PENDEKATAN MATEMATIKA REALISTIK PADA PEMBELAJARAN PECAHAN DI SMP

Oleh: Marsigit
Sebagaimana telah kita ketahui, Matematika Realistik menekankan kepada konstruksi dari konteks benda-benda konkrit sebagai titik awal bagi siswa guna memperoleh konsep matematika. Benda-benda konkret dan obyek-obyek lingkungan sekitar dapat digunakan sebagai konteks pembelajaran matematika dalam membangun keterkaitan matematika melalui interaksi sosial. Benda-benda konkrit dimanipulasi oleh siswa dalam kerangka menunjang usaha siswa dalam proses matematisasi konkret ke abstrak. Siswa perlu diberi kesempatan agar dapat mengkontruksi dan menghasilkan matematika dengan cara dan bahasa mereka sendiri. Diperlukan kegiatan refleksi terhadap aktivitas sosial sehingga dapat terjadi pemaduan dan penguatan hubungan antar pokok bahasan dalam struktur pemahaman matematika.
Menurut Hans Freudental matematika merupakan aktivitas insani (human activities) dan harus dikaitkan dengan realitas. Dengan demikian ketika siswa melakukan kegiatan belajar matematika maka dalam dirinya terjadi proses matematisasi. Terdapat dua macam matematisasi, yaitu: (1) matematisasi horisontal dan (2) matematisasi vertikal. Matematisasi horisontal berproses dari dunia nyata ke dalam simbol-simbol matematika. Proses terjadi pada siswa ketika ia dihadapkan pada problematika yang kehidupan / situasi nyata. Sedangkan matematisasi vertikal merupakan proses yang terjadi di dalam sistem matematika itu sendiri; misalnya: penemuan strategi menyelesaiakn soal, mengkaitkan hubungan antar konsep-konsep matematis atau menerapkan rumus/temuan rumus.
Kita dapat menelaah Bilangan Pecah dalam pembelajaran matematika SMP melalui 2 (dua) sisi yaitu kedudukan formal Bilangan Pecah dalam konteks kurikulum dan silabus, dan kajian substantif bilangan pecah itu sendiri. Di dalam Pedoman Pengembangan KTSP disebutkan bahwa dalam pembelajaran matematika dapat dimulai dengan pengenalan masalah yang sesuai dengan situasi (contextual problem). Dengan mengajukan masalah kontekstual, peserta didik secara bertahap dibimbing untuk menguasai konsep matematika. Tujuan pembelajaran bilangan pecahan di SMP dapat disebutkan sebagai berikut:
1. Memecahkan masalah kontekstual dan menemukan konsep bilangan pecah dari masalah kontekstual yang dipecahkan.
2. Memahami konsep bilangan pecah, menjelaskan keterkaitan antar konsep dan mengaplikasikan konsep bilangan pecah, secara luwes, akurat, efisien, dan tepat, dalam pemecahan masalah
3. Menggunakan penalaran pada pola dan sifat, melakukan manipulasi dan membuat generalisasi tentang bilangan pecah.
4. Mengomunikasikan konsep dan penggunaan bilangan pecah
5. Memiliki sikap menghargai kegunaan bilangan pecah dalam kehidupan sehari-hari.
Standar Kompetensi yang berkaitan dengan pembelajaran pecahan adalah agar siswa memahami sifat-sifat operasi hitung bilangan dan penggunaannya dalam pemecahan masalah. Dengan Materi Pokok berupa Bilangan Bulat dan Bilangan Pecah maka diharapkan dapat dicapai menggunakan 2 (dua) Kompetensi Dasar yaitu: Melakukan operasi hitung bilangan pecahan, dan menggunakan sifat-sifat operasi hitung bilangan pecahan dalam pemecahan masalah.
Tipe realistik mempunyai ciri pendekatan buttom-up dimana siswa mengembangkan model sendiri dan kemudian model tersebut dijadikan dasar untuk mengembangkan matematika formalnya. Ada dua macam model yang terjadi dalam proses tersebut yakni model dari situasi (model of situation) dan model untuk matematis (model for formal mathematics). Di dalam realistik model muncul dari strategi informal siswa sebagai respon terhadap masalah real untuk kemudian dirumuskan dalam matematika formal, proses seperti ini sesuai dengan sejarah perkembangan matematika itu sendiri.
Berikut merupakan contoh pengembangan Masalah Realistik berkaitan dengan Bilangan Pecahan: Suatu Bahan Diskusi Untuk Para Guru
1. Pecahan dan bentuknya
Diskusikan seberapa jauh anda dapat menggunakan ilustrasi atau gambar sebagai sarana agar siswa dapat menggali atau menemukan konsep dan bentuk pecahan?
2. Pecahan Sederhana
Buatlah masalah kontekstual yang dapat menunjang pembelajaran Pecahan Sederhana !
3. Membandingkan Pecahan
Diskusikan bagaimana mengembangkan alat peragayang cukup memadai agar siswa mampu membandingkan pecahan? Jelaskan bagaimana menggunakannya 5. Mengurutkan Pecahan-pecahan
4. Pecahan Desimal
Diskusikan adakah suatu proses yang cukup memadai agar siswa mampu memahami pecahan desimal?
Penulis dapat menyimpulkan bahwa di dalam pembelajaran Bilangan Pecahan melalui pendekatan Realistik kiranya dapat disimpulkan bahwa:
1. Siswa perlu diberi kesempatan untuk menggali dan merefleksikan konsep alternatif tentang ide-ide bilangan pecahan yang mempengaruhi belajar selanjutnya.
2. Siswa perlu diberi kesempatan untuk menggali dan memperoleh pengetahauan baru tentang bilangan pecahan dengan membentuk pengetahuan itu untuk dirinya sendiri.
3. Siswa perlu diberi kesempatan untuk memperoleh pengetahuan sebagai proses perubahan yang meliputi penambahan, kreasi, modifikasi, penghalusan, penyusunan kembali dan penolakan.
4. Siswa perlu diberi kesempatan untuk memperoleh pengetahuan baru tentang bilangan pecahan yang dibangun oleh siswa untuk dirinya sendiri berasal dari seperangkat ragam pengalaman
5. Siswa perlu diberi kesempatan untuk memahami, mengerjakan dan mengimplementasikan bilangan pecahan.
Guru perlu merevitalisasi diri sehingga:
1. Mendudukan dirinya sebagai fasilitator
2. Mampu mengembangkan pembelajaran secara interaktif
3. Mampu memberikan kesempatan kepada siswa untuk secara aktif.
4. Mampu mengembangkan kurikulum dan silabus dan secara aktif mengaitkan kurikulum dengan dunia riil, baik fisik maupun sosial.
5. Mampu mengembangkan skenario pembelajaran:
a. Skema Interaksi: Klasikal, Diskusi Kelompok, Kegiatan Individu
b. Skema Pencapaian Kompetensi: Motivasi, Sikap, Pengetahuan, Skill, dan Pengalaman

BAHAN BACAAN
......... 2003. The PISA 2003 Assessment Framework- Mathematics, Reading, Science and Problem solving Knowledge and Skill.
Koeno Gravemeijer. 1994. Developing Realistics Mathematics Education. Utrecht: CD  Press.
Marsigit, dkk, 2007, Matematika SMP Kl VII, Bogor: Yudistira
Sutarto Hadi. 2002. Effective Teacher Profesional Development for Implemention of Realistic Mathematics Education in Indonesia. Disertasi. Enschede: PrintPartners Ipskamp

Contoh Hasil Observasi Penelitian Kelas Pembelajaran Matematika

Hasil Pengamatan Kelas Tentang Metode Matematika Menurut Versi Katagiri, Terhadap Aktivitas Belajar Matematika SMP Kelas II Dalam Mempelajari Luas dan Volume Tabung, Bola dan Kerucut, Tahun 2006
Peneliti/Pengamat: Marsigit, Mathilda Susanti, Elly Arliani
a. Metode Matematika Jenis Problem Formation and Comprehension
1) Apakah siswa melakukan ABSTRAKSI
Abstraksi adalah mencari kesamaan-kesamaan untuk memperoleh bentuk atau sifat yang paling sederhana yang akan menjadi obyek Mathematical Thinking
Misal Abstraksi:
Dengan ABSTRAKSI maka tentang KUBUS, hanya dipelajari tentang UKURAN dan BENTUK nya saja (bukan warna, bahan, harga, dan sifat-sifat yang lain)
Jawab:
- Siswa melakukan abstraksi terhadap Model Tabung, Bola dan Kerucut
- Model Tabung---abstraksi---unsur-unsur tabung: alas, tinggi, selimut, volum tabung, luas selimut.
- Model Bola---abstraksi----unsur-unsur bola: jari-jari bola, diameter, selimut bola, volum bola, luas selimut
- Model Kerucut---abstraksi---unsur-unsur kerucut: alas kerucut, puncak, tinggi, selimut, volum kerucut
- KONSEP KERUCUT dianggap sebagai SEGITIGA
- KERUCUT adalah bangun yang berbentuk SEGITIGA
- KERUCUT adalah SEGITIGA YANG DI DALAMNYA TERDAPAT RUANG.
2) Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan ABSTRAKSI
Jawab:
Cara melakukan Abstraksi:
- Pengamatan terhadap MODEL PERAGA
- Dengan membandingkan Contoh Benda dalam kehidupan se-hari-
- hari---dengan Model Bangun/Alat Peraga---dan Gambar
- Istila-istilah yang digunakan untuk melakukan abstraksi:
o Mendefinisikan bangun dengan Kalimat sehari-hari: Tabung adalah benda yang bermanfaat untuk menyimpan pensil; Bola adalah benda yang menyerupai jeruk; Kerucut adalah benda yang terdiri dari lingkaran-lingkaran yang semakin ke-atas semakin kecil; Kerucut adalah segitiga yang mempunyai ruang dan lengkungan.
o Menggunakan Istilah pada unsure-unsur:
- unsur-unsur tabung: alas, tinggi, selimut, volum tabung, luas selimut.
- unsur-unsur bola: jari-jari bola, diameter, selimut bola, volum bola, luas selimut
- unsur-unsur kerucut: alas kerucut, puncak, tinggi, selimut, volum kerucut
3) Keadaan yang bagaimana siswa melakukan ABSTRAKSI
Keadaan yang menyebabkan siswa melakukan ABSTRAKSI:
- Setelah guru memberi pertanyaan
- Setelah guru memberi kesempatan melakukan kegiatan kelompok
- Ketika mengerjakan LKS
4) Apakah siswa melakukan IDEALISASI
IDEALISASI adalah (1) menganggap sempurna sifat yang ada, atau (2) menetapkan tujuan atau keadaan untuk dicapai Misal : (1)lurus sempurna, datar sempurna, dst; (2) harus begini, harus begitu dsb
Jawab:
- Siswa melakukan Idealisasi Semu ?
- Mengarah pada bentuk ideal yang ditentukan guru
- Siswa tidak mempermasalahkan Alat peraga yang cacat.
- Siswa belum bisa mengkritisi tentang Model Kerucut jika terbuat dari Baja Tebal. Dapat menyimpulkan akibatnya (idealisasi) jika di ajak diskusi oleh Peneliti
- Idealisasi pada syarat berlakunya rumus Volume Kerucut= 1/3 Volume tabung, yaitu bahwa DIAMETER ALAS KERUCUT dan ALAS TABUNG harus persis sama; dan TINGGI KERUCUT dan TINGGI TABUNG harus persis sama (siswa mengecek/melakukan idealisasi dengan pengamatan)
5) Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan IDEALISASI
- Dengan cara melakukan konfirmasi kepada guru apakah kegiatan sudah sesuai dengan yang diharapkan guru.
- Dengan bertanya kepada siswa yang lain.
- Dengan membetulkan pendapat siswa yang lain.
- Dengan pengamatan pada MODEL PERAGA
6) Keadaan yang bagaimana siswa melakukan IDEALISASI
- Jika mengalami keraguan/kesulitan
- Jika di tanya guru
7) Apakah siswa menggunakan GAMBAR/Model Matematika/ALAT PERAGA untuk menyatakan gagasan matematika? Jika “ya” sebut dan gambarlah serta sebut pula tujuannya.
a. Siswa menggunakan alat peraga untuk menyatakan UNSUR-UNSUR TABUNG, BOLA dan KERUCUT
b. Menggunakan Tinggi Tabung sebagai LEBAR persegi panjang pembentuk selimut.
c. Menggunakan KELILING LINGKARAN ALAS sebagai PANJANG persegi panjang pembentuk selimut.
8) Apakah siswa menggunakan ANGKA/BILANGAN/LAMBANG MATEMATIKA/Model Matematika/ALAT PERAGA untuk menyatakan gagasan matematika? Jika “ya” sebutkanlah dan sebut pula tujuannya.
Jawab:
a. Siswa menyatakan KONSEP KERUCUT dengan KONSEP dan LINGKARAN.
b. Siswa mendefinisikan KERUCUT sebagai BANGUN YANG TERDIRI DARI GARIS LURUS, LINGKARAN ATAU KURVA LENGKUNG.
c. Siswa mendefinisikan KERUCUT sebagai SEGITIGA YANG BERDIRI DI ATAS LINGKARAN.
d. Menyatakan LUAS dengan L, JARI-JARI dengan r, TINGGI dengan t
e. Menyatakan LUAS PERMUKAAN TABUNG =2 phi r (r+t)
f. Menyatakan PERMUKAAN BOLA dengan 4 phi r^2
g. Menyatakan VOLUME KERUCUT dengan 1/3 phi r^2 t
9) Apakah siswa melakukan PENYEDERHANAAN Konsep Matematika?
Jawab:
Siswa melakukan penyederhanaan konsep sbb:
a. KONSEP KERUCUT dianggap sebagai SEGITIGA
b. KERUCUT adalah bangun yang berbentuk SEGITIGA
c. KERUCUT adalah SEGITIGA YANG DI DALAMNYA TERDAPAT RUANG.
d. Penyederhanaan RUMUS/PERHITUNGAN :
4 P r^2= 4 x 22/7 x 10,5 x 10,5 = …
10) Dengan cara apa saja dan sebutkan ISTILAH/LAMBANG/Model Matematika/Alat Peraga yang digunakan untuk melakukan PENYEDERHANAAN
Siswa melakukan penyederhanaan konsep dengan cara:
a. Pengamatan terhadap MODEL PERAGA
b. Melakukan operasi hitung dari penjabaran rumus
11) Keadaan yang bagaimana siswa melakukan PENYEDERHANAAN
a. Jika diberi pertanyaan
b. Jika diberi kesempatan bekerja di dalam kelompok
12) Apakah siswa membuat CONTOH Konsep Matematika?
Jika “Ya” sebutkan contoh-contoh itu.
Jawab:
a. LUAS PERMUKAAN TABUNG =2 phi r (r+t)
b. PERMUKAAN BOLA dengan 4 phi r^2
c. VOLUME KERUCUT dengan 1/3 phi r^2 t
13) Keadaan yang bagaimana siswa mampu membuat CONTOH
Keterangan: Contoh Positif, Contoh Negatif, Secara Lisan, Secara Tertulis, Secara Langsung, Secara Tidak Langsung.
Jawab:
Ketika di beri pertanyaan
Ketika mengerjakan soal
b. Metode Matematika Jenis Establishing a Perspective
1. Apakah siswa melakukan ANALOGI terhadap Prosedur/Langkah Matematika?
Analogi adalah menerapkan suatu prosedur yang sama untuk keadaan atau tujuan yang berbeda.
JAwab:
ya
a. SELIMUT TABUNG di analoginak dengan LUAS PERSEGIPANJANG
b. LILITAN BOLA di analogikan dengan LUAS PERMUKAAN BOLA
c. MODEL GEOMETRI di analogikan dengan BENDA SEKITAR misal, Kerucut dengan Topi, dsb
2. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan ANALOGI
Jawab:
a. Menutup permukaan Bola
b. Menutup permukaan Tabung
c. Membuka lagi
d. Melilit
3. Keadaan yang bagaimana siswa melakukan ANALOGI
a. Ketika mengerjakan LKS
b. Ketika bekerja di kelompok
4. Apakah siswa membuat CATATAN/KETERANGAN terhadap Prosedur/Langkah Matematika?
Jawab:
Ya
5. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan CATATAN/KETERANGAN?
Jika “Ya” sebutkan CATATAN/KETERANGAN tersebut.
Jawab:
Membetulkan rumus
6. Keadaan yang bagaimana siswa Membuat CATATAN/KETERANGAN
Jawab:
Setelah memperoleh konfirmasi dari anggota kelompom yang lain atau dari Ibu Guru
c. Metode Matematika Jenis Executing Solutions
1. Apakah siswa melakukan kegiatan INDUKSI
Keterangan: Induksi dapat berupa: menemukan pola, menemukan rumus,
Jawab:
a. Bagi siswa yang belum tahu melakukan induksi untuk menemukan rumus.
b. Bagi siswa yang sudah tahu, melakukan induksi untuk reconfirm rumus yang telah mereka ketahui (induksi semu)
c. Induksi dilakukan untuk menemukan rumus
2. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan INDUKSI
Jawab:
a. Luas permuk Tabung, induksi dilakukan dengan :
- Pengamatan model tabung
- Manipulasi model Tabung
- Menggambar komponen tabung: lingkaran bawah, lingkaran atas, dan bagian tengah tabung
- Menentukan luas masing-masing komponen
- Menjumlahkan luas masing-masing komponen
Catatan: ada siswa melakukan kekeliruan dengan mengalikan luas.
b. Luas permuk. Bola, induksi dilakukan dengan:
- Pengamatan model Bola
- Melakukan lilitan menutup muka setengah Bola dengan tali
- Memikirkan bahwa panjang lilitan = luas permukaan setengah bola
- Lilitan pada bola digunakan untuk menutup luas daerah lingkaran
- Menemukan bahwa Luas permukaan setengah bola = dua kali luas lingkaran
- Menemukan bahwa luas permukaan bola = 4 kali luas lingkaran.
c. Volume kerucut, induksi dilakukan dengan :
- pengamatan model
- praktek mengisi penuh tabung dengan volume kerucut
- (Terjadi kesalahan prosedur di dalam praktek, karena tidak teliti membaca petunjuk yg dibuat guru)
3. Keadaan yang bagaimana siswa melakukan INDUKSI
Jawab:
a. Ketika mengerjakan LKS
b. Ketika bekerja di dalam kelompok
4. Apakah siswa melakukan kegiatan DEDUKSI
Keterangan: Deduksi dapat berupa: mengerjakan contoh,
Jawab:
a. Bagi siswa yang belum tahu rumusnya, tidak melakukan deduksi untuk menemukan rumus
b. Bagi siswa yang sudah tahu rumusnya, melakukan deduksi untuk reconfirm rumus yang telah mereka ketahui
c. Mereka semua melakukan deduksi bahwa rumus yang mereka temukan berlaku untuk semua (Tabung, Bola, Kerucut); dan ditunjukkan dengan mengerjakan soal yang diberikan oleh gurunya.
5. Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan untuk melakukan DEDUKSI
Jawab:
a. Mengerjakan soal
6. Keadaan yang bagaimana siswa melakukan DEDUKSI
Jawab:
a. Bekerja di dalam kelompok
b. Mengerjakan soal
c. Diberi pertanyaan lisan
7. Apakah siswa menggunakan GAMBAR/Model Matematika/Alat Peraga untuk menyelesaikan soal matematika? Jika “ya” sebut dan gambarlah serta sebut pula tujuannya.
Jawab:
a. Menggunakan Alat Peraga untuk menyelesaikan soal
b. Tetapi tanpa alat peraga juga bisa
8. Apakah siswa menggunakan ANGKA/BILANGAN/LAMBANG MATEMATIKA/Model Matematika/Alat Peraga untuk menyelesaikan soal matematika? Jika “ya” sebutkanlah dan sebut pula tujuannya.
Jawab:
a. Menggunakan Alat Peraga untuk menyelesaikan soal
b. Tetapi tanpa alat peraga juga bisa
d. Metode Matematika Jenis Logical Organization
1) Apakah siswa mempertanyakan KEBENARAN suatu konsep matematika?
Jika “Ya” sebutkan KEBENARAN dari konsep-konsep yang mana?
Jawab: Ya
a. Menanyakan benarkah selimut tabung = bentuk persegi panjang
b. Lilitan Bola kepanjangan
c. Volume pasir pada Tabung kelebihan/kekurangan (kemudian di jelaskan oleh guru beberapa factor penyebabnya a.l. siswa kurang teliti; ukuran tidak tepat, dan pasir tercampur kerikil, dsb)
d. Menanyakan benarkah Volume Tabung = 3 x Volume Kerucut
e. Menanyakan benarkah Luas muka bola = 4 kali luas lingkaran
2) Dengan cara apa saja dan sebutkan ISTILAH-ISTILAH matematika yang digunakan siswa untuk mencek KEBENARAN konsep matematika?
Jawab: Ya
Dengan cara mengajukan pertanyaan kepada Guru/ Siswa yang lain
3) Keadaan yang bagaimana siswa melakukan ceking terhadap KEBENARAN suatu konsep?
Jawab:
Setelah praktek
Pada diskusi kelompok
4) Apakah siswa menggunakan ISTILAH/LAMBANG/NOTASI/MODEL MATEMATIKA/ALAT PERAGA secara BENAR atau SALAH?
Jawab:
a. Terjadi kesalahan prosedur di dalam praktek, karena tidak teliti membaca petunjuk yg dibuat guru, misal:
- mengisi tabung lebih dulu
- melilitkan dengan tali yang menumpuk
b. Salah dalam menulis rumus
c. Salah dalam menemukan rumus.
5) Sebutkan ISTILAH/LAMBANG/NOTASI secara BENAR atau SALAH?
- Salah dalam menulis rumus Karen Kurang teliti
- Salah konsep, mestinya dijumlahkan luasnya, tetapi dikalikan
- Salah menulis rumus L permuk tabung = 2 phi r (r + t) benar = 2 phi r^2 t (salah)
6) Keadaan yang bagaimana siswa menggunakan ISTILAH/LAMBANG/NOTASI/MODEL MATEMATIKA/ALAT PERAGA secara BENAR atau SALAH?
Jawab:
Dalam diskusi kelompok dan presentasi

Matematika Ditinjau Dari Berbagai Sudut Pandang

Resensi buku “The Philosophy of Mathematics Education”, karya Paul Ernest
Oleh: Marsigit
Para absolutis teguh pendiriannya dalam memandang secara objektif kenetralan matematika, walaupun matematika yang dipromosikan itu sendiri secara implisit mengandung nilai-nilai. Abstrak adalah suatu nilai terhadap konkrit, formal suatu nilai terhadap informal, objektif terhadap subjektif, pembenaran terhadap penemuan, rasionalitas terhadap intuisi, penalaran terhadap emosi, hal-hal umum terhadap hal-hal khusus, teori terhadap praktik, kerja dengan fikiran terhadap kerja dengan tangan, dan seterusnya. Setelah mendaftar macam-macam nilai di atas maka pertanyaannya adalah, bagaimana matematisi berpendapat bahwa matematika adalah netral dan bebas nilai ? Jawaban dari kaum absolutis adalah bahwa niai yang mereka maksud adalah nilai yang melekat pada diri mereka yang berupa kultur, jadi bukan nilai yang melekat secara implisist pada matematika. Diakui bahwa isi dan metode matematika, karena hakekatnya, membuat matematika menjadi abstrak, umum, formal, obyektif, rasional, dan teoritis. Ini adalah hakekat ilmu pengetahuan dan matematika. Tidak ada yang salah bagi yang kongkrit, informal, subyektif, khusus, atau penemuan; mereka hanya tidak termasuk dalam sains, dan tentunya tidak termasuk di dalam matematika (Popper, 1979 dalam Ernest, 1991: 132).
Yang ingin ditandaskan di sini adalah bahwa pandangan kaum absolutis, secara sadar maupun tak sadar, telah merasuk ke dalam matematika melalui definisi-definisi. Dengan perkataan lain, kaum absolutis berpendapat bahwa segala sesuatu yang sesuai dengan nilai-nilai di atas dapat diterima dan yang tidak sesuai tidak dapat diterima. Pernyataan-pernyataan matematika dan bukti-buktinya, yang merupakan hasil dari matematika formal, dipandang dapat melegitimasikan matematika. Sementara, penemuan-penemuan matematika, hasil kerja para matematisi dan proses yang bersifat informal dipandang tidak demikian. Dengan pendekatan ini kaum absolutis membangun matematika yang dianggapnya sebagai netral dan bebas nilai. Dengan pendekatan ini mereka menetapkan kriteria apa yang dapat diterima dan tidak diterima. Hal-hal yang terikat dengan implikasi sosial dan nilai-nilai yang menyertainya, secara eksplisit, dihilangkannya. Tetapi dalam kenyataannya, nilai-nilai yang terkandung dalam hal-hal tersebut di atas, membuat masalah-masalah yang tidak dapat dipecahkan. Hal ini disebabkan karena mendasarkan pada hal-hal yang bersifat formal saja hanya dapat menjangkau pada pembahasan bagian luar dari matematika itu sendiri.
Jika mereka berkehendak menerima kritik yang ada, sebetulnya pandangan mereka tentang matematika yang netral dan bebas nilai juga merupakan suatu nilai yang melekat pada diri mereka dan sulit untuk dilihatnya. Dengan demikian akan muncul pertanyaan berikutnya, siapa yang tertarik dengan pendapatnya ? Inggris dan negara-negara Barat pada umumnya, diperintah oleh kaum laki-laki berkulit putih dari kelas atas. Keadaan demikian mempengaruhi struktur sosial para matematisi di kampus-kampus suatu Universitas, yang kebanyakan didominasi oleh mereka. Nilai-nilai mereka secara sadar dan tak sadar terjabarkan dalam pengembangan matematika sebagai bagian dari usaha dominasi sosial. Oleh karena itu agak janggal kiranya bahwa matematika bersifat netral dan bebas nilai, sementara matematika telah menjadi alat suatu kelompok sosial. Mereka mengunggulkan pria di atas wanita, kulit putih di atas kulit hitam, masyarakat strata menengah di atas strata bawah, untuk kriteria keberhasilan penguasaan pencapaian akademik matematikanya.
Suatu kritik mengatakan, untuk suatu kelompok tertentu, misalnya kelompok kulit putih dari strata atas, mungkin dapat dianggap matematika sebagai netral dan bebas nilai. Namun kritik demikian menghadapi beberapa masalah. Pertama, terdapat premis bahwa matematika bersifat netral. Kedua, terdapat pandangan yang tersembunyi bahwa pengajaran matematika juga dianggap netral. Di muka telah ditunjukkan bahwa setiap pembelajaran adalah terikat dengan nilai-nilai. Ketiga, ada anggapan bahwa keterlibatan berbagai kelompok masyarakat beserta nilainya dalam matematika adalah konsekuensi logisnya. Dan yang terakhir, sejarah menunjukkan bahwa matematika pernah merupakan alat suatu kelompok masyarakat tertentu. Kaum ‘social constructivits’ memandang bahwa matematika merupakan karya cipta manusia melalui kurun waktu tertentu. Semua perbedaan pengetahuan yang dihasilkan merupakan kreativitas manusia yang saling terkait dengan hakekat dan sejarahnya. Akibatnya, matematika dipandang sebagai suatu ilmu pengetahuan yang terikat dengan budaya dan nilai penciptanya dalam konteks budayanya.Sejarah matematika adalah sejarah pembentukannya, tidak hanya yang berhubungan dengan pengungkapan kebenaran, tetapi meliputi permasalahan yang muncul, pengertian, pernyataan, bukti dan teori yang dicipta, yang terkomunikasikan dan mengalami reformulasi oleh individu-individu atau suatu kelompok dengan berbagai kepentingannya. Pandangan demikian memberi konsekuensi bahwa sejarah matematika perlu direvisi.
Kaum absolutis berpendapat bahwa suatu penemuan belumlah merupakan matematika dan matematika modern merupakan hasil yang tak terhindarkan. Ini perlu pembetulan. Bagi kaum ‘social constructivist’ matematika modern bukanlah suatu hasil yang tak terhindarkan, melainkan merupakan evolusi hasil budaya manusia. Joseph (1987) menunjukkan betapa banyaknya tradisi dan penelitian pengembangan matematika berangkat dari pusat peradaban dan kebudayaan manusia. Sejarah matematika perlu menunjuk matematika, filsafat, keadaan sosial dan politik yang bagaimana yang telah mendorong atau menghambat perkembangan matematika. Sebagai contoh, Henry (1971) dalam Ernest (1991: 34) mengakui bahwa calculus dicipta pada masa Descartes, tetapi dia tidak suka menyebutkannya karena ketidaksetujuannya terhadap pendekatan infinitas. Restivo (1985:40), MacKenzie (1981: 53) dan Richards (1980, 1989) dalam Ernest (1991 : 203) menunjukkan betapa kuatnya hubungan antara matematika dengan keadaan sosial; sejarah sosial matematika lebih tergantung kepada kedudukan sosial dan kepentingan pelaku dari pada kepada obyektivitas dan kriteria rasionalitasnya. Kaum ‘social constructivist’ berangkat dari premis bahwa semua pengetahuan merupakan karya cipta. Kelompok ini juga memandang bahwa semua pengetahuan mempunyai landasan yang sama yaitu ‘kesepakatan’. Baik dalam hal asal-usul maupun pembenaran landasannya, pengetahuan manusia mempunyai landasan yang merupakan kesatuan, dan oleh karena itu semua bidang ilmu pengetahuan manusia saling terikat satu dengan yang lain. Akibatnya, sesuai dengan pandangan kaum ‘social constructivist’, matematika tidak dapat dikembangkan jika tanpa terkait dengan pengetahuan lain, dan yang secara bersama-sama mempunyai akarnya, yang dengan sendirinya tidak terbebaskan dari nilai-nilai dari bidang pengetahuan yang diakuinya, karena masing-masing terhubung olehnya.
Karena matematika terkait dengan semua pengetahuan dari diri manusia, maka jelaslah bahwa matematika tidaklah bersifat netral dan bebas nilai. Dengan demikian matematika memerlukan landasan sosial bagi perkembangannya (Davis dan Hers, 1988: 70 dalam Ernest 1991 : 277-279). Shirley (1986: 34) menjelaskan bahwa matematika dapat digolongkan menjadi formal dan informal, terapan dan murni. Berdasarkan pembagian ini, kita dapat membagi kegiatan matematika menjadi 4 (empat) macam, di mana masing-masing mempunyai ciri yang berbeda-beda:
a. matematika formal-murni, termasuk matematika yang dikembangkan pada Universitas dan matematika yang diajarkan di sekolah;
b. matematika formal-terapan, yaitu yang dikembangkan dalam pendidikan maupun di luar, seperti seorang ahli statistik yang bekerja di industri.
c. matematika informal-murni, yaitu matematika yang dikembangkan di luar institusi kependidikan; mungkin melekat pada budaya matematika murni.
d. matematika informal-terapan, yaitu matematika yang digunakan dalam segala kehidupan sehari-hari, termasuk kerajinan, kerja kantor dan perdagangan.
Dowling dalam Ernest (1991: 93), berdasar rekomendasi dari Foucault dan Bernstein, mengembangkan berbagai macam konteks kegiatan matematika. Dia membagi satu dimensi model menjadi 4 (empat) macam yaitu : Production (kreativitas), Recontextualization (pandangan guru dan dasar-dasar kependidikan), Reproduction (kegiatan di kelas) dan Operationalization (penggunaan matematika). Dimensi kedua dari pengembangannya memuat 4 (empat) macam yaoitu: Academic (pada pendidikan tinggi), School (konteks sekolah), Work (kerja) dan Popular (konsumen dan masyarakat).
Dengan memasukkan berbagai macam konteks matematika, berarti kita telah mengakui tesis D’Ambrosio (1985: 25) dalam ‘ethnomathematics’ nya. Tesis tersebut menyatakan bahwa matematika terkait dengan aspek budaya; secara khusus disebutkan bahwa kegiatan-kegiatan seperti hitung-menghitung, mengukur, mendesain, bermain, berbelanja, dst. Merupakan akar dari pengembangan matematika. Dowling dalam Ernest (1991: 120) mengakui bahwa pandangan demikian memang agak kabur; kecuali jika didukung oleh pembenaran tradisi matematika.
DAFTAR PUSTAKA
Ernest, P., 1991, The Philosophy of Mathematics Education, London : The Falmer Press.

Apakah motivasi itu?

Oleh: Bindra, D and Stewart, J
Di resensikan oleh : Marsigit
Menurut McDougall, motivasi bersumber dari naluri yang terdalam diri manusia. Secara langsung maupun tidak langsung, tindakan manusia ditentukan oleh unsur naluriahnya. sedangkan menurut Freud, sifat dasar naluriah (instinct) manusia berasal dari energi atau potensi seseorang untuk melakukan sesuatu. Sehingga naluri manusia juga mempunyai obyek atau sasaran. Dalam rangka mencapai obyek itulah maka manusia dikatakan mempunyai tujuan. Perlu diketahui bahwa obyek yang dimaksud tidaklah harus berada diluar diri manusia, tetapi bisa berada dalam diri manusia itu sendiri. Naluri itu sendiri juga mempunyai sumber yang disebut sebagai proses dibawah sadar dari organ atau tubuh manusia. Tidaklah jelas apakah proses ini merupakan hasil dari suatu reaksi kimia atau sesuatu yang dihasilkan atau sesuatu yang menghasilkan energi. Yang jelas diketahui adalah naluri yang berbeda-beda dikendalikan oleh kemampuan jiwa/pikiran untuk suatu kegiatan yang berbeda-beda pula.
Lorenz, mengembangkan teori tentang keadaan naluriah berdasarkan model produksi energi. Energi akan dihasilkan berdasarkan adanya stimulus penyebab. Dari sini maka lahirlah teori stimulus-respon. Namun model energi ini telah juga mendapatkan kritik dari Hinde yang menyatakan bahwa teori demikian dapat menyesatkan kita, karena kita pun tidak akan secara jelas bahwa energi yang dimaksud itu merupakan proses atau hasil, energi kegiatan atau energi fisik belaka. Bagaimanapun tentang keadaan naluriah, Tinbergen berusaha untuk mengembangkan teori bagaimana naluri manusia dapat dikoordinasikan secara hirarkhi. Menurutnya, keadaan naluri manusia mempunyai dua aspek, pertama keadaan yang berkaitan dengan keinginan dan kedua, keadaan yang berkaitan dengan pemenuhan kebutuhan.
Nissen, mengembangkan teori bahwa naluri manusia merupakan daya dorong untuk melakukan suatu kegiatan. Dia menggolongkan daya dorong menjadi 5 (lima) berdasarkan kategori bagaimana suatu aktivitas akan dilaksanakan: a) jika daya dorong relatif lemah dan tidak mempunyai arah maka seseorang akan melakukan kegiatan secara acak atau random atau tanpa pilih, b) jika daya dorong berasal dari tingkatan yang lebih dalam lagi, maka kita katakan bahwa seseorang telah melakukan aktivitas berdasar nalurinya, c) jika hasil belajar merupakan daya dorong maka akan menghasilkan pola kegiatan yang kompleks, d) jika daya dorong didasarkan atas pertimbangan tertentu maka akan menghasilkan suatu kegiatan yang mempunyai tujuan, dan e) jika tidak termasuk ke empat di atas maka naluri itu cukup dikatakan sebagai “potensi”.
Reference:
Semua nama yang tersebut dalam paragraph di atas merupakan penulis bagian dari buku “ Motivation” Karya Dalbir Bindra dan Jane Stewart, Penguin Modern Psychology, Penguin Book, London

KANT’S CONCEPTS OF MATHEMATICS

By Marsigit
Kant argued that mathematics is a pure product of reason, and moreover is thoroughly synthetical.1 Next, the question arises: Does not this faculty, which produces mathematics, as it neither is nor can be based upon experience, presuppose some ground of cognition a priori, 2 which lies deeply hidden, but which might reveal itself by these its effects, if their first beginnings were but diligently ferreted out? 3However, Kant found that all mathematical cognition has this peculiarity: it must first exhibit its concept in a visual intuition and indeed a priori, therefore in an intuition which is not empirical, but pure. Without this mathematics cannot take a single step; hence its judgments are always visual, viz., intuitive; whereas philosophy must be satisfied with discursive judgments from mere concepts, and though it may illustrate its doctrines through a visual figure, can never derive them from it. 4
On the other hand, Kant claimed that empirical intuition enables us without difficulty to enlarge the concept which we frame of an object of intuition, by new predicates, which intuition itself presents synthetically in experience; while pure intuition does so likewise, only with this difference, that in the latter case the synthetical judgment is a priori certain and apodeictical, in the former, only a posteriori and empirically certain; because this latter contains only that which occurs in contingent empirical intuition, but the former, that which must necessarily be discovered in pure intuition. 5 The next step, Kant questioned: "How is then it possible to intuit [in a visual form] anything a priori?" ; however, according to Kant, as an intuition is such a representation as immediately depends upon the presence of the object, it seems impossible to intuit from the outset a priori, because intuition would in that event take place without either a former or a present object to refer to, and by consequence could not be intuition.6
Kant then argued that the intuitions which pure mathematics lays at the foundation of all its cognitions and judgments which appear at once apodictic and necessary are Space and Time. 7 Accordingly, because mathematics must first have all its concepts in intuition, and pure mathematics in pure intuition, hence, mathematics must construct them. 8 Geometry is based upon the pure intuition of space; and, arithmetic accomplishes its concept of number by the successive addition of units in time; and pure mechanics especially cannot attain its concepts of motion without employing the representation of time. Kant stressed that both representations, however, are only intuitions; for if we omit from the empirical intuitions of bodies and their alterations (motion) everything empirical, or belonging to sensation, space and time still remain, which are therefore pure intuitions that lie a priori at the basis of the empirical. 9 Therefore, Kant concluded that pure mathematics, as synthetical cognition a priori, is only possible by referring to no other objects than those of the senses, in which, at the basis of their empirical intuition lies a pure intuition (of space and of time) which is a priori. Kant claimed that this is possible, because the latter intuition is nothing but the mere form of sensibility, which precedes the actual appearance of the objects, in that it, in fact, makes them possible; and yet this faculty of intuiting a priori affects not the matter of the phenomenon 10
Kant illustrated that in ordinary and necessary procedure of geometers, all proofs of the complete congruence of two given figures come ultimately to this that they may be made to coincide; which is evidently nothing else than a synthetical proposition resting upon immediate intuition, and this intuition must be pure, or given a priori, otherwise the proposition could not rank as apodictically certain, but would have empirical certainty only. 11 Kant further claimed that everywhere space has three dimensions, and that space cannot in any way have more, is based on the proposition that not more than three lines can intersect at right angles in one point. Kant argued that drawing the line to infinity and representing the series of changes e.g. spaces travers by motion can only attach to intuition, then he concluded that the basis of mathematics actually are pure intuitions; while the transcendental deduction of the notions of space and of time explains, at the same time, the possibility of pure mathematics. 12
In the Remark I, Kant elaborated that pure mathematics, and especially pure geometry, can only have objective reality on condition that they refer to objects of sense. But in regard to the latter the principle holds good, that our sense representation is not a representation of things in themselves but of the way in which they appear to us. Hence it follows, that the propositions of geometry are not the results of a mere creation of our poetic imagination, and that therefore they cannot be referred with assurance to actual objects; but rather that they are necessarily valid of space, and consequently of all that may be found in space, because space is nothing else than the form of all external appearances, and it is this form alone in which objects of sense can be given. (Immanuel Kant Prolegomena To Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Remark 1, 287) Sensibility, the form of which is the basis of geometry, is that upon which the possibility of external appearance depends. Therefore these appearances can never contain anything but what geometry prescribes to them. The space of the geometer would be considered a mere fiction, and it would not be credited with objective validity, because we cannot see how things must of necessity agree with an image of them, which we make spontaneously and previous to our acquaintance with them. 13
But if this image, or rather this formal intuition, is the essential property of our sensibility, by means of which alone objects are given to us, and if this sensibility represents not things in themselves, but their appearances: we shall easily comprehend, and at the same time indisputably prove, that all external objects of our world of sense must necessarily coincide in the most rigorous way with the propositions of geometry; because sensibility by means of its form of external intuition, viz., by space, the same with which the geometer is occupied, makes those objects at all possible as mere appearances. 14
Because the space of the geometer is exactly the form of sensuous intuition which we find a priori in us, and contains the ground of the possibility of all external appearances (according to their form), and the latter must necessarily and most rigidly agree with the propositions of the geometer, which he draws not from any fictitious concept, but from the subjective basis of all external phenomena, which is sensibility itself. 15
Because it would be absurd to base an analytical judgment on experience, as our concept suffices for the purpose without requiring any testimony from experience, Kant concluded that Empirical judgments are always synthetical, e.g. “That body is extended” is a judgment established a priori, and not an empirical judgment. And also, for before appealing to experience, we already have all the conditions of the judgment in the concept, from which we have but to elicit the predicate according to the law of contradiction, and thereby to become conscious of the necessity of the judgment, Kant concluded that which experience could not even teach us.16 According to Kant, Mathematical judgments are all synthetical and he argued that this fact seems hitherto to have altogether escaped the observation of those who have analyzed human reason; it even seems directly opposed to all their conjectures, though incontestably certain, and most important in its consequences. Further he claimed that for as it was found that the conclusions of mathematicians all proceed according to the law of contradiction (as is demanded by all apodictic certainty), men persuaded themselves that the fundamental principles were known from the same law. “This was a great mistake”, he said. He then delivered the reason that for a synthetical proposition can indeed be comprehended according to the law of contradiction, but only by presupposing another synthetical proposition from which it follows, but never in itself.17 To support this argument, Kant started to examined the case of addition 7 + 5 = 12. According to him, it might at first be thought that the proposition 7 + 5 = 12 is a mere analytical judgment, following from the concept of the sum of seven and five, according to the law of contradiction. However, accordingly, if we closely examine the operation, it appears that the concept of the sum of 7+5 contains merely their union in a single number, without its being at all thought what the particular number is that unites them. Therefore, he concluded that the concept of twelve is by no means thought by merely thinking of the combination of seven and five; and analyze this possible sum as we may, we shall not discover twelve in the concept. Kant suggested that first of all, we must observe that all proper mathematical judgments are a priori, and not empirical. According to him, mathematical judgments carry with them necessity, which cannot be obtained from experience, therefore, it implies that it contains pure a priori and not empirical cognitions.18
We, then, must go beyond these concepts, by calling to our aid some concrete image [Anschauung], i.e., either our five fingers, or five points (as Segner has it in his Arithmetic), and we must add successively the units of the five, given in some concrete image [Anschauung], to the concept of seven; hence our concept is really amplified by the proposition 7 + 5 = I 2, and we add to the first a second, not thought in it. Ultimately, Kant concluded that arithmetical judgments are therefore synthetical, and the more plainly according as we take larger numbers; for in such cases it is clear that, however closely we analyze our concepts without calling visual images (Anscliauung) to our aid, we can never find the sum by such mere dissection. 19
Similarly, Kant argued that all principles of geometry are no less analytical. He illustrated that the proposition “a straight line is the shortest path between two points”, is a synthetical proposition because the concept of straight contains nothing of quantity, but only a quality. He claimed that the attribute of shortness is therefore altogether additional, and cannot be obtained by any analysis of the concept; and its visualization [Anschauung] must come to aid us; and therefore, it alone makes the synthesis possible.20 Kant confronted the previous geometers assumption which claimed that other mathematical principles are indeed actually analytical and depend on the law of contradiction. However, he strived to show that in the case of identical propositions, as a method of concatenation, and not as principles, e. g., a=a, the whole is equal to itself, or a + b > a, the whole is greater than its part. He then claimed that although they are recognized as valid from mere concepts, they are only admitted in mathematics, because they can be represented in some visual form [Anschauung].21

Notes
1. A synthetic proposition is a proposition that is capable of being true or untrue based on facts about the world - in contrast to an analytic proposition which is true by definition. (From Wikipedia, the free encyclopedia)
2. A priori knowledge is propositional knowledge that can be had without experience. It is usually contrasted with a posteriori knowledge, which requires experience. Mathematics and logic are usually considered a priori disciplines. The natural and social sciences are usually considered a posteriori disciplines (From Wikipedia, the free encyclopedia)
3. Immanuel Kant, Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect. 6.
4. This observation on the nature of mathematics gives us a clue to the first and highest condition of its possibility, which is, that some pure intuition [reine Anschauung] must form its basis, in which all its concepts can be exhibited or constructed, in concreto and yet a priori. If we can locate this pure intuition and its possibility, we may thence easily explain how synthetical propositions a priori are possible in pure mathematics, and consequently how this science itself is possible. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sec.7. para. 281
5. Pure intuition [viz., the visualization of forms in our imagination, from which every thing sensual, i.e., every thought of material qualities, is excluded]. Here intuition, being an intuition a priori, is before all experience, viz., before any perception of particular objects, inseparably conjoined with its concept. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect. 7. para. 281
6. Hence it follows: that propositions, which concern this form of sensuous intuition only, are possible and valid for objects of the senses; as also, conversely, that intuitions which are possible a priori can never concern any other things than objects of our senses. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) sec.9 para. 282
7. Space is the form of the external intuition of this sensibility, and the internal determination of every space is only possible by the determination of its external relation to the whole space, of which it is a part (in other words, by its relation to the external sense).
8. If it proceeded in any other way, it would be impossible to make any headway, for mathematics proceeds, not analytically by dissection of concepts, but synthetically, and if pure intuition be wanting, there is nothing in which the matter for synthetical judgments a priori can be given. Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible? Sect.10 para. 283
9. Hence they can never be omitted, but at the same time, by their being pure intuitions a priori, they prove that they are mere forms of our sensibility, which must precede all empirical intuition, or perception of actual objects, and conformably to which objects can be known a priori, but only as they appear to us. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sec.10 para. 283.
10. Phenomenon is the sense-element in it, for this constitutes that which is empirical), but its form, viz., space and time. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect.11 para 284
11. In that case, it could only be said that it is always found to be so, and holds good only as far as our perception reaches. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect. 12 para 285
12. Without some such deduction its truth may be granted, but its existence could by no means be understood, and we must assume "that everything which can be given to our senses (to the external senses in space, to the internal one in time) is intuited by us as it appears to us, not as it is in itself." (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Sect.12 para. 285
13. It would be quite otherwise if the senses were so constituted as to represent objects as they are in themselves. (Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Remark I, para 287
14. Immanuel Kant Prolegomena to Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Remark 1 , para. 287
15. In this and no other way can geometry be made secure as to the undoubted objective reality of its propositions against all the intrigues of a shallow Metaphysics, which is surprised at them [the geometrical propositions], because it has not traced them to the sources of their concepts. (Immanuel Kant Prolegomena To Any Future Metaphysics , First Part Of The Transcendental Problem: How Is Pure Mathematics Possible?) Remark I, para. 288
16. Existence and Reality(Kant e-text reading I) Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2): Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )
17. Kant suggested that first of all, we must observe that all proper mathematical judgments are a priori, and not empirical. According to him, mathematical judgments carry with them necessity, which cannot be obtained from experience, therefore, it implies that it contains pure a priori and not empirical cognitions. (Existence and Reality(Kant e-text reading I) Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2): Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )
18. Existence and Reality(Kant e-text reading I: Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2: Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )
19. ibid.
20. ibid.
21. What usually makes us believe that the predicate of such apodictic judgments is already contained in our concept, and that the judgment is therefore analytical, is the duplicity of the expression, requesting us to think a certain predicate as of necessity implied in the thought of a given concept, which necessity attaches to the concept. But the question is not what we are requested to join in thought to the given concept, but what we actually think together with and in it, though obscurely; and so it appears that the predicate belongs to these concepts necessarily indeed, yet not directly but indirectly by an added visualization [Anschauung]. (Existence and Reality(Kant e-text reading I) Texts For Discussion Page: Prolegomena to Any Future Metaphysics Preamble, Section 2): Kant's argument in support of his view that all properly mathematical judgments are synthetic a priori judgments )

What is Mathematical Thinking and Whay is it Important?

WHAT IS MATHEMATICAL THINKING
AND WHY IS IT IMPORTANT?
By: Kaye Stacey
University of Melbourne, Australia
INTRODUCTION
This paper and the accompanying presentation has a simple message, that
mathematical thinking is important in three ways.
• Mathematical thinking is an important goal of schooling.
• Mathematical thinking is important as a way of learning mathematics.
• Mathematical thinking is important for teaching mathematics.
Mathematical thinking is a highly complex activity, and a great deal has been written and studied about it. Within this paper, I will give several examples of mathematicalthinking, and to demonstrate two pairs of processes through which mathematicalthinking very often proceeds:
• Specialising and Generalising
• Conjecturing and Convincing.
Being able to use mathematical thinking in solving problems is one of the most thefundamental goals of teaching mathematics, but it is also one of its most elusive goals.
It is an ultimate goal of teaching that students will be able to conduct mathematical investigations by themselves, and that they will be able to identify where the mathematics they have learned is applicable in real world situations. In the phrase of the mathematician Paul Halmos (1980), problem solving is “the heart of mathematics”. However, whilst teachers around the world have considerable successes with achieving this goal, especially with more able students, there is always a great need for improvement, so that more students get a deeper appreciation of what it means to think mathematically and to use mathematics to help in their daily and working lives.
MATHEMATICAL THINKING IS AN IMPORTANT GOAL OF SCHOOLING
The ability to think mathematically and to use mathematical thinking to solve problems is an important goal of schooling. In this respect, mathematical thinking will support science, technology, economic life and development in an economy. Increasingly, governments are recognising that economic well-being in a country is underpinned by strong levels of what has come to be called ‘mathematical literacy’(PISA, 2006) in the population. Mathematical literacy is a term popularised especially by the OECD’s PISA program of international assessments of 15 year old students. Mathematical literacy is the ability to use mathematics for everyday living, and for work, and for further study, and so the PISA assessments present students with problems set in realistic contexts. The framework used by PISA shows that mathematical literacy involves many components of mathematical thinking, including reasoning, modelling and making connections between ideas. It is clear then, that mathematical thinking is important in large measure because it equips students with the ability to use mathematics, and as such is an important outcome of schooling.
At the same time as emphasising mathematics because it is useful, schooling needs to give students a taste of the intellectual adventure that mathematics can be. Whilst the highest levels of mathematical endeavour will always be reserved for just a tiny minority, it would be wonderful if many students could have just a small taste of the spirit of discovery of mathematics as described in the quote below from Andrew Wiles, the mathematician who proved Fermat’s Last Theorem in 1994. This problem had been unsolved for 357 years.One enters the first room of the mansion and it’s dark. One stumbles around bumping into furniture, but gradually you learn where each piece of furniture is. Finally, after six months of so, you find the light switch, you turn it on, and suddenly it’s all illuminated.
You can see exactly where you were. Then you move into the next room and spend another six months in the dark. So each of these breakthroughs, while sometimes they’re momentary, sometimes over a period of a day or two, they are the culmination of, and couldn’t exist without, the many months of stumbling around in the dark that precede them. (Andrew Wiles, quoted by Singh, 1997, p236, 237) At the APEC meeting in Tokyo in January 2006, Jan de Lange spoke in detail about the use of mathematics to equip young people for life, so I will instead focus this paper on two other ways in which mathematical thinking is important.
WHAT IS MATHEMATICAL THINKING?
Since mathematical thinking is a process, it is probably best discussed through examples, but before looking at examples, I briefly examine some frameworks provided to illuminate mathematical thinking, going beyond the ideas of mathematical literacy. There are many different ‘windows’ through which the mathematical thinking can be viewed. The organising committee for this conference (APEC, 2006) has provided a substantial discussion on this point. Stacey (2005) gives a review of how mathematical thinking is treated in curriculum documents in Australia, Britain and USA. One well researched framework was provided by Schoenfeld (1985), who organised his work on mathematical problem solving under four headings: the resources of mathematical knowledge and skills that the student brings to the task, the heuristic strategies that that the student can use in solving problems, the monitoring and control that the student exerts on the problem solving process to guide it in productive directions, and the beliefs that the student holds about mathematics, which enable or disable problem solving attempts. McLeod (1992) has supplemented this view by expounding on the important of affect in mathematical problem solving.
In my own work, I have found it helpful for teachers to consider that solving problems with mathematics requires a wide range of skills and abilities, including:
• Deep mathematical knowledge
• General reasoning abilities
• Knowledge of heuristic strategies
• Helpful beliefs and attitudes (e.g. an expectation that maths will be useful)
• Personal attributes such as confidence, persistence and organisation
• Skills for communicating a solution.
Of these, the first three are most closely part of mathematical thinking.
In my book with John Mason and Leone Burton (Mason, Burton and Stacey, 1982), we provided a guide to the stages through which solving a mathematical problem is likely to pass (Entry, Attack, Review) and advice on improving problem solving performance by giving experience of heuristic strategies and on monitoring and controlling the problem solving process in a meta-cognitive way. We also identified four fundamental processes, in two pairs, and showed how thinking mathematically very often proceeds by alternating between them:
• specialising – trying special cases, looking at examples
• generalising - looking for patterns and relationships
• conjecturing – predicting relationships and results
• convincing – finding and communicating reasons why something is true.
I will illustrate these ideas in the two examples below. The first example examines the mathematical thinking of the problem solver, whilst the second examines the mathematical thinking of the teacher. The two problems are rather different – the second is within the mainstream curriculum, and the mathematical thinking is guided by the teacher in the classroom episode shown. The first problem is an open problem, selected because it is similar to open investigations that a teacher might choose to use, but I hope that its unusual presentation will let the audience feel some of the mystery
and magic of investigation afresh.
MATHEMATICAL THINKING IS IMPORTANT AS A WAY OF LEARNING MATHEMATICS
In this section, I will illustrate these four processes of mathematical thinking in the context of a problem that may be used to stimulate mathematical thinking about numbers or as an introduction to algebra. If students’ ability to think mathematically is an important outcome of schooling, then it is clear that mathematical thinking must feature prominently in lessons.Number puzzles and tricks are excellent for these purposes, and in the presentation I will use a number puzzle in a format of the Flash Mind Reader, created by Andy Naughton and published on the internet (HREF1). The Flash Mind Reader does not look like a number puzzle. Indeed its creator writes: We have been asked many times how the Mind Reader works, but will not publish that information on this website. All magicians […] do not give away how their effects work.The reason for this is that it spoils the fun for those who like to remain mystified and when you do find out how something works it's always a bit of a let-down. If you are really keen to find out how it works we suggest that you apply your brain and try to work it out on paper or search further afield. (HREF1). As with many other number tricks, an audience member secretly chooses a number (and a symbol), a mathematical process is carried out, and the computer reveals the audience member’s choice. In this case, a number is chosen, the sum of the digits is subtracted from the number and a symbol corresponding to this number is found from a table. The computer then magically shows the right symbol. The Flash Mind Reader is too difficult to use in most elementary school classes, the target of this conference, but I have selected it so that my audience of mathematics education experts can experience afresh some of the magic and mystery of numbers. As the group works towards a solution, we have many opportunities to observe mathematical thinking in action. Through this process of shared problem solving as we investigate the Flash Mind Reader, I hope to make the following points about mathematical thinking. Firstly, when people first see the Flash Mind Reader, mathematical explanations are far from their minds. Some people propose that it really does read minds, and they may try to test their theory by not concentrating hard on the number that they choose. Others hypothesise that the program exerts some psychological power over the person’s choice of number. Others suggest it is only an optical illusion, resulting from staring at the screen. This illustrates that a key component of mathematical thinking is having a disposition to looking at the world in a mathematical way, and an attitude of seeking a logical explanation. As we seek to explain how the Flash Mind reader works, the fundamental processes of thinking mathematically will be evident. The most basic way of trying to understand a problem situation is to try the Flash Mind Reader several times, with different numbers and different types of numbers. This helps us understand the problem (in this case, what is to be explained) and to gather some information. This is a simple example of specialising, the first of the four processes of thinking mathematically processes. As we enter more deeply into the problem, specialising changes its character. First we may look at one number, noting that if 87 is the number, then the sum of its digits is 15 and 87 – 15 is 72.
Beginning to work systematically leads to evidence of a pattern:
87 8 + 7 = 15 87 – 15 = 72
86 8 + 6 = 14 86 – 14 = 72
85 8 + 5 = 13 85 – 13 = 72
84 8 + 4 = 12 84 – 12 = 72
and a cycle of experimentation (which numbers lead to 72?, what do other numbers lead to?) and generalising follows. Of course, at this stage it is important to note the value of working with the unclosed expressions such as 8+7 instead of the closed 15, because this reveals the general patterns and reasons so much better. Working with the unclosed expression to reveal
structure is an admirable feature of Japanese elementary education.
87 87 – 7 = 80 80 – 8 = 72
86 86 – 6 = 80 80 – 8 = 72
85 85 – 5 = 80 80 – 8 = 72
84 84 – 4 = 80 80 – 8 = 72
It is also worthwhile noting at this point, that although we are working with a specific example, the aim here is to see the general in the specific. This generalising may lead to a conjecture that the trick works because all starting numbers produce a multiple of 9 and all multiples of 9 have the same symbol. But this conjecture is not quite true and further examination of examples (more specialising) finally identifies the exceptions and leads to a convincing argument. In school, we aim for students to be able to use algebra to write a proof, but even before they have this skill, they can be produce convincing arguments. An orientation to justify and prove (at an appropriate level of formality) is important throughout school. If students are to become good mathematical thinkers, then mathematical thinking
needs to be a prominent part of their education. In addition, however, students who have an understanding of the components of mathematical thinking will be able to use these abilities independently to make sense of mathematics that they are learning. For example, if they do not understand what a question is asking, they should decide themselves to try an example (specialise) to see what happens, and if they are oriented to constructing convincing arguments, then they can learn from reasons rather than rules. Experiences like the exploration above, at an appropriate level build these dispositions.
MATHEMATICAL THINKING IS ESSENTIAL FOR TEACHING MATHEMATICS.
Mathematical thinking is not only important for solving mathematical problems and for learning mathematics. In this section, I will draw on an Australian classroom episode to discuss how mathematical thinking is essential for teaching mathematics. This episode is taken from data collected by Dr Helen Chick, of the University of Melbourne, for a research project on teachers’ pedagogical content knowledge. For other examples, see Chick, 2003; Chick & Baker, 2005, Chick, Baker, Pham & Cheng, 2006a; Chick, Pham & Baker, 2006b). Providing opportunities for students to learn about mathematical thinking requires considerable mathematical thinking on the part of teachers. The first announcement for this conference states that a teacher requires mathematical thinking for analysing subject matter (p. 4), planning lessons for a specified aim (p. 4) and anticipating students’ responses (p. 5).These are indeed key places where mathematical thinking is required. However, in this section, concentrate on the mathematical thinking that is needed on a minute by minute basis in the process of conducting a good mathematics lesson. Mathematical thinking is not just in planning lessons and curricula; it makes a difference to every minute of the lesson.The teacher in this classroom extract is in her fifth year of teaching. She stands out in Chick’s data as one of the teachers in the sample exhibiting the deepest pedagogical content knowledge (Shulman, 1986, 1987). Her pupils are aged about 11 years, and are in Grade 6. This lesson began by reviewing ideas of both area and perimeter. We will examine just the first 15 minutes.The teacher selected an open and reversed task to encourage investigation and mathematical thinking. Students had 1cm grid paper and were all asked to draw a rectangle with an area of 20 square cm. This task is open in the sense that there are multiple correct answers, and it is ‘reversed’ when it is contrasted to the more common task of being given a rectangle and finding its area. The teacher reminded students that area could be measured by the number of grid squares inside a shape. In terms of the processes of mathematical thinking, the teacher at this stage is ensuring that each student is specialising. They are each working on a special case, and coming to know it well, and this will provide an anchor for future discussions and generalisations. I make no claim that the teacher herself analyses this move in this way. As the teacher circulated around the room assisting and monitoring students, she came to a student who asked if he could draw a square instead of a rectangle. In the dialogue which follows, the teachers’ response highlighted the definition of a rectangle, and she encouraged the student to work from the definition to see that a square is indeed a rectangle.
S: Can I do a square?
T: Is a square a rectangle?
T: What’s a rectangle?
T: How do you get something to be a rectangle? What’s the definition of a rectangle?
S: Two parallel lines
T: Two sets of parallel lines … and …
S: Four right angles.
T: So is that [square] a rectangle?
S: Yes.
T: [Pause as teacher realises that student understands that the square is a rectangle, but
there is a measurement error] But has that got an area of 20?
S: [Thinks] Er, no.
T: [Nods and winks]
Other responses to this student would have closed down the opportunity to teach him about how definitions are used in mathematics. To the question “Can I do a square?”, she may have simply replied “No, I asked you to draw a rectangle” or she might have immediately focussed on the error that led the student to ask the question. Instead she saw the opportunity to develop his use of definitions. When the teacher realised that the student had asked about the square because he had made a measurement error, she judged that this was within the student’s own capability to correct, and so she simply indicated that he should check his work. In the next segment, a student showed his 4 x 5 rectangle on the overhead projector, and the teacher traced around it, confirmed its area is 20 square cm and showed that multiplying the length by the width can be used instead of counting the squares, which many students did. In this segment, the teacher demonstrated that reasoning is a key component of doing mathematics. She emphasised the mathematical connections between finding the number of squares covered by the rectangle by repeated addition (4 on the first row, 4 on the next, …) and by multiplication. In her classroom, the formula was not just a rule to be remembered, but it was to be understood. The development of the formula was a clear example of ‘seeing the general in a special case’. The formula was developed from the 4 x 5 rectangle in such a way that the generality of the argument was highlighted. The teacher paid further attention to generalisation and over-generalisation at this point, when a student commented: ‘That’s how you work out area – you do the length times the width’. The teacher seized on this opportunity to address students’ tendency to over-generalise, and teased out, through a short class discussion, that LxW only works for rectangles.
S1: That’s how you work out area -- you do the length times the width.
T: When S said that’s how you find the area of a shape, is he completely correct?
S2: That’s what you do with a 2D shape.
T: Yes, for this kind of shape. What kind of shape would it not actually work for?
S3: Triangles.
S4: A circle.
T: [With further questioning, teases out that LxW only applies to rectangles]
In the next few minutes, the teacher highlighted the link between multiplication and area by asking students to make other rectangles with area 20 square centimetres. Previously all students had made 4 x 5 or 2 x 10, but after a few minutes, the class had found 20 x 1, 1 x 20, 10 x 2, 2 x 10, 4 x 5 and 5 x 4 and had identified all these side lengths as the factors of 20. Making links between different parts of the mathematics curriculum characterises her teaching. Then, in another act of generalisation, the teacher begins to move beyond whole numbers:
T: Are there any other numbers that are going to give an area of 20? [Pauses, as if
uncertain. There is no response from the students at first]
T: No? How do we know that there’s not?
S: You could put 40 by 0.5.
T: Ah! You’ve gone into decimals. If we go into decimals we’re going to have heaps,aren’t we?
After these first 15 minutes of the lesson, the students found rectangles with an area of 16 square centimetres and the teacher stressed the important problem solving strategy of working systematically. Later, in order to contrast the two concepts of area and perimeter, students found many shapes of area 12 square cm (not just rectangles) and determined their perimeters.
Even the first 15 minutes of this lesson show that considerable mathematical thinking on behalf of the teacher is necessary to provide a lesson that is rich in mathematical thinking for students. We see how she draws on her mathematical concepts, deeply understood, and on her knowledge of connections among concepts and the links between concepts and procedures. She also draws on important general mathematical principles such as
• working systematically
• specialising – generalising: learning from examples by looking for the
general in the particular
• convincing: the need for justification, explanation and connections
• the role of definitions in mathematics.
Chick’s work analyses teaching in terms of the knowledge possessed by the teachers. She tracks how teachers reveal various categories of pedagogical content knowledge (Shulman, 1986) in the course of teaching a lesson. In the analysis above, I viewed the lesson from the point of view of the process of thinking mathematically within the lesson rather than tracking the knowledge used. To draw an analogy, in researching a students’ solution to a mathematical problem, a researcher can note the mathematical content used, or the researcher can observe the process of solving the problem. Similarly, teaching can be analysed from the “knowledge’ point of view, or analysed from the process point of view. For those us who enjoy mathematical thinking, I believe it is productive to see teaching mathematics as another instance of solving problems with mathematics. This places the emphasis not on the static knowledge used in the lesson asabove but on a process account of teaching. In order to use mathematics to solve a problem in any area of application, whether it is about money or physics or sport or engineering, mathematics must be used in combination with understanding from the area of
application. In the case of teaching mathematics, the solver has to bring together expertise in both mathematics and in general pedagogy, and combine these two domains of knowledge together to solve the problem, whether it be to analyse subject matter, to create a plan for a good lesson, or on a minute-by-minute basis to respond to students in a mathematically productive way. If teachers are to encourage mathematical thinking in students, then they need to engage in mathematical thinking throughout the lesson themselves.
References
APEC –Tsukuba (Organising Committee) (2006) First announcement. InternationalConference on Innovative Teaching of Mathematics through Lesson Study. CRICED, University of Tsukuba.
Chick, H. L. (2003). ‘Pre-service teachers’ explanations of two mathematical concepts’Proceedings of the 2003 conference, Australian Association for Research in Education.From: http://www.aare.edu.au/03pap/chi03413.pdf
Chick, H.L. and Baker, M. (2005) ‘Teaching elementary probability: Not leaving it to chance’, in P.C. Clarkson, A. Downton, D. Gronn, M. Horne, A. McDonough, R. Pierce & A. Roche (eds.) Building Connections: Theory, Research and Practice. (Proceedings of the 28th annual conference of the Mathematics Education Research Group of Australasia), MERGA, Sydney, pp. 233-240.
Chick, H.L., Baker, M., Pham, T., and Cheng, H. (2006a) ‘Aspects of teachers’ pedagogical content knowledge for decimals’, in J. Novotná, H. Moraová, M. Krátká, & N.Stehlíková (eds.), Proc. 30th conference e International Group for the Psychology of Mathematics Education, PME, Prague, Vol. 2, pp. 297-304.
Chick, H.L., Pham, T., and Baker, M. (2006b) ‘Probing teachers’ pedagogical content knowledge: Lessons from the case of the subtraction algorithm’, in P. Grootenboer, R.Zevenbergen, & M. Chinnappan (eds.), Identities, Cultures and Learning Spaces (Proc.29th annual conference of Mathematics Education Research Group of Australasia),MERGA, Sydney, pp. 139-146.
Halmos, P. (1980). The heart of mathematics. American Mathematical Monthly, 87(7), 519– 524.HREF1 CyberGlass Design - The Flash Mind Reader. http://www.cyberglass.biz Accessed 28 November 2006.
Mason, J. Burton, L. and Stacey, K. (1982) Thinking Mathematically. London: Pearson.(Also available in translation in French, German, Spanish, Chinese, Thai (2007))
McLeod, D.B. (1992) Research on affect in mathematics education: a reconceptualisation.In D.A. Grouws, Ed., Handbook of research on mathematics teaching and learning, (pp.575–596).New York: MacMillan, New York.
PISA (Programme for International Student Assessment) (2006) Assessing Scientific,Reading and Mathematical Literacy. A Framework for PISA 2006. Paris: OECD.
Schoenfeld, A. (1985) Mathematical Problem Solving. Orlando: Academic Press.
Shulman, L.S. (1986) Those who understand: Knowledge growth in teaching, EducationalResearcher 15 (2), 4-14.
Shulman, L.S. (1987) Knowledge and teaching: Foundations of the new reform, HarvardEducational Review 57(1), 1-22.
Singh, S. (1997) Fermat’s Enigma, New York: Walter
Stacey, K. & Groves, S. (1985) Strategies for Problem Solving. Lesson Plans forDeveloping Mathematical Thinking. Melbourne: Objective Learning Materials.
Stacey, K. & Groves, S. (2001) Resolver Problemas: Estrategias. Madrid: Lisbon.
Stacey, Kaye (2005) The place of problem solving in contemporary mathematics curriculum documents. Journal of Mathematical Behavior 24, pp 341 – 350.

Mathematical Thinking and How to Teach It?

Mathematical Thinking and How to Teach It
By Shigeo Katagiri
Translated of the rewritten version from Shikgeo Katagiri (2004)., Mathematical Thinking and How to Teach It. Meijitosyo Publishers, Tokyo. Copyright of English version has CRICED, University of Tsukuba. All rights reserved.
Table of Contents
Chapter 1 The Aim of Education and Mathematical Thinking
Chapter 2 The Importance of Teaching to Cultivate Mathematical Thinking
2.1 The Importance of Teaching Mathematical Thinking
2.2 Example 1: How Many Squares are There?
Chapter 3 The Meaning of Mathematical Thinking and How to Teach It
3.1 Characteristics of Mathematical Thinking
3.2 Substance of Mathematical Thinking
List of Types of Mathematical Thinking
I. Mathematical Attitudes
II. Mathematical Thinking Related to Mathematical Methods
III. Mathematical Thinking Related to Mathematical Contents
Chapter 4 Detailed Discussion of Mathematical Thinking Related to Mathematical Methods
Chapter 5 Detailed Discussion of Mathematical Thinking Related to Mathematical Substance
Chapter 6 Detailed Discussion of Mathematical Attitudes Chapter 7 Questions for Eliciting Mathematical Thinking
Chapter 1
The Aim of Education and Mathematical Thinking
1. The Aim of Education: Scholastic Ability From the Perspective of “Cultivating Independent Persons”
School-based education must be provided to achieve educational goals. “Scholastic ability” becomes clear when one views the aim of school-based education.
The Aim of School Education
The aim of school education is described as follows in a report by the Curriculum Council: “To cultivate qualifications and competencies among each individual school child, including the ability to find issues by oneself, to learn by oneself, to think by oneself, to make judgments independently and to act, so that each child or student can solve problems more skillfully, regardless of how society might change in the future.”
This guideline is a straightforward expression of the preferred aim of education.
The most important ability that children need to gain at present and in the future, as society, science, and technology advance dramatically, is not the ability to correctly and quickly execute predetermined tasks and commands, but rather the ability to determine for themselves what they should do, or what they should charge themselves with doing.
Of course, the ability to correctly and quickly execute necessary tasks is also necessary, but from now on, rather than adeptly imitating the skilled methods or knowledge of others, the ability to come up with one’s own ideas, no matter how small, and to execute one’s own independent, preferable actions (ability full of creative ingenuity) will be most important. This is why the aim of education from now on is to instill the ability (scholastic ability) to take these kinds of actions. Furthermore, this is something that must be instilled in every individual child or student. From now, it will be of particular importance for each individual school child to be able to act independently (rather than the entire class acting independently as a unit). Of course, not every child will be able to act independently at the same level, but each school child must be able to act independently according to his or her own capabilities. To this end, teaching methods that focus on the individual are important.
2. The Scholastic Ability to Think and Make Judgments Independently Is Mathematical Thinking
– Looking at Examples –
The most important ability that arithmetic and mathematics courses need to cultivate in order to instill in students this ability to think and make judgments independently is mathematical thinking. This is why cultivation of this “mathematical thinking” has been an objective of arithmetic and mathematics courses in Japan since the year 1950. Unfortunately, however, the teaching of mathematical thinking has been far from adequate in reality.
One sign of this is the assertion by some that “if students can do calculations, that is enough.”
The following example illustrates just how wrong this assertion is:
Example: “Bus fare for a trip is 4,500 yen per person. If a bus that can seat 60 people is rented out, however, this fare is reduced by 20% per person. How many people would need to ride for it to be a better deal to rent out an entire bus?”
This problem is solved in the following manner:
Solving Method When a bus is rented – One person’s fee: 4,500×0.8=3,600
In the case of 60 people: 3,600×60=216,000
With individual tickets, the number of people that can ride is
216,000÷4,500=48 (people).
Therefore, it would be cheaper to rent the bus if more than 48 people ride.
Sixth-graders must be able to solve a problem of this level. Is it sufficient, however, to solve this problem just by being able to do formal calculation (calculation on paper or mental calculation, or the use of an abacus or calculator)? Regardless of how skilled a student is at calculation on paper, and regardless of whether or not a student is allowed to use a calculator at will, these skills alone are not enough to solve the problem. The reason is that before one calculates on paper or with a calculator, one must be able to make the judgment “what numbers need to be calculated, what are the operations that need to be performed on those numbers, and in what order should these operations be performed?” If a student is not able to make these judgments, then there’s not much point in calculating on paper or with a calculator. Formal calculation is a skill that is only useful for carrying out commands such as “calculate this and this” (a formula for calculation) once these commands are actually specified. Carrying out these commands is known as “deciding the operation.” Therefore, “deciding the operation” for oneself in order to determine which command is necessary to “calculate this and this” is an important skill that is indispensable for solving problems.
Deciding the operation clearly determines the meaning of each computation, and decides what must be done based on that meaning. This is why “the ability to clarify the meanings of addition, subtraction, multiplication, and division and determine operations based on these meanings” is the most important ability required for computation.
Actually, there is something more important – in order to correctly decide which operations to use in this way, one must be able to think in the following manner “I would like to determine the correct operations, and to do so, I need to recall the meanings of each operation, and think based on these meanings.” This thought process is one kind of mathematical thinking.
Even if a student solves the group discount problem as described above, this might not be sufficient to conclude that he truly understood the problem. This is why it is important to “change the conditions of the problem a little” and “consider whether or not it is still possible solve the problem in the same way.” These types of thinking are neither knowledge nor a skill. They are “functional thinking” and “analogical thinking.”
For instance, let’s try changing one of the conditions by “changing the bus fare from 4,500 yen to 4,000 yen.” Calculating again as described above results in an answer of 48 people (actually, a better way of thinking is to replace 4,500 above with 4,000 – this is analogical thinking). In this way, one should gain confidence in one’s method of solving the problem, as one realizes that the result is the same: 48 people.
The above formulas are expressed in a way that is insufficient for students in fourth grade or higher. It is necessary to express problems using a single formula whenever possible. When these formulas are converted into a single formula based on this thinking, this is the result:4,500×(1-0.2) ×60÷4,500
When viewed in this form, it becomes apparent that the formula is simply 60×(1-0.2)
What is important here is the idea of “reading the meaning of this formula.” This is important “mathematical thinking regarding formulas.” Reading the meaning of this formula gives us: full capacity × ratio
For this reason, even if the bus fare changes to 4,000 yen, the formula 60×0.8=48 is not affected. Furthermore, if the full capacity is 50 persons and the group discount is 30%, then regardless of what the bus fare may be, the problem can always be solved as “50×0.7=35; the group rate (bus rental) is a better deal with 35 or more people.” This greatly simplifies the result, and is an indication of the appreciation of mathematical thinking, namely “conserving cogitative energy” and “seeking a more beautiful solution.”
Students should have the ability to reach the type of solution shown above independently. This is a desirable scholastic ability that includes the following aims:
• Clearly grasp the meaning of operations, and decide which operations to use based on this understanding
• Functional thinking
• Analogical thinking
• Expressing the problem with a better formula
• Reading the meaning of a formula
• Economizing thought and effort (seeking a better solution)
Although this is only a single example, this type of thinking is generally applicable. In other words, in order to be able to independently solve problems and expand upon problems and solving methods, the ability to use “mathematical thinking” is even more important than knowledge and skill, because it enables to drive the necessary knowledge and skill.
Mathematical thinking is the “scholastic ability” we must work hardest to cultivate in arithmetic and mathematics courses.
3. The Hierarchy of Scholastic Abilities and Mathematical Thinking
As the previous discussion makes clear, there is a hierarchy of scholastic abilities. When related to the above discussion, and limited to the area of computation (this is the same as in other areas, and can be generalized), these scholastic abilities enable the following (from lower to higher levels):
1. The ability to memorize methods of formal calculation and to carry out these calculation
2. The ability to understand the rules of calculation and how to carry out formal calculation
3. The ability to understand the meaning of each operation, to decide which operations to use based on this understanding, and to solve simple problems
4. The ability to form problems by changing conditions or abstracting situations
5. The ability to creatively make problems and solve them
The higher the level, the more important it is to cultivate independent thinking in individuals. To this end, mathematical thinking is becoming even more and more necessary.
Chapter 2
The Importance of Teaching to Cultivate Mathematical Thinking
2.1 The Importance of Teaching Mathematical Thinking
As we found in the previous chapter, the method of thinking is the center of scholastic ability. In arithmetic and mathematics courses, mathematical thinking is the center of scholastic ability. However, in Japan, in spite of the fact that the improvement of mathematical thinking was established as a goal more than 45 years ago, the teaching of mathematical thinking is by no means sufficient.
One of the reasons that teaching to cultivate mathematical thinking does not tend to happen is, teachers are of the opinion that students can still learn enough arithmetic even if they don’t teach in a way to cultivate the students’ mathematical thinking. In other words, teachers do not understand the importance of mathematical thinking.
The second reason is that, in spite of the fact that mathematical thinking was established as a goal, teachers do not understand what it really is. It goes without saying that teachers cannot teach what they themselves do not understand.
Therefore, we shall start out by explaining how important the teaching of mathematical thinking is.
A simple summary follows.
Mathematical thinking allows for:
(1) An understanding of the necessity of using knowledge and skills
(2) Learning how to learn by oneself, and the attainment of the abilities required for independent learning
(1) The Driving Forces to Pursue Knowledge and Skills
Mathematics involves the teaching of many different areas of knowledge, and of many skills. If children are simply taught to “use some knowledge or skill” to solve problems, they will use that knowledge or skill. In this case, however, children will not realize why they are being told to use such a knowledge or skill. Also, when new knowledge or skills are required for problem solving and students are taught what skill to use, they will be able to use that skill to solve the problem, but they will not know why this skill must be used. Students will therefore fail to understand why the new skill is good.
What is important is “how to realize” which previously learned knowledge and skills should be used. It is also important to “sense the necessity of” and “perceive the need or desirability of using” new knowledge and skills.
Therefore, it is necessary for something to act as a drive towards the required knowledge and skills. Children first understand the benefits of using knowledge and skills when they possess and utilize such a drive. This leads them to fully acquire the knowledge and skills they have used.
Mathematical thinking acts as this drive.
(2) Achieving Independent Thinking and the Ability to Learn Independently
Possession of this driving force gives students an understanding of how to learn by themselves.
Cultivating the power to think independently will be the most important goal in education from now on, and in the case of arithmetic and mathematics courses, mathematical thinking will be the most central ability required for independent thinking. By mastering this skill even further, students will attain the ability to learn independently.
The following specific example serves to clarify this point further.
2.2 Example: How Many Squares are There? This instructional material is appropriate for fourth-grade students.
1. The Usual Lesson Process
This is usually taught in the following way (T refers to the teacher, and C the children):
T: There are both big and small squares here. Let’s count how many squares there are in total.
T: (When the children start counting) First, how many small squares are there? C: 25.
T: Which squares are the second smallest?
C: (Indicate the squares using two by two segments)
T: Count the number of those squares.
T: Which squares are the next biggest size, and how many are there?
The questions continue in this manner in order of size. In each case, the teacher asks one child the number, and then asks another child if this number is correct. Alternatively, the teacher might recognize the correctness of the number, and comment “yes, that’s the right number.”
The teacher has the children count squares in order of size, and then has the children add the numbers together to derive the grand total.
2. Problems with This Method
a) When the teacher instructs children to count squares based on size, the children do not realize for themselves that they should sort the squares into groups. As a result, the children do not understand the need to sort, or the thinking behind sorting.
b) The number of squares of each size is determined either by the majority of the children’s answers, or based on the teacher’s approval. These methods are not the right way of determining the correct answer. Correctness must be determined based on solid rationales.
c) Also, if instruction regarding this problem ends this way, children will only know the answer to this particular problem. The important things they must grasp, however, are what to focus upon in general, and how to think about problems of this nature.
Teachers should, therefore, follow the following teaching method:
3. Preferred Method
(1) Clarification of the Problem – 1
The teacher gives the children the previous diagram. T: How many squares are there in this diagram?
C: 25 (many children will probably answer this easily).
Some children will probably respond with a larger number.
The children come up with the answer 25 after counting just the smallest squares. Those who think the number is higher are also considering squares with more than one segment per side. This is the source of the issue, which is not about the correct answer, but the vagueness of the mathematical problem.
The teacher should then have the children discuss “which squares they are counting when they arrive at the number 25,” and inform them that “this problem is vague and does not clearly state which squares need to be counted.” The teacher concludes by clarifying the meaning of the problem, saying “let’s count all the squares, of every different size.”
(2) Clarification of the Problem – 2
First, the teacher lets all the children count the squares independently. Various answers will be given when the teacher asks for totals, or the children may become confused while counting. The children will realize that most of them (or all of them) have failed to count correctly. It is then time to think of a way of counting that is a little better and easier (this becomes a problem for the children to solve).
(3) Realizing the Benefit of Sorting The children will realize that the squares should be sorted and counted based on size. The teacher has the children count the squares again, this time sorting according to size.
(4) Knowing the Benefit of Encoding
Once the children are finished counting, the teacher asks them to give their results. At this point, when the teacher asks “how many squares are there of this size, and how many squares are there of that size...” he/she will run into the problem of not being able to clearly indicate “which size.”
At this point, naming (encoding) each square size should be considered. It is important to make sure that the children realize that calling the squares “large, medium, and small” is not preferable because this naming system is limited. However, the children learn that naming the squares in the following way is a good system, as they state each number.
Squares with 1 segment 25
Squares with 2 segment 16
Squares with 3 segment 9
Squares with 4 segment 4
Squares with 5 segment 1
Total 55
(5) Judging the Correctness of Results More Clearly, Based on Solid Rationale The correctness or incorrectness of these numbers must be elucidated, so have one child count the squares again in front of the entire class. The student will probably count the squares while tracing each one, as shown to the right. This will result in a messy diagram, and make it hard to tell which squares are being counted. Tracing each square is inconvenient, and will make the students feel their counting has become sloppy.
(6) Coming up with a More Accurate and Convenient Counting Method
There is a counting method that does not involve tracing squares. Have the students discover that they can count the upper left vertex (corner) of each square instead of tracing, in the following manner: place the pencil on the
upper left vertex and start to trace each square in one’s head, without moving the pencil from the vertex.
By using this system, it is possible to count two-segment squares as shown in the diagram to the right, by simply counting the upper left vertices of each square. This counting method is easier and clearer.
This method takes advantage of the fact that “squares and upper left vertices are in a one-to-one relationship.” In other words, in the case of two-segment squares, once a square is selected, only one vertex will correspond to that square’s upper left corner. The flip side of this principle is that once a point is selected, if that point corresponds to the upper left corner of a square, then it will only correspond to a single square of that size. Therefore, while sorting based on size, instead of counting squares, one can also count the upper left corners.
Instead of counting squares, this method “uses a functional thinking by counting the easy-to-count upper left vertices, which are functionally equivalent to the squares (in a one-to-one relationship).”
(7) Expressing the Number of Squares as a Formula
When viewed in this fashion, the two-segment squares shown in the diagram have the same number as a matrix of four rows by four columns of dots. When one realizes that this is the same as 4×4, it becomes apparent that the total number of squares is as follows: 5×5+4×4+3×3+2×2+1×1 (A)
Students will understand that it’s a good idea to think of ways to devise different expression methods, and to express problems as formulas.
(8) Generalizing
This makes generalization simple. In other words, consider what happens when “the segment length of the original diagram is increased by 1 to a total of 6.” All one needs to do is to add 6×6 to formula (A) above. Thus, the thought process of trying to generalize, and the attempt to read formulas is important.
(9) Further Generalization
For instance (for students in fifth grade or higher), when this system is applied to other diagrams, such as a diagram constructed entirely of rhombus, how will this change the formula? (Answer: It will not change the above formula at all.)
By generalizing to see the case of parallelograms (as long as the counting involves only parallelograms that are similar to the smallest parallelogram, the diagram can be seen in the same way), the true nature of the problem becomes clear.
4. Mathematical Thinking is the Key Ability
Here What kind of ability is required to think in the manner described above? First, what knowledge and skills are required? The requirements are actually extremely simple: Understanding the meaning of “square,” “vertex,” “segment,” and so on The ability to count to around 100
The ability to write the problem as a formula, using multiplication and addition Possession of this understanding and skills, however, is not enough to solve the problem. An additional, more powerful ability is necessary. This ability is represented by the underlined parts above, from (1) to (9):
Clarification of the Meaning of the Problem
Coming up with an Convenient Counting Method
Sorting and Counting
Coming up with a Method for Simply and Clearly Expressing How the Objects Are Sorted Encoding
Replacing to Easy-to-Count Things in a Relationship of Functional Equivalence
Expressing the Counting Method as a Formula
Reading the Formula Generalizing
This is mathematical thinking, which differs from simple knowledge or skills.
It is evident that mathematical thinking serves an important purpose in providing the ability to solve problems on one’s own as described above, and that this is not limited to this specific problem. Therefore, the cultivation of a number of these types of mathematical thinking must be the aim of this class.
Chapter 3
The Meaning of Mathematical Thinking and How to Teach It Characteristics of Mathematical Thinking
Although we have examined a specific example of the importance of teaching that cultivates mathematical thinking during each hour of instruction, for a teacher to be able to teach in this way, he must first have a solid grasp of “what kinds of mathematical thinking there are.” After all, there is no way a person could teach in such a way as to cultivate mathematical thinking without first understanding the kinds of mathematical thinking that exist. Let us consider the characteristics of mathematical thinking.
1. Focus on Sets
Mathematical thinking is like an attitude, as in it can be expressed as a state of “attempting to do” or “working to do” something. It is not limited to results represented by actions, as in “the ability to do,” or “could do” or “couldn’t do” something.
For instance, the states of “working to establish a perspective” and “attempting to analogize, and working to create an analogy” are ways of thinking. If, on the other hand, one has no intention whatsoever of creating an analogy, and is told to “create an analogy,” he/she might succeed in doing so due to having the ability to do so, but this does not mean that he/she consciously thought in an analogical manner.
In other words, mathematical thinking means that when one encounters a problem, one decides which set, or psychological set, to use to solve that problem.
2. Thinking Depends on Three Variables
In this case, the type of thinking to use is not determined by the problem or situation. Rather, the type of thinking to use is determined by the problem (situation), the person, and the approach (strategy) used. In other words, the way of thinking depends on three variables: the problem (situation), the person involved, and the strategy.
Two of these involve the connotative understanding of mathematical thinking. There is also denotative understanding of the same.
3. Denotative Understanding
Concepts are made up of both connotative and denotative components. One method which clarifies the “mathematical thinking” concept is a method of clearly expressing connotative “meaning.” Even if the concept of mathematical thinking is expressed with words, as in “mathematical thinking is this kind of thing,” this will be almost useless when it comes to teaching, because even if one understands the sentences that express this meaning, this does not mean that they will be able to think mathematically.
Instead of describing mathematical thinking this way, it should be shown with concrete examples. At a minimum, doing this allows for the teaching of the type of thinking shown. In other words, mathematical thinking should be captured denotatively.
4. Mathematical Thinking is the Driving Force Behind Knowledge and Skills Mathematical thinking acts as a guiding force that elicits knowledge and skills, by helping one realize the necessary knowledge or skills for solving each problem. It should also be seen as the driving force behind such knowledge and skills.There is another type of mathematical thinking that acts as a driving force for eliciting other types of even more necessary mathematical thinking. This is referred to as the “mathematical attitude.”
3.2 Substance of Mathematical Thinking
It is important to achieve a concrete (denotative) grasp of mathematical thinking, based on the fundamental thinking described in section 3.1. Let us list the various types of mathematical thinking.
First of all, mathematical thinking can be divided into the following three categories:
II. Mathematical Thinking Related to Mathematical Methods
III. Mathematical Thinking Related to Mathematical Contents
Furthermore, the following acts as a driving force behind the above categories: I. Mathematical Attitudes
Although the necessity of category I was mentioned above, further consideration as described below reveals the fact that it is appropriate to divide mathematical thinking into II and III. Mathematical thinking is used during mathematical activities, and is therefore intimately related to the contents and methods of arithmetic and mathematics. Put more precisely, a variety of different methods is applied when arithmetic or mathematics is used to perform mathematical activities, along with various types of mathematical contents. It would be accurate to say that all of these methods and types of contents are types of mathematical thinking. It is because of the ways of thinking that the existence of these methods and types of contents has meaning. Let us focus upon these types of contents and methods as we examine mathematical thinking from these two angles.
For this reason, three logical categories can be derived.
Specific details are provided below.
List of Types of Mathematical Thinking
I. Mathematical Attitudes
1. Attempting to grasp one’s own problems or objectives or substance clearly, by oneself
(1) Attempting to have questions
(2) Attempting to maintain a problem consciousness
(3) Attempting to discover mathematical problems in phenomena
2. Attempting to take logical actions
(1) Attempting to take actions that match the objectives
(2) Attempting to establish a perspective
(3) Attempting to think based on the data that can be used, previously learned items, and assumptions
3. Attempting to express matters clearly and succinctly
(1) Attempting to record and communicate problems and results clearly and succinctly
(2) Attempting to sort and organize objects when expressing them
4. Attempting to seek better things
(1) Attempting to raise thinking from the concrete level to the abstract level
(2) Attempting to evaluate thinking both objectively and subjectively, and to refine thinking
(3) Attempting to economize thought and effort
II. Mathematical Thinking Related to Mathematical Methods
1. Inductive thinking
2. Analogical thinking
3. Deductive thinking
4. Integrative thinking (including expansive thinking)
5. Developmental thinking
6. Abstract thinking (thinking that abstracts, concretizes, idealizes, and thinking that clarifies conditions)
7. Thinking that simplifies
8. Thinking that generalizes
8. Thinking that specializes
9. Thinking that symbolize
10. Thinking that express with numbers, quantifies, and figures
III. Mathematical Thinking Related to Mathematical Contents
1. Clarifying sets of objects for consideration and objects excluded from sets, and clarifying conditions for inclusion (Idea of sets)
2. Focusing on constituent elements (units) and their sizes and relationships (Idea of units)
3. Attempting to think based on the fundamental principles of expressions (Idea of expression)
4. Clarifying and extending the meaning of things and operations, and attempting to think based on this (Idea of operation)
5. Attempting to formalize operation methods (Idea of algorithm)
6. Attempting to grasp the big picture of objects and operations, and using the result of this understanding (Idea of approximation)
7. Focusing on basic rules and properties (Idea of fundamental properties)
8. Attempting to focus on what is determined by one’s decisions, finding rules of relationships between variables, and to use the same (Functional Thinking)
9. Attempting to express propositions and relationships as formulas, and to read their meaning (Idea of formulas)
Once the student has written part of the number table, he/she can induce that “it is possible to move from one multiple of 8 to another by going down one row, and then left two columns.” When stated the same way for multiples of 4, it is also possible to induce that “it is possible to move from one multiple of 4 to another by going down one row, and then left two columns.”
Then considering “why it is possible to make this simple statement” and “whether or not it is still possible to state this for numbers over 99, and why this is the case” is deductive thinking. Next, consider what to base an explanation of this on. One will realize at this point that it is possible to base this on how the number table is created. This is also deductive thinking, and is based upon the following.
Since this number table has 10 numbers in each row, “going one position to the right increases the number by one, and going one position down increases the number by ten.” Based upon this, it is evident that going down one position always adds 10, and going left two positions always subtracts 2. Combining both of these moves always results in an increase of 8 (10-2=8). Therefore, if one adds 8 to a multiple of 4 (or a multiple of 8), the result will always be a multiple of 4 (8). This explains what is happening.
By achieving results with one’s own abilities in this way, it is possible to gain confidence in the correctness of one’s conclusion, and to powerfully assert this conclusion. Always try to explain the truth of what you have induced, and you will feel this way. Also, think about general explanations based on clear evidence (the creation of the number table). This is deductive thinking.
Example 2: Deductive thinking is not just used in upper grades, but is used in lower grades as well.
Assume that at the start of single-digit multiplication in 3rd grade, the problem “how many sheets of paper would you need to hand out 16 sheets each to 8 children” is presented. When the children respond with “16×8,” the teacher could run with this response and say, “all right, let’s consider how to find the answer to this.”
This is not adequate, however. The students must be made to thoroughly understand the fundamental reasoning behind the solution. It is important that students independently consider “why this is the way the problem is solved.”
The child will probably explain the problem by saying that “in this problem, eight 16s are added: 16+16+16+16+16+16+16+16.” This is based on the meaning of multiplication
Chapter 7
Questions for Eliciting Mathematical Thinking
Teaching should focus on mathematical thinking. Teachers need to first think of how they can help children appreciate and gain the ability to use mathematical thinking. When children get stuck, rather than helping them directly with useful knowledge and skills, teachers must prepare a way to teach the mathematical thinking required to elicit the knowledge and skill and moreover to teach the attitude that leads to this thinking methods. Also, this assistance must be of a general nature, and must be applicable to many different situations. Assistance should take a form that is frequently helpful when one focuses upon it. This is because this kind of assistance is useful in many different situations. By repeatedly providing it, a student can grow accustomed to this type of mathematical thinking. This kind of assistance is not something taught directly, but something that should be used by children themselves to overcome problems. Therefore, this assistance should take the form of questions.
It goes without saying that the goal of teaching based on these kinds of questions is for children to gain the ability to ask these questions of themselves, and to learn how to think for themselves.
Questions related to mathematical thinking and attitudes must be posed based on a perspective of what kinds of questions must be asked. This must be considered in advance. Questions must be created so that the problem solving process elicits mathematical thinking and attitudes. The following offer a list of question analyses designed to cultivate mathematical thinking, based on a consideration of these kinds of questions. In other words, this question analysis list is comprised of questions derived from the main types of mathematical thinking used at each stage of the problem solving process.
The A questions on this list deal with mathematical attitudes, with the stage indicated as “A11” and so on. Questions related to mathematical thinking related to mathematical methods are marked with M, and questions related to mathematical ideas are marked with I. Types of thinking corresponding to the question are given in parentheses ( ).
[List of Questions Regarding Mathematical Thinking]

Questions Regarding Mathematical Attitudes
A11 What kinds of things (to what extent) are understood and usable? (Clarifying the problem)
A12 What is needed to understand, and can this be stated clearly? (Clarifying the problem)
A13 What kinds of things (from what point) are not understood? What does one want to find? (Clarifying the problem)
A14 Does anything seem strange? (A questioning attitude)
Questions Regarding Thinking Related to Methods
M11 What is the same? What is shared? (Abstraction)
M12 Clarify the meaning of the words and use them by oneself. (Abstraction)
M13 What (conditions) are important? (Abstraction)
M14 What types of situations are being considered? What types of situations are being proposed? (Idealization)
M15 Use figures (numbers) for expression. (Diagramming, quantification)
M16 Replace numbers with simpler numbers. (Simplification)
M17 Simplify the conditions. (Simplification)
M18 Give an example. (Concretization)
Questions Regarding Thinking Related to Contents
I11 What must be decided? (Functional)
I12 What kinds of conditions are not needed, and what kinds of conditions are not included? (Functional)

Questions Regarding Mathematical Attitudes
A21 What kind of method seems likely to work? (Perspective)
A22 What kind of result seems to be possible? (Perspective)
Questions Regarding Thinking Related to Methods
M21 Is it possible to do this in the same way as something already known? (Analogy)
M22 Will this turn out the same thing as something already known? (Analogy)
M23 Consider special cases. (Specialization)
Questions Regarding Thinking Related to Contents
I21 What should one consider this based on (what unit)? (Units, sets)
I22 What seems to be the approximate result? (Approximation)
I23 Is there something else with a similar meaning (properties)? (Expressions, operations, properties)

Questions Regarding Mathematical Attitudes
A31 Try using what is known (what will be known). (Logic)
A32 Are you approaching what you seek? (Logic)
A33 Can this be said clearly? (Clarity)
Questions Regarding Thinking Related to Methods
M31 What kinds of rules seem to be involved? Try collecting data. (Induction)
M32 Think based on what is known (what will be known). (Deduction)
M33 What must be known before this can be said? (Deduction)
M34 Consider a simple situation (using simple numbers or figures). (Simplification)
M35 Hold the conditions constant. Consider the case with special conditions. (Specialization)
M36 Can this be expressed as a figure? (Diagramming)
M37 Can this be expressed with numbers? (Quantification)
Questions Regarding Thinking Related to Ideas
I31 Think based on units (points, etc.). (Units)
I32 What unit (what scope) should be used for thinking? (Units, sets)
I33 Think based on the meaning of words (words used to express methods, or methods themselves). (Expressions, operations, properties)
I34 Try following a predetermined procedure (calculations). (Algorithms)
I35 What is this (formula or symbol) expressing? (Formulas, expressions)
I36 Can I express this in a formula? (Formulas)

Questions Regarding Mathematical Attitudes
A41 Why is this (always) correct? (Logical)
A42 Can this be said more accurately? (Accuracy)