In Singapore, many students who choose to enroll in the five polytechnics are poor at math. Before the end of the session, participants reviewed ways to continue the discussion after ICME 12. Note: A link to the DG 2 website, including a full description of the discussion group and all papers submitted, is available on the MCG website .).

Many participants commented on the quality of the discussions, and all were stimulated by sharing ideas. Linear Algebra is one of the most important subjects in the education of mathematicians, scientists, engineers and economists. She found that the majority of students felt comfortable in the symbolic world, but struggled with formal definitions and statements.

Names of participants and photos from the two sessions can be found at http://matrix.skku.ac.kr/2014-Album/ICME12-DG3-report-v1.htm. Wenger (1998), who coined the term "community of practice" (CoP), asserts that in order for a community to be recognized as a CoP, a combination of three characteristics, cultivated in parallel, is necessary: (i) Domain : A CoP is identified by a common area of interest; (ii) Community: A CoP consists of members who are engaged in common activities and discussions, help each other, share information and build relationships that enable them to learn from each other; (iii) Practice: Members of a CoP are practitioners.

## Triggers and Needs for CoPs to Be Formed — Theory and Practice

However, the third quality is still largely lacking in many communities of mathematics teachers, as few develop a shared repertoire of resources. Even the communities of mathematics teachers that develop such resources usually expect community leaders to put them together for the benefit of the entire community. In light of the above, DG4 focused on issues related to the formation of a mathematics teacher's CoP (MTCoP) and their ongoing management from both theoretical and practical points of view.

## Forming, Running and Sustaining an Effective MTCoP

In the second session, Bjørn Smestad and Kathy Clark introduced some good examples of the use of the history of mathematics in teaching, including some from online sources, as an introduction to the discussions on questions 2 and 3. The discussion group consisted of about 25 people from around the world, with a good mix of familiar faces in the HPM community and newcomers. We hope that the dissemination of ideas for this purpose will increase in the coming years.

Thus, mathematics teachers can be understood as epistemological authorities in the classroom, but also as co-researchers of unknown fields, both mathematical and cultural; and as ringleaders in the mathematical circus. How a macro-educational view of the human subject can be reflected in the mathematics classroom and in the education of mathematics teachers. The film is available at: http://www.youtube.com/watch?v=ArKY2y_ve_U and the script for the animated film was published in The Philosophy of Mathematics Education Journal no.

The model has three dimensions and these three dimensions are intertwined in the learning process. Masami Isoda (University of Tsukuba) indicated that there are a few traditions in the Japanese teaching of mathematics. In recent years, among other things, we could observe three important trends in the development of these software tools: (1) Designers of research-oriented software products began to include functions and support for educational purposes; at the same time, teaching-oriented software has become increasingly powerful, so that their use in some research is increasing; (2) The distinction between different types of software is beginning to blur as many products integrate features from other types of software; (3) The computing platforms are diversifying; with the appearance of smartphones, tablets and interactive whiteboards (IWB) in recent years, as well as online services such as Wolfram Alpha that challenge the design and development of mathematics software.

In the first session, Zsolt Lavicza and Balazs Koren outlined the goals of the session and set up a schedule for presentations and discussions. The session also started with the presentation of Ulli Kortenkamp, Germany, who highlighted the problems and processes in the development of mathematical software. The project contributed to the involvement of technology in the curricula in many schools around the world.

How we can deal with infrastructure problems in schools, especially in less developed regions of the world. The focus group organizers were also interested in the extent of teacher retention issues in different countries, as well as their local and global impact on mathematics education. In the US, 50% of teachers leave within their first 5 years of teaching, and the teaching expectancy for beginning math teachers is 1 year.

How does the transition from a math teacher to a math teacher pedagogue take place and what do we gain or lose with the transition. These were grouped into five themes, summarized below, which formed the basis for discussion in the second session.

## The Nature of the Knowledge Needed by MTEs

What measures/criteria exist for successful mathematics teacher education and how do they relate to MTE knowledge? To what extent and in what ways is knowledge necessary for teaching mathematics for MTEs. What contribution can its understanding make to our work and to mathematics education more broadly.

There was a wide range of experience and expertise on the topic, with many participants acknowledging that they had not seriously considered the knowledge of MTEs before attending the discussion group. The discussion in session 1 focused on areas 1, 3 and 4 and ended with the participants writing down one or more questions the participants had about the knowledge of MTEs.

Different Types of Mathematics Teacher Educators and Implications for the Knowledge Needed

Research Methodologies/Approaches

Acquisition of Knowledge for Mathematics Teacher Education

## The Importance of Research in This Area

Scientific "aha-Erlebnis" and "problem solving" are not enough, so how can we encourage a creative mathematical approach. In the 1980s there has been a worldwide effort to make problem solving a central focus of the school mathematics curriculum since the publication of Polly's book on mathematical problem solving in 1954. No attempt to teach problem solving within the school curriculum seems to be successful.

Based on the teaching and research experience of the organizing team, we think that problem solving should still be the direction for teaching mathematics in schools. As such, this discussion group is proposed to identify the practices in teaching problem solving in school mathematics classrooms across different parts of the world, and how these practices are linked to success. As a result of the publication of Polya's book on solving mathematics problems in 1954, the National Council of Teachers of Mathematics and the worldwide educational reforms in school mathematics recommended the study of problem solving at all levels of the mathematics curriculum.

The reform documents indicate that problem solving should be the central focus of these mathematics curricula. As a result of these reforms, problem solving has taken up a major focus in school mathematics curricula worldwide. Problem solving had been identified as both a main goal of teaching and a main activity in mathematics teaching and learning.

How much curriculum time is spent on problem solving compared to the other components of the mathematics curriculum. In addition, problem solving is one of the fundamental skills that students will carry with them throughout their lives and use long after they leave school. In order to improve the young student's mathematical knowledge and procedures, it is important that the teacher knows how to teach problem solving and how to approach problem solving teaching.

They should also recognize problem solving as the main goal of teaching and the main activity in teaching and learning mathematics. What is the general opinion about the importance of solving mathematical problems among school teachers. Although global education reforms in school mathematics have recommended the study of problem solving at all levels of the mathematics curriculum, problem solving remains an unfamiliar concept to most school mathematics teachers.

An example of the latter is the MProSE project based in Singapore. A student's written work on a problem can be used to help evaluate progress in problem solving.