**Mathematics Education Around the World:**

**Bridging Policy and Practice**

**International Seminar**

**IAS/Park City Mathematics Institute**

*The International Seminar is funded by the Wolfensohn Family Foundation and the Institute for Advanced Study and has received funding from the Bristol-Myers Squibb Foundation, and the International Commission on Mathematical Instruction.*

**History**

**MISSION:** To bring together teams of two educators-a university mathematics educator or policy-maker and a secondary teacher-from a small number of nations representing a cross-section of the regions of the world to discuss major issues in mathematics education policy and practice. The seminar goals are to:

- promote open discussion of issues affecting the mathematics education policies and practices of each nation,
- identify common issues faced across national contexts,
- identify common sources of direction and support for efforts to address problems, and
- search for common solutions to related problems

Funded by the Wolfensohn Family Foundation, the Bristol-Myers Squibb Foundation, and the International Commission on Mathematical Instruction.

**Summer 2013**: for more in-depth information and briefs, **click here**

__Theme__

The focus was on the teaching and learning of transformational geometry and implications for teacher preparation and development. Discussions and presentations related to the general questions: Where do transformations in geometry fit in your country's curriculum? Is there a role for geometry with transformations in your geometry curriculum?

In particular participants responded to the following:

- What is the status of geometric transformations in the K-12 curriculum in your school?
- What is the role of technology (both potential and how it is actually used in classrooms) in developing understanding of geometric transformations?
- Do you agree or disagree with the statement that the concept of transformation can be a unifying concept in geometry? Explain.
- How are prospective teachers prepared for teaching about geometric transformations in your country?
- How do these topics fit into a tertiary mathematics program at the introductory level?

__Countries__

China, Finland, Honduras, Slovenia, South Africa, and the United States

__Emerging Issues/Central Themes__

Major discussion points included the presence of geometry in the secondary curriculum around the world and how it is presented.

The participants worked together to establish consensus on various issues that emerged in the course of the discussions and, in working groups, produced three short policy briefs that present their collective views.

__2013 Briefs__

- Geometric Transformations Sequence
- The Use of Dynamic Geometry Software in the Teaching and Learning of Geometry through Transformations
- Connecting Transformational Geometry and Transformations of Functions

**Summer 2012**: for more in-depth information and briefs, **click here**

The International Seminar did not officially take place during the summer of 2012 because the Twelfth International Congress on Mathematical Education (ICME) was held in Seoul, South Korea. All participants from the past International Seminars were invited to meet and to join a special sharing group session entitled: Park City Mathematics Institute International Collective Seminars. This session was designed and accepted by the program committee of ICME-12 to allow all participants to share experiences in mathematics education they have had in their countries since attending the PCMI International Seminar. In particular, participants were asked to reflect on experiences directly affected by attendance and working in the PCMI seminars. Additionally, the session was used as a follow-up to the work of ICME-11 and the discussion group on cooperation of mathematics educators on an international basis and to recommend outline changes to be to a book about the work of the PCMI International seminar.

**Summer 2011**: for more in-depth information and briefs, **click here**__Theme__

The focus was on the teaching and learning of complex numbers and implications for teacher preparation and development. Discussions and presentations related to the general questions: Where do complex numbers fit into your country's curriculum? Is there a role for complex numbers in the study of geometry with transformations?

In particular participants responded to the following:

- Complex numbers and their usage are taught in many countries in the solving of quadratic equations. Is this common in your country? If so, how and when does it occur? If not, when and how are solutions of quadratics handled?
- The study of complex numbers sometimes focuses on the arithmetic of complex numbers at the secondary level (high school). Is this common in your country? If not, when and how are the numbers introduced? What obstacles or misconceptions do students encounter when learning about complex numbers?
- How do prospective teachers encounter complex number in their teacher preparation programs? How extensively is it used and for what levels of teachers?
- What applications of complex numbers does your country teach (such as DeMoive's Theorem and polar coordinates)? How do these topics fit into a tertiary mathematics program at the introductory level?

__Countries__

Canada, Finland, Ghana, Honduras, Indonesia, Slovenia, South Korea, and the United States

__Emerging Issues/Central Themes__

Major discussion points included the decreased presence of complex numbers in the secondary curriculum around the world and whether or not this should happen. In some countries, the topic of complex numbers has been removed from national or provincial curricula.

Issues emerging from the conversations formed the basis for short jointly written policy briefs on topics of mutual concern related to the theme. The consensus of the seminar was there was value to introducing and teaching complex numbers in secondary schools. The 2011 briefs deal with a brief history of the complex numbers, how complex numbers should be used in the curriculum in a learning progression, interconnections between algebraic and geometric approaches to complex numbers, and how teachers should be prepared to use the topic.

__2011 Briefs__

*Reflections on the History of Complex Numbers**Integrating Algebra and Geometry with Complex Numbers**A Learning Progression for Complex Numbers**Complex Numbers in Teacher Education: Connecting Mathematics and Pedagogy*

**Summer 2010**: for more in-depth information and briefs, **click here**__Theme__

The focus was on the teaching and learning of recursion and mathematical induction and implications for teacher preparation and development. Discussions and presentations related to the general questions: Where does recursion fit into your country's curriculum? What is the role of technology in teaching and applying recursive techniques?

In particular participants responded to the following:

- Recursion is a common tool used in work with spreadsheets and especially with arithmetic and geometric progressions. Is this common in your country? If so, how and when does it occur? If not, when and how are arithmetic and geometric progressions introduced?
- Recursion is sometimes used as an introduction to mathematical induction at the secondary level (high school). Is this common in your country? If so, how and when does it occur? If not, when and how is each topic introduced?
- How does recursion fit with geometry/algebra in your country? If it does not, do you expect that it would at some point in the future?
- How do prospective teachers encounter recursion in their teacher preparation programs? How extensively is it used and for what levels of teachers?

__Countries__

Cambodia, Canada, Denmark, Ghana, Korea, Israel, Peru, and the United States

__Emerging Issues/Central Themes__

Major discussion points centered on the use of technology and the importance of recursion and induction in different countries. The use of these topics was very uneven from the represented countries and discussions included how and it they might be more incorporated into the curriculum. The model for reasoning that developed was related to how proof by mathematical induction, which was not seen in most countries, could be used.

Issues emerging from the conversations formed the basis for short jointly written policy briefs on topics of mutual concern related to the theme. The 2010 briefs deal with the "what, how and why" of recursion and mathematical induction and their place in the high school curriculum and in teacher preparation.

__2010 Briefs__

*Multiple Perspectives on the Important Concepts for Understanding Recursion and Induction**The Power of Recursion and Induction**A Model for Reasoning with Recursion and Mathematical Induction in School Mathematics*

**Summer 2009**: for more in-depth information and briefs, **click here**__Theme__

The focus was *functions:* the teaching of functions and teacher preparation for the teaching of functions, including implications for the use of technology and the mathematical knowledge needed by teachers for working with functions.

Discussions and presentations related to the general questions:

How does the teaching of functions take place in the classrooms in your country; when do students start learning about functions; and when do concepts become formalized?

In particular participants responded to the following:

- Are Dynamic Geometry Environments being used in your country and if so, how does this use relate to the way functions are treated in your country's school curriculum?
- How does the curriculum in your country deal with multiple representations of function (tables, graphs, symbols)? Is there support in your country for the claim that the core concept of function is not really represented by any of the multiple representations?
- How is function defined and what is its role in your school curriculum? What are some of the variations of that role as students progress through the curriculum?
- Is function being used as a foundational concept and an organizing principle in your curriculum and if so, how?

__Countries__

Australia, Cambodia, Denmark, Israel, Namibia, Peru, Vietnam, and the United States

__Emerging Issues/Central Themes__

Major discussion centered on the use of technology and on the role of functions in the school mathematics curriculum.

Issues emerging from the conversations formed the basis for short jointly written policy briefs on topics of mutual concern related to the theme of functions. The 2009 briefs deal with the relationship of function to the curriculum, teacher preparation and technology.

__2009 Briefs__

*The Place of Functions in the School Mathematics Curriculum**Teacher Professional Development in the Teaching and Learning of Functions**Assets and Pitfalls to Using Technology in Teaching and Learning Functions*

**Summer 2008**: for more in-depth information and briefs, **click here**

The International Seminar did not officially take place during the summer of 2008 because the Eleventh International Congress on Mathematical Education (ICME) was held in Monterrey, Mexico. All participants from the past International Seminars were invited to meet and to join Discussion Group 6: The nature and roles of international co-operation in mathematics education. One highlight of that section was the participation of Michèle Artigue, Past-President, International Commission on Mathematical Instruction.

**Summer 2007**: for more in-depth information and briefs, **click here**__Theme__

This seminar focused on mathematics teacher education with a strong emphasis on preparing teachers to promote mathematical problem solving, reasoning, and proof. Standards for reasoning, and proof vary across countries with a strong historical background to have reasoning and proof based in geometry.

__Countries__

Australia, Columbia, Mexico, Namibia, the Netherlands, Turkey, Vietnam and the United States of America.

__Emerging Issues/Central Themes__

There are similarities and differences in the ways in which reasoning and proof are implemented in the mathematics curricula of different countries. Among the considerations emerging were:

- The meaning of reasoning and proof
- When and how students learn the concepts and when the concepts become formalized
- The effect of technology on what and how reasoning and proof are taught
- The mathematical content and didactical knowledge teachers need to teach reasoning and proof
- Needed research from an international level on teaching reasoning and proof

Most of the participating countries demonstrated diversity in their approach to these topics, both with respect to the school curriculum and to the teacher education and professional development programs. Variety ranged from little emphasis on reasoning and proof prior to secondary school Geometry to frequent problem-based learning in elementary school. Preparation for teachers ranged from little or no mention of these topics (except to the extent that they are already imbedded in the official curriculum) to intense focus in the teacher education program and frequent professional development opportunities in these areas.

__Briefs__

Participants produced three short policy briefs that present their collective views on

- The Nature and Role of Reasoning and Proof,
- Conditions for the Effective Teaching and Learning of Reasoning and Proof, and
- Assessment of Reasoning and Proof.

**Summer 2006**: for more in-depth information and briefs, **click here**__Theme__

This seminar focused on mathematics teacher education with a strong emphasis on preparing teachers to promote mathematical problem solving, reasoning, and proof.

__Countries__

Cameroon, Germany, Mexico, Pakistan, Poland, Singapore, Uganda, and USA, along with a commentator/discussant from France and the from the USA.

__Emerging Issues/Central Themes__

There are similarities and differences in the ways in which problem solving is conceptualized and implemented in the primary and secondary mathematics curricula of different countries. In several of the participating countries, the standards for problem solving, reasoning, and proof contained in the National Council of Teachers of mathematics (NCTM, 1989, 2000) standards have influenced how these topics are defined and when they appear in the curriculum.

Most of the participating countries demonstrated diversity in their approach to these topics, both with respect to the school curriculum and to the teacher education and professional development programs. Variety ranged from little emphasis on reasoning and proof prior to secondary school Geometry to frequent problem-based learning in elementary school. Preparation for teachers ranged from little or no mention of these topics (except to the extent that they are already imbedded in the official curriculum) to intense focus in the teacher education program and frequent professional development opportunities in these areas.

Issues that influence whether mathematics teachers emphasize problem solving, reasoning, and proof in their classrooms include:

- teacher's level of mathematical knowledge, which must be high to support dynamic interpretation and flexible responses to students mathematical ideas.
- availability of materials for use in problem solving, as examples of the types of reasoning teachers should aim to promote, and as models of informal and formal proof that are appropriate for students at different grade levels.
- Infrastructure that supports high levels of peer interaction and teacher-student interaction, such as sufficient classroom space for setting small groups and class sizes that are small enough to allow for adequate monitoring and guidance.

__Proceedings__

The report, *Bridging Policy and Practice in the Context of Reasoning and Proof Statements on: Problem-Solving in the School Mathematics Curriculum; Preparation of Teachers for Teaching Problem Solving, Reasoning and Proof; and Conditions for Teachers to Engage in Problem Solving and Reasoning in Their Classrooms* is available by clicking on Summer 2006.

**Summer 2005**: for more in-depth information and briefs, **click here**__Theme__

To consider the meaning of mathematical literacy in an international context and examine the connection between definitions of mathematical literacy and views on the mathematical knowledge needed by teachers in order to teach well. Participants also explored the ways in which teacher preparation programs in the seven countries approach the task of instilling the types of mathematical knowledge for teaching that support mathematical literacy among all students.

__Countries__

Chile, Germany, Iran, Russia, Singapore, Uganda, and the United States

__Emerging Issues/Central Themes__

We can learn a great deal by examining what works in other nations, but educators in each country must attend to issues of context (both in the originating country and the destination country) when trying to decide if a particular practice that works abroad will work at home. Simply adopting a practice because it had a desired outcome abroad may introduce a destabilizing element into the education system at home. Care must be taken to adapt exogenous practices to the local social, cultural, and economic context rather than impose such practices on systems and peoples to which those practices are ill-suited.

Regional conferences that are attended by mathematics teachers, mathematics teacher educators, and mathematicians can help to close gaps in communication and promote better alignment of secondary and tertiary mathematics education.

The training and working conditions for teachers in some countries place them at a significant disadvantage with regard to fulfilling national expectations for students' mathematical proficiency and increasingly globalized expectations for mathematical literacy. Providing teacher preparation experiences and working conditions that adhere to some internationally-proven standards could significantly improve the teaching and learning environments in many countries.

There are similarities and differences in the ways in which mathematical literacy is defined in different countries. In some cases, these definitions are borrowed from other countries. In other cases, they are indexed to the local/national economy. International assessments that attempt to measure mathematical literacy define it in a particular way and produce results that support and confirm the definition and measured outcomes in some countries and conflict with those in others. What would it mean to be mathematically literate from a multi-national point of view? What mathematics would such an individual know, and what mathematics-related activities would that individual be able to do?

__Proceedings__

The report, *Bridging Policy and Practice: Statements on Adapting/Adopting Best Practices, Establishing Regional PCMI Seminars, International Recommendations for National Standards and Norms Concerning Teachers' Preparation and Working Conditions, and Mathematical Literacy for All Students* is available by clicking on Summer 2005.

**Summer 2004**: for more in-depth information and briefs, **click here**

The International Seminar did not officially take place during the summer of 2004 because the Tenth International Congress on Mathematical Education (ICME) was held in Copenhagen, Denmark. All participants from the past three summers were invited to take part in two PCMI-related sessions that were held as part of ICME. In addition, the classroom teacher from Ecuador, Luis Fernandez, was attended the PCMI High School Teachers Program (HSTP) immediately following ICME. Mr. Fernandez took part in the HSTP courses, had informal discussions with the US participants, and made a formal presentation to the group on teaching in Ecuador. He also consulted with PCMI leaders on how to take what he learned at HSTP and adapt it in ways that would be most useful in his country.

__Proceedings__

The brief report on the PCMI sessions held during ICME is available by click on Summer 2004.

**Summer 2003**: for more in-depth information and briefs, **click here**__Theme__

To examine the preparation and professional development of teachers with respect to both mathematical content knowledge and how to teach that content. To deepen the personal contact among policy makers and teachers from the selected countries representing diverse educational systems and cultures by sharing examples of "best practice" from pre-service and in-service programs.

__Countries__

Cameroon, Ecuador, Iran, Japan, Northern Ireland, Romania, New Zealand, and the United States

__Emerging Issues/Central Themes__

Support and help for beginning teachers is critical but is rarely provided in a systematic way in most of the countries.

Issues of communication, resources, and educational infrastructure are problematic in developing countries, directly impacting what it is even possible to do with teachers.

Grounding the conversation in examples was productive. It was particularly helpful to participate in a teaching laboratory observation.

**Summer 2002**: for more in-depth information and briefs, **click here**

__Theme__

To consider goals, content, and delivery of pre-service and in-service education for mathematics teachers, as well as the policies that govern these in each nation. To identify common issues faced across national contexts and identify pre-service and in-service programs and practices that work well in a particular nation, and may work well in others.

__Countries__

Brazil, Egypt, France, India, Japan, Kenya, Sweden, and the United States

__Emerging Issues/Central Themes__

Teacher preparation varies widely depending on the country, particularly with respect to criteria for entering candidates and the amount and level of preparation needed to complete the program.

In-service opportunities are typically uncoordinated and scattered, and usually not mandated. Even when they are required, in-service activities are not necessarily situated in a coherent program (Japan is a notable exception).

Several countries are trying to reform their teacher pre-service programs. One example is a program design that has a common sequence of courses for all students prior to specialization.

Teacher education programs respond differently to diverse cultures within a country. Some countries provide specific courses on these cultures and their way of life as part of the work toward certification. Others focus on assimilating everyone, regardless of culture, into the mainstream culture.

The use of instructional technology in mathematics is not well thought out or enacted in many areas. Gaps among policy, theory, and practice are common.

__Proceedings__

The report, *Bridging Policy and Practice: A Focus on Teacher Preparation. Reflections from the 2002 Park City Mathematics Institute International Panel on Policy and Practice in Teacher Education, can be found by clicking on Summer 2002.*

**Summer 2001**: for more in-depth information and briefs **click here**__Theme__

To consider issues and challenges faced by mathematics educators from very different contexts and identify common strategies for, and approaches to, meeting those challenges. Of particular interest were challenges, strategies, and approaches regarding national standards, teacher education, content case studies, reform efforts, depth versus breadth in curriculum, mathematics education for all students, and the role and status of mathematics education as a profession.

__Countries__

Brazil, Egypt, France, India, Japan, Kenya, Sweden, and the United States

__Emerging Issues/Central Themes__

Teacher preparation emerged as a primary area of interest and concern across the countries and, as a result, became the theme for the subsequent seminar (2002).

The second theme related to the relationship between national curricula and teacher autonomy regarding classroom practice. A major issue with regard to this theme was that the standardizing effect of national curricula seemed to be mitigated by the varying ways in which teachers implemented the curriculum in their classrooms.

The third theme related to the highly influential nature of mathematics education reforms in the United States. Efforts to improve mathematics education in the US are applauded and used as models or guidelines by educators from other countries.

Regarding the fourth theme of tradition versus reform in mathematics education, participants from several countries noted that tradition related to historical approaches to learning certain procedures or concepts.

Countries varied widely on the mission and purpose of mathematics education in terms of access and equity for all students. The role of schools in society differed markedly across countries. In a significant number of countries, mathematics education combines a national system and a private tutoring system. The private system enables teachers to earn a living wage and enables students who can afford the expenses to receive a higher quality education than is available to most students.

__Proceedings__

The report, Mathematics Education Around the World: Bridging Policy and Practice Reflections from the 2001 Park City Mathematics Institute International Panel on Policy and Practice in Mathematics Education, can be found by clicking on Summer 2001.