Common Sense Concrete for Concreting Abstract Mathematics

This paper is the reflection of the article of Koeno Gravemeijer that titles How Concrete is Concrete?  Koeno Gravemeijer is a realistic mathematics education expert from Freudenthal Institute, Utrecht University, Nederland.

Mathematics is abstract, so most of the students think that mathematics is not easy. For making the learning of mathematics easier, the teachers introduce tactile or visual models. They think it is able to make the abstract mathematics concrete. However, Gravemeijer will show that our common way of making things concrete for the students does not work because we help students to make a connection with what we know, not what they know.

The first example that will be shown by Gravemeijer is about how the student of grade 6 solved the comparing fraction problem. The context is about a bakery that would cut banquet bars (a sort of large cookies) to order. They were given paper strips of a given length to enact the cutting process. Then they were asked to cut the banquet bars into eight equal pieces, or six, or ten, and so forth. After they asked to use similar strips to compare 1/3 and 2/6, they solved this problem by comparing the lengths of pieces produced in the two different divisions-either dividing by three or dividing by six. However, students concluded that 1/3 was not equal to 2/6.

Gravemeijer analized the answer of the students then he found that the way students cut the strips was not very precise. It made the answer of the student is like that. It means that the tactile representations do not support an insightful solution. Furthermore, in this case, we may denote the two ways in which ‘concrete’ can be understood as either “material concrete” or as “common sense concrete”. The students’ point of view is sometimes different with teacher’s point of view. In this case, the teacher see the relation between 1/3 and 2/6 in the paper cutting, but the students just see the blocks with various sizes.

So, how is the better one? We prefer to help students to reason how 1/6 relates to 1/3. First, a bar is divided into three parts and each piece into halves next, to get six equal pieces. Then, they will understand that 1/6 is a half of 1/3 and they able to concluse that 2/6 equal to 1/3.

equality

The second example is about addition tasks for the student of grade 1. She is Auburn. When Auburn counting on 16 + 9, she got 25. She also know that if she had 16 cookies and another 9 added, she would have 25 cookies. But, when she filled the worksheet, she wrote 15. When she was interviewed why the answers are different, she said that she didn’t know. Lets look at a part of that worksheet below.

concrete

If we analize the worksheet, we able to know why Auburn’s answer is 15. She just imitated the example from the worksheet. She thought that the mathematics of the worksheets seem different with the world of everyday-life experience. She will not be inclined to use everyday-life knowledge to make sense of ‘school math’ problems. Gravemeijer argue that the knowledge gap between teachers and students is too big to make this work. And manipulatives cannot bridge this gap.

So, what is the better? We could use arithmetic rack to fasilitate the students for communicating and scaffolding.

artmtk rck

In conclution, we may try to follow Freudenthal’s adagio that ‘mathematics should start and stay within common sense’ by trying to foster the growth of what is common sense for the students.

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Percentage Bar As The Powerful Model for Learning Percentage

This article is the reflection paper of the article of Frans van Galen and Dolly van Eerde that titles Solving Problems with The Percentage Bar. It was publised in IndoMS JME, Volume 4 Number 1, 1 January 2013, page 1-8.  Frans van Galen and Dolly van Eerde are realistic mathematics education experts from Freudenthal Institute, Utrecht University, Nederland.

Percentage is one of the matematics material that is taught from grade 4 until 6 in primary school. We can imagine, how often this material is taught. However, when a group of IMPOME students carried out a study in grade 7 on students’ understanding of percentages, they were surprised because there are only four students from 14 students who they tested were able to give a correct answer on the following problem :

bike problem

Although the students had a certain area to be a scrap paper, they just used it to wrote calculation. None of them used it to draw model to clarify the relation between the given numbers. It was obvious that most of the students didn’t know a systematics procedure for working with percentages.

A week later, they interviewed the students about the way they solved some of these problems. They find that there are four students’ ways to solve the problem.

  • Converting the percentage into a decimal number and multiplied, for example : 0,15 x 600
  • Multiply with fractions, for example :

soal 1

  • Trying to make proportions equal, for example :

soal 2

  • Guided by the numbers in the problem, for example : 600 is divided by 15

Eventhought two first solutions are true, but they were not consistent and made calculation error. While, the way solution number 3 made the students were confused.

In a teaching experiment, the students were taught the use of the percentage bar.The researchers has 2 approaches.

  • Calculating via 10%. The problem is like that “if the price of the handphone is € 300 and you just pay € 240, how much the percentage of the discount?” The answer of the student is shown below.

PBar

First, student calculated 50% and 25% of € 300, but that didn’t lead to a discount of € 60. As a next step, she started with 10% and found € 60 by doubling.

  • Calculating via 1%. The problem is about the downloading the computer file. “How many Mb if someone download 11% of 600Mb?” The student answer is shown below.

PBar 2

Student’s strategy is 50% equals 300, 25% equals 150 and 10% equals 60. 11% as 10% + 1%.

Although teaching experiment by percentage bar was carried out just one lesson, the result confirm that the percentage bar is a powerful model that deserves a central place in the teaching of the percentage. It helps the students to make a representation for themselves of the relations between what is given and what is asked, offers scrap paper for the intermediate steps in the calculation process, and offers a natural entry to calculating via 1%.

Ayo, siapa lagi yang bisa nebak ????

bayi ketawaAnak-anak mulai suntuk untuk belajar???? Ayo kita refresh dulu… Kalo kita merasa senang, otak kita masuk kondisi alfa lho, yang memungkinkan kita untuk lebih mudah belajar…. Hal ini sesuai dengan apa yang disampaikan oleh salah satu pakar pendidikan, yaitu Bapak Munif Chatib dalam bukunya Gurunya Manusia. So, gak ada salahnya kita sisipkan tebak-tebakan lucu di sela-sela pelajaran. Eits, tapi inget, tetap yang sopan…cz kita orang yang berkarakter INDONESIA! Ini ni, contoh tebakan-tebakannya….

  1. Ada bebek 10 di kali 2 jadi berapa?………….8, soalnya yg 2 lagi maen di kali
  2. Hewan apa yang namanya 2 huruf?…………U dan g
  3. Apa yang mempunyai kaki enam dan bisa terbang?………..Tiga ekor burung
  4. Punya delapan kaki tapi yang dipakai cuma empat?………………Seekor kuda yang sedang ditunggangi dua orang!
  5. Bagaimana Membedakan Zebra Jantan Dengan Betina?………………………Zebra Jantan Aslinya Berwarna Hitam Garis – Garisnya Putih, Zebra Betina Aslinya Berwarna Putih Garis – Garisnya Hitam
  6. Gimana caranya 5 ekor gajah naek sedan? ………….2  di depan 3 di belakang
  7. Bola apa yang mirip kucing? ……………Bola emon
  8. Ada dua orang bapak dan dua orang anak, mereka pergi ke hutan untuk berburu kancil, sayangnya mereka hanya dapat tiga ekor, tetapi waktu kembali ke rumah, masing-masing membawa satu ekor. Mungkinkah ? Jawaban disertai alasan! …………………..Mungkin aja, karena yang pergi berburu hanya 3 orang yaitu seorang kakek, seorang ayah dan seorang anak
  9. Ada berapa hurufkah dalam abjad?…………..Ada 5 (lima) a-b-j-a-d
  10. Selalu diam di pojok tapi bisa keliling dunia?……………..Perangko

Mau tebak-tebakan lebih banyak lagi ? ayo, klik disini……….

<a href=”http://mulanovich.blogspot.com/2012/07/114-teka-teki-kocak.html&#8221; target=”_blank”>114 Tebak-Tebakan Kocak</a>

Inspirasi Kartu Permainan Matematika dari Applet

Sebagai pendidik yang kreatif, tentu banyak cara yang bisa dilakukan untuk membuat pembelajaran matematika menjadi menarik. Salah satunya melalui game atau permainan. Dari game online maupun offline yang saya postingkan sebelumnya, kita dapat mengambil inspirasi dari game-game tersebut kemudian mengkreasikannya dalam bentuk kartu atau lembaran kertas. Selain tidak membutuhkan koneksi internet maupun ketersediaan komputer, game yang disajikan dalam bentuk kartu atau lembaran kertas ini dapat dimainkan kapan saja, dimana saja, dan oleh lebih banyak anak (tentu kalo kartunya dicetak banyak).

Salah satu game online maupun offline yang bisa disajikan dalam lembaran kertas adalah pathuku. Cara membuatnya sebagai berikut :

Langkah 1

Buka tampilan game pathuku seperti pada langkah yang saya posting sebelumnya

https://umipuji.wordpress.com/2013/10/29/belajar-matematika-sambil-nge-game/

Langkah 2

Cara manual :

Menggambar manual berdasarkan tampilan game yang tersedia atau bisa mengkreasikannya sendiri

Copy paste dari tampilan computer :

  1. Klik print screen pada keyboard
  2. Ctrl C
  3. Buka Paint
  4. Ctrl V
  5. Crop gambar
  6. Ctrl C
  7. Ctrl V pada Ms. Word atau Corel atau Paint
  8. Atur penempatan gambar

Aturan permainannya sama dengan game online maupun offline nya, hanya saja disini kita gunakan pensil atau pena untuk menggambarkan rute yang dilewati dan tidak ada batasan waktunya. Salah satu kelemahan game yang disajikan dalam kertas ini, yaitu hanya bisa dipakai sekali (kecuali menggunakan pensil dan bersedia menghapus bekas goresannya…).

Selain Pathuku, masih banyak lagi game online yang bisa disajikan dalam bentuk kartu atau lembaran kertas. Sebagai contoh lainnya, mari klik alamat berikut.

1. Tic Tac Go Penjumlahan

http://heningarea.blogspot.com/2013/09/tic-tac-go-penjumlahan-paper-based.html

2. Mencari Harta Karun

http://rudilyas.wordpress.com/2013/10/10/pulau-kemarau-pulau-harta-karun-paper-based-applet/

3. Factor Game

http://lauradorindira.wordpress.com/2013/10/14/implementation-of-factor-game-paper-based-applet/

4. Broken Calculator

http://ratiihayu.blogspot.com/2013/10/adopting-idea-of-applet-software-broken.html

5. Number Pattern

http://husnulkhuluq91.wordpress.com/2013/10/10/lets-play-math-colourful-necklace-for-learning-number-pattern-a-paper-based-applet/

Selamat mencoba dan berkreasi….

How Fast Your Multiplication Ability?

For making mathematics interesting for the student, especially for the students of class 5A SD IBA Palembang, I introduced some kinds of online and offline games. They are Penguin Jump, Pathuku, and Tux of Math. Penguin Jump and Pathuku have been described in https://umipuji.wordpress.com/2013/10/29/belajar-matematika-sambil-nge-game/ and Tux of Math have been described in https://umipuji.wordpress.com/2013/11/09/tux-of-math-game-matematika-yang-mengasyikkan/.

Beside for making mathematics interesting for the student, Penguin Jump and Tux of Math can be used not only for checking multiplication speed but also for increasing multiplication speed. We know that students are very interested in games, so they will increase their ability of multiiplication if they replays this games.

First, I asked the student how fast their multiplication ability. Then I gave challenge for them to check their ability by playing the games. When I tried that games for them, they were very enthusiastic. They played Penguin Jump first then Tux of Math. You can look at the picture that show that condition.

aplet

Picture 1. Enthusiastic of The Student To Play The Game

One of the student sat down and operated the computer alternately, while others gave instruction for him/her. They could play that games quickly and succeeded to win this games. When I asked them what their opinion about that games is, all of them answered that that games are very interesting.  They wanted to know where they can download or play that games, so I gave them the address of that games. In my mind, I think that I succeed to introduce that games for them.

Tux of Math : Game Matematika yang Mengasyikkan

Operasi hitung tambah, kurang, kali, dan bagi pada bilangan bulat positif maupun negatif merupakan kemampuan dasar  matematika yang perlu dikuasai anak. Hal ini dikarenakan materi ini akan terus dipakai hampir selama siswa bersekolah maupun dalam kehidupan sehari-hari siswa. Jika siswa belum tuntas menguasai materi ini, bisa dibayangkan betapa sulitnya mereka mengikuti pelajaran selanjutnya. Sayangnya, padatnya materi pelajaran di SD terkadang membuat guru kesulitan untuk menuntaskan materi tersebut. Alih-alih mengajarkan kecepatan berhitung siswa, terkadang mengajarkan konsep dasar saja terasa waktunya masih kurang. Namun, jangan khawatir, selain kurikulum pendidikan yang mulai berubah (materi bisa tidak sepadat dahulu), ada juga fasilitas yang bisa kita manfaatkan untuk mengasah kemampuan berhitung siswa. Salah satunya yaitu dengan memanfaatkan game matematika yang bisa didownload melalui internet.

Salah satu game matematika yang cocok untuk mengasah kemampuan operasi hitung dasar pada bilangan bulat adalah Tux of Math. Tux of Math merupakan game debian dari Linux yang bisa didownload secara gratis di internet. Salah satu alamat download nya sebagai berikut :

http://sourceforge.net/projects/tuxmath/

Game ini menyajikan misi ruang angkasa yang dijalankan oleh agen pinguin untuk melindungi iglo atau rumah pinguin dari gempuran meteor. Gempuran meteor-meteor yang berisi berbagai macam operasi hitung dapat dicegah apabila agen dapat menemukan kunci pemecah meteor tersebut. Jika kunci telah diketikkan pada alat yang tersedia, sang agen akan segera menembakkannya pada batu meteor. Jika kunci benar, meteor akan pecah. Begitu pula sebaliknya.

tux math

Gambar 1. Tampilan Tux of Math

Meskipun game ini paling cocok dimainkan oleh siswa SD, tetapi tetap mengasyikkan jika dimainkan oleh orang dewasa karena game ini disajikan dengan kecepatan yang berlevel-level dari sangat pelan hingga sangat cepat.

Selain kecepatan yang berlevel-level, macam soalnya pun berlevel-level. Kita dapat memilih untuk bermain sendiri (Play Alone) maupun bersama teman (Play with Friends). Berikut ini saya tuliskan macam-macam soal pada game ini.

A. Factoroids

(menembak bilangan-bilangan yang memiliki faktor yang telah ditentukan)

B. Math Command Training Academy

Berisi 58 macam operasi hitung pada bilangan bulat (positif dan negatif) yang disajikan berlevel-level. Pada bagian ini, kita masih bisa memilih materi mana saja yang akan kita mainkan. Level-levelnya sebagai berikut :

  • Number Typing (Pengetikan bilangan, sesuai untuk anak kelas 1)
  • Addition : 1 to 3
  • Addition : 0 to 5
  • Addition : Sums to 10
  • Addition : Sums to 15
  • Addition : Sums to 20
  • Addition : Two Digit Numbers
  • Addition : Missing Numbers
  • Substraction : 0 to 10
  •  Substraction : 0 to 20
  • Substraction : Two Digit Numbers
  • Addition and Substraction Review
  • Multiplication : 0 to 3
  • Multiples of 2
  • Multiples of 3
  • Multiples of 4
  • Multiples of 5
  • Multiplication : 0 to 5
  • Multiples of 6
  • Multiples of 7
  • Multiplication : 0 to 7
  • Multiples of 8
  • Multiples of 9
  • Multiples of 10
  • Multiplication : 0 to 10
  • Multiples of 11 and 12
  • Multiplication : 0 to 12
  • Multiples of 13, 14, and 15
  • Multiplication : 0 to 15
  • Multiplication : Missing Numbers
  • Divition by 2
  • Divition by 3
  • Divition by 4
  • Divition by 5
  • Divition : 1 to 5
  • Divition by 6
  • Divition by 7
  • Divition by 8
  • Divition by 9
  • Divition by 10
  • Divition by 1 to 10
  • Divition by 11 and 12
  • Divition : 1 to 12
  • Divition by 13, 14, and 15
  • Divition : 1 to 15
  • Multiplication and Divition Review

 (Bilangan negatif mulai diajarkan di kelas 4 semester 2)

  • Typing Negative Numbers
  • Substraction : Negative Answer
  • Adding Negatives to Positives
  • Adding Positives to Negatives
  • Substracting Negatives From Positives
  • Substracting Positives From Negatives
  • Adding Negatives to Negatives
  • Substracting Negatives From Negatives
  • Negative and Positive Numbers Review

(Perkalian dan pembagian dengan bilangan negatif mulai diajarkan di kelas 5 semester 1)

  • Multiplication and Divition of Negatives
  • Multiplication of Positives and Negatives
  • Divition of Positives and Negatives

C. Math Command Fleet Missions

Pada bagian ini, jika kita gagal pada level tertentu, kita akan kembali ke level paling dasar. Berikut level-levelnya.

Session 1

  1. Round 1-Pengetikan
  2. Round 2-Penjumlahan 1- 20
  3. Round 3-Penjumlahan 1-40 (ada yang melibatkan penyimpanan)

Session 2

  1. Round 1-Pengurangan tanpa meminjam
  2. Round 2-Pengurangan 1-20
  3. Round 3-Penjumlahan dan Pengurangan 1-20

Session 3

  1. Round 1-Perkalian 1-100
  2. Round 2-Penjumlahan dan Perkalian
  3. Round 3-Penjumlahan, Pengurangan, Perkalian pada Bilangan Bulat Positif dan Negatif

Dst…

D. Play Arcade Game, meliputi :

  1. Space codet – Penjumlahan 1-10
  2. Scout – Penjumlahan dan Pengurangan 1-10
  3. Ranger – Penjumlahan, Pengurangan, Perkalian, dan Pembagian pada Bilangan Bulat Positif
  4. Ace – Penjumlahan, Pengurangan, Perkalian, dan Pembagian pada Bilangan Bulat Positif dan Negatif
  5. Commando – Operasi Hitung Campuran pada Bilangan Bulat Positif dan Negatif (cocok untuk kelas 6)

E. Play Custom Game 

Meliputi penjumlahan, pengurangan, perkalian, dan pembagian pada bilangan bulat positif

Mengasyikkan bukan? Selamat mencoba…. Oya, bisa ikut diskusi Tux of Math at tuxmath-devel@list.sourceforge.net

Alamat Download Game Matematika

Pada postingan sebelumnya, saya menuliskan tentang game matematika online beserta alamatnya. Tetapi, bagi temen-temen yang mengalami kesulitan untuk akses internet setiap saat, bisa mendownload beberapa game berikut :

Yang lainnya menyusul ya… Maklum, modem lagi lambat bener….