy

83

workers

will

be

empowered to reflect on the assumptions

of

those decision-

making processes in their workplace that depend on the implicit mathematics

of the economic structures in which the workplace

is

embedded. An ethnoma-

thematical perspective does not automatically imply that those applications of

mathematics which are used as a way of supporting privilege and power, for

example by governments to justify political decisions,

will

be

discussed.

The mathematical practices of different cultural groups can become relevant

in educational contexts that focus on problems of regional interest, particularly

problems of practical and material importance, carried

out

in interdisciplinary

projects. Learning

is

then conceived of as having a closer relation to action,

stressing the active nature of students' participation in the learning process.

Ideally, the aims

of

the students involved in a project match those of the teachers

and of the community in which the school

is

situated. Borba (1995) describes a

project on fund raising for soccer games that he carried

out

with children from

the slums in Brazil. The knowledge produced in school can

be

brought into the

community, an educational process in which local knowledge (of the students

and of other participants) interacts with more global knowledge (of the teachers).

The orientation of such projects toward action (as opposed to problems that

aim at developing understanding in mathematics) can foster a confrontation of

different practices that leads to a critical evaluation of mathematical practices

with respect to the goals of the action.

Knijnik (2000) provides an example from her work with settlers of the Landless

People's Movement (MST) in Brazil where the practices

of

production and sale

of melon crops were 'naturally' changed through the process of confrontation

and translation of different forms of knowledge. She argues that a 'ghettoisation

process' (Grignon, 1989) would occur

if

the pedagogical process were limited to

the recovery of the native knowledge that has the consequence of reinforcing

social inequalities.

Not

unpacking

or

further developing the mathematics can

also have the effect of disempowering individuals by excluding them from aca-

demic mathematics, which means excluding them from career options. Similarly,

Moses and Cobb (2001) argue that mathematical knowledge, especially in alge-

bra,

is

the key to the future of disenfranchised communities (in the USA) because

economic access depends on these skills.

Another strand of ethnomathematical research consists

of

uncovering the

latent mathematical content that

is

hidden in traditional artefacts of indigenous

people. The artisan who, for example, weaves baskets cannot be said to do

mathematics, but those who discovered the techniques may have been involved

in some kind

of

mathematical thinking.

It

is

proposed

that

these artefacts can

be incorporated into a multicultural curriculum as a starting point for a mathe-

matical exploration. For example, pattern designs can

be

used to introduce

transformation geometry,

or

basketry for developing concepts of number theory

(Gerdes, 1999). This can be seen as an expression

of

opposition to the initiation

into a practice (that

of

Western mathematics) owned

by

groups who otherwise

oppress them. Thus it

is

a means of avoiding a break in cultural identity, a

break that might occur when Western curricula and textbooks are imported into

84

Jablonka

developing countries.

It

is

hoped that students will realise that mathematics may

originate from their own culture and experience.

But mathematical exploration may lead to a level of mathematical abstraction

that

is

not necessarily of interest, either for the person engaged in a distinct

ethnomathematical practice or for the evaluation

or

improvement of her practice.

Consequently, the appreciation of a student's cultural background runs the risk

of being counterbalanced by the implicit appreciation of academic mathematics

that

is

used for re-interpreting ethnomathematical practices

or

artefacts, as long

as they are valued only as a springboard from which to develop this kind of

mathematics. The fact that the situations and practices are relevant to the

students does not automatically imply that the mathematical exploration of

these practices

is

also relevant to them.

On

the other hand, ethnomathematics includes a challenge to the traditional

history of so-called Western mathematics that devalues, if not ignores, the

contributions of cultures outside Europe. Telling the socio-cultural histories of

mathematics, including the histories of the dissemination of mathematical prac-

tices through schooling, can form an important component of developing 'ethno-

mathematical literacy' (see, for example, Joseph, 1992). This may help in gaining

insight into the culture and into mathematics, thereby building an awareness of

the socio-cultural embeddedness of the development of mathematics and math-

ematics education. Finding similarities in, and differences between, the use of

mathematics in

our

own culture and in other cultures can be a means for

reflecting on how

we

make sense of

our

social actions (Gellert, 2000). Examining

mathematical talk in languages different from the Indo-European tradition - a

tradition in which most academic mathematics has developed - can create an

awareness of differences in conceptions of quantity, relationships and space and

show how supposedly universal concepts may

be

culturally defined (Barton,

1996).

The recognition that all cultures have developed some kind of mathematical

activities may, on the other hand, serve to show that the logico-operationallevel

of mathematics

is

not determined

by

culture and social needs, but consists of

cognitive mechanisms which have become fixed in the course of evolution (Rav,

1993). This would qualify mathematics as a pan-cultural universal activity that

makes it a privileged form for representing and explaining the natural world

(cf.

Bishop, 1988).

While ethnomathematics stresses the importance of the cultural backgrounds

of the students it assumes that there are no cultural conflicts in classrooms

related to these backgrounds. But it

is

questionable whether cultures can be

assumed to

be

compatible and to be in harmony with themselves. The reverse

question of how mathematics determines the socio-cultural environment and the

problem of developing a competency for understanding and evaluating social or

material technologies that are based upon mathematics - an important compo-

nent of mathematical literacy -

is

not problematised in the research related to

ethnomathematics

(cf.

Vithal & Skovsmose, 1997).

MathematicaL

Literacy

85

3.3.

Mathematical Literacy

for

Social

Change

Whereas the focus of ethnomathematics

is

cultural identity, within the realm

of

critical pedagogy mathematics education is to be viewed as a project with a

political vision aiming

at

critical citizenship. Mathematical literacy then

is

a

competency for re-interpreting parts

of

reality

and

participating in a process

of

pursuing a different reality.

Such a conception comprises a critique of the function

of

school mathematics:

school mathematics

is

utilised as a means

of

societal reproduction by implicitly

teaching patterns of behaviour

and

by reproducing inequalities along class,

ethnic and gender lines.

It

tends to exclude those groups

of

students

that

are

already marginalised. This exclusion

is

related to the notions of ability and

understanding

that

are informed by objectivistic philosophies of school math-

ematics (Frankenstein, 1989) and by the implicit view

of

the ideal student as a

young intellectual scholar

(cf.,

for example, Teese, 2000, for

an

analysis focusing

on

secondary schools in Victoria, Australia). Consequently, many students do

not

view mathematical knowledge as something that can be created

and

owned

by themselves. Basing a curriculum

upon

an

alternative vision calls for changing

the mathematical content as well as the social relations

that

are established by

traditional teaching methods.

Critical mathematics education assumes

that

students may use mathematical

knowledge for analysing critical features

of

societal realities (Skovsmose &

Nielsen, 1996).

One

strand

of

realising this vision

is

using problems to which

mathematics

is

applied in the classroom

that

have the potential

of

sensitising

students to social problems

and

of

helping them to articulate their interests as

citizens. Thus, mathematics

is

seen as an

important

tool for uncovering

and

communicating aspects of reality that are

of

social

or

political interest, especially

in challenging societal inequities. One important function of mathematics within

this vision

of

mathematical literacy refers to the use of basic statistical

data

and

statistical questions to deepen one's understanding

of

particular issues and to

change people's perceptions

of

those issues (Frankenstein & Powell; 1989, Shan

& Bailey, 1991). These may be literacy rates

of

men

and

women, infant mortality

rates and

life

expectancy,

or

data

about

unemployment

and

national income.

However, the putative objectivity

and

accuracy

of

numerical descriptions is often

used to obscure ideological connotations in the course of public political discus-

sions. Consequently it

is

questionable whether these seemingly neutral descrip-

tions

of

particular issues can be deconstructed by generating alternative

'objective' perceptions by means of mathematical language. This aspect points

to another important ingredient

of

mathematical literacy.

'Critical mathematical literacy' includes the ability to understand

and

critically

evaluate statistical

data

and

arguments that are presented by others, that

is,

to

understand the mathematics

of

political knowledge (Frankenstein, 2000). The

issue

at

stake

is

the extent to which mathematical awareness can contribute to

this ability.

Pimm

(1990) argues

that

(traditional) mathematics teaching may

even conflict with the development

of

political awareness because the power of

86

Jablonka

mathematical problem solving

is

due to its level of abstraction, to its 'de-

meaning', while political thinking demands a focus on the particular, metaphoric

content of a problem.

Problems for developing critical mathematical literacy need not be restricted

to statistics. They comprise, for example, comparing energy requirements of

household appliances, modelling risks of transportation, comparing tariffs and

charges, analysing arguments of employers about wage costs

and

looking for

relationships between energy and famine. However, the application

ofmathemat-

ical and scientific knowledge in the course of interdisciplinary projects runs the

risk of simplifying the process of mathematical modelling. Consequently ques-

tions about the accuracy and assumptions of the models developed in the course

of projects have to be introduced.

In essence, the argument here

is

that there

is

a contradiction built into a

conception that stresses the potential of using mathematics as a tool for gaining

critical consciousness by representing

or

modelling personal-social problems.

The problems are to be seen as exemplary with respect to their social and

political relevance

or

they are to be of immediate relevance to the students. But

this does not imply that they are exemplary with respect to different practices

of using mathematics.

3.4. Mathematical Literacy

for

Environmental Awareness

Conceptions of mathematical and scientific literacies are linked not only to the

individual's capacity to solve personal and local problems, but also to global

environmental problems (NSF,

1994;

see, for example, the International

Environmental Education Programme of UNESCO). Listed as global problems

are, for example, food and water resources, population growth, atmosphere and

climate, energy shortages, and pollution. The

OECD's

definition of mathematical

literacy

(see

'Mathematical literacy for developing human capital') suggests -

referring to the individual as an informed citizen - that the problems should be

drawn, for example, from the context of pollution, traffic safety

or

population

growth. Within such an approach, environmental problems are added as an

important field of applying mathematics, in the form of

data

and results of

mathematical models from environmental reports. These attempts are, however,

questionable, not least because environmental problems are interdisciplinary

in nature.

Environmental education

is

rather an aim than a subject.

It

is

a process of

clarifying concepts and developing certain skills and attitudes for understanding

the interrelationship between man and the natural environment. The overall

goal

is

to develop a basis for action in the light of environmental problems.

Mathematics plays a twofold role with respect to environmental problems.

It

is,

on the one hand, used as a language for (re)formulating important biological

and physical concepts.

On

the other hand, it

is

used as a tool for modelling

environmental problems, for example, to run simulations of complex systems

or

even to gain theoretical insights into ecological systems. Mathematics as a

Mathematical Literacy

87

language of physics

is

involved in all relevant definitions, for example, that of

energy and energy transformation. Computing gross energy versus net energy

and comparing efficiency factors of different transformation processes from

different resources involves sophisticated mathematics,

and

there exist various

models for calculating the time of depletion of non-renewable energy resources.

Mathematical models of ecosystems (that

is

plants, animals and micro-organisms

in an arbitrary defined zone in their dynamic interaction with the non-living

components) describe the dynamic interrelationship between species. Many

so-called environmental problems deal with population.

It

is

argued that local

or

global over-population causes over-exploitation and mathematics plays an

important role in all kinds of population studies. In addition, mathematics

is

always involved in studying possible links between physical aspects and economy,

whether

by

prices, taxes, rents

or

waste charges.

However, mathematics

is

itself an essential constituent of technological devel-

opment. Mathematics and science are the core of those disciplines that originally

were considered as a basis for social advance which

is

linked to liberation from

moral constraints. The accompanying values of rationality and objectivity were

and still are associated with technological progress and industrialisation leading

to an improvement of living conditions

(cf.

the discussion in the section on

mathematical literacy for developing human capital). But this scientific optimism

has been dampened

by

the recognition of the dangers of advanced technologies

and the ecological crisis as well as the effects of technological applications in

military research. In addition, the importation of technology into developing

countries often continues their exploitation rather than bringing social advance.

Thus it

is

debatable whether applying mathematics to problems that are the

'unwelcome concomitants'

of

technological solutions

is

appropriate.

As

long as mathematics

is

not conceived as intrinsically linked to destructive

technological developments, the problem

is

viewed only as one of control over

the fields to which it

is

applied. This

is

a serious problem because mathematicians

as individuals

or

as groups usually do not evaluate, manage and control the

technological transformation of their products.

On

the other hand, the

de-humanising effect can be viewed

as

immanent in the causal-logical nature of

mathematics. Consequently it

is

argued that mathematical literacy involves an

attempt

at

changing the perception of mathematics towards a more human view

in the hope that this may eventually even lead to the development of new forms

of mathematics.

Fischer (1993), for example, argues that mathematics should be oriented more

towards problem description than to final solutions. This

is

to define a new role

for mathematics as a discipline, following the suggestion (or prediction) that

classical computational mathematics

will

be taken over by computer science. He

argues that examples of this new orientation towards incorporating the human

factor do already exist. These involve systems analysis as

part

of

strategic

modelling, the use of a variety of alternative measures in economics as a means

of presenting and communicating ideas, and applications of mathematics in

psychology and sociology. Other areas where new developments have been

88

Jablonka

witnessed are exploratory

data

analysis

and

the modelling

of

dynamic(al) sys-

tems. The use of computers has already lead to introducing experimental methods

and

visual reasoning into mathematical research (Dreyful3, 1993).

For

dealing

with huge numbers of variables

and

with non-linear problems simulation-models

are developed. It is commonly argued

that

these models are more 'realistic'. The

strength

of

these new tools

is

inferred from the fact that they can

be

applied

directly to systems

and

phenomena without relying

on

a system of theoretical

concepts for describing them. This leads in return to a lack

of

generality

of

these models.

Similarly,

D'

Ambrosio (1994)

and

Fusaro (1995) argue for the development

of

a new kind

of

mathematics that would

be

more suitable for dealing with

environmental problems. The approach

of

traditional (academic) mathematics

(e.g.,

ecosystems, ecological modelling)

is

labelled as clinical

and

not interdis-

ciplinary. What

is

required

is

a radical change in outlook, to overcome the

homocentric bias

of

modern civilisation

(cf.

Kreith, 1993),

and

this should be a

special concern

of

mathematical scientists. The phrase 'Environmental

Mathematics' is introduced to label the ways in which these new kinds

of

mathematics are to be distinguished from traditional approaches. The mathemat-

ical content comprises arguments underpinned by mathematical visualisations,

qualitative mathematics

that

is

characterised as not aiming

at

an analytical

solution but serving as thought experiments,

and

computational mathematics,

which includes the use

of

simulation packages, graphing calculators

and

spread-

sheets. Linked with these supposedly new forms

of

mathematics

is

a pedagogy

that

stresses interdisciplinary project

work

and

should be driven by environmen-

tal commitment and engagement. The main goal of introducing 'Environmental

Mathematics'

is

to raise the environmental awareness of (future) mathematicians.

Some

of

these recent developments in academic mathematics are frequently

interpreted as a revolution of the characteristic ways of mathematical thinking,

not so much linked to the values

of

rationality and objectivity,

but

to openness

and

creativity.

As

opposed to traditional perceptions

of

mathematical modelling,

it

is

argued, these new forms

of

applications do not obscure the fact that values

and

interests drive them.

It

can be objected that this

is

true for all applications

of

mathematics, the difference being only the extent to which it

is

recognised

and

whether this

is

viewed as strength

or

weakness. Thus, it

is

doubtful whether

the given examples are new forms of mathematics rather than new interpretations

of,

or

new epistemological perspectives on mathematics. The mere existence

of

new mathematical forms for representing system relations does not automatically

imply that there

is

a canon

of

'new' mathematical techniques ready to be applied

to environmental

and

ecological problems. Particularly when referring to the

theory

of

chaotic dynamical systems, this shows the boundaries of

our

under-

standing

of

systems with complicated dynamical properties by means of relatively

simple representations.

In addition, complexity

is

also a property of some

of

the new tools, for

example, computer simulations, and there are no well-established criteria for

evaluating their output. Bool3-Bavnbeck (1991) points

out

that technological

MathematicaL

Literacy

89

applications

of

simulation-models cause a risk because there

is

a lack

of

theoretical comprehension and they are insensitive to the limits

of

validity

of

existing empirical knowledge.

The development

of

mathematical knowledge

is

embedded into a larger net-

work

of

human activities. Thus, as Restivo (1993) argues, as a social institution

modern mathematics itself can be viewed as a social problem

of

modern society.

Therefore it seems unreasonable

that

mathematics could

be

changed into alterna-

tive forms by educational reformers independently of some broader social

changes. Consequently there

is

the risk

that

the celebration

of

these new forms

of mathematical knowledge will be used by some people to mask a tech-

nocratic attitude.

Mathematical literacy for future citizens who are aware

of

global environmen-

tal problems can come into conflict with mathematical literacy for cultural

identity. D'Ambrosio (1994) reports from working in Caribbean communities

that the concern for immediate challenges (such as running

out

of

firewood)

takes priority over

that

for global environmental problems.

It

seems reasonable

to address such a conflict by uncovering and discussing the relationship between

poverty, depleting resources

and

economic constraints rather than by educating

an

environmental awareness among children

and

parents.

3.5. Mathematical

Literacy

for

Evaluating Mathematics

Critical pedagogy, for example, as articulated by Giroux (1989), focuses on

citizenship education and comprises the aim

that

students should learn

about

the structural

and

ideological forces

that

influence

and

restrict their lives.

Mathematics

is

a component

of

these forces in many ways.

As

a school subject

it establishes notions of learning

and

ability that are related to social class

(Dowling, 1991).

As

a part

of

the 'number-language'

of

public political discus-

sions it serves a central function as a tool for justifying all kinds of decisions.

As

a basis for material and social technologies it has a formatting power

(Skovsmose, 1994; Keitel, Kotzmann,

& Skovsmose, 1993; Keitel, 1997).

Thus a conception of mathematical literacy, particularly for critical citizenship

in

an

economically advanced technological society, comprises the aims

of

being

prepared to interpret information presented in a more

or

less scientific way, to

educate for an awareness

of

applications

that

affect society,

and

to develop a

consciousness

of

the limits

of

reliability of mathematical models. These compo-

nents of mathematical literacy refer more to the citizen as 'consuming' rather

than as developing mathematics.

It

can be said that the school context does

not

contribute a lot to these components of mathematical literacy

(cf.

also Steen,

2001; for a general discussion

of

arguments for incorporating applications see

Blum, 1991).

Critical mathematics education (see for example, Skovsmose & Nielsen, 1996)

involves as

an

important concern the fact

that

mathematics itself has to be

considered as a problematic technology

that

colonises the lifeworld. A problem

of

material technologies that are based

upon

mathematics

is

caused by the fact

90

JabLonka

that the technological transformation of academic mathematical knowledge

is

a

process embedded in a highly specialised division of labour. The transformation

is

mediated by several disciplines so that it

is

not easily seen through.

In

addition,

the mathematics involved

is

in general too sophisticated for incorporating tech-

nologically relevant examples

of

applications (as from engineering

or

geodesics)

into school mathematics. There

is

a lack of accounts that provide examples and

explain the principles (instead of the detailed mathematics) of these applications.

Many popular books and classroom materials aim to show the fascination, the

richness and unreasonable power of applications of mathematics, but they

do

not contain unbiased information about their social relevance and they lack a

discussion of the conditions and the consequences of their implementation. Some

attempts of providing readers for students that explain important examples from

engineering mathematics (as for example,

Maal3

& Schloglmann, 1993) have

been criticised for not containing enough mathematics.

In

aiming

at

mathematical literacy for critical citizenship, given that most

students will become 'consumers' of more

or

less explicit mathematics, discussions

about the evaluation of applications

of

mathematics can be introduced into the

classroom. Keitel, Kotzmann and Skovsmose (1993,

p.

271) develop a six-step-

model for producing reflective knowledge that can be used to frame such a

discussion.

However, the critical a posteriori evaluation

of

applications of mathematics

can be very demanding because there

is

no way of objectivising the criteria for

evaluation (Jablonka, 1996). An important aim

is

to overcome biased uncritical

or

misleading interpretations that result from a transmission of the prestige of

some well-founded mathematical applications that have withstood scrutiny in

many situations. This may result in

an

exaggerated optimism about the universal

applicability of mathematics, as, for example, reflected in the conception of

mathematical literacy of the

OECD

(see

the section on 'Mathematical literacy

for developing human capita!'). However, the recognition that all applications

of mathematics are value-driven can - on the one hand - lead to abandoning

well established criteria for evaluation

(cf.

the discussion reviewed in the section

on

mathematical literacy for environmental awareness),

or

- on the other hand

- inform an unjustified rejection of the use of mathematics.

Consequently, an important task before introducing a critical discussion,

is

to

search for categories of different practices that can

be

labelled as referring to

mathematics. Another task

is

to analyse developments that result in the colonisa-

tion of society with 'implicit mathematics' (Keitel, Kotzmann,

& Skovsmose,

1993), that

is

the mathematics incorporated into machines or other, including

social, technologies. Keitel (1997) argues that numeracy has to involve the

reflective knowledge necessary to analyse and evaluate this process. The following

examples refer to the explicit use of mathematics rather than to implicit

mathematics.

3.5.1.

Reasoning with condensed measures and indexes

One popular practice

is

the custom of using highly condensed measures and

complicated indexes (such as the poverty line, purchasing power parity in US-$,

Mathematical Literacy

91

GNP

per capita, unemployment rates, wage ratio, productivity, profitability,

price index for cost-of-living, share indexes, corruption index, etc.) as objectivising

arguments for pushing through distinct claims of (economic) policy. The ideologi-

cal connotation may be positive when there

is

a need to overcome prejudices.

However, figures relating to the macro-level of socio-economic structures are

likely to be misinterpreted, especially by readers who lack relevant cultural

knowledge.

To understand how such measures relate to perceptions of reality, it

is

impor-

tant to know which

data

are used and how these

data

are standardised and

aggregated. An essential condition for the

use

of

measures for comparisons

across regions

or

countries

is

that

all sources use the same methodology. But

even

if

the reliability of a measure

is

high, there remain difficult questions about

its validity.

A critical discussion of statements

that

contain such measures as, for example,

the rate of inflation, may involve checking the results of the calculations, but

these are usually correct. What

is

more important

is

asking for the reasons why

the weighted average

is

used and looking for alternative ways of calculating a

measure. This includes comparing different results when using different measures

(see

the classroom examples given by Frankenstein, 2000). The definition of the

concept purported to

be

measured may turn

out

to

be

inadequate. A mathemati-

cal inquiry into the principles of the construction of indices helps in recognising

that many of these measures do not meet the criteria one might ask for in

constructing an index

(see

Herget,

1984,

for an analysis

of

the price index for

cost-of-living).

A different question refers to the quality and origin of the

data

that

are used.

Usually it

is

very hard to find data from alternative sources and to gain some

information about the process of measurement. A lot of possibly demanding

questions can be asked concerning the estimation of consequences of different

kinds of errors in order to determine the range of values of the measured

quantities to guarantee a given accuracy of the results. These questions introduce

the perspective of applied mathematics from science and engineering. Referring

to economic data, Morgenstern (1973) observed

that

the measuring error of

economic data that were

well

defined and carefully collected was about ten

percent.

What are the implications of basing decisions

on

this sort of calculation? Why

are these measures used and accepted? Davis (1989) describes the process by

which mathematisations, particularly those referring to social

life,

are established,

broken and re-established, by using the metaphor of social contract based on

the willingness of the community to accept them. Thus the application of math-

ematics

is

a public enterprise that serves to regulate social relationships. Its

normative function can

be

interpreted according to the spirit of the time

(as

divine, deriving from logic

or

from empirical knowledge). However, it can

be

argued that the 'state of nature', that

is

assumed prior to the contract between

people and their mathematics by the metaphor of social contract,

is

already a

92

Jablonka

consequence of taking into public ownership, for example in the form of 'real

abstractions' (Sohn-Rethel, 1978).

The frame that

is

already established by the institutionalisation of its predeces-

sors often limits changing

or

improving such mathematisations. Qualitatively

alternative measures seem to be impossible because the frame reflects the organi-

sational principles of the economic system.

The introduction of the System for Integrated Environmental and Economic

Accounting (SEEA) developed by the United Nations and World Bank serves

as an instructive

example of this process (see Statistics Canada,

1998;

United

Nations, 1993; United Nations,

1999;

United Nations, n.d.). The aim

is

to

incorporate ecological criteria into economic thinking by monitoring the environ-

mental changes caused by economic activities as a basis for integrated economic

and environmental policies. The question

is

how definitions, classifications, con-

cepts of environment and resource accounting can be linked to the traditional

systems of national accounting, while

at

the same time leaving the central

framework and the basic concepts of these systems unmodified. The SEEA

focuses on parts of the environment that are absorbed into the economy, rather

than on outputs from the economy to the environment. It introduces additional

concepts in terms of physical

data

on

environmental cost and capital, or adjusted

concepts of cost and capital by incorporating values of the physical data. It

is

argued that valuation in monetary terms would give a link to common valuation.

From the concept of sustainable development it follows that use has to be

valued on the basis of the costs for fully maintaining the natural capital. This

means, to value actual degradation and depletion. This

is

similar to valuing

consumption of fixed capital to measure services of man-made capital using

actual market replacement costs. But are these the maintenance costs? So the

costs are actual

or

hypothetical cost

data

for something restored

or

avoided,

such as for maintaining the services (protection costs)

or

for mitigation of damage

(to health). The philosophy

is

that

is

it possible to determine prices without

markets. Use

is

measured in terms of influence

or

availability of capital for usage

and without a known effect the cost would

be

zero.

Measuring in physical terms causes similar problems of quantifying qualitative

concepts. What relevant constituents should be chosen? How should they be

measured? What about their relative importance? How can they be combined

into one indicator of quality? How can a condensed version be derived if one

component

is

measured in ppm, another in units of volume

or

tons? How can

the problem of counting something twice be avoided? How can a measure of

the available volume of water

or

air be defined?

The main interrelationship built into this model

is

the relation of a part of a

natural asset to a special economic activity. The

data

structure of the physical

accounts looks very similar to that of one of the monetary accounts. It

is

doubtful

whether drawing on the similarities leads to an integration of economic and

ecological thinking if this means understanding the dynamics of the process of

exchange between man and the biophysical or natural environment. The time-

scale

of

ecological problems

is

different from that of economic accounts and the

Mathematical Literacy

93

spatial distribution

is

ignored in statistical figures. The model involves only one-

dimensional characterisations.

It

is

based on assumptions about the availability

of resources

or

about the effects of certain economic activities. But these will

change

in

the course of time and the environment itself may affect economic

activity. Even if it were possible to calculate certain damages, the question "who

is

to compensate whom?" would remain,

and

this has to do with the status

of

property within the economic system.

The institutionalisation of such a system may lead to the impression that the

given figures are objective measures of the quality of the environment.

If

no

alternative valuations are shown, the fragility of the measures

is

not visible any

more. Another instance of a misleading interpretation of a national accounts

aggregate

is

the common practice of taking the

GDP

as a measure for prosperity.

The discussion of the SEEA shows that it

is

impossible to reconstruct all the

assumptions that inform the construction of a condensed measure from the given

figures. The mathematics involved

is

elementary, but the issue of defining the

basic concepts and the problems of measurement

and

aggregation can only be

discussed

by

drawing on reasonably specialised knowledge.

As

for environmental accounting for integrated economic and environmental

policies, in some cases it

is

not the absence of empirical knowledge,

but

the

political and economical conditions that tend to prohibit an action, even if a

solution

is

known. This sheds light on the importance of discussing the circum-

stances and reasons for ecological, social

or

political problems if they are to be

taken seriously. This includes a discussion of different perceptions of a problem

and thus conflicting criteria for its solution.

A mathematically literate adult should be aware of the danger of the substitu-

tion of political, philosophical, social and juridical arguments by numerical

arguments that rely on complicated measures. This does not mean that these

arguments cannot be based on measures referring to socio-economic issues,

but

they have to be complemented

by

relevant local, cultural and political knowledge.

3.5.2.

Formalising transactions

The example above shows that mathematical descriptions are not restricted to

representing a piece of observed or constructed reality, whether natural

or

artificial. Mathematics

is

also used to formalise procedures of distributing power

or

money, calculations of earnings

or

costs, and some other regularities found

in social actions.

Modes of accounting, book keeping, distributing votes

or

calculating interests

are examples which can be analysed in terms of the traditional norms that are

encapsulated in the algorithms. The societal conditions that framed the develop-

ment of these calculations cannot easily be reconstructed from their mathematical

representations. They have become seemingly natural parts of reality. Damerow

et al. (1974) demonstrate the resulting effect of 'realised abstractions'

(see

also

Keitel, Kotzmann, & Skovsmose, 1993) on social structures

by

analysing double-

entry bookkeeping. Many examples can be found on a more local level, mostly

referring to the modes of distributing budgets.

94

JabLonka

3.5.3.

Reasoning with Platonic models

Another practice of using mathematics

is

that of reasoning

by

means of models,

which

do

not have a relationship to empirical

data

because they involve variables

that cannot

be

measured. Thus, they refer to some non-empirical 'Platonic'

reality. This

is

not a problem as long as no different claims are made.

The Quaker Pacifist L.F. Richardson (1919) used a simple model for a battle

between two armies as a basis for arguing that no blood need

be

shed because

the outcome of the battle can be calculated beforehand. He was fully aware of

the fact that this application

is

different from those used

in

physics, when

he wrote:

"In this essay a very different use

is

made of mathematical symbols. The

successive formulae are not usually deduced from those which precede.

Rather each formula has been mentally compared with the miscellaneous

facts known to the author, and the succeeding formula

is

often an improve-

ment, a higher synthesis in the Hegelian sense, and not a deduction

....

Indeed on account of the difficulty of defining the fundamental quantities,

there remains a general vagueness, which may scandalise some of those who

have been trained in the exact sciences, but which, in the author's opinion,

does not deprive the formulae of meaning, interest and suggestiveness."

(p.67)

This

is

in sharp contrast to the claims made in teaching materials and popular

articles when similar models are introduced, for example, when referring to game

theoretic model of group behaviour it

is

proposed that:

'These

results may be of value for interpreting historical trends, but also

for an informed restructuring of corporations, trade unions, governments

and other social groups."

(Glance

& Hubermann,

1994,

p.

37)

Or

in a textbook on mathematical modelling it

is

said that:

"Game theory offers very interesting possibilities for the analysis of voting

power. ... Until recently, people believed that the number of votes a player

controlled was directly proportional to his voting power."

(Swetz

& Hartzler,

1991,

p.

60)

The claim

is

misleading because these models involve too many restricting

conditions so their domain

of

validity does not exist

or

cannot even

be

con-

structed. This does not mean

that

the heuristic value of such a model has to be

questioned.

It

might

be

a tool for exactifying concepts

or

reorganising them,

or

for identifying key points.

On

the other hand, it may

be

a formulation of

something already known, a rhetorical artefact (Davis

& Hersh, 1986), or the

consequence of a certain style

or

fashion.

Mathematical Literacy

95

There are examples of very sophisticated mathematical models that involve

many variables, but it

is

impossible to gain the data (as numerical values). This

is

a different reason for the lack of a link between empirically given (observed

or

constructed) reality and the model. Holzwarth and Weyer (1992) developed,

for example, a model of the spread

of

AIDS, which contains 1650 equations and

2.7

millions (formal) parameters in order to predict mortality for

life

insurance

companies. The complexity of the model

is

due to incorporating different 'risk

groups' and age groups. But the parameters could not be estimated because the

data were not available. This model turned out to be accurate for

an

ex-post

facto prediction of the cases that were documented (which could have been done

with a simple model), but it was totally inadequate for extrapolation. Such

models are nevertheless used as a basis for arguments about the development

of a population.

Confusion about the status of a mathematical description may arise when

inferring from the fact that a model fits empirical data that the theoretical

assumptions are valid. The population model developed by Franc;ois Verhulst

(1840) may serve as an example. He based the assumptions about population

growth on theoretical arguments. The model contains a (constant) parameter

for the net proportionate excess of births over deaths and a quadratic term

representing competition for resources:

P(n

+

1)

= k •

P(n)

- c .

P(n)2.

Translating

this model into the language of calculus leads to the logistic curve resulting in

the pleasant diagnosis that population would stabilise some day (as long as the

number of people

at

the beginning

is

smaller than that of the stable level). This

model gave a relatively accurate prediction of the population of the United

States from 1840 until

1940.

But this fact does not mean that the theoretical

assumptions of the model are valid. The parameters cannot be interpreted

in

the way that was intended by Verhulst for there were waves of immigration and

a civil war during that period.

3.5.4.

Constructing surface-models

In contrast to models based on a theory that

is

itself mathematised and that

provides a generative mechanism for a class of systems (for example, when

finding the geostationary orbit for a satellite) there are many examples of ad-hoc

modelling of the performance of a system by arbitrary fitting. Research has

shown that many pupils who are asked to solve tasks as the following: "Sabine

is

11,

her brothers Jens and Klaus are 8 and

6.

How old

is

their father?" give

an answer involving a calculation like 8 x 6 -

11

=

37.

This can be interpreted

as an ad-hoc construction that refers to the given data and also matches the

experience about ages of fathers. But in simple cases like this, the practice of

constructing a surface-model

is

commonly labelled as non-rational.

The following example

is

based on a collection of data about delinquents. The

formula (developed by

lB.

Copas, Department of Statistics, University of

Warwick, UK) gives the probability S of a person lapsing back into crime:

S=31-A-C+75

J--g-

+K

(F+

5)

96

Jablonka

The

data

that

are needed

is

the age

A,

the number

of

offenses

C,

the number

of

convictions g

and

the number

of

years after the first one

F.

K depends

on

the

specialisation

of

the person into a distinct type of crime and it has to be looked

up in a list.

This formula does not do any

harm

as long as it

is

not used by probation

officers as a means

of

prediction. Constructing a formula may

be

motivated by

the wish to overcome judgment based on mere subjective impressions,

but

it

runs the risk of being interpreted as the expression of a law of nature.

BooB-Bavnbeck, Bohle-Carbonell and

Pate

(1988) give some more serious

examples of the risk of 'feasibility over control' from technological contexts.

Using

an

ad-hoc-construction can be due to the fact that the construction

and

implementation of a theory-based model

is

too expensive

and

time consuming,

that

it turns

out

to be too complex,

or

that

a theory does not exist.

Within some theories a simplification

or

refinement of a mathematical model

is possible without changing its overall structure. In general, this indicates that

the phenomenon

is

well

understood in terms

of

empirical

and

theoretical

knowledge.

3.5.5. Numerology

There are still many involved in the study

of

hidden meanings of numbers and

their supposed influence on

human

life.

These meanings might relate to supernat-

ural agencies

or

to a hidden spiritual order. Numerological practices often

involve the use

of

knowledge

that

is

available only to the initiated.

It

is

to affect

the world in ways

that

could be described as magic.

One common and ancient practice (the Greek isopsephia and the cabalist

gematria) involves the reading

of

significance into numerical equalities between

words

or

phrases after assigning numerical values to the letters.

The following example, which has proved to generate interesting discussions

among students,

is

from a

German

magazine for young women.

It

supposes a

relationship between the

full

name

of

a person and personality traits, such as

talents, emotions, strengths and weaknesses. There are nine distinct personalities

that are described

in

detail.

After

an

introduction to the principles of numerology the following algorithm

is

introduced. A given table assigns numbers from 1 to

22

to single

and

distinct

groups of letters of the alphabet. Then one has to calculate the sum resulting

from the name, if married from both names. From this number the sum

of

digits

has to be subtracted and the resulting number divided by nine. To this number

one (referred to as the 'factor of personality') has to be added.

If

this algorithm

gives a number larger than nine, the sum

of

the digits

is

the result.

A mathematical investigation of the algorithm shows the reason for the distinct

steps.

Other

algorithms can be constructed for different numbers of personality

structures. One can invent names

that

result in personality traits

that

are desir-

able,

or

investigate the distribution

of

the nine characters. Such an activity

inevitably engages students

in

a discussion of the criteria for reasonable applica-

tions

of

mathematics. An evaluation from a mathematical

or

scientific perspective

Mathematical Literacy

97

leads, of course, to the conclusion

that

practices like this are to be labelled as

irrational. However, one can ask the question as to what causes its appeal. This

can open up a discussion of the function

of

other versions

of

pseudo science,

which are

not

as easily seen through.

4.

CONCLUSIONS

There are many ways in which the relationship between school mathematics

and

out-of-school mathematics can be analysed

and

constructed. Conceptions

of mathematical literacy draw on this relationship because they are

about

the

individual's ability to use the mathematics they are supposed to learn

at

school.

The relationship can be constructed by using examples

of

everyday practices,

already re-contextualised from the point

of

view of (academic) mathematics, for

developing de-contextualised mathematical skills. The underlying assumption

that mathematical concepts represent essential features

of

these practices gives

rise to a conception

of

mathematical literacy in terms

of

(higher-order) mathe-

matical skills

that

are applicable to all kinds

of

measuring, estimating and

calculating problems (see

Mathematical literacy for developing human capital).

To avoid privileging (Western) academic mathematical knowledge

and

those

who master it, a bridge may be built by incorporating ethnomathematical

practices

or

ordinary everyday knowledge into school mathematics.

It

was

argued

that

this runs the risk

of

either limiting mathematical literacy to the

recovery of local knowledge

or

of still (albeit only implicitly) privileging academic

mathematics by using it for re-interpreting the ethnomathematical practices (see

Mathematical literacy

for

cultural identity). Another

attempt

at

overcoming the

dominance of academic mathematics in the curriculum

is

using mathematics as

a critical tool for addressing problems that are of social

or

political relevance

(see 'Mathematical literacy for social change').

As

far as this includes the decon-

struction

of

the objectivity

of

claims based

on

statistics

and

mathematics by

putting forward alternative statistical and mathematical arguments, the implica-

tions

of

such a perspective are, in the end, incoherent.

In

the light

of

global

environmental problems

and

scientific fallacies, which are conceived as partly

caused by technological interventions based

on

traditional mathematics, a

few

writers consistently suggest

that

mathematics itself should be developed into

more suitable alternative forms (see

Mathematical literacy for environmental

awareness).

This runs the risk

of

abandoning well-established principles

of

con-

structing, evaluating

and

validating scientific assertions.

It

was argued that mathematical literacy focussing on citizenship also refers

to critically evaluating aspects

of

the culture of the students. This culture

is

more

or

less permeated by practices that involve mathematics. A mathematically

literate adult should know examples

of

technologically relevant applications

of

mathematics, be able to decode popular texts

that

contain mathematics and to

participate in political discussions

that

draw

on

statistics

and

results from

mathematical models. Thus, the ability to understand and evaluate different

98

lab

Lanka

practices that involve mathematics

is

an important component of mathematical

literacy.

An

essential assumption here

is

that it

is

possible to distinguish applica-

tions of mathematics in terms of consistency, connectiveness, complexity, compre-

hensiveness, embeddedness in a theory, and linkage to observed

or

constructed

reality.

The ability to evaluate critically can neither be considered as mathematical,

nor automatically follows from a high level of mathematical knowledge.

Consciousness of the values and perceptions of mathematical knowledge associ-

ated with distinct mathematical practices and their history can compensate to a

large extent for a lack of detailed expert knowledge. Introducing critical discus-

sions,

as

proposed here, means introducing a new discourse into school math-

ematics that will eventually establish a new practice of out-of-school mathematics

of informed citizens.

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... Developing ML requires positive dispositions toward using mathematics and an appreciation of mathematics and its benefits (Jablonka, 2003). Positive dispositions involve willingness and confidence to engage with mathematics. ...

... ML is about recognising the power and risk when issues are expressed numerically and to critically consider the contexts, mathematical knowledge and tools involved. Mathematically literate individuals recognise the role mathematics plays in culture and society, for example, how mathematical information and practices can be used to persuade, manipulate, disadvantage or shape opinions about social or political issues (Jablonka, 2003). Hence, they know and can use efficient methods and evaluate the results obtained (Goos et al., 2014). ...

• Oda Heidi Bolstad

Worldwide, there has been an increased emphasis on enabling students to recognise the real-world significance of mathematics. Mathematical literacy is a notion used to define the competencies required to meet the demands of life in modern society. In this article, students' encounters with mathematical literacy are investigated. The data comprises interviews with 22 students and observations of 16 mathematics lessons in three grade 9 classes in Norway. The analysis shows that students' encounters with mathematical literacy concern specific mathematical topics and contexts from personal and work life. Students' encounters with ML in school is characterised by an emphasis on developing mathematical knowledge within the school context.

... Mathematical digital literacy is the ability to understand and use information related to mathematical material in various formats from various sources that are presented through computers and especially through internet media (Kilpatrick, 2001). According to Jablonka (2003), mathematical digital literacy is a constellation of knowledge, skills, and competencies mathematics needed to thrive in a learning culture dominated by technology. So that the application of mathematical digital literacy provides opportunities for interaction, literacy of interesting reading sources, various reference materials, communication, and problem-solving (Kissane, 2009). ...

• Aidha Rosalia Agustin
• Widodo Winarso

The phenomenon of procrastination behavior in problem-solving and mathematical digital literacy still occurs among students. This study aimed to analyze the profile of students' academic procrastination behavior in problem-solving and mathematical digital literacy-purposive sampling technique involving 19 students of one junior high school in Indonesia. Academic procrastination behavior is obtained through qualitative research with a case study approach using the Tuckman Procrastination Scale (TPS) instrument, digital literacy scale, mathematical problem-solving ability tests, and semi-structured interviews as well as data analysis with four steps of research, namely data collection, data reduction, data presentation, and concluding. Most of the students (84.2%) experienced academic procrastination in solving mathematical problems and mathematical digital literacy (21%). There are six kinds of problem-solving and mathematical digital literacy student profiles in the high, moderate, and low category.

... They need to learn mathematics in a way that corresponds with current and future challenges and demands, helping them to become constructive, committed, and reflective citizens. One approach to making mathematics more meaningful is including a paradigm of mathematical literacy as part of mathematics education (Jablonka, 2003;Machaba, 2018;Masal & Yılmazer, 2014;Niemi et al., 2018;Uzunboylu et al., 2012;Vithal & Bishop, 2006). The PISA framework described mathematical literacy and highlighted that learners should identify and understand the role of mathematics in today's world (OECD, 2006). ...

The aim of this study was to uncover how digital storytelling advances students' self-efficacy in mathematics learning and what kinds of learning experiences contribute to self-efficacy. Four Chinese classes with 10- to 11-year-old students ( N = 121) participated in the project. The mathematics learning theme was geometry. Quantitative data was collected with questionnaires. The qualitative data was based on teachers' and students' interviews and observations. Both data sets showed that the students' self-efficacy increased significantly during the project. The most important mediator was students' perception of the meaningfulness of mathematics learning; digital storytelling enhanced the students' ability to see mathematics learning as useful. They became more confident that they could learn mathematics and understand what they had learned. They also felt more confident in talking with their classmates about mathematical concepts. The role of self-efficacy was twofold: it supported students' learning during the project and it increased due to meaningful mathematics learning experiences.

... Bireyin, matematiğin gerçek hayattaki önemini fark etmesinde ve önüne çıkan problemlerin üstesinden gelmesinde, matematiksel süreç becerilerini kullanabilmesinde, matematik okuryazarlığının önemi büyüktür (Gellert, 2004). Bu açıdan bakıldığında, her bireye öğrenme sürecinde matematik okuryazarlığı kazandırmanın gerekli olduğu sonucuna ulaşılmaktadır (Jablonka, 2003 (Bansilal, Webb ve James, 2015) çalışmaların yapıldığı görülmektedir. Uluslararası literatür incelendiğinde, matematik okuryazarlık üzerine çok fazla çalışmanın olduğu ve farklı ülkelerde matematik okuryazarlığın karşılaştırıldığı çalışmaların da yapıldığı göze çarpmaktadır. ...

... Mathematical literacy generally refers to an individual's capacity for the study and application of mathematics. This includes the cognitive and executive components responsible for performing mathematical operations, from simple calculations to complex mathematical judgments (Jablonka, 2003). ...

The use of maps as a complex source of geographical information requires a certain level of mathematical literacy. The lack of such literacy can cause severe failures in map use and the development of map skills. Therefore, this paper aims to contribute to the discussion about the difficulties in using quantitative thematic maps (specifically choropleth maps and proportional symbol maps), which may result from insufficient level of mathematical literacy at the lower secondary level of education. The paper is structured into two studies: Study 1 focuses on the continuity of mathematics and geography curricula (employing methods of expert cognitive walkthrough and content analysis), while Study 2 examines the relationship between achieved mathematical literacy and map skills (using two achievements tests and a questionnaire). The findings show that the continuity of curricula often fails and that map skills development precedes the development of mathematical literacy. The identified inappropriate chronology might have important consequences, since the correlation of mathematical literacy with the level of thematic map use skills proves to be statistically significant. Their relationship is significant in all aspects of map use (map reading, analysis, and interpretation) and in the use of both types of quantitative thematic maps examined in the study. The results should be of interest to geography teachers, teacher trainers, and curriculum leaders on the national and school levels.

... The problem found is confidence, satisfaction, and confidence, both for researchers and students. For example: (1) McCoy [10] reported from multiple regression analysis it was found that knowledge of certain mathematical contents of prospective teacher students significantly increased during lecture methods/contents; general mathematics content knowledge, self-confidence, and expectations of mathematics teaching outcomes also increased during college; and mathematical content knowledge is significantly correlated with selfconfidence but does not significantly predict the growth of self-confidence experienced during the semester of lecture, (2) Learning content knowledge is a reliable and valid measure of mathematics satisfaction for prospective teacher students; satisfaction instrument is an important stage as a measure of knowledge of prospective teachers; satisfaction instruments assess the design of pedagogy for prospective teachers and teachers to think mathematically [11]; [12], and [13] concluded that significant changes were in the beliefs of prospective teacher students who tended towards conceptual views or thinking. ...

• Mohamad Rif'at
• Soleh Khalimi

The purpose is to implement action research in mathematics teaching and learning process. The questions addressed are: (1) Is the teaching and learning process compatible with the management standard? And (2) how does the teaching and learning process influence the ability to handle varies needs of students? The research procedure is the researcher observed by the lecturers of mathematics through survey about the knowledge and experience based on the standard. The implementation directed at improving student learning outcomes and the improvements to the didactical, methodical and pedagogical components. Through the research activity: (1) directly involved in providing action; (2) to learn and apply the knowledge and thoughts; (3) get social measures; (4) openly justified biases to students; (5) get a permanent effect on the groups; and (6) to contribute to the desired changes.The data is qualitative and quantitative and analyzed descriptively. The results are: (a) the compatible with teaching standard, i.e., more benefit, practical, valid and reliable and (b) the teaching and learning process accomodated to vary of the student needs.Keywords: action research; compatible; didactical; pedagogical; methodical

... The problem found is confidence, satisfaction, and confidence, both for researchers and students. For example: (1) McCoy [10] reported from multiple regression analysis it was found that knowledge of certain mathematical contents of prospective teacher students significantly increased during lecture methods/contents; general mathematics content knowledge, self-confidence, and expectations of mathematics teaching outcomes also increased during college; and mathematical content knowledge is significantly correlated with self-confidence but does not significantly predict the growth of self-confidence experienced during the semester of lecture, (2) Learning content knowledge is a reliable and valid measure of mathematics satisfaction for prospective teacher students; satisfaction instrument is an important stage as a measure of knowledge of prospective teachers; satisfaction instruments assess the design of pedagogy for prospective teachers and teachers to think mathematically [11]; [12], and [13] concluded that significant changes were in the beliefs of prospective teacher students who tended towards conceptual views or thinking. ...

• Safarunita Wahyuni Reski
• Mastiah Mastiah
• Yumi Sarassanti

This study aims to determine differences in readingcomprehension skills in Indonesian subjects before and after using the Method Question Read Recited Review Survey (SQ3R) for students in the SDN 05 Pemuar. This research was conducted based on the background of the problem of teacher-centered learning processes, this is what makes students less active and look passive in the classroom, therefore students have difficulty absorbing the material and causing students' reading skills to understand the story text is less. Through this research the researcher hopes to be able to improve students' reading comprehension skills, especially in learning Indonesian in class III SDN 05 Pemuar. This research is an experimental study using thePretest-Posttest One-Group Design. The population in this study included all third gradestudents of SDN 05 PEMUAR with a sample of 14 people consisting of 9 male students and 5 female students. The sample used in this study is a saturated sample, therefore all members of the population are sampled. The preliminary data of this study usedavaluepretest with an average of 61.75 and an averagescore of posttest70.35.Based on the results of research tests of reading comprehension of students' understanding shows the value of t count of 49.571 and t table of 42.048 so that Ho is rejected and Ha is accepted because t count from t table. So it can be concludedthat with the Method it Question Read Recited Review Survey (SQ3R) has been proven effective against students' reading comprehension abilities, especially in class III SDN 05 Pemuar. Theresultsposttest show students' reading comprehension ability using the Survey Question Read Recited Review Method (SQ3R) higher than before being treated with the Method Question Read Recited Review Survey (SQ3R).Keywords: comprehension reading skills, Indonesian and Question Read Recited ReviewMethod(SQ3R). Abstrak:Penelitian ini bertujuan untuk mengetahuiperbedaankemampuanmembacapemahamanpada matapelajaran Bahasa Indonesiasebelum dan sesudahmenggunakan MetodeSurvey Question Read Recited Review (SQ3R) pada siswa SDN 05 PEMUAR. Penelitianinidilaksanakanberdasarkanlatarbelakangmasalah proses belajarberpusat pada guru, inilah yang membuatsiswakurangaktif dan terlihatpasif di dalamkelasmakadariitusiswakesulitanmenyerapmateri dan menyebabkankemampuanmembacasiswadalammemahamiteksceritakurang. Melaluipenelitianinipenelitiberharapmampumeningkatkankemampuanmembacapemahamansiswakhususnyadipembelajaran Bahasa Indonesia di kelas III SDN 05 Pemuar. Penelitianiniadalahpeneletianeksperimendengan menggunakan design One-Group Pretest-Posttest.Populasidalampenelitianinimencakupseluruhsiswakelas III SDN 05Pemuardenganjumlahsampel 14 orang yang tediridari 9 orang siswalaki-laki dan 5 orang siswaperempuan. Sampel yang dipakaidalampenelitianiniadalahsampeljenuh, makadariitusemuaanggotapopulasidijadikansampel. Data awalpenelitianinimenggunakannilaipretestdengan rata-rata 61,75 dan rata-rata nilaiposttest70,35.Berdasarkanhasilpenelitianteskemampuanmembacapemahamansiswamenunjukannilai t hitungsebesar 49,571 dan t tabelsebesar 42,048 sehingga Ho ditolak dan Ha diterimakarena t hitungdari t tabel. JadidapatdisimpulkanbahwadenganMetodeSurvey Question Read Recited Review (SQ3R) terbuktiefektifterhadapkemampuanmembacapemahamansiswakhususnya di kelas III SDN 05 Pemuar. Hasil posttestmenunjukankemampuanmembacapemahamansiswadenganmenggunakanMetodeSurvey Question Read Recited Review (SQ3R) lebihtinggidarisebelumdiberiperlakuandenganMetodeSurvey Question Read Recited Review (SQ3R). Kata Kunci: kemampuanmembacapemahaman, Bahasa Indonesia danMetodeSurvey Question Read Recited Review (SQ3R).

... Matematinis raštingumas -atvirkščiai: konkretus, realaus konteksto, priklausantis nuo visuomenės, politinis, aproksimuojantis, nenuspėjamas". Konkretumo matematinio raštingumo sampratai suteikė E. Jablonka [8] analizė (taip pat žiūrėti [9]). Jos teigimu tenka pripažinti, kad matematinis raštingumas negali būti apibrėžtas matematikos žinių terminais. ...

• Rimas Norvaiša

We discuss different alternatives of the content of school mathematics. According to the prevalent public opinion in Lithuania school mathematics can be oriented either to the academic mathematics or to the applications of mathematics. In reality the second alternative means lowering of the level of teaching in the hope that school mathematics will be accessible to all students. While the content oriented to the academic school mathematics is accessible only to gifted students. In this article we describe a middle alternative content which we call school mathematics based on mathematical reasoning. We argue that such school mathematics serves all students and makes acquaintance with mathematical reasoning and with applications of mathematics to the real world. Reasoning makes mathematics reasonable for all.

• Martin Braund

The COVID-19 pandemic has resulted in unprecedented amounts of information communicated to the public relating to STEM. The pandemic can be seen as a 'wicked problem' defined by high complexity, uncertainty and contested social values requiring a transdisciplinary approach formulating social policy. This article argues that a 'Critical STEM Literacy' is required to engage sufficiently with STEM knowledge and how science operates and informs personal health decisions. STEM literacy is necessary to critique government social policy decisions that set rules for behaviour to limit the spread of COVID-19. Ideas of scientific, mathematical and critical literacy are discussed before reviewing some current knowledge of the SARS-CoV-2 virus to aid interpretation of the examples provided. The article draws on experience of the pandemic in the United Kingdom (UK), particularly mathematical modelling used to calculate the reproductive rate (R) of COVID-19, communication of mortality and case data using graphs and the mitigation strategies of social distancing and mask wearing. In all these examples, there is an interaction of STEM with a political milieu that often misrepresents science as activity to generate one dependable truth, rather than through careful empirical validation of new knowledge. Critical STEM literacy thus requires appreciation of the social practices of science such as peer review and assessment of bias. Implications of the pandemic for STEM education in schools requiring critical thinking and in understanding disease epidemiology in a global context are discussed.

The modern teaching/learning environment is, like never before, rich with digital teach-ing/learning technologies and tools that are becoming part of children's daily lives. Background: In Lithuania, virtual teaching/learning platforms (environments for mathematics, knowledge of nature , history, and language practice) in primary education became more widely used approximately three years ago after the implementation and application of the virtual teaching/learning platform EDUKA. The purpose of this study was to establish the effect of the virtual teach-ing/learning platform EDUKA on the learning outcomes of primary-grade students in the subject of mathematics. Methods: In this study, a pre-test/middle-test/post-test experimental strategy was used to avoid any disruption of educational activities due to the random selection of children in each group. Mathematical diagnostic progress tests (MDPTs) are an objective way to measure skills and abilities. The MDPTs were divided into two sections: the tasks were allocated according to performance levels and the content, as well as fields of activity and cognitive skills. The assessment of all areas of activity was based on the primary school children's performance (i.e., unsatisfactory, satisfactory, basic, and advanced). Results: An analysis of the results of the MDPTs showed that, across the seven possible tasks, both male and female seven-year-old children achieved satisfactory results (results were observed between groups) (post-test: control gathering (CG) 5.10; test gathering (EG) 5.04; p = 0.560), basic results (post-test: CG 6.28; EG 6.42; p = 0.630), and advanced results (post-test: CG 1.90; EG 2.27; p = 0.025). The differences between the pre-test and post-test advanced (p = 0.038) and the pre-test and post-test basic (p = 0.018) levels were found to increase. Conclusions: It was found that intensively integrating the virtual learning platform EDUKA into formal education specifically in the subject of mathematics-had a significant impact on primary school chil-dren's mathematical performance. In addition, after the experiment, a statistically significant difference was found (p < 0.05) in primary school children with higher levels. The intervention in the experimental group (i.e., integration of the virtual learning platform into the formal mathematics learning process) had a positive impact on access to mathematics. Students' math learning achievements were positive in progressive mathematics.

• Lindsey M. Jesnek

Non-traditional student enrollment, especially at community colleges, has markedly risen in the last ten years due to national unemployment rates, the current economic climate, and employer demand for computer-literate employees. While university instructors struggle to constantly adapt their course materials to incorporate updates in software modules, various online learning systems, and consumer gadgets, they must also troubleshoot the obstacles inherent in their changing class rosters. Functioning under the definition of non-traditional as students over the age of 25 who are often first-generation college enrollees, displaced from their previous careers due unforeseen layoffs, or desperate to update their rum by earning an advanced certification or degree in order to ensure job security, this paper examines the lagging response of higher education institutions to appropriately manage the widening digital divide. The clear dissonance between typical non-traditional student computer competency and typical traditional student computer competency specifically informs this examination. In response to the amalgamated complications revealed in the non-traditional students charge to function successfully within a technologically-driven university environment, practical application strategies in the form of pre-enrollment computer competency placement testing and the implementation of required, degree-credit introductory computer courses must be established as a national initiative in order to formalize the concerted effort needed to encourage the overall academic success of non-traditional students nationwide.

Mathematical Literacy Grade 12 Textbook Pdf

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