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

The 15th ICMI Study on the Professional Education and Development of Teachers of Mathematics Ruhama Even Weizmann Institute of Science

ICMI Studies • ICMI - International Commission on Mathematical Instruction. • The series of ICMI Studies was launched in the mid-1980s. • Aim - • to investigate themes of particular significance to contemporary mathematics education, • to provide an up-to-date presentation and analysis of the state-of-the-art concerning a theme.

The 15th ICMI Study on the Professional Education and Development of Teachers of Mathematics Three phases: • Dissemination of a Discussion Document announcing the Study and inviting contributions. • Study Conference, held in Brazil, 15-21 May 2005. • Publication of the Study Volume– a Report of the Study’s achievements, products and results.

Scope and Focus of the Study Two main strands: • Strand I: Teacher Preparation and the Early Years of Teaching • Strand II: Professional Learning for and in Practice

Strand I: Teacher Preparation and the Early Years of Teaching • Structure of teacher preparation • Recruitment and retention • Curriculum of teacher preparation: • What is the nature of the diversity that is most pressing within a particular context? • How are teachers prepared to know mathematics for teaching? • The early years of teaching • Most pressing problems of preparing teachers • History and change in teacher preparation

Strand II: Professional Learning for and in Practice • The strand’s central focus is rooted in two related and persistent challenges of teacher education and professional development: • The role of experience in learning to teach. • The divide between formal knowledge and practice.

Strand II: Professional Learning for and in Practice The central question: • How can teachers learn for practice, in and from practice?

How can teachers learn for practice, in and from practice? • Most teachers report that they learned to teach “from experience” - but experience is not always a good teacher. • Moreover, teacher education often seems remote from the work of teaching mathematics, and professional development does not necessarily draw on or connect to teachers’ practice.

Strand II of the Study asks: How mathematics teachers’ learning may be better structured to support learning in and from professional practice - • at the beginning of teachers’ learning, • during the early years of their work, • as they become more experienced.

Central questions include • What sorts of learning seem to emerge from the study of practice? • In what ways are practices of teaching and learning mathematics made available for study? • What kinds of collaboration are practiced in different countries? • What kinds of leadership help support teachers’ learning from the practice of mathematics teaching? • What are crucial practices of learning from practice? • How does language play a role in learning from practice?

What sorts of learning seem to emerge from the study of practice? • What do teachers learn from different opportunities to work on practice – their own, or others’? • In what ways are teachers learning more about mathematics, about students' learning of mathematics, and about the teaching of mathematics, as they work on experiences in practice? • What seems to support the learning of content? • In what ways are teachers learning about diversity, about culture, and about ways to address the important problems that derive from social and cultural differences in particular settings?

In what ways are practices of teaching and learning mathematics made available for study? • How is practice made visible and accessible for teachers to study it alone or with others? • How is "practice" captured or engaged by teachers as they work on learning in and from practice? (e.g., video, journals, lesson study, joint research, observing one another and taking notes)

What kinds of collaboration are practiced in different countries? • How are teachers organized in schools (e.g., in departments)? • What forms of professional interaction and joint work are engaged, supported, or used?

What kinds of leadership help support teachers’ learning from the practice of mathematics teaching? • Are there roles that help make the study of practice more productive? • Who plays such roles, and what do they do? • What contribution do such people make to teachers’ learning from practice?

What are crucial practices of learning from practice? • What are the skills and practices, the resources and the structures that support teachers’ examination of practice? • How have ideas such as “reflection,”“lesson study,” and analysis of student work been developed in different settings? • What do such ideas mean in actual settings, and what do they involve in action?

How does language play a role in learning from practice? • What sort of language for discussing teaching and learning mathematics –– professional language –– is developed among teachers as they work on practice?

Example 1: Research in the field • From Adler and Jaworski’s plenary on “The state of research in mathematics teacher education and how it needs to develop” • Jill Adler (South Africa) - Chair of the ICME-10 Survey of research on mathematics teacher education and development • Barbara Jaworski (England/Norway) - Editor in Chief of the Journal of Mathematics Teacher Education

ICME-10 Survey of research on mathematics teacher education and development • 282 publications were surveyed (1999-2003)

ICME-10 Survey of research on mathematics teacher education and development Three major claims: • Small-scale qualitative research predominates; little is large scale or quantitative. • Most teacher education research is conducted by teacher educators studying the teachers with whom they are working (researching own practices or programs). • Research in countries where English is the national language dominates the literature.

How the field needs to develop (Adler & Jaworski) We have to expand the field in terms of: • Large-scale studies • Longitudinal studies • Policy studies • Studies across diverse contexts (cross national, language and culture) • Methodology and theory

How the field needs to develop (cont.) Critical stance within teacher educator researcher practice: • TEs need awareness that they develop practice as much as teachers do • We need research into the assumptions on which TE practice is built • We need to investigate relationships between teacher education, teacher learning, and pupils’ learning

Example 2: Examination of a specific TE activity • From Ruhama Even’s presentation on “Developing and integrating knowledge and practice: examination of two activities”

Nature and quality of school mathematics tasks Multi-stage activity: • What is a good problem in school mathematics?

Multi-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.

What is a good problem in school mathematics? - Round 1 A good problem should - • be clearly phrased • stimulate thought • be interesting • be connected to everyday life • not require previously-learned knowledge • cause conflict • not be frustrating

Examination of a special case Add squares so that the figure would have a perimeter of 18. • What is the least number of squares that must be added? • What is the greatest number of squares that can be added? • What if we allow the dimensions to be any real number? • What would be a convincing argument for a junior-high school student?

Perimeter of 18 problem • If the area is increased, the perimeter might remain unchanged or even decrease. • By adding one area unit, the perimeter could change by either +2, 0 , -2, or -4 units. • If the small squares’ side is a whole number, the 4x5 rectangle has the greatest area. • If the small squares’ side is a real number, the 4.5x4.5 square has the greatest area.

What is a good problem in school mathematics? - Round 2 • Problem situation: meaningful context. • Time spent on a problem: investigation of a "big problem". • Different solutions. • Mathematical connections among different representations and different domains. • Different levels of presentation and solution are possible. • ...

Conduction of a class activity Difficulties • Time limitation (traditional, insecurity). • No good problems available (criteria checklist, ready-made problem illegitimate, materials limitation, inability to see through the problem into the activity).

Reflective summary Helping students develop as problem solvers: • "Work on this problem encouraged students to ask additional questions and to expand their investigation. They also became motivated to deal with problem situations."

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

Reflective summary Exploration is important: • Still, there were some who focused only on the exploratory aspect of the activity and did not pay much attention to the mathematics involved.

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

Reflective summary Not topic specific: • “The content of the problem is broad and is not narrowly tied to the mathematical topic currently dealt with in class. The problem can be connected to various topics and approached in various representations.” • Could represent a mature view of mathematics, in contrast to viewing mathematics as a collection of unrelated topics, concepts, and procedures.

Reflective summary Not topic specific and no need for previously-learned knowledge: • May indicate that a ‘good problem’ was still not regarded as an integral part of the mathematics curriculum. • A different approach: Students worked on a standard trig problem using any tools they wanted. All solutions were based on rather advanced previously-learned mathematics knowledge, but not necessarily related to last week's instruction.

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

How and what can teachers learn for practice, in and from practice? The activity “What is a good problem?” - • develops richer, more empowering vision about good mathematical activities • enables the enactment of this new knowledge and its continuous development in practice • offers the opportunity to work on solving an authentic problem of teaching mathematics, and to study closely an important teaching practice • makes what was previously assumed and taken for granted, questionable and examinable

How and what can teacher educators learn for practice, in and from practice? • Knowledge development • Practice development • Know-tice (integration of knowledge and practice) development • Theory development

Structure of the Study Volume • Setting the stage for the Study • Initial teacher education (Strand I) • Learning in and from practice (Strand II) • Resources for teacher development, professional discussions, or interactions with policymakers • Key issues for research in the education and professional development of teachers of mathematics • Concluding commentary • CD to represent ideas that would be more interactive

Deborah Loewenberg Ball (Co-Chair IPC), USA Ruhama Even (Co-Chair IPC), ISRAEL Jo Boaler, USA Chris Breen, SOUTH AFRICA Frédéric Gourdeau, CANADA Marja van den Heuvel-Panhuizen, NETHERLANDS Barbara Jaworski, NORWAY Gilah Leder, AUSTRALIA Shiqi Li, CHINA Romulo Lins (Chair of the Local Organising Committee), BRAZIL João Filipe Matos, PORTUGAL Hiroshi Murata, JAPAN Jarmila Novotna, CZECH REPUBLIC Aline Robert, FRANCE Bernard R. Hodgson, Secretary-General of ICMI, CANADA Hyman Bass, President of ICMI, USA International Programme Committee

The Study Website http://www-personal.umich.edu/~dball/icmistudy15.html

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