Why should We Change Mathematics Teaching?

This is the question that I proposed to my teacher in one of my course. In the discussion, we discussed about the percentage and its relationship with decimal, fractions, ratio and proportion. I argue that we have all been taught about percents and other topics of mathematics in the traditional way, with an emphasis on learning the correct calculation procedures, and very little or not on understanding. Still we all understand what percentages, fractions and decimals are and how they are related. So why should we change the way these topics are taught? Or in any case, why should we not teach the calculation procedures first and let understanding develop later? These questions are the paraphrased ones done by my teacher.

It is an issue that we should discuss, because we hear this argument quite often. Then, my teacher proposed his own question relating to my own question: what arguments do we have for wanting to change the way mathematics is taught traditionally? What data could be used to argue against teaching the procedures, in stead of teaching for understanding?

Mathematics as a human activity

I try to give an answer to my own question, why should we change the way these topics are taught? In my perspectives, I think we have to change the way mathematics has been taught because when we teach students in the traditional way of teaching, I can be really sure that they will not make sense of what they have been learned. Moreover, as Freudenthal said in his wonderful book, Revisiting Mathematics Education, mathematics is a human activity and therefore it must be connected to reality, stay close to children and be relevant to society in order to be of human value. So, in this sense, the focal point is not mathematics as a ready-made subject full of procedures, formulas or whatsoever. Instead, the focal point is on the activity. Education should gives students a guided opportunity to reinvent the mathematics.

why should we not teach the calculation procedures first and let understanding develop later?
I think, if we do it in this way, students will not develop their understanding of mathematics. Instead, what they get from this way of teaching is that they know the procedures and formulas but they do not know how to use it in their life. It can be seen from the test such as PISA or TIMSS, that many Indonesian students fail to answer questions that dealing with complex situations and asking for advance mathematical thinking and reasoning. Even, I argue that students will not develop their understanding after they know the procedures and formulas. It is because, they will always looking for the formulas and procedures when they want to solve the problems.
what arguments do we have for wanting to change the way mathematics is taught traditionally? What data could be used to argue against teaching the procedures, in stead of teaching for understanding?
As Freudenthal said that learning processes, or at least part of them, can be more essential than their products (Cited from Revisiting Mathematics Education book). The focus of the teaching of mathematics is the process of learning it, not the product. In RME, the use of context is very important. In contrast with the traditional way of teaching, RME use the context problem both as a source of learning and to apply mathematical concepts. The students, in RME class, can develop mathematical tools and understanding while working on context problems. In the first time, they develop strategies or models that are closely related to the context. Later on, certain aspects of the context situation can become more general which means that the context can get more or less the character of a model and as such can give support for solving other but related problems (http://www.fisme.science.uu.nl/en/rme/).
One of a good example is the experience of a college professor who experienced the change in her son’s understanding of mathematics after following the RME class in the Netherlands. You can read her story in this link: http://mathematicallysane.com/realistic-math-makes-sense-for-students/

An Analysis of Misunderstanding in Mathematics: The Case of Meanings of Equal Sign

Introduction

Many mathematical symbols and notations are figured routinely by students in learning mathematics in the school contexts. Those symbols and notations are mostly faced by students, especially when they learn about arithmetic and algebra. In fact, many students are struggling to understand the meaning of those mathematical symbols and notations, and sometimes lead them to the misunderstandings (e.g. Kiran, 1981). Realizing this issue, many researchers and experts from many different fields and backgrounds have been trying to find out what kinds of misunderstandings that happen in the students’ thinking in learning mathematics and how to deal with them.

Furthermore, misunderstandings about mathematical symbols and notations are also happened in the case of equal sign (or more holistically, equality). Question about how students understand the equality symbol have largely been discussed by many researchers and experts from many different fields persist through elementary schools to high schools and colleges (e.g. Jones and Pratt, 2007; Hattikudur and Alibali, 2010). Based on those extensive researches focusing in this issue, many students do not interpret the equal sign, as an equivalence symbol. They misunderstand about the meaning of the equal sign. Moreover, as will be seen, understanding the equal sign as an equivalence relation does not seem to come easily to the students. The purpose of this essay is to analyze about misunderstandings of the equal sign among elementary and secondary school children.

Constructivist Approaches are Compatible with Human Cognitive Architecture: A Response to Kirschner, Sweller, and Clark (2006)

Introduction

Learning is much more than memorizing. Learning refers to the acquisition of knowledge through interactions with, and observation of, the physical word and the creatures that inhabit it (Ashman & Conway, 1997). In order to really understand and be able to apply knowledge, students must work to solve problems, to discover things for themselves, and to struggle with ideas. The question of how to help students learn particular knowledge, skills, and concepts that will be useful in their life is at the core of the argument presented by Kirschner, Sweller, and Clark (2006). The authors compare minimally guided instructions with instructional approaches that provide direct instructional guidance of the student learning process. They define minimally guided instruction as ”one in which learners, rather than being presented with essential information, must discover or construct essential information for themselves” and then inversely define direct instruction as “providing information that fully explains the concepts and procedures that students are required to learn as well as learning strategy support that is compatible with human cognitive architecture” (p. 1).

In their argument, Kirschner Sweller, and Clark (2006) affirm that minimal guided instruction approaches are less effective and efficient than fully guided instruction approaches because they ignore the structures that constitute human cognitive architecture. On the contrary to this, they put a strong emphasis on direct, strong instructional guidance, as an effective and efficient way to teach students. By referring to several studies concerning the efficacy of direct instruction (e.g., Klahr & Nigam, 2004), they claim that students learn more deeply from strongly guided instruction than from constructivist or discovery approaches. Opposing this claim, Kuhn and Dean (2006) have found that direct instruction does not work so well for robust acquisition or for maintenance knowledge over time.

Am I Teaching Well? The Searching for The Ground rules of Effective Math Teachers

Presently, many practitioners, researchers and educators have given deep attention to the development and enhancement of teaching mathematics and mathematics education world-wide. Yet, many problems still exist when we are looking to the classroom and see how math teachers do their job. Teaching mathematics is always being a challenging issue to be explored, and there is no easy-single recipe for helping all the teachers to become effective and professional. In this essay, however, I try to make a very brief explanation on how teachers should teach mathematics by considering theory and several aspects of teaching mathematics in the classroom.

In order to answer this question, I propose three premises underpin what the idea I want to explain. Firstly, I argue that teachers as professional ones have to understand well about the contents of mathematics, the development of the students and pedagogical aspects of mathematics. Teachers have to know deeply about the topics that s/he is going to teach and be able to formulate the knowledge with meaningful contexts in their teaching approach. Effective mathematics teaching requires creativity and flexibility of the teachers to propose certain questions that can help the students to think. The teachers also have to understand the strength and weakness of the learners. They have to know the development of students’ understanding of mathematics and what kind of help that s/he can offer in order to guide the learners and prevent them from misunderstandings. Moreover, the teachers also need to understand the big ideas, strategies and models involving in the certain topics of mathematics. By having these kinds of things in mind, the teachers will be able to select and use appropriate tools and materials that can engage the students to develop their mathematical understanding.

Permen, Tepuk Tangan dan Pembelajaran KPK & FPB

Laporan ini memaparkan tentang proses belajar mengajar Kelipatan Persekutuan Terkecil (KPK) dan Faktor Persekutuan Terbesar (FPB) pada kelas V di SD Negeri 98 Palembang, Sumatera Selatan, Indonesia dengan menggunakan pendekatan Realisitic Mathematics Education (RME). Pembelajaran ini melibatkan 33 siswa dalam dua hari.

Pada hari pertama, saya bertindak sebagai guru di kelas. Proses belajar mengajar berlangsung selama 60 menit yang dibantu oleh Ibu Maryani dan Novita Sari untuk mendokumentasikan dan membantu siswa untuk mengikuti pembelajaran. Pada pertemuan ini, kami memperkenalkan konsep Faktor Persekutuan Terbesar (FPB) kepada siswa dengan memberikan masalah realistik.

Pada pertemuan kedua, di hari kedua, Novita Sari bertindak sebagai guru dan dibantu oleh Ibu Maryani dan saya untuk membimbing siswa dan mendokumentasikan proses belajar mengajar. Pada kesempatan tersebut, kami memperkenalkan konsep Kelipatan Persekutuan Terkecil (KPK) dengan menggunakan masalah realistik sebagaimana yang kami lakukan pada pertemuan pertama.

Dengan menggunakan pendekatan RME tersebut, kami memulai pembelajaran dengan sebuah masalah yang berasal dari lingkungan siswa sendiri, sesuai dengan materi pembelajaran, membimbing siswa menemukan sendiri konsep KPK dan FPB, serta memberikan kesempatan kepada siswa untuk saling berinteraksi satu sama lain.

Menghitung Semua Kemungkinan (Sebuah Strategi Pemecahan Masalah)

Mempertimbangkan semua pilihan dapat menjadi cara yang efektif untuk memecahkan sebuah masalah. Meskipun mungkin ada beberapa hal dimana strategi ini bukanlah merupakan prosedur yang paling canggih, strategi tersebut mungkin adalah strategi yang paling mudah digunakan, karena strategi tersebut tidak terlalu abstrak. Namun, dengan menghitung semua kemungkinan merupakan hal yang krusial dalam menggunakan strategi ini. Jika kita tidak mempunyai langkah-langkah pengaturan untuk menghitung semua kemungkinan, strategi tersebut sering kali tidak berhasil dengan baik. Hal ini terlihat dalam aplikasi matematika dan contoh-contoh dalam kehidupan sehari-hari yang lebih rumit dengan menggunakan strategi ini.

Kita sering kali menggunakan strategi pemecahan masalah ini dalam kehidupan sehari-hari tanpa menyadari bahwa strategi ini sebenarnya telah kita gunakan. Misalnya anda diundang untuk datang pada suatu pertemuan di hotel yang berjarak sekita 150 mil. Cara yang paling banyak digunakan oleh orang untuk memutuskan jalan terbaik untuk pergi ke pertemuan tersebut adalahh dengan mendaftar semua kemungkinan jenis perjalanan (misalnya kereta, pesawat, mobil, bus, helikopter, dll) yang dapat digunakan. Baik tertulis maupun secara mental, dan kemudian memilih metode yang paling efisien dengan mengeliminasi atau memilih secara langsung (disebabkan oleh waktu, biaya, dll).

Landskape Pembelajaran Matematika

Masih ingat dengan tulisan sebelumnya tentang kelas Madeline? Klo belum, saya menyarankan kepada para pembaca untuk membaca tulisan sebelumnya berjudul “matematika atau matematisasi” agar tidak mengalami kebingungan.

Secara historis, para pembuat kurikulum tidak menggunakan kerangka pengembangan seperti yang digunakan oleh Madeline ketika mereka menyusun sebuah standar kurikulum, seperti halnya mereka tidak melihat matematika sebagai suatu proses matemamatisasi – sebagai aktivitas.  Mereka menggunakan kerangka pembelajaran berdasarkan akumulasi konten mata pelajaran.  Mereka menganalisis struktur matematika dan menggambarkan tujuan-tujuan pembelajaran seperti sebuah garis. Kemampuan-kemampuan dan ide-ide kecil diasumsikan terakumulasi kedalam konsep-konsep (Gagne 1965; Bloom dkk, 1971). Sebagai contoh, ide sederhana tentang pecahan dianggap sesuai bagi siswa jika mereka diajarkan dengan cara menunjukkan bagian yang diarsir dari keseluruhan suatu bentuk atau dengan pola blok – blok. Selanjutnya, di kelas tiga, kesamaan pecahan kemudian diperkenalkan, dan berlanjut sampai pada kelas lima dan enam, operasi pada pecahan. Tahapan perkembangan hanya dipertimbangan dalam hal hubungannya dengan konten: dari konsep-konsep dan kemampuan-kemampuan sederhana sampai pada yang kompleks.

“Matematika atau Matematisasi”

Merupakan sebuah kebenaran bahwa tujuan pengajaran adalah untuk membantu murid belajar. Namun, proses belajar dan mengajar pada masa lampau lebih sering dianggap sebagai dua hal yang terpisah. Mengajar adalah apa yang dilakukan oleh guru. Ia dianggap sebagai orang yang mengetahui mata pelajaran yang diajarkannya dan harus mampu menjelaskannya dengan baik. Sedangkan murid diposisikan sebagai orang yang belajar. Mereka diharapkan untuk belajar dengan keras, mempraktekkan, dan mendengarkan agar dapat memahami mata pelajaran yang mereka pelajari. Jika mereka tidak belajar, itu adalah kesalahan mereka. Mereka mempunyai kesulitan dalam belajar, membutuhkan remedial, khawatir yang berlebihan, dan pemalas. Bahkan ketika kita berbicara tentang perkembangan, biasanya menyangkut tentang mengevaluasi murid untuk melihat apakah dari segi perkembangan, mereka siap menerima pembelajaran dari guru.

Menariknya, dalam beberapa bahasa, belajar dan mengajar merupakan bahasa yang sama. Dalam Bahasa Belanda misalnya, perbedaan antara belajar dan mengajar hanyalah pada kata depannya. Kata kerjanya sama. Leren aan berarti mengajar; leren van berarti belajar. Ketika belajar dan mengajar begitu dekat, maka akan terintegrasi dalam kerangka pembelajaran: mengajar akan dilihat sebagai sesuatu yang begitu dekat dengan belajar, bukan hanya dalam bahasa dan pikiran, tetapi juga dalam tindakan. Jika proses belajar tidak terjadi, maka tidak akan ada proses mengajar. Tindakan belajar dan mengajar adalah dua hal yang tidak dapat terpisahkan. Continue reading

Bermain Geometri dengan Puzzle “Hatching The Egg”

Kita semua sudah mengetahui bahwa anak-anak begitu senang dan menikmati permainan. Pengalaman telah membuktikan bahwa permainan dapat menjadi aktivitas pembelajaran yang sangat produktif. Ketika kita ingin menggunakan permainan dalam pembelajaran matematika, maka seorang pendidik atau guru harus mampu membedakan antara permainan dan aktivitas. Sebagai salah satu bentuk permainan yang banyak digemari oleh anak-anak, puzzle dapat menjadi salah satu alternatif bagi guru dalam mengajarkan matematika khususnya bagi anak-anak. Dengan mengajarkan matematika melalui aktivitas bermain, anak-anak dapat lebih aktif dan merasa senang melakukannya. Selain itu, rasa ingin tahu mereka semakin meningkat seiring dengan meningkatnya tingkat kesulitan permainan tersebut.

Salah satu bentuk puzzle yang dapat digunakan dalam pembelajaran matematika adalah Hatching The Egg. Sesuai dengan namanya, permainan puzzle ini erat kaitannya dengan telur dan burung. Sudah merupakan hal yang diketahui oleh banyak orang bahwa burung berasal dari telur. Oleh karena itu, sudah pasti tidak mengejutkan lagi jika sebuah puzzle telur yang dipisah-pisah ternyata dapat disusun kembali menjadi salah satu bentuk burung yang ada di bawah ini:

Counting by Rhyme (Mathematics Poem by Annie M.G. Schmidt)

By the end of the second year of kindergarten, the children should be able to count to at least ten and should be able to use this skill for making reasonable estimates, for ordering, comparing and determining, as well as for manipulating whole number quantities. They should, for example, be able to solve the puzzles set out in the rhyme below by calculation as well as by rhyming, or be able to check the result by using various counting procedures— provided that the sweets mentioned in the rhyme are actually present.