Philosophy and History of Mathematics KMP 6073

Hak Cipta © Noor Shah Saad, Hak Cipta Terperlihara

Categories of The Knowledge Base

• Content knowledge • General pedagogical knowledge• Pedagogical contents knowledge (PCK)• Curriculum knowledge• Knowledge of learners and their characteristics• Knowledge of educational context• Knowledge of educational ends, purpose, and value and their

philosophical and historical grounds

Shulman (1986) – 7 categories of knowledge

Categories of The Knowledge Base

• Content knowledge • General pedagogical knowledge• Pedagogical contents knowledge (PCK)• Curriculum knowledge• Knowledge of learners and their characteristics• Knowledge of educational context• Knowledge of educational ends, purpose, and value and

their philosophical and historical grounds

Shulman (1986) – 7 categories of knowledge

Philosophy & History of Mathematics

• Falsafah Matematik- Falsafah- Matematik

- Falsafah Pendidikan Matematik - Falsafah Pendidikan Kebangsaan - Falsafah Matematik * Logikisme * Intuisisme * Formalisme

PART IPART I

• Content knowledge (Subject matter knowledge)- Substantive structure:

Knowledge of the major facts, concepts, principles within a field and the relationships among them.

- Syntactic structure: The nature of enquiry in the field, and how

new knowledge is introduced and accepted in that community, history of the knowledge in the discipline

Philosophy & History of Mathematics

• Cabang-cabang Falsafah - Metafizik - Etika - Logik - Epistemologi

PART IPART I

Philosophy & History of MathematicsPART II• Sejarah Matematik

- Matematik Islam - Matematik Barat - Matematik Cina - Matematik India

Pengajian tentang falsafah tidak lengkap tanpa unsur sejarah yang mencorak perkembangannya.

Philosophy & History of Mathematics

• Falsafah- Falsafah Pendidikan- Falasafah Pendidikan Matematik

- Falsafah Matematik - Falsafah Pendidikan Kebangsaan

• Cabang-cabang Falsafah - Metafizik - Etika - Logik - Epistemologi

• Bidang-bidang falsafah - Logikisme - Intuisisme - Formalisme

PART IPART I

Philosophy of Education (Falsafah Pendidikan)

Mazhab Perenialisme

Mazhab Esensialisme

Mazhab Progresivisme

Mazhab Rekonstruksionisme

Falsafah Pendidikan

• Mazhab Perenialisme

- Tokoh falsafah RM Hutchins & Mortimer Adler.- Falsafah tertua, tradisional dan paling konservatif.- Dipengaruhi oleh fahaman idealisme dan realisme- Pegangan bahawa ilmu yang terkumpul sejak awal peradaban manusia harus dijadikan asas pendidikan.

- Prinsip mazhab ini:

- pendidikan adalah ke arah mendapatkan kebenaran seterusnya memahami kebenaran tersebut.

- pendidikan adalah latihan bagi membentuk dan memperkembangkan mental dan intelektual.

- pendidikan untuk menyelaras manusia kepada kebenaran kerana kebenaran adalah untuk kehidupan manusia.

Kurikulum pendidikan – berpusatkan mata pelajaran dan sama untuk semua pelajar.

- mata pelajaran bahasa, ilmu kemanusiaan, kesusasteraan,matematik, sejarah, doktrin- doktrin serta maklumat daripada buku-buku

ternama

- guru berautoriti, ilmu dan pengetahuannya tidak boleh dipersoalkan - kaedah pengajaran Socratic: secara lisan, kuliah dan perbincangan

Mazhab Esensialisme

- Tokoh falsafah William C. Bagley, Thomas Briggs, Frederick Breed, Issac L. Kandel

- dikategorikan sebagai falsafah tradisional & konservatif- Pendidikan adalah melalui pengetahuan teras yang terkumpul

dan berkekalan buat beberpa lama- Pengetahuan bersifat teras, wajar disampaikan kepada pelajar, merangkumi kemahiran, sikap dan nilai yang sesuai

- Prinsip falsafah ini:

- pendidikan bertujuan untuk menyampaikan warisan budaya dan sejarah- ilmu pengetahuan harus diamalkan supaya berperanan

produktif - pembelajaran adalah usaha dan aplikasi bersungguh- sungguh- guru memainkan lebih banyak berperanan dalam pengajaran

Kurikulum:

- berpusatkan kepada mata pelajaran dan lebih fokus kepada masa hadapan

- Kemahiran 3M (membaca, menulis dan mengira)

- peringkat menengah – ditekankan kepada mata pelajaran matematik, sains, ilmu kemanusian, bahasa dan kesusasteraan

- penguasaan fakta dan konsep yang asas bagi setiap disiplin amat mustahak

- Guru: * sebagai pakar (role model), * kelas di bawah pengaruh dan kawalan guru, * mengetahui bidang pembelajaran seperti psikologi

kanak-kanak, proses pembelajaran dan pedagogi

- Sekolah: * badan yang mengekal, memelihara dan menyebar warisan budaya dan sejarah

* tempat memperoleh pengetahuan, kemahiran, sikap dan nilai yang membantu menjadi anggota masyarakat yang berguna.

• Mazhab Progresivisme

- Tokoh: Charles Pierce, Charles Darwin, William James, William Kilpatrick, John Dewey

- Sekolah sebagai unit masyarakat bersifat demokratik

- Corak individu bergantung kepada persekitaran dan masyarakat

- Pendidikan perlu secara semula jadi dan mengikut tabii pelajar serta alam sekitar

- Prinsip:

- Pendidikan sebagai proses mencorak manusia mengikut persekiataran sedia ada- Pelajar aktif dalam aktiviti pembelajaran- Guru sebagai pembimbing dalam proses p & p- Ciri demokratik perlu wujud di sekolah- Pendidikan perlu secara semula jadi dan mengikut tabii pelajar serta alam sekitar

- Kurikulum:

- Berasaskan kepada pengalamnan, peribadi dan sosial pelajar- Berteraskan subjek sains sosial- Merentas disiplin di mana penyelesaian masalah melibatkan aktiviti:

- Kemahiran berkomunikasi- Kemahiran menggunakan logik matematik- Proses inkuari

- Merentas disiplin di mana penyelesaian masalah melibatkan aktiviti:

- Peranan Guru:

– peranan berbeza, sebagai pembimbing pelajar - Pembimbing pelajar untuk berfikir, meninjau dan

membuat penerokaan - sebagai pemandu kepada aktiviti pelajar

Mazhab Rekonstruksionisme

- Tokoh falsafah John Dewey, Goerge S. Counts, Harlord O. Rugg

- Lanjutan fahaman progresivisme

- Pembentukan msyarakat amat perlu dilakukan dengan segera - perkembangan teknologi perlu ada order baru dalam masyarakat

- Pelajar:

- perlu disedarkan dengan masalah sosial, ekonomi, politik,

teknologi- dilatih memperolehi kemahiran mengatasi masalah

- Guru: - menyedarkan pelajar kepada masalah yang dihadapi manusia - membantu pelajar mengenal pasti masalah - mempunyai kemahiran membantu pelajar - menggalakan pencapahan pemikiran

- Prinsip:

- pendidikan bertujuan untuk pembaharuan perkara-perkara yang terdapat dalam sistem sosial.

- sekolah sebagai permulaan kepada pembaharuan yang perlu dalam sistem sosial

- Kurikulum:

- pelbagai masalah sosial, ekonomi, politik, masalah peribadi dan sosial yang dihadapi oleh pelajar

- disusun secara organisasi berstruktur dalam disiplin sains sosial - proses siasatan atau penyelidikan sainstifik sebagai kaedah

penyelesaian masalah

- Sekolah: - agensi utama bagi mengerakkan perubahan sosial, politik, ekonomi dalam masyarakat

Thank You

Philosophy & History of Mathematics

• Falsafah- Falsafah Pendidikan- Falasafah Pendidikan Matematik

- Falsafah Matematik - Falsafah Pendidikan Kebangsaan

• Cabang-cabang Falsafah - Metafizik - Etika - Logik - Epistemologi

• Bidang-bidang falsafah - Logikisme - Intuisisme - Formalisme

PART IPART I

Falsafah

• Philosophy of Education

• Philosophy of Mathematics Education

• Philosophy of Mathematics

• Philosophy of National Education

Falsafah Matematik (Philosophy of Mathematics)

Satu cabang falsafah membincangkan matematik dari sudutSatu cabang falsafah membincangkan matematik dari sudut ontologi dan epistemologi ontologi dan epistemologi

Is the branch of philosophy whose task is to reflect on and Is the branch of philosophy whose task is to reflect on and account for the nature of mathematics.account for the nature of mathematics.

Addresses such as Addresses such as - What is the basic for mathematical knowledge?What is the basic for mathematical knowledge?- What is the nature of mathematical truth?What is the nature of mathematical truth?- What characterizes the truths of mathematics?What characterizes the truths of mathematics?- What is the justification for their assertion?What is the justification for their assertion?- Why are the truths of mathematics necessary truths? Why are the truths of mathematics necessary truths?

• The Role of The Philosophy of Mathematics

to provide a systematic, andto provide a systematic, and absolutely secure foundation for mathematical knowledge absolutely secure foundation for mathematical knowledge that is for mathematical truth.that is for mathematical truth.

Paul Ernest (1988) menyenaraikan tiga amalan guru matematik, iaitu:

a)Sistem kepercayaan guru tentang matematik, pengajaran dan pembelajaran matematik;b)Konteks sosial bagi situasi pengajaran matematik; danc)Aras refleksi yang dibuat oleh guru

Philosophy of Mathematics

Bidang-bidang falsafahBidang-bidang falsafah

• Logisisme (Logicium)Logisisme (Logicium)

• Intuisisme (Intuitionism)Intuisisme (Intuitionism)

• Formalisme (Formalism)Formalisme (Formalism)

• Perkembangan fahaman falsafah – bagaimana & apa yang dimaksud dengan falasafah

• Tokoh-tokoh yang terlibat

• Prinsip-prinsip bidang-bidang falsafah

• Contoh-contoh:

– logik– Theorem– Lemma– Aksiom (postulat) – proposisi yang sangat asas dan tidak dibuktikan dalam sesuatu sistem– Proof of theory

Principia Mathematica - 1903 (A.N. Whitehead & B. Russell)

Aksiom-aksiom logik dihurai dengan terperinci

Digunakan dalam penakulaan menghasilkan teorem-teorem matematik

- Huraian tentang aksiom-aksiom logik bermula dengan penerangan tentang sebutan-sebutan tak tertakrif seperti usulan, fungsi usulan, percanggahan, dan ‘dan’

- Aksiom-aksiom mengandungi banyak sebutan-sebutan tak tertakrif.

– Aksiom adalah anggapan asas dalam sesuatu landasan ilmu yang kebenarannya terswadalil dan tidak memerlukan pembuktian lagi

– Aksiom menjadi tapak yang paling asas dalam pembinaan sesuatu cabang ilmu matematik.

– Aksiom tidak perlu dibuktikan, diterima secara intuisi dan perasaan dalam insan sebab kebenarnya tidak boleh dibuktikan secara analsis tetapi diterima seadanya oleh semua fikiran yang waras.

Principia Mathematica

Contoh pernyatan logik dijadikan aksiom

Setiap usulan yang diimplikasikan oleh usulan yang benar juga ialah usulan yang benar; jika kesimpulan sesuatu implikasi itu benar, maka antesedennya benar ataupun kesimpulannya benar; dan ‘jika anteseden itu benar atau anteseden itu benar, maka anteseden itu benar’

Contoh:

‘Minah seorang wanita’ merupakan satu usulan

Fungsi usulan ialah ‘M seorang wanita’ iaitu pernyataan yang tidak mengkhususkan kepada Minah sahaja walaupun boleh mewakili Minah.

Jika U ialah usulan ‘Minah’ seorang wanita, maka ~ U ialah percanggahan U iaitu pernyataan ‘Tidak benar Minah seorang wanita ataupun ‘Minah bukan seorang wanita’.

Hubungan ‘dan’ – menghubungkan U benar, R benar, iaitu kedua-duanya semestinya benar Hubungan ‘atau’ – menghubungkan U dengan R, U V R

Thank You

Falsafah

• Philosophy of Education

• Philosophy of Mathematics Education

• Philosophy of Mathematics

• Philosophy of National Education

Falsafah Pendidikan Matematik (Philosophy of Mathematics Education)

• Falsafah Pendidikan Matematik

(The Philosophy of Mathematics Education)

- Isu tentang ‘what is the nature mathematics’

- ‘most perfect of all science’ (Lakatos, 1986)

- ‘queen of all science’ (McGinnis, Randy, Shama, McDuffie, Huntley, King & Watanabe, 1996)

- ‘a science in its own right’ (Mura, 1995)

• Schwab’s (1961) – issues related to mathematics curriculum

• Subject (mathematics)• The learner of mathematics• The mathematics teacher• The milieu of teaching (relationship of

mathematics teaching and learning)its aims

• Society in general

• Stephen Brown (1995) – philosophy of

mathemaatics of education focus on:

• Philosophy applied to or of mathematics

education?

• Philosophy of mathematics applied to

mathematics education or to education

in general?

• Philosophy of education applied to

mathematics education?

• Issues of the philosophy of mathematics education

1. What is mathematics?

• How can its unique characteristics be accommodated in a philosophy?

• What philosophy of mathematics have been developed?

• What is their impacts on teaching and learning of mathematics?

• What is the rationale for picking out certain elements of mathematics for schooling?

• How can mathematics be conceptualized and transformed for education purpose?

• What values and goal are involved?• How do mathematicians work and create new

mathematical knowledge?• How does history of mathematics relate to the

philosophy of mathematics?• Is mathematics changing as new methods and

information and communication technologies emerge?

2. How does mathematics relate to society?

• How does mathematics education relate to society?

• What are the aims of mathematics education? (ex: the aims of

mathematics teaching?)

• Are these aims valid?

• Whose aims are they? For Whom?

• Based on which values?

•The aims of teaching, and learning of mathematics?

• How does mathematics contribute to the overall goals of society and education?

• What is the role of the teaching and learning of mathematics in promoting or hindering social justice conceived in terms of gender, race, class, ability and critical citizenship?

• How is mathematics viewed and perceived in society?

• What impact does this have on education?

• What is the relationship between mathematics and society?

• To whom is mathematics accountable?

3. What is learning (mathematics)?

– What assumptions, possibly implicit, underpin views of learning mathematics?

– Are these assumptions valid?– Which epistemologies and learning theories are

assumed?– What are constructivist, and other theories of learning

mathematics? – Do they have any impact on classroom practice?

– What are constructivist, and other theories of learning mathematics?

– Do they have any impact on classroom practice?– What elements of learning mathematics are valuable?– How can they be and should they be assessed?– What is the role of the learner?– How important are affective dimensions including

attitudes, beliefs and values in learning mathematics?– What is mathematical ability and how can it be

fostered?

4. What is teaching (mathematics)?

– What theories and epistemologies underlie the teaching of mathematics?

– What assumptions, possibly implicit, do mathematics teaching approaches rest on?

– Are these assumptions valid?– What means are adopted to achieve the aims of

mathematics education?– Are the ends and means consistent?– What methods, resources and techniques are, have

been, and might be, used in the teaching of mathematics?

– What theories underpin the use of different information and communication technologies in teaching mathematics?

– What is it to know mathematics in satisfaction of the aims of teaching mathematics?

– How can teaching and learning of mathematics be evaluated and assessed?

– What is the role of the teacher?– What mathematical knowledge does the teacher need?– What impact do the teacher’s beliefs, attitudes and personal

philosophies of mathematics have on practice?– How should mathematics teachers be educated?– What is the difference between educating, training and developing

mathematics?– What is the role of research in mathematics teaching and the

education of mathematics teachers?

5. What is the status of mathematics education as knowledge field?

– What is the basis of mathematics education as a field of knowledge?

– Is mathematics educations a discipline, a field of enquiry, an interdisciplinary area, a domain of extra-disciplinary applications, or what?

– What is its relationship with other disciplines such as a philosophy, sociology, psychology, linguistics, etc.?

– How do we come to know in mathematics education?– What is the basis for knowledge claims in research in

mathematics education?

– What research methods and methodologies are employed and what is their philosophical basis and status?

– How does the mathematics educations research community judge knowledge claims?

– What standards are applied?– What is the role and function of the researcher in

mathematics education?– What is the status of theories in mathematics

educations?– What is the impacts of research in mathematics

education on other disciplines?– Can the philosophy of mathematics education have any

impact on teaching and learning mathematics, on research in mathematics education, or other disciplines?

• ‘The way we learn and teach mathematics within the classroom and the school environment (Southweel, 1999)• Platonist – ‘just entity out there waiting to be discovered’ Then it will be enough for schools to present the curriculum instruction as a mere collection of facts, definitions and algorithms.

• Teaching mathematics would be like just transmitting an immutable body of knowledge that students have to accept as a perennial facts without any reasoning.

• Polya (1986), mathematics is both demonstration and creation. Demonstration is achieved by proofs while creation consists of plausible reasoning that includes guessing

• Mathematical methods – are not perfect and cannot claim absolute truth. Mathematics truth is not absolute but relative because in fact truth is time dependent (Grabiner, 1986) and space dependent (Wilder, 1986).

• Mathematical methods are also space dependent because different people cultures have different ways of doing and validating their mathematical knowledge (Ascher, 1991).

1. Philosophy Overall

2. Philosophy of Mathematics

3. Philosophy of Mathematics

Education

MATHEMATICS EDUCATION

Application

Different applications of Philosophy to Mathematics Education

Pendidikan Matematik (Mathematics Education)

• Aims of Mathematics Education

The aims of mathematics education can be a hotly contested area, especially when new curricula are being developed or disseminated through a national or regional education system.

Ernest (1991): Identified 5 different groups • Industrial Trainer aims – ‘’back-to-basics’: numeracy and

social training in obedience (authoritarian)

• Technological Pragmatist aims – useful mathematics to the

appropriate level and knowledge and skill certification

(industry-centred)

• Old Humanist aims – transmission of the body of mathematical

knowledge (mathematics-centred)

• Progressive Educator aims – Creativity, self-realisation through

mathematics (child-centred)

• Public Educator aims – critical awareness and democratic

citizenship via mathematics (social justice centred)

• Matlamat Pendidikan Matematik

• Menguasai kemahiran matematik

(Pendukung Behaviourisme yang berlandaskan Falsafah

Realisme)

• Menguasai kemahiran untuk memperoleh, memproses,

menyimpan, dan menggunakan maklumat matemaik

dengan berkesan

(Pendukung pendekatan pemprosesan maklumat yang

berlandaskan falsafah Realisme)

• Pembinaan kekuatan matematik oleh semua pelajar

(Pendukung Konstruktivisme berlandaskan falsafah

pragmatisme)

• Falsafah Pendidikan Kebangsaan

“Pendidikan di Malaysia adalah suatu usaha yang berterusan ke arah lebih memperkembangkan potensi individu secara menyeluruh dan bersepadu untuk melahirkan insan yang seimbang dan harmonis dari segi intelek, rohani, emosi dan jasmani berdasarkan kepercayaan dan kepatuhan kepada Tuhan. Usaha ini adalah bertujuan untuk melahirkan warganegara Malaysia yang berilmu pengetahuan, berketerampilan, berakhlak mulia, bertanggungjawab dan berkeupayaan mencapai kesejahteraan diri serta memberi sumbangan terhadap keharmonian dan kemakmuran keluarga, masyarakat dan negara”

(PPK, KPM, 2003)

• Matlamat

Kurikulum Matematik Sekolah Menengah bertujuan untuk membentuk individu yang berpermikiran matematik dan berketerampilan mengalikasikan pengetahuan matematik dengan berkesan dan bertanggungjawab dalam menyelesaikan masalah dan membuat keputusan, supaya berupaya menangani cabaran dalam kehidupan harian bersesuaian dengan perkembangan sains dan teknologi (PPK, KPM 2003

Thank You