Along with language, mathematics has always been at the core of education in all civilized societies. In the school context, as Latterell (2005) said that most students (and many adults) view mathematicians, and even students who are good in mathematics, as probably smart, but socially inept. Being good in mathematics is not something many students strive to be.

Mathematics education researchers try to offer solutions for this case and other problems in the teaching and learning mathematics. In fact, mathematics education is not just simply a discipline or a body of knowledge, but much more than that, it comprises things that people do. Now the didactics and the design of mathematics education become more and more develop. The focus is on theory of mathematics education. This paper explains a comparison between two approaches in mathematics education.

**Lesson Study**

Lesson study is a collaboration-based teacher professional development approach that originated in Japan (Murata, 2011). Lesson study gain an international attention in the past decade and in 2002 it was one of the foci for the Ninth Conference of International Congress on Mathematics Education (ICME) held by International Commission on Mathematical Instruction (ICMI). Began in the late 19th century in Japan, lesson study refers to a process in which teachers progressively strive to improve their teaching methods by working with other teachers to examine and critique one another’s teaching techniques.

Lesson study places teachers at the center of the professional activity with their interests and a desire to better understand student learning based on their own teaching experiences. The idea is simple: teachers organically come together with a shared question regarding their students’ learning, plan a lesson to make student learning visible, and examine and discuss what they observe (Murata, 2011).

The process of lesson study consists of preparation class, actual class, and class review sessions. The preparation class begins with finding and selecting materials relevant to the purpose of the class and trying this into lesson plan. Actual class is that teacher taught based on the teaching plan or lesson plan devised and the class is observed by many teachers, sometimes joined by university instructors and supervisor from board of education. After the class, a review session is held for all observers together with the teacher. In that session, each participant expresses their own opinions, experiences or asks questions about the problems in the class as well as the students learning activities. The purpose of review session is to find out ways to improve teaching and learning process in the classroom and what is exactly happening there.

According to Murata (2011), there are some of the characteristics of lesson study, that is:

- Lesson study is centered around teachers’ interests: Teachers’ interests are central to their professional development. Lesson study goals should be something teachers feel is important to investigate and relevant to their own classroom practice.
- Lesson study is student focused: Lesson study is about student learning. At any part of the lesson study cycle, the activities should focus teachers’ attention to student learning and its connections to lessons/teaching.
- Lesson study has a research lesson: Teachers have shared physical observation experiences (in some special cases, video may be used in place of the live lessons, but this is not recommended), that provide opportunities for teachers to be researchers.
- Lesson study is a reflective process: Lesson study provides plenty of time and opportunities for teachers to reflect on their teaching practice and student learning, and the knowledge gained from and for the reflective practice should be shared in some format with the larger teaching and educational communities.
- Lesson study is collaborative: Teachers work interdependently and collaboratively in lesson study.

Other professional development programs like action research incorporate many of the characteristics of lesson study. However, lesson study as the live research lesson is something unique that apart from that activity. The live research lesson creates a unique learning opportunity for teachers.

In Japan, lesson study has been widely used for over a century. Lesson study works effectively to connect theory and practice in Japan. While in the United States (and other parts of the world) lesson study is mainly known as a small, school-based collaboration, typically in the subject area of mathematics, it comes in many different shapes and sizes in Japan. There is small and school-based lesson study as well as large-scale, national-level lesson study (Murata, 2011).

When new educational approaches (e.g., problem-based math instruction, cooperative learning) or a new content of instructions are implemented, large-scale lesson study is important to be used in order to make teachers across different schools understand of what it means in their respective classrooms.

As Professor Hattori remarks later in this book, “Lesson Study does not refer just to in-school training (or, in our words, simply to observing another teacher’s lesson). It is a process by which teachers of mathematics at several schools in the same community work together to research teaching materials, develop teaching plans (lesson plans) and practice teaching lessons (Isoda, 2007).

**Realistic Mathematics Education**

Realistic Mathematics Education (hereafter RME), is a new approach to mathematics education developed in The Netherlands. The development of RME and its ground educational theory still continues until recently. Freudenthal’s view of mathematics as a human activity plays an important role in the development of RME. According to Freudenthal, mathematics must be connected to reality, stay close to children and be relevant to society in order to be of human value (Van den Heuvel-Panhuizen, 1996).

The main activity in mathematics education, based upon Freudenthal’s view of mathematics, is mathematizing. When setting ‘mathematizing’ as a goal for mathematics education, this can involve mathematizing mathematics and mathematizing reality (Gravemeijer, 1994). In Freudenthal’s view, mathematizing is closely related to level-raising which is obtained when we do features that characterize mathematics such as generality, certainty, exactness, and brevity.

The idea for making mathematizing as the key process in mathematics education is at least based upon two resons. Firstly, mathematizing is not only mathematicians’ activity but also familiarizes the students with a mathematical approach to deal with everyday life situations. Secondly, mathematizing is closely related to the idea of reinvention. Freudenthal advocates that mathematics education organized as a process of guided reinvention, where students can experience a (to some extent) similar process as the process by which mathematics was invented (Gravemeijer, 1994).

Later on, Adri Treffers’s doctoral dissertation titled *Three Dimensions*, supervised by Freudenthal, formulated the idea of two types of mathematization; he pronounced “horizontal” mathematization, related to the applied aspect of mathematics and “vertical” mathematization, related to the pure aspect of mathematics. Although this distinction seems to be free from ambiguity, Freudenthal stated that it does not mean that the difference between these two forms of mathematization is clear cut and they are equal value.

The influence of RME has been enormous around the world. Many countries such as South Africa and Indonesia, even big countries like USA, have adopted and implemented RME theory in their education systems.

To sum up, I pen down saying that, both of these approaches have raised in educational context, particularly in mathematics education. Knowing them are undeniable thing, especially for the teachers. Both of those approaches also can be implemented together in the classroom. This idea can be elaborated more by mathematics education researchers or the teachers. Hopefully, through this article, the Japanese approach and the Dutch approach to mathematics education are no longer the terra incognita for the entire stake holders in mathematics education, notably the math teachers.

**References:**

Gravemeijer, K. 1994. *Developing Realistic Mathematics Education.* Utrecht: CD – B Press/Freudenthal Institute.

Isoda, M. 2007. *Japanese Lesson Study in Mathematics, Its Impact, Diversity and Potential for Educational Development*. Singapore: World Scientific Publishing C. Pte. Ptd.

Latterell, C. 2005. *Math Wars, A Guide for Parents and Teachers*. United States of America: Praeger Publishers.

Murata, A., Hart, L. and Alston, A. 2011. *Lesson Study Research and Practice in Mathematics Education. *New York: Springer.

Van den Heuvel-Panhuizen, M. 1996. *Assessment and Realistic Mathematics Education*. Utrecht: CD – B Press/Freudenthal Institute.

Van den Heuvel-Panhuizen, M. 2000. *Mathematics Education in The Netherlands: A guided Tour*. Freudenthal Institute Cd-rom for ICME9. Utrecht: Utrecht University.

Wittmann, E. 2005. *Freudenthal 100 symposium, Realistic Mathematics Education, past and present*. Nieuw Archief voor Wiskunde Journal Vol 5 December 2005.

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