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Philosophy and History of Mathematics(Falsafah dan Sejarah Matematik)
TMP 6073
Prof. Madya Dr. Hj. Noor Shah Hj. Saad
Pensyarah Pendidikan Matematik/
Pengarah Pusat Kecemerlangan & Pembangunan Insan, UPSI
Hak Cipta Noor Shah Saad, Hak Cipta Terperlihara
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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
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Content knowledge (Subject matter knowledge)
- Substantive structure:
Knowledge of the major facts, concepts, principles withina 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 theknowledge in the discipline
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Knowledge of Educational Ends, purposes, andvalues, and their philosophical and historicalgrounds.
Phil h Hi f
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Philosophy & History ofMathematics
Falsafah Matematik- Falsafah- Matematik- Falsafah Pendidikan Matematik- Falsafah Pendidikan Kebangsaan
Cabang-cababang Falsafah- Metafizik- Etika- Logik
- Epistemologi
Bidang-bidang falsafah- Logikisme- Intuisisme
- Formalisme
PART I
Phil h & Hi f
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Philosophy & History ofMathematics
Falsafah Matematik
- Falsafah- Matematik- Falsafah Pendidikan Matematik- Falsafah Pendidikan Kebangsaan
Cabang-cababang Falsafah- Metafizik- Etika
- Logik- Epistemologi
Bidang-bidang falsafah- Logikisme- Intuisisme- Formalisme
Sejarah Matematik- Matematik Islam- Matematik Barat- Matematik Cina- Matematik India
Pengajian tentang falsafah tidak lengkap tanpa unsur sejarah yang
bernah mencorak perkembangnya.
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Falsafah
- Berasal daripada sebtuan Latin philosophia bermaksud Cinta kepadakebenaran
- Eropah Barat ilmu fizik (Mathematical Principles of naturalPhilosophy)
- Peradaban Islam falsafah dimaksudkan sebagai ilmu warisanpemikiran Yunani, khususnya Aristotle, Plato dan Neo-Platoisme yangdisaring dalam acuan islam.Ibn Sina, al-Kindi, al-Haytham & al-Farabi ahli falsafah Islam.
- Ilmu yang berkaitan dengan pemikiran yang menghubungkan alamtabii dan alam ketuhanan.
- Zaman moden, pengjian tentang alam tabii disebut sebagaisains tabii iaitu sebagai ilmu fizik. Manakala metafizik adalah ilmuberkaitan dengan ketuhanan.
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Matematik
- Bilakah matematik bermula?Tiada jawapan yang tepat tetapi matematik muncul seawalkemunculan insan di muka bumi.
- Koleksi ilmu yang dibina berdasarkan aksiom, berbentuk kuantitatifatau kualitatif, yang mengkaji hubungan antara kedua-dua aspektersebut yang dibina secara deduksi atau induksi.
- aspek kuantitatif dan kualitatif ini berkembang secara selari di manafaktor pengerak kajian matematik bukan bermula dengan aspek
kuantitatif tetapi bermula dengan falsafah dan aspek kualitatif.
- Matematik adalah ilmu untuk mencari hakikat kebenaran.
- Matematik adalah ilmu perantaraan antara kuantitatif ke aspekkualitatif, mengetahui hakikat diri dan pencipta (Euklid).
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Falsafah Pendidikan Matematik(The Philosophy of Mathematics Education)
- Isu tentang what is the nature of 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)
The way we learn and teach mathematics within the classroom and theschool environment (Southweel, 1999)
Platonistjust entity out there waiting to be discovered
Then it will be enough for schools to present the curriculum instruction as amere collection of facts, definitions and algorithms.
Teaching mathematics would be like just transmitting an immutable body ofknowledge that students have to accept as a perennial facts without anyreasoning.
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Polya (1986), mathematics is both demonstration and creation.
Demonstration is achieved by proofs while creation consists ofplausible reasoning that includes guessing
Mathematical methods are not perfect and cannot claim absolutetruth. Mathematics truth is not absolute but relative because in facttruth is time dependent (Grabiner, 1986) and space dependent(Wilder, 1986).
Mathematical methods are also space dependent because differentpeople cultures have different ways of doing and validating theirmathematical knowledge (Ascher, 1991).
Schwabs (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
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Stephen Brown (1995) philosophy of mathemaatics ofeducation focus on:
Philosophy applied to or of mathematics education?
Philosophy of mathematics applied to mathematicseducation or to education in general?
Philosophy of education applied to mathematicseducation?
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1. Philosophy Overall
2. Philosophy of Mathematics
3. Philosophy of Education
MATHEMATICSEDUCATION
Application
Different applications of Philosophy to MathematicsEducation
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Issues of the philosophy of mathematics education
1. What is mathematics?
How can its unique characteristics be accommodated in aphilosophy?
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 ofmathematics 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?
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Issues of the philosophy of mathematics education
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 orhindering 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?
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Issues of the philosophy of mathematics education
3. What is learning (mathematics)?
What assumptions, possibly implicit, underpin views oflearning 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 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?
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Issues of the philosophy of mathematics education
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 teachers 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?
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Issues of the philosophy of mathematics education
5. What is the status of mathematics education as knowledgefield?
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 jugde 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 oonother disciplines?
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Falsafah Pendidikan Kebangsaan
Pendidikan di Malaysia adalah suatu usaha yangberterusan ke arah lebih memperkembangkan potensiindividu secara menyeluruh dan bersepadu untukmelahirkan insan yang seimbang dan harmonis dari segi
intelek, rohani, emosi danjasmani berdasarkankepercayaan dan kepatuhan kepada Tuhan. Usaha iniadalah bertujuan untuk melahirkan warganegaraMalaysia yang berilmu pengetahuan, berketerampilan,berakhlak mulia, bertanggungjawab dan berkeupayaanmencapai kesejahteraan diri serta memberi sumbangan
terhadap keharmonian dan kemakmuran keluarga,masyarakat dan negara
(PPK, KPM, 2003)
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Falsafah Pendidikan Kebangsaan
Pendidikan di Malaysia adalah suatu usaha yangberterusan ke arah lebih memperkembangkan potensiindividu secara menyeluruh dan bersepadu untukmelahirkan insan yang seimbang dan harmonis dari segi
intelek, rohani, emosi danjasmani berdasarkankepercayaan dan kepatuhan kepada Tuhan. Usaha iniadalah bertujuan untuk melahirkan warganegaraMalaysia yang berilmu pengetahuan,berfikir secara kritisdan kreatif, berketerampilan, berakhlak mulia,bertanggungjawab dan berkeupayaan mencapai
kesejahteraan diri serta memberi sumbangan terhadapkeharmonian dan kemakmuran keluarga, masyarakatdan negara
(PPK, KPM, 2003)
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Matlamat
Kurikulum Matematik Sekolah Menengah bertujuan untuk membentukindividu yang berpermikiran matematik dan berketerampilanmengalikasikan pengetahuan matematik dengan berkesan danbertanggungjawab dalam menyelesaikan masalah dan membuatkeputusan, supaya berupaya menangani cabaran dalam kehidupanharian bersesuaian dengan perkembangan sains dan teknologi (PPK,KPM 2003
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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 orregional education system.
Ernest (1991): Identified 5 different groups
Industrial Trainer aimsback-to-basics: numeracy and social training in
obedience (authoritarian)
Technological Pragmatist aimsuseful mathematics to the appropriate level and
knowledge and skill certification (industry-centred)
Old Humanist aimstransmission of the body of mathematical knowledge
(mathematics-centred)
Progressive Educator aims
Creativity, self-realisation through mathematics(child-centred)
Public Educator aimscritical awareness and democratic citizenship via mathematics
(social justice centred)