Numeracies, mathematics
and values – a culturalperspective.
Alan J. Bishop
Monash University
Melbourne
Australia

11. Introduction - numeracy in a global
cultural context
Global citizenship has many implications formathematics education.
Australia is predominantly a migrant country.
More than one in four of Australia’s population
was born overseas.
They came from more than 150 countries.
In Japan the construct of ‘culture’ also hasstrong meanings, and therefore important
implications for numeracy education.
2

Three main constructs in mathematics
education:
curriculum, teaching, and learning, embedded
in social, cultural and political contexts.
Focus here on two main areas:
culturally and socially-based numeracies, andvalues in numeracy teaching.

3

2. The challenge of culturally-based
mathematical knowledge.
In the last 20 years, two main strands ofresearch have developed inmathematics education.
Strand 1. Ethnomathematics

4Cultural roots:
Western Mathematics is one form amongmany.
Wasan is another in ancient Japan.
Interactions between mathematics andlanguage:.
Languages and other symbol systems are
the carriers of mathematical ideas andvalues in different cultures andsocieties.
5Human interactions:
Focus on mathematical activities and practices insociety.
Roles of other people in mathematics education.
Values and beliefs:
Any mathematical activity involves values, beliefs
and personal choices.
Current curriculum structures show mathematics
as culture and value free.
6

Usually we are preparing an elite who will
study mathematics at university.
But this is inappropriate when
considering numeracy.
Majority of the population debarred fromdiscussion, critique, of the underlying
mathematical constructs used intoday’s highly mathematically formatted
and structured society (Skovsmose,1994).
7

Research needs: how to makemathematics curricula more culturally
and socially responsive?
A numeracy-based curriculum is one
way to emphasise the societal andcultural aspects of mathematics.

8

Strand 2. The idea of ‘context’ within which themathematical practice is situated.
“Situated mathematical practice” (Kirshner &Whitson, 1997; Zevenbergen, 2000).
Based on studies of mathematical activities
occurring outside school (e.g. Abreu et al.,
2002; Lave & Wenger, 1991)
These practices are the roots of numeracy
education.

9

The context of school is different fromthe contexts outside school wherenumeracy practices occur.
How can we ensure that the school
context doesn’t ignore the outsideschool numeracy contexts?

10How to develop a numeracy curriculumfor schools, to satisfy
1. the needs of a modern democraticsociety for having fully educated andpolitically contributing adults, andalso
2. the needs of the higher mathematicalinstitutions which still wantcompetent, and in some cases
brilliant, mathematicians?
113. Numeracies and mathematical theory
Numeracy is being defined in many differentways:¾ as a part of mathematics, often as simple orbasic arithmetic¾ as far more than just arithmetic¾ as linked to literacy rather than tomathematics¾ as mathematical literacy¾ as quantitative literacy12I believe we need a conceptualisation ofnumeracy which meets three criteria:
it recognises the existence of many
numeracies
it clarifies the links between
numeracies and mathematics, andit clarifies the educational task facing
those responsible for numeracy
education.
13

The cultural history of (Western)
Mathematics shows us that its essence
is its generality
whereasthe ethnomathematics literature
indicates that numeracies are all about
particular practices, and they also
include meanings, beliefs andconceptualisations

14

Both Mathematics and numeracies arepowerful forms of knowledge in theirown contexts.
The context of Mathematics is theabstract world of the theorist.
The context of numeracy is thepragmatic world of theethnomathematical practitioner.
15

So, the relationship between them isbetween theory and practices, withMathematics being the theory andnumeracies being the practices.
Mathematics explains, theorises, andclarifies the rationales underlying thosepractices, and also gets applied throughvarious developments of those practices.
This conceptualisation meets the threecriteria above.
16

The question “Why does any particular
numeracy practice work?” is the key toclarifying the educational task.
Learners bring many numeracy, andethnomathematical, practices to theireducation, mainly from their families
but also from their wider society
contacts – friends, media, other adults
etc.
17But the naive learner lacks anyunderstanding of the Mathematicaltheories that help to explain thosepractices.
Without the Mathematical theories theylack the tools to understand, analyseand critique those practices.

18Without these tools they becometrapped by those practices, just as
most adults currently are, without theunderstanding to question them, andperhaps even develop alternativepractices.

19

There are many ‘why’ questions that canbe asked of school students within thecurrent currulum, such as these simple
ones:
Why do the many different practices forcounting, addition, subtraction etc. allsucceed?
It is because of the underlying
mathematical structures of thosealgorithms.
20Why do two negative numbers when
multiplied together give a positivenumber?
It is because of the wish to keep therules of number theory consistent
when dealing with positive andnegative integers.
21

Why can you divide one fraction byanother by ‘turning it upside down andmultiplying’?
Because multiplying the numerator anddenominator of a fraction by the same
number doesn‘t change the value ofthat fraction.
22

Why does a three-legged stool always
balance, while a four-legged one
doesn’t?
Because 3 points determine a plane.

23

Asking these ‘why’ questions may seem
inappropriate to some, especially since
the demise of proof from the school
curriculum in many countries.
Proofs are the ultimate mathematicalexplanation. But it is not necessary or
appropriate to learn proofs of theorems
just for examinations.
24

Far more important is to understand the role ofthe process of ‘proving’ in the development
and education of the numerate andMathematically literate student.
‘Why’ questions can be asked at any level ofMathematics and numeracy practice, andwithin Mathematics itself the answers tothose questions will often be found byproving25

My conceptualisation:
Numeracies are culturally, socially, andhistorically determined, and practicallypowerful.
Mathematics has developed culturally ina different way, and in a different
context, as an extremely powerful wayof theorising, explaining, and extending
our knowledge of the world.
26

Some implications for numeracy/Mathematics
school curricula:
These curricula need to recognise, and to bebuilt around, the many numeracy practices
which the learners have learnt at home and in
the wider society.
Not all the curricular content can bedetermined centrally. Much must bedetermined locally within a national orregional framework.
27

These curricula need also to introduce
learners to the many numeracy practices
which powerful others in society practise andwhich can impinge on their lives in crucialways.
These numeracy practices need to beclarified, theorised, and critiqued in terms of
the Mathematical theories behind them.
This is to reverse the usual approach ofteaching the Mathematical theory followed by
its applications.
28

The particular curricular challenge istherefore how to create a sensible
framework which can structure bothnational and local
numeracy/Mathematical curricula.

29

4. The challenge of ‘hidden values’ innumeracy and Mathematics teaching.
Many people still think Mathematics isuniversal and (thus) culture-free, andtherefore also value-free.
If one asks “What values are you transmittingor promoting when you teachMathematics?”
one is often met with a blank look.
30

In parallel curriculum developments suchas Science and Technology in Society,values are taken as serious curricularcontent.
Values teaching and learning does occur
in the teaching and learning ofMathematics and numeracy, as it doesin any educational encounter.

31But values are never addressed
explicitly in mathematics classrooms,
which implies that any values learningwhich takes place there must happen
implicitly.
The values being learnt are notnecessarily the ones which teachers
would ideally hope for.
32

Why is there little explicit values
education in Mathematics classrooms,
and very little explicit addressing ofvalues in the general discourse about
Mathematics and numeracy education?

33

Why do Mathematics educators know solittle about values in this context?
Is it because much teaching of values isdone by modelling, by imitation, and byother implicit, rather than explicit,
methods which we fail to see andrecognise?

34

Values in Mathematics education appear
to have the role of cultural ‘hidden
persuaders’ (Bishop, 1990).
Could it be that it is the implicit nature ofvalues teaching in Mathematics which
has continued to feed the myth thatmathematics is value-free?
35

Papers and articles referring to our earlyresearch in this area can be found at
this web-site:
http://www.education.monash.edu.au/centres/
scienceMTE/vamppublications.html

36Values in Mathematics and Science
Education publications
Publications and useful links on values inmathematics education, and related topics
Related publications on values teaching in(mathematics) education
Publications and useful links on values inscience education, and related topics

37

There are two main kinds of values whichteachers seek to convey: the general
educational and the Mathematical.
Many examples of these general
educational values are explicitly statedin official documents in countries
which take the explicit teaching ofvalues seriously:

38

Indonesia's 'Panca Sila'
Malaysia's 'Nilai-Nilai Murni'
Singapore's 'Shared Values'
Thailand's recent educational reform
documents
BUT
How can we begin to understand the ideaof Mathematical values?

39Some particular values have developed
through the history of Western
Mathematics (Bishop, 1988) and they
may well apply to some numeracies
also.
My analysis gives the following six sets
of values, conceptualised as three
dimensions of the value ‘space’:40

1 Rationalismemphasises argument, reasoning, logicalthinking and explanations. Rationalism
concerns theory, and theorising.
2 Empiricism
emphasises ideas in Mathematics asfixed, and rather like objects, with asymbolised form. Mathematicalpractices are also seen to be fixed andimmutable.
41

3 Control
emphasises the power of Mathematicalknowledge through the mastery of rules,facts, procedures and established criteria. It
also promotes feelings of security in thisknowledge.
4 Progressemphasises the ways that Mathematical ideas
grow and develop, through alternative
theories and the questioning of existingideas. It also promotes the values ofindividual creativity.
42

5 Opennessemphasises the democratisation ofknowledge, through demonstrations,proving and individual explanations.
6 Mysteryemphasises the abstractness, andmystique of Mathematical ideas as wellas the dehumanised nature ofMathematical knowledge.
43

In this conceptual ‘space’ each value is pairedwith its complementary partner, as follows:
Rationalism with Empiricism, which is about
the ideology of Mathematical knowledge
Control with Progress, about the relationshipof Mathematical knowledge with society, andOpenness with Mystery, about how
individuals relate to Mathematical knowledge.
44

Seah’s (1999) research on values in textbooks
confirmed my earlier hypothesis (Bishop,1988) that the values being emphasised in
Mathematics classrooms were mainly those
associated with Empiricism, Control andMystery.
These three values are just those which wouldunderlie an approach which emphasises themastery of routines, resulting in secureknowledge, which is treated as fixed andpassed on from generation to generation.

45

What are missing are the complementary
partners ofRationalism, Progress and Openness,
which would enable learners toexperience the more theoretical,
progressive and democratising aspectsof a Mathematical education.
So how could this situation bechanged?
46

Generally, the research literature indicatesthat values education involves two essentialcomponents.
1. The explicit development of awareness ofcertain values, be they general educational
values or more Mathematical values as
described above, and2. the explicit awareness and development ofalternatives, choices and choosing,preferences, and consistency (see forexample, the advice of Raths et al.,1987).
47

So numeracy teaching must ensure:¾ the full range of values are beingreflected in the teaching, and¾ the numeracy and Mathematics
learning context is a place of explicit
choices and choosing for the learners.
48

Teachers of numeracy and Mathematics
should be presenting their learners withactivities that encourage them to makechoices, and to discuss and compare theirindividual choices about, for example,
the various approaches to solving problems,
the different criteria for judging the worth ofsolutions,
the Mathematical interpretations of numeracy
practices, andthe wider relevance of the numeracy
practices and the Mathematical models beingtaught.
49

For example,
a task such as ‘Describe and compare threedifferent proofs of the Pythagorean theorem’
would inevitably engage secondary school
students in discussing the values associatedwith proving.
The choosing of symbols with which torepresent Mathematical ideas and numeracy
practices.
50

Even the simple act of presenting different
problem-solving solutions to be comparedand contrasted by the learners stimulates theideas of choice, criteria and therefore values.
A general teaching strategy would be always
to take time, following some problem-solving
activity, to discuss what happened, why, andwhat has been learnt which could help withthe choices next time.

51

This is not a natural thing for learners todo, nor is it at present a typical thing
for teachers to encourage, but this iswhere teachers can focus learners’
direct attention on:
the awareness of values;
the clarification of values;
the conflict between different values,and eventhe critique of values.
52

6. Concluding remarks
Complex issues surround numeracyand Mathematics education.
There are many implications here forthe practice of numeracy education,
Also there is scope for increased
research on the whole area ofnumeracy education.
53

In most countries it is currently anunder-funded and under-supported
research area.
It should be better supported because
it is already clear that research into
ethnomathematics and other sociocultural aspects of numeracy andMathematics, are revealing the promiseof more democratic approaches toMathematics education.
54

This is a promise which all countries
desperately need fulfilling if they are todevelop a more critically aware
citizenry.
It is also what our global community
needs55