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:

1. 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.
2. 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.
3. 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.
4. 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.
5. 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.

Biografi Singkat

Hans Freudenthal lahir di Luckenwalde, sebuah kota kecil di provinsi Prussian, Brandenburg, terletak antara Berlin dan Frakfurt, pada tanggal 17 September 1905. Kedua orangtuanya bernama Elisabeth Ehmann dan Joseph Freudenthal. Ayahnya bekerja sebagai guru agama di sebuah komunitas kecil yahudi di daerahnya. Rumah tempat dimana Hans Freudenthal lahir dijadikan sebagai synagogue (tempat beribadah dan belajar kaum yahudi). Hans Freudenthal kecil memulai pendidikannya di sebuah sekolah negeri didaerahnya bernama Friedrichsschule, yang kemudian diubah menjadi Reformrealgymnasium. Meskipun minatnya lebih cenderung kepada hal tentang kemanusiaan dan kesusasteraan, Hans Freudenthal memilih untuk belajar tentang matematika dan fisika dan kemudian kuliah di Berlin university pada tahun 1923.

Profesor matematika yang ada di universitas tersebut pada saat itu diantaranya adalah Erhard Schmidt, Richard von Mises, Issai Schur, dan Ludwig Bieberbach dan beberapa dosen muda lainnya seperti John von Neumann, Heinz Hopf, and Karl Löwner yang memperkenalkan kepada Hans Freudenthal tentang intuitionism. Ketertarikannya tentang intuitionism semakin dalam setelah bertemu dengan Luitzen E.J. Brouwer pada suatu semester di musim dingin pada tahun 1926. Kemudian pada musim panas tahun 1927, Hans Freudenthal pindah ke Paris dan menjadi asisten dosen bagi Hadamard, Gaston Julia, dan Émile Picard. Setelah kembali ke Berlin, dia mendapat posisi sebagai Hilfsassistent di seminar matematika, membuatnya berkonsentrasi menyusun disertasi doktor. Bidang kajian penelitiannya adalah topologi yang pada saat itu sedang berkembang pesat sebagai sebuah cabang ilmu baru. Pembimbingnya adalah Hopf, dan ketua promotor disertasinya adalah Bieberbach. Hans Freudenthal lulus ujian doktor pada tahun 1930, dan disertasinya tentang ruang topologi dan group topologi menjadikan penelitian di bidang tersebut terkenal.

Setelah lulus ujian doktor, Hans Freudenthal menerima tawaran Brouwer untuk datang ke Amsterdam University dan menjadi asistennya di bidang matematika karena ketertarikannya di bidang topologi, meskipun ketertarikan Brouwer di bidang topologi tidak lagi seperti sebelumnya. Kemudian, Hans Freudenthal menjadi fellow assistant bagi Witold Hurewicz, yang juga seorang dosen yang menerima penghargaan sebagai Privaat-docenten pada tahun 1931 pada bidang aljabar, teori group, teori pengukuran, analisis kompleks, topologi dan operasi linear. Pada tahun 1936, Hurewicz pidah ke Amerika dan pada tahun 1937 dua orang professor di universitas tersebut pensiun sehingga Hans Freudenthal dan Heyting mengambil alih tugas mengajar mereka  yang kemudian di promosikan sebagai professor.

Pada tahun 1932, Hans Freudenthal menikah dengan Suzanne (Suus) Johanna Lutter, yang kuliah dibidang bahasa Jerman dan kesusasteraan di University of Amsterdam. Selama proses kekuasaan Nazi di Belanda, Suzanne membantu Hans Freudenthal bertahan hidup. Setelah perang dunia II, Suzanne tertarik mengubah pendidikan di sekolah dan memperkenalkan ide dari seorang pakar pendidikan Jerman bernama Peter Petersen.

Pada tahun 1930an, Hans Freudenthal terlibat untuk pertama kalinya dalam mengedit sebuah jurnal. Brouwer yang dipecat oleh Hilbert pada tahun 1929 sebagai Editor Board untuk jurnal Mathematische Annalen karena beberapa konflik, kemudian menerbitkan jurnal international bernama Compositio Mathematica. Pada awal publikasinya di tahun 1934, Hans Freudenthal menangani jurnal tersebut di bawah bimbingan Brouwer. “Meskipun Hans Freudenthal hanya mengumpulkan edisi lengkap dari junrla tersebut sesuai dengan kesanggupannya. Walaupun Brouwer adalah Penanggungjawab editor, sebagian besar jurnal tersebut dikerjakan dan diselesaikan oleh Hans Freudenthal (van Dalen/Remmert 2006, 1088, 1090). Hans Freudenthal pun tidak pernah menjadi member of the Board, yang mungkin disebabkan karena dia tidak mempunyai gelar Professor. Pada tahun 1940, setelah pendudukan Nazi di Belanda berakhir, jurnal tersebut berhenti di terbitkan. Setelah perang, ketika jurnal Compositio mulai terkenal tanpa Brouwer, Hans freudenthal kemudian menjadi member of its Editorial Board.

Pada perang dunia ke dua, pasukan tempur Jerman melakukan penyerangan pada bulan Mei 1940. Hans freudenthal, seperti orang pada umumnya, dihentikan kegiatannya di universitas pada bulan Desember. Selama proses penghentian aktivitas di kampus, Hans Freudenthal tetap bekerja namun semakin hari situasi keamanan semakin memburuk. Pada suatu hari, meskipun dikenal sebagai non-Aryan (Aryan adalah sebutan bagi orangkulit putih dari Eropa utara), Hans Freudenthal terlindungi karena menikah dengan “Suus”, istrinya yang berkebangsaan Belanda. Pada tahun 1942, dia jatuh sakit selama 6 minggu di penjara Gestapo ketika dia ditemukan mempunyai passpor Jerman meskipun tanpa stempel “J”. Penyakit berbahaya yang dideritanya membuatnya tidak direkrut menjadi pekerja di kemah-kemah selama beberapa wakt., Namun, pada bulan Mei 1944 dia dipenjarakan di sebuah kamp di Havelte, di daerah timur Belanda. Pada Bulan September 1944, istrinya, “Suus” berhasil membantunya keluar dari kamp, dan kembali ke Amsterdam dimana dia bisa menghirup udara kebebasan.

Setelah akhir perang dunia ke-II, pada bulan Mei 1945, Freudenthal memulai lagi aktivitasnya di University of Amsterdam. Pada tahun 1946, the faculty of sciences at Utrecht State University menawarkan kepadanya jabatan professor di bidang geometry yang kemudian diterimanya dan pindah ke Utrecht. Dia menduduki posisi tersebut sampai pensiun pada tahun 1976. Tugas mengajar geometri membuatnya mengubah arah penelitiannya, dan kemudian minat penelitiannya pada hubungan antara geometri dan grup simetrinya. Pada saat itulah awal penelitianya beralih ke bidang Lie Groups. Bahkan pada tahun 1972, dia menjadi pendiri jurnal bernama Geometriae Dedicata, dimana dia sebagai Chief Editor sampai tahun 1981.

Jumlah publikasinya sangat banyak; daftar yang ada di bagian bawah tulisan ini hanya sebagian kecilnya saja. Sebagian besar penghargaan yang diraihnya ada di bidang matematika seperti the theory of ends in topology, the suspension theorems, a spectral theorem for Riesz spaces, the algebraic characterization of the topology of the real semi-simple Lie groups, work on the characters of the semi-simple Lie groups, octonion planes dan bidang kajian geometri yang lain yang berhubungan dengan the exceptional Lie groups.

Kontribusinya bukan hanya di bidang matematika, tapi juga di bidang fisafat dan sejarah matematika. Walaupun, minat awalnya bukan matematika tapi Kesusasteraan. Dia adalah seorang pembaca kesusateraan klasik yang tekun. Dia membuat diari dan menulis puisi, novel, librettos dan cerita anak-anak. Sebagian besar karya ini belum dipublikasikan. Dia bahkan ikut berpartisipasi dalam beberapa kontes kesusasteraan. Josette Adda menyebutnya sebagai “Homo Universalis”.

Kontribusi di Bidang Pendidikan

Hans freudenthal dikenal telah tertarik di bidang pendidikan matematika ketika mulai kuliah di Berlin. Namun, baru pada tahun 1950an ia benar-benar menekuninya. Dia menjadi orang yang menolak dengan tegas pendekatan strukturalistik “New Math” dan ketika institut IOWO (Instituut voor de Ontwikkeling van het Wiskunde Onderwijs) didirikan di Utrecht, pendiri arah baru pendidikan matematika, Pendidikan Matematika Realistik (Realistic Mathematics Education) yang terkenal, ia mendapat perhatian dunai internasional khususnya berkaitan dengan PISA (Programme for International Student Assessment).

Sejak mendirikan kembali ICMI (International Commission on Mathematical Instruction), Hans Freudenthal mendapat posisi yang berpengaruh baik secara nasional di Belanda maupun di dunia internasional. Dia menjadi salah seorang anggota pemimpin komite nasional Belanda, IMUK. Dia mengatur dan menerbitkan beberapa laporan nasional.

Freudenthal sering mengungkapkan penyesalannya tidak bisa hadir di konferensi di Royaumont pada tahun 1959 yang di buat oleh OECD yang menjadi pendiri gerakan “New Math”. Sejalan dengan keinginannya untuk bisa berhasil membendung kemunculan “New Math”, dia kemudian memunculkan polemik besar dalam menangkal efek negatif pengajaran matematika dari New Math.

Pada tahun 1963, Freudenthal menjadi anggota the Executive Committee of ICMI sampai tahun 1966, namun pada tahun 1967 sampai 1970, Freudenthal terpilih sebagai presiden ICMI dan kemudian menjadi ex-officio member pada tahun 1971 – 1978. Selama menjadi presiden ICMI, Freudenthal berhasil memperkenalkan inovasi-inovasi yang meyakinkan untuk melahirkan komunitas internasional pendidik matematika. Tidak puas dengan jurnal L’Enseignement Mathématique yang sejak tahun 1908 sebagai bagian dari ICMI, namun sejak 1920 secara esensial adalah jurnal matematika murni, pada tahun 1968 Freudenthal menginisiasi jurnal baru bernama Educational Studies in Mathematics (ESM), yang kemudian mendunia dan ditujukan sebagai jurnal khusus penelitian pendidika matematika. Diterbitkan oleh penerbit Belanda, Reidel kemudian berubah menjadi Kluwer dan sekarang diterbitkan oleh Springer, New York. Jurnal tersebut sukses dan sekarang menjadi salah satu jurnal internasional utama di bidang pendidikan matematika.

Inovasi penting lainnya dari Hans Freudenthal adalah berasal dari kecilnya tempat bagi pendidikan matematika dalam kongres matematikawan internasional. Hal ini tidak sesuai dengan perkembangan yang pesat di bidang pendidikan matematika untuk membicarakan masalah dan ide-ide di bidang tersebut. Hal ini membuat Freudenthal menyusun kongres internasional pertama di bidang pendidikan matematika bernama ICME (International Congress on Mathematical Education) di Lyon, Prancis pada tahun 1969. Sejak saat itu, ICME menjadi kongres 4 tahunan dan menjadi kegiatan penting dalam bidang pendidikan matematika. Proceeding pertama diterbitkan sebagai edisi khusus ESM.

Setelah masa jabatannya sebagai presiden ICMI berakhir, Freudenthal kemudian membuat inovasi baru yang lain, awalnya semua sesuai dengan konteks Belanda, namun kemudian menjadi relevan di seluruh dunia. Pada tahun 1971, didirikan IOWO (the institute for Development of Mathematics Education) di Utrecht University, dimana Hans Freudenthal sebagai direktur. Pendirian institut untuk mewadahi penelitian dan pengembangan di bidang pendidikan matematika sudah dipersiapkan sejak tahun 1961 oleh CMLW (Commissie Modernisering Leerplan Wiskunde). Walaupun dengan banyak halangan birokrasi, institut tersebut tetap berdiri dan menjalankan fungsinya. Hans Freudenthal menjadi direktur sampai tahun 1980. Ketika untuk pertama kalinya dibubarkan, institut tersebut ditransformasi menjadi OW&OC ((Onderzoek Wiskunde-onderwijs en Onderwijs Computercentrum) dan akhirnya berubah menjadi Freudenthal Institute pada tahun 1991.

Tidak cukup sampai disitu, Hans Freudenthal juga berperan penting dalam mendirikan pengembangan baru di dalam dunia pendidikan matematika, bidang kajian psikologi dalam pembelajaran matematika yang disebut PME (the International Group for the Psychology of Mathematics Education) didirikan pada kongres ICME 3 di Karlsruhe, kemudian mulai mengatur kongres pertamanya. Kongres pertama disusun oleh Freudenthal dan dilaksanakan di Utrecht pada tahun 1976.

Sebagai salah satu warisan dari Freudenthal, hasil kajian institut tersebut berperan penting dalam menyebarluaskan pendekatan khusus di dalam pengajaran matematika, diaman Freudenthal pertama kali mengembangkannya bersama cucunya Bastiaan, pendekatan yang dikenal dengan nama Realistic Mathematics Education (RME), yang kemudian secara tak disangka dikenal diseluruh dunia. Akibat dari kritikan dari berbagai kalangan tentang asumsi-asumsi konseptual dari kajian evaluasi TIMSS, sebuah ide baru akhirnya dikembangkan dengan nama PISA. Dan ide baru ini, literasi matematika, berdasarkan pada hasil kerja Freudenthal Institut dan pendekatannya yang dikenal dengan nama Pendidikan Matematika Realistik.

Pada tahun 2002, komite eksekutif ICMI membuat dua penghargaan dibidang penelitian pendidikan matematika, salah satunya diberi nama Hans Freudenthal Award, untuk program penelitian pendidikan matematika yang berkelanjutan selama 10 tahun terakhir.

Beberapa hasil karya Hans Freudenthal

H. FREUDENTHAL. 1958. Logique mathématique appliquée. Paris: Gauthier-Villars.

H. FREUDENTHAL (Ed.). 1958. Report on methods of initiation into geometry. Groningen: Wolters ([Reports]/International Commission on Mathematical Instruction/Subcommittee for the Netherlands, 3).

H. FREUDENTHAL (Ed.). 1961. The concept and the role of the model in mathematics and natural and social sciences: proceedings of the colloquium …/organized at Utrecht, January 1960, Dordrecht, Reidel (Synthese library, [3]).

H. FREUDENTHAL (Ed.). 1962. Algebraical and Topological Foundations of Geometry: Proceedings of a Colloquium held in Utrecht, August 1959, Oxford, Pergamon Pr.

H. FREUDENTHAL (Ed.). 1962. Report on the relations between arithmetic and algebra in mathematical education up to the age of 15 [fifteen] years. Groningen: Wolters ([Reports]/International Commission on Mathematical Instruction/Subcommittee for the Netherlands, 5).

H. FREUDENTHAL. 1966. The language of logic. Amsterdam: Elsevier. Dutch original, Exacte logica.

H. FREUDENTHAL. 1965. Probability and statistics. Amsterdam: Elsevier. Dutch original, Waarschijnlijkheid en statistiek.

H. FREUNDENTHAL. 1968. Mathematik in Wissenschaft und Alltag. München: Kindler.

H. FREUDENTHAL, H. DE VRIES. 1969. Linear Lie Groups, New York: Academic Press.

H. FREUDENTHAL. 1973. Mathematics as an Educational Task. Dordrecht: Reidel.

H. FREUDENTHAL (Ed.). 1975. Les applications nouvelles des mathématiques et l’enseignement secondaire, Conférences du 3me séminaire organisé par la C.I.E.M. à Echternach. juin 1973, Esch-sur-Alzette (Gr. D. de Luxembourg), Victor.

H. FREUDENTHAL. 1978. Weeding and sowing: preface to a science of mathematical education. Dordrecht: Reidel.

H. FREUDENTHAL. 1983. Didactical phenomenology of mathematical structures. Dordrecht: Reidel.

H. FREUDENTHAL. 1987. Berlin 1923 – 1930: Studienerinnerungen. Berlin: de Gruyter.

H. FREUDENTHAL. 1987. Schrijf dat op, Hans. Knipsels uit een leven. Amsterdam: Meulenhoff.

H. FREUDENTHAL. 1991. Revisiting mathematics education: China lectures. Dordrecht: Kluwer Academic Publ.

Beberapa Tulisan tentang Hans Freudenthal

DIRK VAN DALEN. 1991. Freudenthal and the foundations of mathematics, Nieuw Arch. Wiskunde: 4, 9(2), 137-143.

DIRK VAN DALEN. V. REMMERT 2006. The birth and youth of Compositio Mathematica: ‘Ce périodique foncièrement international’. Compositio Mathematica: 142, 1083-1102.

HANS TER HEEGE (Ed.). 2005. Freudenthal 100: speciale editie ter gelegenheid van de honderdste geboortedag van Professor Hans Freudenthal, Utrecht, Freudenthal Institut, Universiteit. Nieuwe wiskrant: 25,1, Panama-Post, 24, 3.

LEEN L. STREEFLAND (Ed.). 1993. The legacy of Hans Freudenthal. Dordrecht: Kluwer.

HENK J.M. BOS. 1993. ‘The bond with reality is cut’ – Freudenthal on the foundations of geometry around 1900. In The legacy of Hans Freudenthal. Dordrecht: Kluwer, 51-58.

FERDINAND D. VELDKAMP. 1985. In honor of Hans Freudenthal on his eightieth birthday. Geometriae Dedicata: 19(1), 2-5.

FERDINAND D. VELDKAMP. 1991. Freudenthal and the octonions, Nieuw Arch. Wiskunde: 4, 9(2), 145-162.

JOSETTE ADDA. 1993. Une lumière s’est éteinte: H. Freudenthal, homo universalis. In The legacy of Hans Freudenthal. Dordrecht: Kluwer, 9-19.

HENK J.M. BOS. 1992. In memoriam: Hans Freudenthal (1905-1990). Historia Mathematica: 19(1), 106-108.

JAN P. HOGENDIJK. 1991. In memoriam: Hans Freudenthal (1905-1990). Archives Internationales d’Histoire des Sciences: 41(127), 353-354.

ANTONIE F. MONNA. 1992. Werner Fenchel and Hans Freudenthal, Nieuw Arch. Wiskunde: 4, 10(1-2), 111-114.

ANTONIE F. MONNA. 1993. Supplementary note: ‘Werner Fenchel and Hans Freudenthal’, Nieuw Arch. Wiskunde: 4, 11(2), 171.

KARL STRAMBACH, F.D. VELDKAMP. 1991. In memoriam Hans Freudenthal. Geometriae Dedicata: 37(2), 119.

WILLEM T, VAN EST. 1993. Hans Freudenthal (17 September 1905 – 13 October 1990). In The legacy of Hans Freudenthal. Dordrecht: Kluwer, 59-69.

WILLEM T, VAN EST. 1999. Hans Freudenthal: 17 September 1905 – 13 October 1990. In History of topology. Amsterdam: 1009-1019.

FERDINAND D. VELDKAMP. 1991. Hans Freudenthal: 1905-1990″. Notices of the American Mathematical Society: 38(2), 113-114.

References:

Gravemeijer, K., Terwel, J. 2000. Hans Freudenthal: A Mathematician on Didactics and CurriculumTheory. Journal of Curriculum Studies: Vol. 32, No 6, Page 777-796.

Niss, M (Ed). 2008. ICME-10 Proceedings. Denmark: IMFUFA, Department of Science, Systems and Models, Roskilde University.

Schubring, G. 2008. Hans Freudenthal: 17 September 1905 Luckenwalde – 13 October 1990 Utrecht. Online published at http://www.icmihistory.unito.it/portrait/freudenthal.php. Accessed in 27 October, 2011.

The development of curriculum in Indonesia has changed many times through the ages. Relating to the issue of mathematics curriculum reform, the government of Indonesia has implemented at least six different mathematics curriculums since the 1970s. These curriculum reforms are Curriculum before 1975, curriculum 1975, curriculum 1984, curriculum 1994, curriculum 1994 revised and Competence based curriculum. Generally speaking, curriculum is defined as the package of a syllabus together with the implementation tools such as textbooks and other resource materials for teacher training. In Indonesian context, although the mathematics syllabus did not change, the mathematics curriculum has changed by the movement of PMRI. Normally, the next interested topic is the design of mathematics curriculum embracing PMRI.

The discussion in this summary was based upon Lee Peng Yee’s article with the same title. The summary covers several important factors in designing curriculum, some practices and the most recent trends.

DESCRIPTIVE VERSUS PRESCRIPTIVE

Typically, there are two types of syllabuses. The first is descriptive and the other is prescriptive. Descriptive means that the syllabus is brief and there is available space for teachers to interpret them so that they were not to be dictated by designers. The other type of syllabus is prescriptive in which items to be excluded would be clearly stated in that syllabus and the outline of topics covered.

It is already known that some of these descriptive and prescriptive syllabuses are moving closer together. Specifically, the descriptive syllabuses that tried to go for subject-specific content, that is, to be more prescriptive. On the other hand, the prescriptive syllabuses went in the opposite direction is to be more descriptive. In the other worlds, they are moving toward the same goal from two opposite ends.

In the syllabus, there are several kinds of strands. Strands such as numbers, algebra, geometry and statistics are commonly happen in the syllabus. Some syllabuses may have more than four strands. Most of syllabuses are spiral in which a topic may be introduced at a lower grade, but it will be revisited again and again at a higher level. There are also countries that adopted non-spiral approach in order to making it easier for teacher training. A syllabus describes about what to teach, and sometimes how to teach, but never why we teach what we teach.

A major event in the history of mathematics education in the past 50 years was the Maths Reforms in the 60s and 70s. Then it was followed by another 20 years of recovery and the era of problem solving. The Maths Reforms has been generally regarded as a failure. However, it is also undeniable that it helped to bring the changes for some developed countries in teacher training and locally-produced textbooks.

TEACHER FACTOR

There is a known fact that a reform can move only as fast as teachers can move. Therefore, training of teachers becomes an important factor in the implementation of a curriculum. It was a common practice to conduct workshops for in-service teachers and seems to be the only way to train teachers to acquire new knowledge in the short period of time. Generally speaking, teachers tend to keep their original way of teaching even though they have been trained to conduct a new approach in their classroom. Intervention seems to be effective way to change the teachers’ habits. Sometimes it works. But, when it does not, it just ends up as a fashion without lasting effect.

STUDENT FACTOR

Student profile has changed as society changed. The environment definitely affects their learning. Therefore, it is very important to know our students when designing a curriculum. Teaching mathematics is not just teaching mathematics alone. It is also a part of training for students to deal with work place.

One of difficult concepts to be explained is negative numbers. In the 70s, there is a great effort was made by textbook writers and teachers to differentiate the negative symbol from the minus sign. But, finally, they gave up. The other example is teaching fractions. Presently, we still teach simple fractions, move on rapidly to decimals, and then come back to fractions again. This is because of the fact that students use calculators nowadays. Because the habit of students has changed, the design of curriculum should also change.

It is already known that student profile is not a top priority in designing curriculum. Understanding the learning habit of our students is as important as selecting the topics to be included in a syllabus.

DIFFERENTIATED SYLLABUS

The essence of issue of Mathematics for All was that all students should study mathematics and up to a certain level. The questions were and still are how to help them and whether they really need it. Every country has its own way in dealing with this problem. For example, Japanese syllabus is a minimum syllabus in the sense that every teacher goes beyond it. Likewise, the U.S. standard is a maximum syllabus in the sense that nobody reaches it. Many other countries adopted differentiated syllabus, in one way or another. It seems that confidence, belief and commitment play an important role in learning, particularly in the Asian culture. Assuming that every students need to learn mathematics; a solution could be differentiated syllabus or differentiated curricula when designing a curriculum.

ASSESSMENT

Firstly, assessment was meant as assessment of learning means we assess what is learned. Later on, people started advocating assessment for learning means we learn from what is learned. In the past two years, people invented the other new term, that is, assessment as learning means assessing is a way of learning.

There is nothing wrong to teach for examination. What is wrong is to teach for examination only.  As far as we can see, examination will stay longer at least in the near future. If examination can lead to learning, it can also lead to the change in learning. Simply, examination plays an important role in designing a curriculum.

MODELLING

The latest change in the classroom is to teach in context and to emphasize on the process. Geometry with proofs and mechanics were two essential subjects in school mathematics before the Maths Reforms. Now Algebra dominates the school mathematics. We lost two rich training grounds to teach thinking process and to teach applications. Unfortunately, there is no replacement so far.

One way to compensate this situation is to incorporate modeling into curriculum. Many countries such as Australia and Germany have done with it. The success of the modeling project depends on how the participants play it.

CONCLUSION

A curriculum is good curriculum only when we have implemented it successfully. In order to be successful, it has to be considered in connection with teachers, students and many other determining factors.  When we review our syllabus, we look around among other countries and compare with them. PMRI with activity-based teaching may have made in-roads into the classroom in Indonesia. The design of locally-produced curriculum will lead the movement of PMRI to become a main stream in the reform of mathematics teaching in particular, and education in general, in Indonesia.

Reference:

Lee, P. Y. (2010). Designing A Mathematics Curriculum. Journal on Mathematics Education IndoMS-JME Volume 1 No. 1. ISSN: 2087-8885.

1. Mengisi aplication form dengan lengkap, download di sini: stuned_form_impome_2012

2. Mengisi CV dengan lengkap,  download di sini: cv-form-neso_2012

3. Fotocopy Kartu Tanda Penduduk (KTP)

4. Pas Photo 4 x6 (1 lembar)

5. Ijazah S1

6. Transkrip nilai dengan nilai IPK minimal 3, 00

7. Sertifikat TOEFL dengan score minimal 500

8. SK CTAB (Surat Keputusan Calon Tenaga Akademik Baru) dari Rektor

Persyaratan di atas dibuat dengan ramgkap 3 ( 1 asli, 2 fotokopi) menggunakan kertas A4 di bundel berdasarkan nomer urut di atas dan di jilid menggunakan plastik mika warna putih (bening).

Mohon tidak melampirkan dokumen yang tidak kami cantumkan di atas.

Semua berkas harap dikirimkan ke:

Martha Metrica, S.E

PMRI – PPPPTK IPA Bandung

Jalan Diponegoro No.12

Bandung

Telp/Fax: 022-4213950/022 -4213949

Paling lambat tanggal 31 Desember 2011, berkas sudah kami terima.

Terima kasih.

Invitation Letter and Administrative Arrangement

Proposal IYF2011

Reccomendation Letter of Tana Toraja Regional Government

Visa Information

IYF 2011 Application Form

for further information, contact this email address: iyf2011committee@gmail.com

A. Introduction
Addition and subtraction problems potentially create many strategies for children to solve. The strategies which can be used are, for example, adding on or counting up, counting back, splitting, removing, taking away, etc. The use of strategy by students depends on the numbers which are available in contexts or questions. Moreover, it is mainly influenced by the understanding of students toward the mathematical idea behind the contexts. When talking about contexts of subtraction, there are mostly three kinds of them which often appear in classroom: subtraction in the form of distance or difference, removal, and comparison. The three kinds of them are categorized as mathematical ideas that students are expected to grasp. In this report I will make report and use the examples in the video Addition and Subtraction Minilessons, Grade pre-3. In the video, Michael Galland, as a teacher and his students are involved in discussion of addition and subtraction problems. The students use many different strategies of some problems which are given by Michael.
B. Observational Question
On this occasion I elicit a question to answer next: How the understanding of mathematical ideas and the numbers available in questions develop mathematical strategy of children to solve problems?
C. Discussion
There are three problems given by Michael shown in the video: 272 – 14, 146 – 12, and 283 – 275. In the first problem, Michael asks Samantha and Louis to answer it. Samantha uses splitting strategy in which she splits 12 becomes 10 and 4, then takes away 10 from 272 resulting 262 and then 4 which finally results 258. Louis also uses similar way as Samantha does; however, he has different answer that is 256. The answer is derived from taking away 2 first, then 10, and 4. However, many other students disagree since the total of number Louis has split is 16. Immediately, Louis changes his answer which is the same as that of Samantha. In the second problem, 146 – 12, there is a student answering the result is 134. Then Maria gives explanation by giving strategy splitting up: removing 10 first and then 2. In the third problem, there are three children participating to solve it, Ian, Daniel, and Colleen. Ian uses strategy counting up in which he jumps from 275 to 280 and uses jump of three to get 283. He comes up with answer 8. Similarly, Daniel also uses Ian’s strategy. On the other hand, Colleen uses jumping back strategy by also splitting up the number. She comes up with answer 8 after removing 275.
D. Analysis
In the first problem, Samantha and Louis seem to know that the idea of subtraction is removal. That is why they take away 14 from 272 and get result 258. The understanding of Maria also in the second problem is the same as that of Samantha and Louis in the previous problem. However, I personally am not quite brave to say that they only know the subtraction as a removal. They choose taking away since 14 and 12 are small numbers which are easy to remove. The most interesting thing is the discussion of the third problem. Ian and Daniel seem to know that mathematical idea behind subtraction can be categorized as distance or difference. Unlike many children, they use adding up from 275 to 283 instead of removing 273 from 283 as Colleen does. The understanding that subtraction also means difference helps Ian and Daniel quickly solve the problem more than Colleen does since 275 is so close to 283. Another mathematical idea which probably exists in Ian and Daniel is subtraction as comparison in which they think to use the same strategy.
E. Conclusion
The understanding of students of big ideas of subtraction really helps them in using effective strategy to solve problems. When students only know that the meaning of subtraction is taking away or removing, they will find easy to count easy question, for example 172 – 14. Because in the example, 14 is such a small number that students will is likely losing their tracks. However, when finding a question, for example 283 – 275, in which they want to remove 275, students are more likely losing their track because 275 is high number. In contrast, when students know the meaning of subtraction can be also distance or difference and comparison, they will use adding on strategy in which it simplifies them to count. For instance, 283 – 275 can be solved by adding on 275 to 8 which results 283. The answer is absolutely the same as the answer when students remove 275 from 283.

reference:

Antonio Chameron et all. Video of Young Mathematicians at Work. Constructing Number Sense, Addition and Subtraction. Heinemann

Gambar 1. Model permainan Turning right/left

Gambar 2. siswa sedang melakukan kegiatan turning right/left

Camping route
Camping route adalah sejenis permainan dimana siswa menggambar Sembilan buah tenda yang masing-masing dihubungkan ke semua tenda lainnya dengan menggambarkan garis untuk menghubungkannya. Siswa lalu menuliskan di kertas jumlah rute yang bisa dilalui apabila terdapat dua buah tenda sampai dengan terdapat Sembilan buah tenda.

Gambar 3. Siswa melakukan kegiatan camping route

Football match
Football match merupakan permainan untuk menemukan berapa banyak kombinasi pertandingan yang bisa digelar apabila terdapat beberapa tim sepakbola dalam jumlah tertentu. Siswa memulai menentukan banyaknya pertandingan apabila terdapat 2 tim hingga 9 tim.

Gambar 4. Siswa melakukan kegiatan football match dengan bimbingan oleh guru

Rotating the Arrow
Permainan memutar lingkaran ini menggunakan komputer yang dimana siswa membagi lingkaran ke dalam beberapa bagian dengan membuat beberapa juring pada lingkaran tersebut. Masing-masing juring mempunyai warna yang berbeda-beda. Pada titik tengah lingkaran, terdapat gambar panah yang dapat diputar melalui tombol perintah di komputer. Siswa dapat mengatur banyaknya bagian dan besar sudut pada masing-masing juring dan selanjutnya menekan tombol perintah untuk memutar gambar panah. Siswa selanjutnya mencatat warna juring yang ditunjuk oleh tanda panah saat tanda itu berhenti.

Gambar 5. siswa melakukan kegiatan rotating arrow

Refleksi
Materi matematika yang berkaitan dengan kegiatan permainan di atas adalah probabilitas dan kombinatorik. Pada akhir kegiatan permainan turning left/right, guru mencatat hasil angka di mana siswa berdiri di baris terakhir. Hasil menunjukkan angka 7,6 dan 5 berturut-turut merupakan angka yang paling sering muncul. Diharapkan siswa bisa menarik kesimpulan mengapa angka-angka tersebut adalah yang terbanyak kemunculannya diantara angka yang lainnya. Tentunya, hal ini membutuhkan kemampuan berpikir dan penalaran siswa akan konsep peluang. Pada kegiatan camping route dan football match, siswa diharapkan menemukan pola pada banyaknya rute jalan atau pertandingan yang bisa diselenggaakan dalam jumlah tertentu. Hal ini membutuhkan kemampuan dalam berpikir tentang konsep kombinatorik. Sedangkan, dalam permainan rotating the arrow, siswa diharapkan mampu memahami dan mengidentifikasi alasan-alasan disbanding seringnya juring atau bagian lingkaran yang lebih besar muncul dibanding juring atau bagian lingkaran yang lebih kecil.
Di tiap-tiap kegiatan yang dilakukan oleh siswa, guru senantiasa mengemukakan pertanyaan kritis kepada siswa agar mereka dapat sampai pada kepercayaan diri atau keyakinan akan jawaban mereka dan mampu mengkomunikasikan gagasan secara efektif.

Pemahaman akan matematika tidak hanya berkenaan dengan penguasaan aksioma, teorema, dan bukti-bukti, melainkan juga proses dalam “melakukan” matematika tidak kalah pentingnya. Proses tersebut bisa meliputi menggunakan heuristik, melakukan kesalahan dalam pengerjaan, ragu, dan menimbulkan miskonsepsi. Proses mengkonstruksi suatu pemahaman matematika dalam hal pendidikan. Sejarah matematika dalam hal ini, merupakan suatu hal yang alami yang bisa dimasukkan ke dalam pembelajaran matematika oleh guru terhadap siswa. Karena sejarah menunjukkan bahwa bagaimana matematikawan terdahulu melakukan proses pengkonstruksian pengetahuan matematika.

Proses menyatukan sejarah ke dalam pendidikan matematika telah digagas sejak lama. Namun, ada beberapa penolakan akan hal tersebut dikarenakan karena beberapa hal berikut ini:

Dari segi filosofi:

1. Sejarah bukanlah matematika, apabila guru ingin mengajarkan sejarah, ajarlah terlebih dahulu matematika lalu mengajarkan sejarahnya.
2. Sejarah matematika bisa membuat siswa bingung dibandingkan memberikan pencerahan dikarenakan kemungkinan penalaran yang digunakan matematikawan terdahulu berbelit-belit.
3. Mempelajari sejarah memungkinkan siswa mempunyai pemahaman yang tak menentu apabila tidak mempunyai pemahaman yang luas terhadap sejarah itu sendiri
4. Banyak siswa yang tidak menyukai sejarah dan bisa menyebabkan mereka juga tidak menyukai matematika
5. Matematika berfungsi untuk mengembangkan pengetahuan di zaman sekarang, lalu untuk apa melihat ke belakang?
6. Mempelajari sejarah matematika berpotensi membuat siswa mempunyai sikap chauvinism yang berlebihan.

Dari segi didaktik:

1. Waktu yang terbatas untuk mengajarkan sejarah ke dalam matematika
2. Sumber atau referensi yang terbatas pada guru
3. Kemampuan mengajarkan sejarah pada guru matematika masih kurang
4. Tidak adanya sistem penilaian yang jelas untuk menilai pemahaman siswa terhadap sejarah matematika

referensi:

John Fauvel and Jan Van Maanen. 2002. History in Mathematics Education. ICMI Study Series

The Support of Big Ideas toward Mathematical Strategies

A. Introduction
Addition and subtraction problems potentially create many strategies for children to solve. The strategies which can be used are, for example, adding on or counting up, counting back, splitting, removing, taking away, etc. The use of strategy by students depends on the numbers which are available in contexts or questions. Moreover, it is mainly influenced by the understanding of students toward the mathematical idea behind the contexts. When talking about contexts of subtraction, there are mostly three kinds of them which often appear in classroom: subtraction in the form of distance or difference, removal, and comparison. The three kinds of them are categorized as mathematical ideas that students are expected to grasp. In this report I will make report and use the examples in the video Addition and Subtraction Minilessons, Grade pre-3. In the video, Michael Galland, as a teacher and his students are involved in discussion of addition and subtraction problems. The students use many different strategies of some problems which are given by Michael.

B. Question
On this occasion I elicit a question to answer next: How the understanding of mathematical ideas and the numbers available in questions develop mathematical strategy of children to solve problems?

C. Discussion
There are three problems given by Michael shown in the video: 272 – 14, 146 – 12, and 283 – 275. In the first problem, Michael asks Samantha and Louis to answer it. Samantha uses splitting strategy in which she splits 12 becomes 10 and 4, then takes away 10 from 272 resulting 262 and then 4 which finally results 258. Louis also uses similar way as Samantha does; however, he has different answer that is 256. The answer is derived from taking away 2 first, then 10, and 4. However, many other students disagree since the total of number Louis has split is 16. Immediately, Louis changes his answer which is the same as that of Samantha. In the second problem, 146 – 12, there is a student answering the result is 134. Then Maria gives explanation by giving strategy splitting up: removing 10 first and then 2. In the third problem, there are three children participating to solve it, Ian, Daniel, and Colleen. Ian uses strategy counting up in which he jumps from 275 to 280 and uses jump of three to get 283. He comes up with answer 8. Similarly, Daniel also uses Ian’s strategy. On the other hand, Colleen uses jumping back strategy by also splitting up the number. She comes up with answer 8 after removing 275.

D. Analysis
In the first problem, Samantha and Louis seem to know that the idea of subtraction is removal. That is why they take away 14 from 272 and get result 258. The understanding of Maria also in the second problem is the same as that of Samantha and Louis in the previous problem. However, I personally am not quite brave to say that they only know the subtraction as a removal. They choose taking away since 14 and 12 are small numbers which are easy to remove. The most interesting thing is the discussion of the third problem. Ian and Daniel seem to know that mathematical idea behind subtraction can be categorized as distance or difference. Unlike many children, they use adding up from 275 to 283 instead of removing 273 from 283 as Colleen does. The understanding that subtraction also means difference helps Ian and Daniel quickly solve the problem more than Colleen does since 275 is so close to 283. Another mathematical idea which probably exists in Ian and Daniel is subtraction as comparison in which they think to use the same strategy.

E. Conclusion
The understanding of students of big ideas of subtraction really helps them in using effective strategy to solve problems. When students only know that the meaning of subtraction is taking away or removing, they will find easy to count easy question, for example 172 – 14. Because in the example, 14 is such a small number that students will is likely losing their tracks. However, when finding a question, for example 283 – 275, in which they want to remove 275, students are more likely losing their track because 275 is high number. In contrast, when students know the meaning of subtraction can be also distance or difference and comparison, they will use adding on strategy in which it simplifies them to count. For instance, 283 – 275 can be solved by adding on 275 to 8 which results 283. The answer is absolutely the same as the answer when students remove 275 from 283.

To overcome the problem above, the Portuguese Ministry of Education led a project in curricula change of mathematics teaching and learning. One of the aims was definitively the improvement of the attitudes of the students towards mathematics. The main characteristics of the new curricula are the absence of mathematics textbook, which is usually used in classroom as a guide for students and teachers, instead the substitute is prepared by the ministry of education and there are also some resources from teachers as complement, the contents of mathematical learning which emphasize real situation to be more realistic, and the method of learning process in classroom. This essay mainly summarizes the case study of researchers investigating the effectiveness of curricula change in affecting students’ attitude toward mathematics.

B. Question
It is assumed that the purpose of this study is investigating the students’ views and attitudes towards mathematics teaching and learning based on the curricula change.

C. Methodology
The case study take samples of students grade 7 and grade 10 in one school. The study itself has three main phases, preparation, conduction, and writing final report. The data are mainly collected by interviewing teachers and students, observation in classroom, and document analysis.

D. Discussion
This section emphasizes two main parts of overview namely overview of the curriculum experience and overview of mathematics and mathematics class. The former, in the absence of textbooks, students made wide use of their own notebooks. The 7th grade classes were mostly structured around worksheet based activities (containing exercises, conceptual questions, and problems). They were carried out in pairs, but sometimes also in groups of four students, followed by discussions on the blackboard involving the general participation of the class. The 10th grade class tended to be structured around the discussion conducted by the teacher. The main conclusions were written on the blackboard and then readily copied down by the students to their notebooks. It was quite noticeable that one of the teachers highly valued extra classroom activities, which very often were called to the discussion.

In the second overview, 7th grade students, in general terms, were satisfied with their mathematics classes and with the new curriculum. In the beginning, they were concerned because there was no textbook, but now they felt all right without it. The students felt that classes were different. In their view, there was much group work, reports, and investigations; there was more discussion, less work on the blackboard, more work on the notebook. They felt that the new curriculum implied more work and more thinking. 10th grade students felt particularly insecure in the beginning of the school year; how would it be with no textbook? But these students felt that all the disciplines were going all right. In overall, their view toward mathematics is mathematics is a good lesson for them.

E. Conclusion
As the curricula change, researchers find that the students’ view and attitude toward mathematics is relatively better than before.

Reference
Ponte, Joao Pedro. (1990). Students’ Views and Attitudes towards Mathematics Teaching and Learning: A Case Study of a Curriculum Experience. Centro de Investigação em Educação da Faculdade de Ciências University of Lisbon. Lisbon, Portugal

Zan, Rosetta. (2004). Attitude toward Mathematics: Overcoming the Positive/Negative Dichotomy. TMM Monograph. Pisa, Italy