Teachers’ mathematical values for developing Mathematical thinking through Lesson Study

Teachers’ mathematical values Teachers’ mathematical values for developing Mathematical for developing Mathematical

thinking through Lesson Studythinking through Lesson StudyAlan J. BishopAlan J. Bishop

Monash UniversityMonash UniversityMelbourne, AustraliaMelbourne, Australia

<[email protected]><[email protected]>

Mathematical thinking from a Mathematical thinking from a sociocultural perspectivesociocultural perspective

Mathematical thinking sounds like a Mathematical thinking sounds like a psychological issue. psychological issue.

But we can never observe mathematical But we can never observe mathematical thinking.thinking.

We can observe what we assume to be its We can observe what we assume to be its products.products.

We can also observe what conditions and We can also observe what conditions and contexts might have been responsible for contexts might have been responsible for the products of mathematical thinking.the products of mathematical thinking.

Mathematics thinking from a Mathematics thinking from a sociocultural perspectivesociocultural perspective

Because I work in the field of education, I Because I work in the field of education, I prefer to consider mathematical thinking not prefer to consider mathematical thinking not from a psychological perspective but from a from a psychological perspective but from a socio-cultural perspective.socio-cultural perspective.

Three theoretical ideas show how we can Three theoretical ideas show how we can shape our understanding of this perspective, shape our understanding of this perspective, in relation to mathematical thinking. in relation to mathematical thinking.

Lancy’s developmental theory of Lancy’s developmental theory of cognition cognition

David Lancy (1983) in his major cross-David Lancy (1983) in his major cross-cultural study in Papua New Guinea cultural study in Papua New Guinea developed a stage theory:developed a stage theory:

Stage 1 Similar to Piaget’s sensori-motor Stage 1 Similar to Piaget’s sensori-motor and early concrete operational stagesand early concrete operational stages

Genetic programming has its major Genetic programming has its major influence, and socialisation is the key focus influence, and socialisation is the key focus of communication. of communication.

Lancy’s developmental theory of Lancy’s developmental theory of cognition cognition

Stage 2, a later concrete operational stage, Stage 2, a later concrete operational stage, is where enculturation takes over from is where enculturation takes over from socialisation.socialisation.

““Stage 2 has much to do with culture and Stage 2 has much to do with culture and environment and less to do with genetics”.environment and less to do with genetics”.

Different cultures will emphasise different Different cultures will emphasise different knowledge and ideas.knowledge and ideas.

Lancy’s developmental theory of Lancy’s developmental theory of cognitioncognition

Stage 3 concerns the metacognitive level.Stage 3 concerns the metacognitive level. He says:” In addition to developing cognitive and He says:” In addition to developing cognitive and

linguistic strategies, individuals acquire ‘theories’ linguistic strategies, individuals acquire ‘theories’ of language and cognition.”of language and cognition.”

Different cultural groups emphasise different Different cultural groups emphasise different ‘theories of knowledge’.‘theories of knowledge’.

Piaget’s ‘formal operational’ stage is one such Piaget’s ‘formal operational’ stage is one such theory of knowledge emphasised in Western theory of knowledge emphasised in Western culture. Confucian Heritage Cultures emphasise culture. Confucian Heritage Cultures emphasise other theories of knowledge.other theories of knowledge.

Lancy’s developmental theory of Lancy’s developmental theory of cognitioncognition

The theories of knowledge represent the The theories of knowledge represent the ideals and values lying behind the actuial ideals and values lying behind the actuial language or symbols developed by a language or symbols developed by a cultural group. cultural group.

Thus it is in Stages 2 and 3 that values are Thus it is in Stages 2 and 3 that values are inculcated in the individual learners.inculcated in the individual learners.

Billett’s (1998) analysis of the social Billett’s (1998) analysis of the social genesis of knowledge. genesis of knowledge.

Stephen Billett’s analysis categorises the Stephen Billett’s analysis categorises the social genesis of knowledge in 5 levelssocial genesis of knowledge in 5 levels

1.1. Socio-historic knowledgeSocio-historic knowledge 2.2. Socio-cultural practiceSocio-cultural practice 3.3. The community of practice in the The community of practice in the

classroomclassroom 4.4. Microgenetic developmentMicrogenetic development 5.5. Ontogenetic developmentOntogenetic development

Billett’s (1998) analysis of the social Billett’s (1998) analysis of the social genesis of knowledge.genesis of knowledge.

Socio-historic knowledgeSocio-historic knowledge is knowledge is knowledge coming from the history of the society.coming from the history of the society.

Socio-cultural practiceSocio-cultural practice is described by is described by Billett as historically derived knowledge Billett as historically derived knowledge transformed by cultural needs, together with transformed by cultural needs, together with goals, techniques, and norms to guide goals, techniques, and norms to guide practice.practice.


Of special interest to us is the Of special interest to us is the The The community of practice in the classroom.community of practice in the classroom.

Billett defines this category as “particular Billett defines this category as “particular sociocultural practices shaped by a complex sociocultural practices shaped by a complex of circumstantial social factors (activity of circumstantial social factors (activity systems), and the norms and values which systems), and the norms and values which embody them.”embody them.”


Microgenetic and Ontogenetic developmentMicrogenetic and Ontogenetic development

are concerned with individuals’ personal are concerned with individuals’ personal histories, and ‘moment by moment’ histories, and ‘moment by moment’ constructions of actions and values.constructions of actions and values.

Bishop’s socio-cultural dimension Bishop’s socio-cultural dimension and its levelsand its levels

My research context has been in the field of culture, My research context has been in the field of culture, and especially with considering mathematics as a and especially with considering mathematics as a form of cultural knowledge.form of cultural knowledge.

When we are considering how to develop values in When we are considering how to develop values in relation to mathematical thinking, I relation to mathematical thinking, I believe we believe we need to keep in mind the socio-cultural dimension need to keep in mind the socio-cultural dimension of mathematics education.of mathematics education.

This dimension influences the values of This dimension influences the values of mathematical thinking at five levels. mathematical thinking at five levels.

Bishop’s socio-cultural dimension and its Bishop’s socio-cultural dimension and its levelslevels

1.1. Cultural level – the overarching culture of the Cultural level – the overarching culture of the people, their language, their mathematics, their people, their language, their mathematics, their core valuescore values

2.2. Societal level – the social institutions of the Societal level – the social institutions of the society, their goals, and their values regarding society, their goals, and their values regarding mathematicsmathematics

3.3. Institutional level – the educational institutions’ Institutional level – the educational institutions’ values and the place of mathematics within them values and the place of mathematics within them

4.4. Pedagogical level – the teachers’ values and Pedagogical level – the teachers’ values and decisions, the classroom culture of mathematical decisions, the classroom culture of mathematical thinking thinking

5.5. Individual level – individual learners’ values and Individual level – individual learners’ values and goals regarding mathematics goals regarding mathematics

A synthesis of the three theoriesA synthesis of the three theories

In the rest of this talk, I will assume that my In the rest of this talk, I will assume that my ideas about values regarding mathematical ideas about values regarding mathematical thinking are:thinking are:

1.1. Concerned with developing metacognitionConcerned with developing metacognition

2.2. Located within the socio-cultural dimensionLocated within the socio-cultural dimension

3.3. Focused on the community of practice in Focused on the community of practice in the classroom.the classroom.

Values and mathematics educationValues and mathematics education

Firstly I realised that it was necessary to distinguish Firstly I realised that it was necessary to distinguish between three sets of values: between three sets of values:

Mathematical values: values which have developed Mathematical values: values which have developed as the subject has developed within our history and as the subject has developed within our history and cultureculture

General educational values: values associated with General educational values: values associated with the norms of the particular society, and of the the norms of the particular society, and of the particular educational institutionparticular educational institution

Mathematics educational values: values embedded Mathematics educational values: 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

My research approach to values and mathematical My research approach to values and mathematical thinking has been to focus on thinking has been to focus on mathematicalmathematical values, and on the actions and choices values, and on the actions and choices concerning them.concerning them.

I have used White’s (1959) three component I have used White’s (1959) three component analysis and terminology:analysis and terminology:

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 values – Mathematical values –

IdeologicalIdeological Valuing Rationalism means:Valuing Rationalism means:

emphasising argument, reasoning, logical emphasising argument, reasoning, logical analysis, and explanations. analysis, and explanations.

Do you encourage your students to argue in your classes? Do you have debates? Do you emphasise mathematical proving? Do you show the students examples of proofs from history (for example, different proofs of Pythagoras' theorem)?

Mathematical values – Mathematical values –

IdeologicalIdeological

Valuing Objectism means:Valuing Objectism means:emphasising objectifying, concretising, emphasising objectifying, concretising, symbolising, and applying the ideas of symbolising, and applying the ideas of mathematics.mathematics. Do you encourage your students to invent their own symbols and terminology before showing them the 'official' ones? Do you use geometric diagrams to illustrate algebraic relationships? Do you show them different numerals used by different cultural groups in history? Do you discuss the need for simplicity and conciseness in choosing symbols?

Mathematical values -Mathematical values -SentimentalSentimental

Valuing Control means:Valuing Control means:emphasising the power of mathematical and scientific emphasising the power of mathematical and scientific

knowledge through mastery of rules, facts, procedures knowledge through mastery of rules, facts, procedures and established criteria. and established criteria.

Do you emphasise not just 'right' answers, but also the Do you emphasise not just 'right' answers, but also the checking of answers, and the reasons for other answers checking of answers, and the reasons for other answers not being 'right'? Do you encourage the analysis and not being 'right'? Do you encourage the analysis and understanding of why routine calculations and algorithms understanding of why routine calculations and algorithms 'work'? Do you always show examples of how the 'work'? Do you always show examples of how the mathematical ideas you are teaching are used in society mathematical ideas you are teaching are used in society

Mathematical values -Mathematical values -SentimentalSentimental

Valuing Progress means:Valuing Progress means: emphasising the ways that mathematical and emphasising the ways that mathematical and

scientific ideas grow and develop, through scientific ideas grow and develop, through alternative theories, development of new alternative theories, development of new methods and the questioning of existing ideasmethods and the questioning of existing ideas

Do you emphasise alternative, and non-routine, Do you emphasise alternative, and non-routine, solution strategies together with their reasons? solution strategies together with their reasons? Do you encourage students to extend and Do you encourage students to extend and generalise ideas from particular examples? Do generalise ideas from particular examples? Do you stimulate them with stories of mathematical you stimulate them with stories of mathematical developments in history?developments in history?

Mathematical values -Mathematical values -SociologicalSociological

Valuing Openness means:Valuing Openness means: emphasising the democratisation of knowledge, emphasising the democratisation of knowledge,

through demonstrations, proofs and individual through demonstrations, proofs and individual explanations. explanations.

Do you encourage your students to defend and Do you encourage your students to defend and justify their answers publicly to the class? Do justify their answers publicly to the class? Do you encourage the creation of posters so that you encourage the creation of posters so that the students can display their ideas? Do you the students can display their ideas? Do you help them create student math newsletters, or help them create student math newsletters, or web-pages, where they can present their ideas? web-pages, where they can present their ideas?

Mathematical values -Mathematical values -SociologicalSociological

Valuing Mystery means:Valuing Mystery means:emphasisingemphasising the wonder, fascination, and the wonder, fascination, and

mystique of mathematical ideas. mystique of mathematical ideas.

Do you tell them any stories about Do you tell them any stories about mathematical puzzles in the past, about mathematical puzzles in the past, about for example the 'search' for negative for example the 'search' for negative numbers, or for zero? Do you stimulate numbers, or for zero? Do you stimulate their mathematical imagination with their mathematical imagination with pictures, artworks, images of infinity etc.?pictures, artworks, images of infinity etc.?

Values, mathematical thinking and Values, mathematical thinking and lesson studylesson study

Lesson study is an excellent method for Lesson study is an excellent method for studying the development of values in the studying the development of values in the classroom. classroom.

In our VAMP project we already used a In our VAMP project we already used a version of lesson study, but without trying to version of lesson study, but without trying to affect the teachers’ plans for their lessons.affect the teachers’ plans for their lessons.


1.1. The teachers told us before the lessons The teachers told us before the lessons what values they thought they were going what values they thought they were going to develop.to develop.

2.2. We observed and recorded the lessonsWe observed and recorded the lessons

3.3. We interviewed the teachers after the We interviewed the teachers after the lessons to have them explain what they lessons to have them explain what they thought they had achieved.thought they had achieved.


For a full lesson study of mathematical For a full lesson study of mathematical thinking values, it would be necessary to thinking values, it would be necessary to plan together with the teachers what values plan together with the teachers what values they would try to develop.they would try to develop.

The teaching ideas earlier would be very The teaching ideas earlier would be very appropriate for this.appropriate for this.

Have the experiment go over a group of Have the experiment go over a group of lessons. Values could hardly be developed lessons. Values could hardly be developed in one lesson. in one lesson.

Conclusions for researchConclusions for research

1. With any design and development research it is essential 1. With any design and development research it is essential to have good theories to support and structure the work.to have good theories to support and structure the work.

2. Mathematical thinking has been studied in many ways, 2. Mathematical thinking has been studied in many ways, but in relation to values it is important to consider it as an but in relation to values it is important to consider it as an aspect of meta-cognition.aspect of meta-cognition.

3. The context for the research should be the classroom, as 3. The context for the research should be the classroom, as it is there that the community of practice significantly it is there that the community of practice significantly influences the meta-cognitive aspects of mathematical influences the meta-cognitive aspects of mathematical thinking.thinking.

4. Equally important is the socio-cultural context as any 4. Equally important is the socio-cultural context as any educational values are embedded in the culture of the educational values are embedded in the culture of the society.society.

Conclusions for researchConclusions for research

5. Lesson study is an excellent research approach 5. Lesson study is an excellent research approach for studying any experimental educational for studying any experimental educational development.development.

6. It is particularly appropriate for studying values 6. It is particularly appropriate for studying values development.development.

7. However there need to be a series of lessons 7. However there need to be a series of lessons studied as values do not develop in the space of studied as values do not develop in the space of one lesson.one lesson.

8. Finally the teachers need special support in this 8. Finally the teachers need special support in this research, as values teaching involves the research, as values teaching involves the teacher’s pedagogical identity, which must be teacher’s pedagogical identity, which must be respected.respected.

Alan J. BishopAlan J. BishopMonash UniversityMonash University

Melbourne, AustraliaMelbourne, Australia<[email protected]><[email protected]>

Thank you for your attention!Thank you for your attention!

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Teachers’ mathematical values for developing Mathematical thinking through Lesson Study