Bishop atsima 16

Culture and values in relationCulture and values in relation
to mathematics educationto mathematics education
Alan J. BishopAlan J. Bishop
Monash UniversityMonash University
MelbourneMelbourne
alaalaan.bishop@monash.eduan.bishop@monash.edu

Culturally-basedCulturally-based
mathematical knowledge:mathematical knowledge:
EthnomathematicsEthnomathematics
research:research:
 Mathematics in traditional societies:Mathematics in traditional societies:
e.g. Papua New Guinea, Africa,e.g. Papua New Guinea, Africa,
South America, AboriginalSouth America, Aboriginal
Australia…Australia…
 Different mathematical histories: e.g.Different mathematical histories: e.g.
India, China, Arabic…India, China, Arabic…
 Children’s outside school societalChildren’s outside school societal
knowledge…knowledge…

Ethnomathematics.Ethnomathematics.
 Human interactions.Human interactions.
Ethnomathematics concernsEthnomathematics concerns
mathematical activities and practicesmathematical activities and practices
in society.in society.
 It thereby draws attention to the rolesIt thereby draws attention to the roles
which people other than teacherswhich people other than teachers
and learners play in mathematicsand learners play in mathematics
education.education.

 Universal mathematical activities:Universal mathematical activities:
 CountingCounting
 LocatingLocating
 MeasuringMeasuring
 DesigningDesigning
 PlayingPlaying
 ExplainingExplaining

 Interactions between mathematicsInteractions between mathematics
and languages.and languages.
 Languages act as the principalLanguages act as the principal
carriers of mathematical ideas andcarriers of mathematical ideas and
values in different cultures.values in different cultures.

 Cultural roots. Ethnomathematics isCultural roots. Ethnomathematics is
making us more aware of the culturalmaking us more aware of the cultural
starting points and different historiesstarting points and different histories
of mathematical development.of mathematical development.

Values and mathematicsValues and mathematics
teachingteaching
 Mathematics educators are increasinglyMathematics educators are increasingly
being challenged about the goals to whichbeing challenged about the goals to which
mathematics education should aim:mathematics education should aim:
technology, societal demands, scientifictechnology, societal demands, scientific
development, economic growth….development, economic growth….
 To a large degree these goals are all aboutTo a large degree these goals are all about
the values we should be inculcating in ourthe values we should be inculcating in our
students.students.

teachingteaching
 Values are the deep affective qualitiesValues are the deep affective qualities
which education aims to foster through thewhich education aims to foster through the
teaching of mathematics.teaching of mathematics.
 Values are not the same as beliefs,Values are not the same as beliefs,
although the two constructs are related, andalthough the two constructs are related, and
there is much research on beliefs inthere is much research on beliefs in
mathematics education but little on values.mathematics education but little on values.

teachingteaching
 One way to relate them is to see values asOne way to relate them is to see values as
‘beliefs in action’, that is, one may hold‘beliefs in action’, that is, one may hold
several beliefs, but when one is faced withseveral beliefs, but when one is faced with
choices it is one’s values which determinechoices it is one’s values which determine
which choice one accepts.which choice one accepts.
 Beliefs are the support or justification forBeliefs are the support or justification for
one’s choices.one’s choices.

teachingteaching
 There is a widespread misunderstanding thatThere is a widespread misunderstanding that
mathematics is a value-free subject, and manymathematics is a value-free subject, and many
parents and policy makers (as well as someparents and policy makers (as well as some
educators) might initially be concerned abouteducators) might initially be concerned about
explicit values education in mathematics.explicit values education in mathematics.
 Whereas it is relatively easy and common in theWhereas it is relatively easy and common in the
teaching of humanities, arts subjects and perhapsteaching of humanities, arts subjects and perhaps
also the sciences to discuss the development ofalso the sciences to discuss the development of
values, this is not the case at present invalues, this is not the case at present in
mathematics teaching.mathematics teaching.

teachingteaching
 What parents, policy makers and othersWhat parents, policy makers and others
should be concerned about, however, is thatshould be concerned about, however, is that
values teaching and learning inevitably goesvalues teaching and learning inevitably goes
on in mathematics classrooms.on in mathematics classrooms.
 This is because whenever teaching takesThis is because whenever teaching takes
place, choices are made, which are basedplace, choices are made, which are based
on, and therefore reveal, certain values.on, and therefore reveal, certain values.

teachingteaching
 From a research perspective, there is only aFrom a research perspective, there is only a
limited understanding at present of whatlimited understanding at present of what
values are being transmitted, and of howvalues are being transmitted, and of how
effectively they are being transmitted.effectively they are being transmitted.
 Perhaps this is because most values appearPerhaps this is because most values appear
to be taught and learnt implicitly rather thanto be taught and learnt implicitly rather than
explicitly in mathematics classrooms.explicitly in mathematics classrooms.

teachingteaching
Therefore there are new research questionsTherefore there are new research questions
which need to be asked such as:which need to be asked such as:
 What values are teachers of mathematicsWhat values are teachers of mathematics
teaching?teaching?
 What are teachers’ understandings of theirWhat are teachers’ understandings of their
own intended values?own intended values?
 To what extent can mathematics teachersTo what extent can mathematics teachers
gain control over their own values teaching?gain control over their own values teaching?

teachingteaching
 What values are students learningWhat values are students learning
from their teachers?from their teachers?
 Are they learning more significantAre they learning more significant
values from their peers than from theirvalues from their peers than from their
teachers?teachers?

teachingteaching
 What values are implicitly andWhat values are implicitly and
explicitly being transmitted or ‘shaped’explicitly being transmitted or ‘shaped’
through curricula and textbooks?through curricula and textbooks?
 How do these values intersect withHow do these values intersect with
the values being transmitted throughthe values being transmitted through
other subjects such as science andother subjects such as science and
information technology?information technology?

teachingteaching
 Is it possible to develop more effectiveIs it possible to develop more effective
mathematics teaching through themathematics teaching through the
values education of teachers, and ofvalues education of teachers, and of
student teachers?student teachers?
 To what extent can teachers beTo what extent can teachers be
helped to teach other values thanhelped to teach other values than
those they currently teach?those they currently teach?

teachingteaching
 Mathematical valuesMathematical values: values which have developed: values which have developed
as the subject has developed within the particularas the subject has developed within the particular
culture.culture.
 General educational valuesGeneral educational values: values associated with: values associated with
the norms of the particular society, and of thethe norms of the particular society, and of the
particular educational institution.particular educational institution.
 Mathematics educational valuesMathematics educational values: values embedded: values embedded
in the curriculum, textbooks, classroom practices,in the curriculum, textbooks, classroom practices,
etc. as a result of the other sets of values.etc. as a result of the other sets of values.

Values and mathematics teachingValues and mathematics teaching
My research approach to these issues has been toMy research approach to these issues has been to
focus onfocus on mathematical valuesmathematical values, and on the actions, and on the actions
and choices concerning them:and choices concerning them:
Mathematical value competencesMathematical value competences
I have found White’s (1959) three component analysisI have found White’s (1959) three component analysis
and terminology very helpful:and terminology very helpful:
 Ideological values: ‘Rationalism’ and ‘Objectism’Ideological values: ‘Rationalism’ and ‘Objectism’
 Sentimental values: ‘Control’ and ‘Progress’Sentimental values: ‘Control’ and ‘Progress’
 Sociological values: ‘Openness’ and ‘Mystery’.Sociological values: ‘Openness’ and ‘Mystery’.

Mathematical value competences -Mathematical value competences -
IdeologicalIdeological
Valuing Rationalism meansValuing Rationalism means emphasisingemphasising
argument, reasoning, logical analysis,argument, reasoning, logical analysis,
and explanations.and explanations.
It concerns theory, and hypothetical andIt concerns theory, and hypothetical and
abstract situations, and promotesabstract situations, and promotes
universalist thinking.universalist thinking.

IdeologicalIdeological
Valuing Objectism meansValuing Objectism means emphasisingemphasising
objectifying, concretising, and applyingobjectifying, concretising, and applying
ideas in mathematics and science.ideas in mathematics and science.
It favours analogical thinking, symbolising, andIt favours analogical thinking, symbolising, and
the presentation and use of data.the presentation and use of data.
It also promotes materialism and determinism.It also promotes materialism and determinism.

SentimentalSentimental
Valuing Control meansValuing Control means emphasising theemphasising the
power of mathematical and scientificpower of mathematical and scientific
knowledge through mastery of rules,knowledge through mastery of rules,
facts, procedures and establishedfacts, procedures and established
criteria.criteria.
It also promotes security in knowledge,It also promotes security in knowledge,
and the ability to predict.and the ability to predict.

SentimentalSentimental
Valuing Progress meansValuing Progress means emphasising theemphasising the
ways that mathematical and scientific ideasways that mathematical and scientific ideas
grow and develop, through alternativegrow and develop, through alternative
theories, development of new methods andtheories, development of new methods and
the questioning of existing ideas.the questioning of existing ideas.
It also promotes the values of individual libertyIt also promotes the values of individual liberty
and creativity.and creativity.

SociologicalSociological
Valuing Openness meansValuing Openness means emphasisingemphasising
the democratisation of knowledge,the democratisation of knowledge,
through demonstrations, proofs andthrough demonstrations, proofs and
individual explanations.individual explanations.
Verification of hypotheses, clearVerification of hypotheses, clear
articulation and critical thinking arearticulation and critical thinking are
also significant.also significant.

SociologicalSociological
Valuing Mystery meansValuing Mystery means emphasisingemphasising thethe
wonder, fascination, and mystique ofwonder, fascination, and mystique of
scientific and mathematical ideas.scientific and mathematical ideas.
It promotes thinking about the origins andIt promotes thinking about the origins and
nature of knowledge and of the creativenature of knowledge and of the creative
process, as well as the abstractness andprocess, as well as the abstractness and
dehumanised nature of scientific anddehumanised nature of scientific and
mathematical knowledge.mathematical knowledge.

Findings from the VAMPFindings from the VAMP
researchresearch
 Teachers find it difficult to discussTeachers find it difficult to discuss
values and mathematics.values and mathematics.
 Teachers have many goals in planningTeachers have many goals in planning
for lessons.for lessons.
www.education.monash.edu.au/centres/sciencemte/www.education.monash.edu.au/centres/sciencemte/
vamp.htmlvamp.html

researchresearch
 The plans include the development ofThe plans include the development of
the individual child, both in acquiringthe individual child, both in acquiring
mathematics content and in gainingmathematics content and in gaining
confidence with doing mathematics,confidence with doing mathematics,
 as well as in showing howas well as in showing how
mathematics can be relevant to themmathematics can be relevant to them
personally as well as to society atpersonally as well as to society at
large.large.

researchresearch
 Mathematics teachers themselves holdMathematics teachers themselves hold
strong values about mathematics and aboutstrong values about mathematics and about
mathematics education.mathematics education.
 Teachers may choose to make explicitTeachers may choose to make explicit
certain mathematics or mathematicscertain mathematics or mathematics
education values or they may ‘show’ themeducation values or they may ‘show’ them
implicitly.implicitly.
 It is easier for teachers to think about andIt is easier for teachers to think about and
recognise the values they are teaching, thanrecognise the values they are teaching, than
to implement new values.to implement new values.

Teachers’ values and practices inTeachers’ values and practices in
mathematics and sciencemathematics and science
 13 primary and 17 secondary teachers13 primary and 17 secondary teachers
volunteered to answer our questionnaires.volunteered to answer our questionnaires.
 Questions 1 and 2 of the questionnaires askQuestions 1 and 2 of the questionnaires ask
for the extent to which particular activitiesfor the extent to which particular activities
are emphasised in practice in the teacher’sare emphasised in practice in the teacher’s
mathematics (and science) classes.mathematics (and science) classes.

Teachers’ values and practices inTeachers’ values and practices in
 Questions 3 and 4 are the questions whichQuestions 3 and 4 are the questions which
concern the teachers’ preferences for theconcern the teachers’ preferences for the
six value clusters described above.six value clusters described above.
 The structure of these two questions is thatThe structure of these two questions is that
each question contains 6 statements to beeach question contains 6 statements to be
ranked by the teachers.ranked by the teachers.

TeachersTeachers’’ values and practices invalues and practices in
 ForFor all the statements in Questions 1all the statements in Questions 1
and 2, we scored the responses as 4and 2, we scored the responses as 4
(for(for ““AlwaysAlways””), 3 (), 3 (““OftenOften””), 2), 2
((““SometimesSometimes””), 1 (), 1 (““RarelyRarely””), and we), and we
also calculated means. We recognisealso calculated means. We recognise
that in doing this we have taken anthat in doing this we have taken an
ordinal scale and treated it as if it wasordinal scale and treated it as if it was
a ratio scale.a ratio scale.

 To facilitate comprehension of the results,To facilitate comprehension of the results,
we have combined the data for Questions 1we have combined the data for Questions 1
and 2, and in the data reported below, forand 2, and in the data reported below, for
example, a teacherexample, a teacher’’s view of the frequencys view of the frequency
of emphasis on Rationalism in his/her classof emphasis on Rationalism in his/her class’’
activities is represented by the mean scoreactivities is represented by the mean score
for the six items relating to that value clusterfor the six items relating to that value cluster
in the two questions.in the two questions.

 Comparisons between the values inComparisons between the values in
mathematics and science for the twomathematics and science for the two
groups of teachers show interestinggroups of teachers show interesting
differences, reflecting their concernsdifferences, reflecting their concerns
with the curriculum and teaching atwith the curriculum and teaching at
their respective levels.their respective levels.
 ‘‘ObjectismObjectism’’ was converted towas converted to
‘‘EmpiricismEmpiricism’’ for this comparativefor this comparative
study.study.

 The primary teachers, concerning Ideology,The primary teachers, concerning Ideology,
prefer Empiricism over Rationalism for bothprefer Empiricism over Rationalism for both
science and mathematics. Their reportedscience and mathematics. Their reported
practices also show this.practices also show this.
 For the Sentimental (attitudinal) dimension,For the Sentimental (attitudinal) dimension,
Control is much less favoured than ProgressControl is much less favoured than Progress
for both subjects, but the practices are veryfor both subjects, but the practices are very
different.different.

 Another main difference is in theAnother main difference is in the
Sociological dimension where OpennessSociological dimension where Openness
and Mystery reverse their positions with theand Mystery reverse their positions with the
two subjects, the first being more favouredtwo subjects, the first being more favoured
than the second in mathematics and thethan the second in mathematics and the
reverse in science.reverse in science.
 However this difference does not translateHowever this difference does not translate
to the practices, with the science practicesto the practices, with the science practices
being ranked much more like thebeing ranked much more like the
mathematics practices.mathematics practices.

For the secondary teachers, concerning theFor the secondary teachers, concerning the
Ideological dimension, they favourIdeological dimension, they favour
Rationalism for mathematics andRationalism for mathematics and
Empiricism for science, disagreeing withEmpiricism for science, disagreeing with
their primary colleaguestheir primary colleagues
 However for the Sentimental dimension, theHowever for the Sentimental dimension, the
secondary teachers largely agree with theirsecondary teachers largely agree with their
primary colleaguesprimary colleagues

For the Sociological dimension, they againFor the Sociological dimension, they again
agree with their primary colleaguesagree with their primary colleagues
favouring Openness for mathematicsfavouring Openness for mathematics
compared with Mystery, and reversing thesecompared with Mystery, and reversing these
for science.for science.
 Indeed Mystery for science is ranked 2 andIndeed Mystery for science is ranked 2 and
4 by the secondary teachers and ranked 24 by the secondary teachers and ranked 2
and 3 by the primary teachers, showing howand 3 by the primary teachers, showing how
significant they consider that aspect to be.significant they consider that aspect to be.

Some conclusions andSome conclusions and
implicationsimplications
 Rationalism, Empiricism and Control areRationalism, Empiricism and Control are
strongly favoured in practice, but the otherstrongly favoured in practice, but the other
three values figure more prominently in thethree values figure more prominently in the
teachersteachers’’ preferences.preferences.
 However, we must remember that the dataHowever, we must remember that the data
are from questionnaires and consist ofare from questionnaires and consist of
teachersteachers’’ reported views of theirreported views of their
preferences and their practices. We do notpreferences and their practices. We do not
know the extent to which their rankings ofknow the extent to which their rankings of
these practice statements reflect their actualthese practice statements reflect their actual
practices.practices.

 But the data for science at the secondaryBut the data for science at the secondary
level, where teachers emphasises otherlevel, where teachers emphasises other
values than mathematics, indicates thevalues than mathematics, indicates the
usefulness of comparing subjects and theirusefulness of comparing subjects and their
values emphases.values emphases.
 Finally one can see that, if the data reportedFinally one can see that, if the data reported
here are valid, the differences show thathere are valid, the differences show that
teachersteachers’’ values in the classroomvalues in the classroom areare
shaped to some extent by the valuesshaped to some extent by the values
embedded in each subject, as perceived byembedded in each subject, as perceived by
them.them.

 This implies that changing teachersThis implies that changing teachers’’
perceptions and understandings of theperceptions and understandings of the
subject being taught may well change thesubject being taught may well change the
values they can emphasise in class.values they can emphasise in class.
 Further if teachers wish to emphasiseFurther if teachers wish to emphasise
values other than those they currentlyvalues other than those they currently
emphasise, it is possible to learn strategiesemphasise, it is possible to learn strategies
from their teaching of other subjects.from their teaching of other subjects.

Bishop atsima 16

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Bishop atsima 16