the 15th icmi study on the professional education and development of teachers of mathematics ruhama...

The 15th ICMI Study on the The 15th ICMI Study on the Professional Education and Professional Education and Development of Teachers of Development of Teachers of

MathematicsMathematics

The 15th ICMI Study on the The 15th ICMI Study on the Professional Education and Professional Education and Development of Teachers of Development of Teachers of

MathematicsMathematics

Ruhama EvenRuhama Even

Weizmann Institute of ScienceWeizmann Institute of ScienceRuhama EvenRuhama Even

Weizmann Institute of ScienceWeizmann Institute of Science

ICMI StudiesICMI StudiesICMI StudiesICMI Studies

ICMI - International Commission on ICMI - International Commission on Mathematical Instruction.Mathematical Instruction.

The series of ICMI Studies was launched The series of ICMI Studies was launched in the mid-1980s. in the mid-1980s.

Aim - Aim - to investigate themes of particular to investigate themes of particular significance to contemporary mathematics significance to contemporary mathematics education, education,

to provide an up-to-date presentation and to provide an up-to-date presentation and analysis of the state-of-the-art concerning analysis of the state-of-the-art concerning a theme.a theme.

ICMI - International Commission on ICMI - International Commission on Mathematical Instruction.Mathematical Instruction.

The series of ICMI Studies was launched The series of ICMI Studies was launched in the mid-1980s. in the mid-1980s.

Aim - Aim - to investigate themes of particular to investigate themes of particular significance to contemporary mathematics significance to contemporary mathematics education, education,

to provide an up-to-date presentation and to provide an up-to-date presentation and analysis of the state-of-the-art concerning analysis of the state-of-the-art concerning a theme.a theme.

The 15th ICMI Study on the The 15th ICMI Study on the Professional Education and Professional Education and Development of Teachers of Development of Teachers of MathematicsMathematics


Three phases: Three phases: Dissemination of a Dissemination of a Discussion Discussion DocumentDocument announcing the Study and announcing the Study and inviting contributions.inviting contributions.

Study ConferenceStudy Conference, held in Brazil, 15-, held in Brazil, 15-21 May 2005. 21 May 2005.

Publication of the Publication of the Study VolumeStudy Volume – a – a Report of the Study’s achievements, Report of the Study’s achievements, products and results. products and results.







Study ConferenceStudy Conference, held in Brazil, , held in Brazil, 15-21 May 2005. 15-21 May 2005.



Study ConferenceStudy Conference, held in Brazil, , held in Brazil, 15-21 May 2005. 15-21 May 2005.


Scope and Focus of the Scope and Focus of the Study Study Scope and Focus of the Scope and Focus of the Study Study

Two main strands:Two main strands:

Strand IStrand I: Teacher Preparation : Teacher Preparation and the Early Years of and the Early Years of TeachingTeaching

Strand IIStrand II: Professional : Professional Learning for and in PracticeLearning for and in Practice

Two main strands:Two main strands:

Strand IStrand I: Teacher Preparation : Teacher Preparation and the Early Years of and the Early Years of TeachingTeaching

Strand IIStrand II: Professional : Professional Learning for and in PracticeLearning for and in Practice

Strand I: Teacher Strand I: Teacher Preparation and the Early Preparation and the Early Years of TeachingYears of Teaching

Strand I: Teacher Strand I: Teacher Preparation and the Early Preparation and the Early Years of TeachingYears of Teaching Structure of teacher preparationStructure of teacher preparation Recruitment and retentionRecruitment and retention Curriculum of teacher preparation:Curriculum of teacher preparation:

What is the nature of the What is the nature of the diversitydiversity that is that is most pressing within a particular context? most pressing within a particular context?

How are teachers prepared to know mathematics How are teachers prepared to know mathematics for teaching? for teaching?

The early years of teachingThe early years of teaching Most pressing problems of preparing Most pressing problems of preparing teachersteachers

History and change in teacher preparationHistory and change in teacher preparation

Structure of teacher preparationStructure of teacher preparation Recruitment and retentionRecruitment and retention Curriculum of teacher preparation:Curriculum of teacher preparation:

What is the nature of the What is the nature of the diversitydiversity that is that is most pressing within a particular context? most pressing within a particular context?

How are teachers prepared to know mathematics How are teachers prepared to know mathematics for teaching? for teaching?

The early years of teachingThe early years of teaching Most pressing problems of preparing Most pressing problems of preparing teachersteachers

History and change in teacher preparationHistory and change in teacher preparation

Strand II: Strand II: Professional Professional Learning for and in Learning for and in PracticePractice

Strand II: Strand II: Professional Professional Learning for and in Learning for and in PracticePractice The strand’s central focus is The strand’s central focus is rooted in two related and rooted in two related and persistent challenges of teacher persistent challenges of teacher education and professional education and professional development: development: The role of experience in learning The role of experience in learning to teach.to teach.

The divide between formal knowledge The divide between formal knowledge and practice. and practice.

The strand’s central focus is The strand’s central focus is rooted in two related and rooted in two related and persistent challenges of teacher persistent challenges of teacher education and professional education and professional development: development: The role of experience in learning The role of experience in learning to teach.to teach.

The divide between formal knowledge The divide between formal knowledge and practice. and practice.



The central question: The central question:

How can teachers learn for How can teachers learn for practice, in and from practice, in and from practice?practice?

The central question: The central question:



How can teachers learn for How can teachers learn for practice, in and from practice, in and from practice?practice? Most teachers report that they Most teachers report that they learned to teach “from experience” - learned to teach “from experience” - but experience is not always a good but experience is not always a good teacher. teacher.

Moreover, teacher education often Moreover, teacher education often seems remote from the work of seems remote from the work of teaching mathematics, and teaching mathematics, and professional development does not professional development does not necessarily draw on or connect to necessarily draw on or connect to teachers’ practice.teachers’ practice.

Most teachers report that they Most teachers report that they learned to teach “from experience” - learned to teach “from experience” - but experience is not always a good but experience is not always a good teacher. teacher.

Moreover, teacher education often Moreover, teacher education often seems remote from the work of seems remote from the work of teaching mathematics, and teaching mathematics, and professional development does not professional development does not necessarily draw on or connect to necessarily draw on or connect to teachers’ practice.teachers’ practice.

Strand II of the Study Strand II of the Study asks: asks: Strand II of the Study Strand II of the Study asks: asks:

How mathematics teachers’ learning may How mathematics teachers’ learning may be better structured to support be better structured to support learning in and from professional learning in and from professional practice - practice -

at the beginning of teachers’ at the beginning of teachers’ learning,learning,

during the early years of their work, during the early years of their work,

as they become more experienced. as they become more experienced.

How mathematics teachers’ learning may How mathematics teachers’ learning may be better structured to support be better structured to support learning in and from professional learning in and from professional practice - practice -

at the beginning of teachers’ at the beginning of teachers’ learning,learning,

during the early years of their work, during the early years of their work,

as they become more experienced. as they become more experienced.

Central questions includeCentral questions includeCentral questions includeCentral questions include

What sorts of learning seem to emerge from the What sorts of learning seem to emerge from the study of practice? study of practice?

In what ways are practices of teaching and In what ways are practices of teaching and learning mathematics made available for study? learning mathematics made available for study?

What kinds of collaboration are practiced in What kinds of collaboration are practiced in different countries? different countries?

What kinds of leadership help support teachers’ What kinds of leadership help support teachers’ learning from the practice of mathematics learning from the practice of mathematics teaching?teaching?

What are crucial practices of learning from What are crucial practices of learning from practice?practice?

How does language play a role in learning from How does language play a role in learning from practice? practice?

What sorts of learning seem to emerge from the What sorts of learning seem to emerge from the study of practice? study of practice?

In what ways are practices of teaching and In what ways are practices of teaching and learning mathematics made available for study? learning mathematics made available for study?

What kinds of collaboration are practiced in What kinds of collaboration are practiced in different countries? different countries?

What kinds of leadership help support teachers’ What kinds of leadership help support teachers’ learning from the practice of mathematics learning from the practice of mathematics teaching?teaching?

What are crucial practices of learning from What are crucial practices of learning from practice?practice?

How does language play a role in learning from How does language play a role in learning from practice? practice?

What sorts of learning seem What sorts of learning seem to emerge from the study of to emerge from the study of practice?practice?

What sorts of learning seem What sorts of learning seem to emerge from the study of to emerge from the study of practice?practice? What do teachers learn from different What do teachers learn from different opportunities to work on practice – their own, opportunities to work on practice – their own, or others’? or others’?

In what ways are teachers learning more about In what ways are teachers learning more about mathematics, about students' learning of mathematics, about students' learning of mathematics, and about the teaching of mathematics, and about the teaching of mathematics, as they work on experiences in mathematics, as they work on experiences in practice? practice?

What seems to support the learning of content? What seems to support the learning of content? In what ways are teachers learning about In what ways are teachers learning about diversity, about culture, and about ways to diversity, about culture, and about ways to address the important problems that derive from address the important problems that derive from social and cultural differences in particular social and cultural differences in particular settings?settings?

What do teachers learn from different What do teachers learn from different opportunities to work on practice – their own, opportunities to work on practice – their own, or others’? or others’?

In what ways are teachers learning more about In what ways are teachers learning more about mathematics, about students' learning of mathematics, about students' learning of mathematics, and about the teaching of mathematics, and about the teaching of mathematics, as they work on experiences in mathematics, as they work on experiences in practice? practice?

What seems to support the learning of content? What seems to support the learning of content? In what ways are teachers learning about In what ways are teachers learning about diversity, about culture, and about ways to diversity, about culture, and about ways to address the important problems that derive from address the important problems that derive from social and cultural differences in particular social and cultural differences in particular settings?settings?

In what ways are practices of In what ways are practices of teaching and learning teaching and learning mathematics made available for mathematics made available for study?study?

In what ways are practices of In what ways are practices of teaching and learning teaching and learning mathematics made available for mathematics made available for study?study? How is practice made visible and How is practice made visible and accessible for teachers to study accessible for teachers to study it alone or with others? it alone or with others?

How is "practice" captured or How is "practice" captured or engaged by teachers as they work engaged by teachers as they work on learning in and from practice? on learning in and from practice? (e.g., video, journals, lesson (e.g., video, journals, lesson study, joint research, observing study, joint research, observing one another and taking notes) one another and taking notes)

How is practice made visible and How is practice made visible and accessible for teachers to study accessible for teachers to study it alone or with others? it alone or with others?

How is "practice" captured or How is "practice" captured or engaged by teachers as they work engaged by teachers as they work on learning in and from practice? on learning in and from practice? (e.g., video, journals, lesson (e.g., video, journals, lesson study, joint research, observing study, joint research, observing one another and taking notes) one another and taking notes)

What kinds of collaboration What kinds of collaboration are practiced in different are practiced in different countries? countries?

What kinds of collaboration What kinds of collaboration are practiced in different are practiced in different countries? countries?

How are teachers organized in How are teachers organized in schools (e.g., in departments)?schools (e.g., in departments)?

What forms of professional What forms of professional interaction and joint work are interaction and joint work are engaged, supported, or used? engaged, supported, or used?

How are teachers organized in How are teachers organized in schools (e.g., in departments)?schools (e.g., in departments)?

What forms of professional What forms of professional interaction and joint work are interaction and joint work are engaged, supported, or used? engaged, supported, or used?

What kinds of leadership help What kinds of leadership help support teachers’ learning support teachers’ learning from the practice of from the practice of mathematics teaching?mathematics teaching?

What kinds of leadership help What kinds of leadership help support teachers’ learning support teachers’ learning from the practice of from the practice of mathematics teaching?mathematics teaching? Are there roles that help make Are there roles that help make the study of practice more the study of practice more productive? productive?

Who plays such roles, and what do Who plays such roles, and what do they do? they do?

What contribution do such people What contribution do such people make to teachers’ learning from make to teachers’ learning from practice?practice?

Are there roles that help make Are there roles that help make the study of practice more the study of practice more productive? productive?

Who plays such roles, and what do Who plays such roles, and what do they do? they do?

What contribution do such people What contribution do such people make to teachers’ learning from make to teachers’ learning from practice?practice?

What are crucial practices of What are crucial practices of learning from practice?learning from practice?What are crucial practices of What are crucial practices of learning from practice?learning from practice?

What are the skills and practices, the What are the skills and practices, the resources and the structures that support resources and the structures that support teachers’ examination of practice? teachers’ examination of practice?

How have ideas such as “reflection,” How have ideas such as “reflection,” “lesson study,” and analysis of student “lesson study,” and analysis of student work been developed in different work been developed in different settings? settings?

What do such ideas mean in actual What do such ideas mean in actual settings, and what do they involve in settings, and what do they involve in action? action?

What are the skills and practices, the What are the skills and practices, the resources and the structures that support resources and the structures that support teachers’ examination of practice? teachers’ examination of practice?

How have ideas such as “reflection,” How have ideas such as “reflection,” “lesson study,” and analysis of student “lesson study,” and analysis of student work been developed in different work been developed in different settings? settings?

What do such ideas mean in actual What do such ideas mean in actual settings, and what do they involve in settings, and what do they involve in action? action?

How does language play a role How does language play a role in learning from practice?in learning from practice?How does language play a role How does language play a role in learning from practice?in learning from practice?

What sort of language for What sort of language for discussing teaching and learning discussing teaching and learning mathematics –– professional mathematics –– professional language –– is developed among language –– is developed among teachers as they work on teachers as they work on practice?practice?

What sort of language for What sort of language for discussing teaching and learning discussing teaching and learning mathematics –– professional mathematics –– professional language –– is developed among language –– is developed among teachers as they work on teachers as they work on practice?practice?

Example 1: Research in the Example 1: Research in the fieldfieldExample 1: Research in the Example 1: Research in the fieldfield

From Adler and Jaworski’s plenary on From Adler and Jaworski’s plenary on “The state of research in mathematics “The state of research in mathematics teacher education and how it needs to teacher education and how it needs to develop”develop”

Jill Adler (South Africa) - Chair of Jill Adler (South Africa) - Chair of the ICME-10 Survey of research on the ICME-10 Survey of research on mathematics teacher education and mathematics teacher education and developmentdevelopment

Barbara Jaworski (England/Norway) - Barbara Jaworski (England/Norway) - Editor in Chief of the Journal of Editor in Chief of the Journal of Mathematics Teacher Education Mathematics Teacher Education

From Adler and Jaworski’s plenary on From Adler and Jaworski’s plenary on “The state of research in mathematics “The state of research in mathematics teacher education and how it needs to teacher education and how it needs to develop”develop”

Jill Adler (South Africa) - Chair of Jill Adler (South Africa) - Chair of the ICME-10 Survey of research on the ICME-10 Survey of research on mathematics teacher education and mathematics teacher education and developmentdevelopment

Barbara Jaworski (England/Norway) - Barbara Jaworski (England/Norway) - Editor in Chief of the Journal of Editor in Chief of the Journal of Mathematics Teacher Education Mathematics Teacher Education

ICME-10 Survey of research on ICME-10 Survey of research on mathematics teacher education and mathematics teacher education and developmentdevelopment


282 publications were surveyed 282 publications were surveyed (1999-2003)(1999-2003)

282 publications were surveyed 282 publications were surveyed (1999-2003)(1999-2003)



Three major claims:Three major claims: Small-scale qualitative research Small-scale qualitative research predominates; little is large scale predominates; little is large scale or quantitative. or quantitative.

Most teacher education research is Most teacher education research is conducted by teacher educators conducted by teacher educators studying the teachers with whom studying the teachers with whom they are working (researching own they are working (researching own practices or programs).practices or programs).

Research in countries where English Research in countries where English is the national language dominates is the national language dominates the literature. the literature.

Three major claims:Three major claims: Small-scale qualitative research Small-scale qualitative research predominates; little is large scale predominates; little is large scale or quantitative. or quantitative.

Most teacher education research is Most teacher education research is conducted by teacher educators conducted by teacher educators studying the teachers with whom studying the teachers with whom they are working (researching own they are working (researching own practices or programs).practices or programs).

Research in countries where English Research in countries where English is the national language dominates is the national language dominates the literature. the literature.

How the field needs to develop How the field needs to develop (Adler & Jaworski)(Adler & Jaworski)How the field needs to develop How the field needs to develop (Adler & Jaworski)(Adler & Jaworski)

We have to expand the field in terms We have to expand the field in terms of:of:

Large-scale studies Large-scale studies

Longitudinal studiesLongitudinal studies

Policy studiesPolicy studies

Studies across diverse contexts Studies across diverse contexts (cross national, language and (cross national, language and culture)culture)

Methodology and theoryMethodology and theory

We have to expand the field in terms We have to expand the field in terms of:of:

Large-scale studies Large-scale studies

Longitudinal studiesLongitudinal studies

Policy studiesPolicy studies

Studies across diverse contexts Studies across diverse contexts (cross national, language and (cross national, language and culture)culture)

Methodology and theoryMethodology and theory

How the field needs to develop How the field needs to develop (cont.)(cont.)How the field needs to develop How the field needs to develop (cont.)(cont.)

Critical stance within teacher Critical stance within teacher educator researcher practice:educator researcher practice:

TEs need awareness that they develop TEs need awareness that they develop practice as much as teachers do practice as much as teachers do

We need research into the assumptions We need research into the assumptions on which TE practice is builton which TE practice is built

We need to investigate relationships We need to investigate relationships between teacher education, teacher between teacher education, teacher learning, and pupils’ learninglearning, and pupils’ learning

Critical stance within teacher Critical stance within teacher educator researcher practice:educator researcher practice:

TEs need awareness that they develop TEs need awareness that they develop practice as much as teachers do practice as much as teachers do

We need research into the assumptions We need research into the assumptions on which TE practice is builton which TE practice is built

We need to investigate relationships We need to investigate relationships between teacher education, teacher between teacher education, teacher learning, and pupils’ learninglearning, and pupils’ learning

Example 2: Examination of a Example 2: Examination of a specific TE activityspecific TE activityExample 2: Examination of a Example 2: Examination of a specific TE activityspecific TE activity

From Ruhama Even’s From Ruhama Even’s presentation on “Developing presentation on “Developing and integrating knowledge and and integrating knowledge and practice: examination of two practice: examination of two activities”activities”

From Ruhama Even’s From Ruhama Even’s presentation on “Developing presentation on “Developing and integrating knowledge and and integrating knowledge and practice: examination of two practice: examination of two activities”activities”

Nature and quality of Nature and quality of school mathematics tasksschool mathematics tasksNature and quality of Nature and quality of school mathematics tasksschool mathematics tasks

Multi-stage activity: Multi-stage activity:

What is a good problem in What is a good problem in school mathematics?school mathematics?

Multi-stage activity: Multi-stage activity:

What is a good problem in What is a good problem in school mathematics?school mathematics?

Multi-stage activityMulti-stage activityMulti-stage activityMulti-stage activity

Acquaintance with theoretical background.

Answering the question: “What is a good problem in school mathematics?”

Examination of a special case.

Re-visiting the question: “What is a good problem in school mathematics?”

Conduction of a classroom activity.

Reflective summary.

Acquaintance with theoretical background.

Answering the question: “What is a good problem in school mathematics?”

Examination of a special case.

Re-visiting the question: “What is a good problem in school mathematics?”

Conduction of a classroom activity.

Reflective summary.

What is a good problem in What is a good problem in school mathematics? - Round school mathematics? - Round 11

What is a good problem in What is a good problem in school mathematics? - Round school mathematics? - Round 11A good problem should - A good problem should - be clearly phrasedbe clearly phrased stimulate thoughtstimulate thought be interestingbe interesting be connected to everyday lifebe connected to everyday life not require previously-learned not require previously-learned knowledgeknowledge

cause conflictcause conflict not be frustratingnot be frustrating

A good problem should - A good problem should - be clearly phrasedbe clearly phrased stimulate thoughtstimulate thought be interestingbe interesting be connected to everyday lifebe connected to everyday life not require previously-learned not require previously-learned knowledgeknowledge

cause conflictcause conflict not be frustratingnot be frustrating

Examination of a special Examination of a special casecaseExamination of a special Examination of a special casecase

What is the least number What is the least number of squares that must be of squares that must be added? added?

What is the greatest What is the greatest number of squares that number of squares that can be added? can be added?

What if we allow the What if we allow the dimensions to be any real dimensions to be any real number?number?

What would be a What would be a convincing argument for a convincing argument for a junior-high school junior-high school student?student?

What is the least number What is the least number of squares that must be of squares that must be added? added?

What is the greatest What is the greatest number of squares that number of squares that can be added? can be added?

What if we allow the What if we allow the dimensions to be any real dimensions to be any real number?number?

What would be a What would be a convincing argument for a convincing argument for a junior-high school junior-high school student?student?

Add squares Add squares so that the so that the figure would figure would have a have a perimeter of perimeter of 18.18.

Perimeter of 18 problemPerimeter of 18 problemPerimeter of 18 problemPerimeter of 18 problem

If the area is increased, the perimeter If the area is increased, the perimeter might remain unchanged or even decrease. might remain unchanged or even decrease.

By adding one area unit, the perimeter By adding one area unit, the perimeter could change by either +2, 0 , -2, or -4 could change by either +2, 0 , -2, or -4 units.units.

If the small squares’ side is a whole If the small squares’ side is a whole number, the 4x5 rectangle has the greatest number, the 4x5 rectangle has the greatest area.area.

If the small squares’ side is a real If the small squares’ side is a real number, the 4.5x4.5 square has the number, the 4.5x4.5 square has the greatest area.greatest area.

If the area is increased, the perimeter If the area is increased, the perimeter might remain unchanged or even decrease. might remain unchanged or even decrease.

By adding one area unit, the perimeter By adding one area unit, the perimeter could change by either +2, 0 , -2, or -4 could change by either +2, 0 , -2, or -4 units.units.

If the small squares’ side is a whole If the small squares’ side is a whole number, the 4x5 rectangle has the greatest number, the 4x5 rectangle has the greatest area.area.

If the small squares’ side is a real If the small squares’ side is a real number, the 4.5x4.5 square has the number, the 4.5x4.5 square has the greatest area.greatest area.

What is a good problem in What is a good problem in school mathematics? - Round school mathematics? - Round 22

What is a good problem in What is a good problem in school mathematics? - Round school mathematics? - Round 22 Problem situation: meaningful context.Problem situation: meaningful context. Time spent on a problem: investigation Time spent on a problem: investigation of a "big problem".of a "big problem".

Different solutions.Different solutions. Mathematical connections among Mathematical connections among different representations and different different representations and different domains.domains.

Different levels of presentation and Different levels of presentation and solution are possible. solution are possible.

......

Problem situation: meaningful context.Problem situation: meaningful context. Time spent on a problem: investigation Time spent on a problem: investigation of a "big problem".of a "big problem".

Different solutions.Different solutions. Mathematical connections among Mathematical connections among different representations and different different representations and different domains.domains.

Different levels of presentation and Different levels of presentation and solution are possible. solution are possible.

......

Conduction of a class Conduction of a class activityactivityConduction of a class Conduction of a class activityactivity

DifficultiesDifficulties

Time limitation (traditional, Time limitation (traditional, insecurity).insecurity).

No good problems available No good problems available (criteria checklist, ready-made (criteria checklist, ready-made problem illegitimate, materials problem illegitimate, materials limitation, inability to see limitation, inability to see through the problem into the through the problem into the activity).activity).

DifficultiesDifficulties

Time limitation (traditional, Time limitation (traditional, insecurity).insecurity).

No good problems available No good problems available (criteria checklist, ready-made (criteria checklist, ready-made problem illegitimate, materials problem illegitimate, materials limitation, inability to see limitation, inability to see through the problem into the through the problem into the activity).activity).

Reflective summaryReflective summaryReflective summaryReflective summary

Helping students develop as problem Helping students develop as problem solvers:solvers:

"Work on this problem encouraged "Work on this problem encouraged students to ask additional students to ask additional questions and to expand their questions and to expand their investigation. They also became investigation. They also became motivated to deal with problem motivated to deal with problem situations."situations."

Helping students develop as problem Helping students develop as problem solvers:solvers:

"Work on this problem encouraged "Work on this problem encouraged students to ask additional students to ask additional questions and to expand their questions and to expand their investigation. They also became investigation. They also became motivated to deal with problem motivated to deal with problem situations."situations."


Exploration is important:Exploration is important:

““Students did not have a prescribed Students did not have a prescribed algorithm for solving the problem. algorithm for solving the problem. Therefore, they did not know how to Therefore, they did not know how to solve the problem at first glance. solve the problem at first glance. They really needed to search and They really needed to search and investigate, to look for patterns investigate, to look for patterns and generalize, to hypothesize and and generalize, to hypothesize and then justify their hypotheses or then justify their hypotheses or refute them, and to check their refute them, and to check their answers."answers."


““Students did not have a prescribed Students did not have a prescribed algorithm for solving the problem. algorithm for solving the problem. Therefore, they did not know how to Therefore, they did not know how to solve the problem at first glance. solve the problem at first glance. They really needed to search and They really needed to search and investigate, to look for patterns investigate, to look for patterns and generalize, to hypothesize and and generalize, to hypothesize and then justify their hypotheses or then justify their hypotheses or refute them, and to check their refute them, and to check their answers."answers."



Still, there were some who focused Still, there were some who focused only on the exploratory aspect of only on the exploratory aspect of the activity and did not pay much the activity and did not pay much attention to the mathematics attention to the mathematics involved.involved.


Still, there were some who focused Still, there were some who focused only on the exploratory aspect of only on the exploratory aspect of the activity and did not pay much the activity and did not pay much attention to the mathematics attention to the mathematics involved.involved.


Attention to student diversity:Attention to student diversity: ““There are many possible solutions to the There are many possible solutions to the problem which vary by approaches and levels. problem which vary by approaches and levels. This makes the problem suitable for a large This makes the problem suitable for a large range of students. Some of these solutions range of students. Some of these solutions require only limited knowledge of mathematics require only limited knowledge of mathematics (e.g., several students found the sum by (e.g., several students found the sum by actually adding the numbers on a calculator); actually adding the numbers on a calculator); other solutions involve deep and broad other solutions involve deep and broad knowledge (e.g., making connections to knowledge (e.g., making connections to algebra and geometry or translating from one algebra and geometry or translating from one representation to another.”representation to another.”

Attention to student diversity:Attention to student diversity: ““There are many possible solutions to the There are many possible solutions to the problem which vary by approaches and levels. problem which vary by approaches and levels. This makes the problem suitable for a large This makes the problem suitable for a large range of students. Some of these solutions range of students. Some of these solutions require only limited knowledge of mathematics require only limited knowledge of mathematics (e.g., several students found the sum by (e.g., several students found the sum by actually adding the numbers on a calculator); actually adding the numbers on a calculator); other solutions involve deep and broad other solutions involve deep and broad knowledge (e.g., making connections to knowledge (e.g., making connections to algebra and geometry or translating from one algebra and geometry or translating from one representation to another.”representation to another.”


Not topic specific:Not topic specific:

““The content of the problem is broad and The content of the problem is broad and is not narrowly tied to the mathematical is not narrowly tied to the mathematical topic currently dealt with in class. The topic currently dealt with in class. The problem can be connected to various problem can be connected to various topics and approached in various topics and approached in various representations.”representations.”

Could represent a mature view of Could represent a mature view of mathematics, in contrast to viewing mathematics, in contrast to viewing mathematics as a collection of unrelated mathematics as a collection of unrelated topics, concepts, and procedures.topics, concepts, and procedures.

Not topic specific:Not topic specific:

““The content of the problem is broad and The content of the problem is broad and is not narrowly tied to the mathematical is not narrowly tied to the mathematical topic currently dealt with in class. The topic currently dealt with in class. The problem can be connected to various problem can be connected to various topics and approached in various topics and approached in various representations.”representations.”

Could represent a mature view of Could represent a mature view of mathematics, in contrast to viewing mathematics, in contrast to viewing mathematics as a collection of unrelated mathematics as a collection of unrelated topics, concepts, and procedures.topics, concepts, and procedures.


Not topic specific and no need for Not topic specific and no need for previously-learned knowledge:previously-learned knowledge:

May indicate that a ‘good problem’ was May indicate that a ‘good problem’ was still not regarded as an integral part of still not regarded as an integral part of the mathematics curriculum.the mathematics curriculum.

A different approach: Students worked on a A different approach: Students worked on a standard trig problem using any tools they standard trig problem using any tools they wanted. All solutions were based on rather wanted. All solutions were based on rather advanced previously-learned mathematics advanced previously-learned mathematics knowledge, but not necessarily related to knowledge, but not necessarily related to last week's instruction.last week's instruction.

Not topic specific and no need for Not topic specific and no need for previously-learned knowledge:previously-learned knowledge:

May indicate that a ‘good problem’ was May indicate that a ‘good problem’ was still not regarded as an integral part of still not regarded as an integral part of the mathematics curriculum.the mathematics curriculum.

A different approach: Students worked on a A different approach: Students worked on a standard trig problem using any tools they standard trig problem using any tools they wanted. All solutions were based on rather wanted. All solutions were based on rather advanced previously-learned mathematics advanced previously-learned mathematics knowledge, but not necessarily related to knowledge, but not necessarily related to last week's instruction.last week's instruction.


"'What is a good problem?' "'What is a good problem?' developed my awareness of developed my awareness of what I am teaching. It also what I am teaching. It also developed my ability and my developed my ability and my knowledge to transform a knowledge to transform a plain problem into a good plain problem into a good one."one."

"'What is a good problem?' "'What is a good problem?' developed my awareness of developed my awareness of what I am teaching. It also what I am teaching. It also developed my ability and my developed my ability and my knowledge to transform a knowledge to transform a plain problem into a good plain problem into a good one."one."

How and what can How and what can teachersteachers learn for practice, in and learn for practice, in and from practice?from practice?

How and what can How and what can teachersteachers learn for practice, in and learn for practice, in and from practice?from practice?The activity “What is a good problem?” The activity “What is a good problem?” - -

develops richer, more empowering vision develops richer, more empowering vision about good mathematical activities about good mathematical activities

enables the enactment of this new knowledge enables the enactment of this new knowledge and its continuous development in practiceand its continuous development in practice

offers the opportunity to work on solving offers the opportunity to work on solving an authentic problem of teaching an authentic problem of teaching mathematics, and to study closely an mathematics, and to study closely an important teaching practice important teaching practice

makes what was previously assumed and taken makes what was previously assumed and taken for granted, questionable and examinablefor granted, questionable and examinable

The activity “What is a good problem?” The activity “What is a good problem?” - -

develops richer, more empowering vision develops richer, more empowering vision about good mathematical activities about good mathematical activities

enables the enactment of this new knowledge enables the enactment of this new knowledge and its continuous development in practiceand its continuous development in practice

offers the opportunity to work on solving offers the opportunity to work on solving an authentic problem of teaching an authentic problem of teaching mathematics, and to study closely an mathematics, and to study closely an important teaching practice important teaching practice

makes what was previously assumed and taken makes what was previously assumed and taken for granted, questionable and examinablefor granted, questionable and examinable

How and what can How and what can teacher teacher educatorseducators learn for practice, learn for practice, in and from practice?in and from practice?

How and what can How and what can teacher teacher educatorseducators learn for practice, learn for practice, in and from practice?in and from practice? Knowledge developmentKnowledge development

Practice developmentPractice development

Know-tice (integration of Know-tice (integration of knowledge and practice) knowledge and practice) developmentdevelopment

Theory developmentTheory development

Knowledge developmentKnowledge development

Practice developmentPractice development

Know-tice (integration of Know-tice (integration of knowledge and practice) knowledge and practice) developmentdevelopment

Theory developmentTheory development





Publication of the Publication of the Study VolumeStudy Volume – a – a Report of the Study’s achievements, Report of the Study’s achievements, products and results.products and results.



Publication of the Publication of the Study VolumeStudy Volume – a – a Report of the Study’s achievements, Report of the Study’s achievements, products and results.products and results.

Structure of the Study VolumeStructure of the Study VolumeStructure of the Study VolumeStructure of the Study Volume

Setting the stage for the StudySetting the stage for the Study Initial teacher education (Strand I)Initial teacher education (Strand I) Learning in and from practice (Strand II)Learning in and from practice (Strand II) Resources for teacher development, professional Resources for teacher development, professional discussions, or interactions with policymakersdiscussions, or interactions with policymakers

Key issues for research in the education and Key issues for research in the education and professional development of teachers of professional development of teachers of mathematicsmathematics

Concluding commentaryConcluding commentary CD to represent ideas that would be more CD to represent ideas that would be more interactiveinteractive

Setting the stage for the StudySetting the stage for the Study Initial teacher education (Strand I)Initial teacher education (Strand I) Learning in and from practice (Strand II)Learning in and from practice (Strand II) Resources for teacher development, professional Resources for teacher development, professional discussions, or interactions with policymakersdiscussions, or interactions with policymakers

Key issues for research in the education and Key issues for research in the education and professional development of teachers of professional development of teachers of mathematicsmathematics

Concluding commentaryConcluding commentary CD to represent ideas that would be more CD to represent ideas that would be more interactiveinteractive

International Programme International Programme

CommitteeCommittee International Programme International Programme

CommitteeCommittee Deborah Loewenberg Ball (Co-Chair IPC), USADeborah Loewenberg Ball (Co-Chair IPC), USARuhama Even (Co-Chair IPC), ISRAELRuhama Even (Co-Chair IPC), ISRAEL

Jo Boaler, USAJo Boaler, USAChris Breen, SOUTH AFRICAChris Breen, SOUTH AFRICAFrédéric Gourdeau, CANADAFrédéric Gourdeau, CANADAMarja van den Heuvel-Panhuizen, NETHERLANDSMarja van den Heuvel-Panhuizen, NETHERLANDSBarbara Jaworski, NORWAYBarbara Jaworski, NORWAYGilah Leder, AUSTRALIAGilah Leder, AUSTRALIAShiqi Li, CHINAShiqi Li, CHINARomulo Lins (Chair of the Local Organising Committee), Romulo Lins (Chair of the Local Organising Committee),

BRAZILBRAZILJoão Filipe Matos, PORTUGALJoão Filipe Matos, PORTUGALHiroshi Murata, JAPANHiroshi Murata, JAPANJarmila Novotna, CZECH REPUBLICJarmila Novotna, CZECH REPUBLICAline Robert, FRANCEAline Robert, FRANCEBernard R. Hodgson, Secretary-General of ICMI, CANADABernard R. Hodgson, Secretary-General of ICMI, CANADAHyman Bass, President of ICMI, USAHyman Bass, President of ICMI, USA

Deborah Loewenberg Ball (Co-Chair IPC), USADeborah Loewenberg Ball (Co-Chair IPC), USARuhama Even (Co-Chair IPC), ISRAELRuhama Even (Co-Chair IPC), ISRAEL

Jo Boaler, USAJo Boaler, USAChris Breen, SOUTH AFRICAChris Breen, SOUTH AFRICAFrédéric Gourdeau, CANADAFrédéric Gourdeau, CANADAMarja van den Heuvel-Panhuizen, NETHERLANDSMarja van den Heuvel-Panhuizen, NETHERLANDSBarbara Jaworski, NORWAYBarbara Jaworski, NORWAYGilah Leder, AUSTRALIAGilah Leder, AUSTRALIAShiqi Li, CHINAShiqi Li, CHINARomulo Lins (Chair of the Local Organising Committee), Romulo Lins (Chair of the Local Organising Committee),

BRAZILBRAZILJoão Filipe Matos, PORTUGALJoão Filipe Matos, PORTUGALHiroshi Murata, JAPANHiroshi Murata, JAPANJarmila Novotna, CZECH REPUBLICJarmila Novotna, CZECH REPUBLICAline Robert, FRANCEAline Robert, FRANCEBernard R. Hodgson, Secretary-General of ICMI, CANADABernard R. Hodgson, Secretary-General of ICMI, CANADAHyman Bass, President of ICMI, USAHyman Bass, President of ICMI, USA

The Study WebsiteThe Study WebsiteThe Study WebsiteThe Study Website

http://www-personal.umich.edu/~dball/http://www-personal.umich.edu/~dball/icmistudy15.htmlicmistudy15.html

http://www-personal.umich.edu/~dball/http://www-personal.umich.edu/~dball/icmistudy15.htmlicmistudy15.html

the 15th icmi study on the professional education and development of teachers of mathematics ruhama...

Documents

icmi study

study volume

study conference

study s achievements

teacher preparation

early years of teaching

professional education

development of teachers