MATHEMATICS - THE SCIENCE OF PATTERNS AND ITS IMPLICATIONS TO TEACHING AND LEARNING

Mathematics is one way of understanding the world and universe.
Patterns are everywhere. Patterns occur in geometry (eg wallpaper/tiles patterns using various geometrical shapes/figures), in music,  in human behavior (eg voting patterns )

What the mathematician does is mainly looking for a  pattern or to examine abstract patterns which give rise to different branches of mathematics - numerical patterns/patterns of numbers and counting  (arithmetic and number theory), patterns of shape, symmetry and regularity (geometry; the mathematics of beauty- transformation), patterns of motion and change (mathematics in motion- calculus), patterns of reasoning and communicating (logic; logical arguments/connections), patterns of chance (probability theory - making prediction etc), patterns of closeness and position (topology), and so on.

Those patterns can either real or imagined, visual or mental, static or dynamic, qualitative or quantitative,
purely theoretical or utilitarian.

Those patterns can arise from the world or phenomena around us, from the depths of space and time
( geometry of the universe; three- dimensional physical universe etc), or from the inner workings of the human mind.

Discuss the implications of maths as a science of patterns to the teaching and learning of mathematics in secondary schools.

KAEDAH UMUM PENYELESAIAN MASALAH

Dalam kehidupan seharian, kita sering menghadapi pelbagai jenis masalah (persoalan) samada berbentuk peribadi, keluarga, masyarakat, ekonomi, politik dsb (kecil atau besar) yg memerlukan kemahiran kita menyelesaikan masalah masalah berkenaan dgn betul dan berkesan. Oleh yg demikian kemahiran menyelesaikan masalah merupakan satu perkara asas yg dapat membantu kita berfungsi dgn lebih berkesan dlm kehidupan ini.

Sesaorang itu dikatakan menghadapi masalah bilamana ia tidak mempunyai penyelesaian serta-merta/segara terhadap sesuatu soalan yg dikemukakan . Terdapat dua syarat utama yg menentukan kewujudan sesuatu masalah kpd sesaorang individu iaitu :

1. mesti terdapat tujuan/matlamat/objektif/perkara yg jelas untuk dicari/dicapai/diselesaikan.
2. mesti terdapat halangan(obstacles) terhadap jalan penyelesaian itu.

Namun demikian kita perlu maklum bahawa bergantung kpd pengalaman dan pengetahuan sedia ada
masing masing sesuatu masalah yg menjadi "masalah" kepada sesaorang individu tidak semestinya menjadi masalah kpd indidividu yg lain. Sebagai contoh selesaikan masalah berikut :

Ahmad mempunyai RM 60. Dia menggunakan 1/4 drp wangnya untuk membeli 2 kg udang.
Berapakah harga sekilogram udang itu?

Secara umumnya terdapat 4 langkah utama ( dlm pendidikan matematik ini dikenali sbg kaedah Polya) bagi menyelesaikan sesuatu masalah :

1. memahami masalah ( tentukan data/maklumat yg ada/diberi - lebih umum lagi apakah punca punca kpd
    masalah berkenaan?) - pemikiran kritis
2. merancang strategi penyelesaian (tentukan kaedah,langkah,formula, teknik, lukis gambarajah, bina jadual,
    bina graf, laksanakan ujikaji, cadangan penyelesaian dsb) - pemikiran kreatif
3. melaksanakan strategi penyelesaian   - pemikiran kreatif
4. menyemak semula jawapan (adakah jawapan yg diperolehi tepat, munasabah dan logik?
     kaedah alternatif ?) - pemikiran kritis 

Walaupun kaedah diatas sering digunakan untuk menyelesaikan pelbagai jenis dan bentuk masalah matematik
tetapi  ianya boleh digunakan sbg kaedah dan proses umum bagi menyelesaikan sebarang masalah dlm konteks yg luas dlm kehidupan seharian.

MANTIK, KBKK DAN PENYELESAIAN MASALAH

Pengenalan

Mantik (logik) dan peranannya :

             -  akar kata/berasal drp bahasa arab "mantiq" bererti percakapan/pertuturan dan fikiran yg benar
                 (ilmu kaedah/cara/teknik/strategi berfikir secara bersistem dan teratur untuk mencari kebenaran
                  dan mengelak drp kesilapan dan kesalahan berfikir)

             -  salah satu cabang bidang falsafah selain epistemologi, metafizik dan aksiologi  (etika dan
                 estetika).
          
            -  dianggap ilmu sepunya/sejagat/ilmu alat bagi seluruh disiplin ilmu, samada falsafah atau bukan
                falsafah
          
             -  melatih kekuatan hujah/kecekapan akal/ketajaman minda/ rasional dlm kehidupan seharian, dlm
                perbincangan, dialog, debat, penghujahan (yg sahih, benar dan tepat), pembuktian/evidens,          
                kemunasabahan, premis/dasar
                proposisi/pernyataan dsb.
           
             -  mendasari ilmu sains dan matematik, proses berfikir secara induktif (kes khusus kpd umum) dan  
                deduktif (kes umum kpd khusus)
          
             - mengembangkan kemahiran berfikir secara kritis dan kreatif, kemahiran aras tinggi (HOTs) serta
               kebolehan penyelesaian masalah dlm pelbagai konteks dan bidang.
        

ETHICAL ISSUES IN SCIENCE AND TECHNOLOGY

Introduction :

The meaning of ethics -  related to main concepts such as good, right, values, obligation, freedom, rationality and choice  (generally amount the same thing as morality of people - the difference is not always clear)

Various kinds of ethics - eg medical ethics, legal ethics, business ethics, political ethics, social ethics,  
                                      science, engineering and technological ethics, computer/ICT ethics etc.

Science, engineering and technological ethics : 2 dimensions

Ethics in science, engineering and technology - refer to the practices of scientist,engineers and technologists  
                                                                      eg honesty, integrity

Ethics of science, engineering and technology - refer to the relation/impact/contribution of scientists,            
                                                                        engineers
                                                                        and technologists  to the society
                                                                        ie improving quality of life, productivity (agricultural,
                                                                        manufacturing etc),
                                                                        economic growth, services  etc
                                                                        - health, safety, environment, green technology, sustainability
                                                                        /re-cyclying etc

Ethical issues and dilemmas/controversial issues related to the development of science and technology
(some examples)

1. risks and benefits - a question of balance? eg drugs, nuclear energy,chemicals etc
2. economic (eg industrial development) - air pollution, water pollution, hazardous wastes etc
3. ICT development - information safety, privacy, piracy etc
4. social issues - poor and rich, availability, rural and urban, exploitation etc
5. political issues - under developed and developed countries-  globalization, military, war etc
6. natural resources (fossil resources eg oil, gas, coal ) - present consumption and future needs (vis- a- vis population growth).
                                                                         

KEMAHIRAN BERFIKIR KRITIS DAN KREATIF (KBKK) DLM MATEMATIK

Pemikiran kritis adalah kebolehan seseorang untuk menganalisis, mentafsir dan menilai sesuatu hujah dgn mendalam, terperinci dan objektif. Antara contohnya spr mencirikan, membanding dan membeza, kebaikan dan keburukan (pros and cons), mengelas/mengkategori/ menyusun/mengumpul, menganalisis idea/konsep, menentukan hubungan antara bahagian - keseluruhan (parts  - whole), menerangkan sebab dan musabab (cause and effect), meneliti/menyiasat andaian/premis, menilai ketepatan, keaslian, kejituan, kebenaran, kesahihan/kebolehpercayaan/kemunasabahan sumber/hujah/definisi/hukum/prinsip/teorem/kesimpulan
/generalisasi/inferens/jawapan dsb - beri contoh contoh dlm matematik.

Ciri paling utama pemikir kritis adalah reaktif ( falsafah - matematik sbg suatu penemuan - mathematics is discovered)


Pemikiran kreatif adalah kebolehan seseorang untuk menjana, membina atau mencipta idea baru/pelbagai/asli, mengembangkan, mensintesis/menggabung/mencantum/menginterasi idea idea sedia ada, menghubungkait/mencari perhubungan/perkaitan antara pembolehubah, membuat hipotesis/inferens/generalisasi, merekabentuk/mereka cipta, inovatif (adapt and modify), imaginatif, mencari kaedah alternatif dlm penyelesaian masalah dan memindahkan pengetahuan dan kemahiran dlm konteks baharu. - beri contoh contoh dlm matematik.

Ciri paling utama pemikir kreatif adalah proaktif (falsafah- matematik sbg sesuatu penciptaan- mathematics is created)

Kedua dua kemahiran kritis dan kreatif diperlukan untuk membuat keputusan dan menyelesaikan masalah.

Membuat keputusan dan menyelesaikan masalah - spr memilih kaedah penyelesaian masalah terbaik drp beberapa alternatif setelah menimbangkan/menilai/menganalisis dgn teliti sebelum membuat pilihan terbaik
kekuatan dan kelemahan sesuatu kaedah.

Bincangkan bagaimana anda dapat meningkatkan kemahiran berfikir kritis dan kreatif ini dlm proses
pengajaran dan pembelajaran matematik di bilik darjah.

PRINCIPLES OF OBE

There four main principles of OBE:

1. Clarity of focus on outcomes

- clear/well defined intended learning outcomes (LOs)/explicitly stated expectations

(competencies)

- using Bloom taxanomy (knowledge, comprehension, application, analysis, synthesis and

evaluation) as a guideline to construct LOs

2. Design backwards

- start with the end in mind (outcomes- ie LOs of the programmes and subjects)

- design curriculum backward to achieve the outcomes

- deliver forward

- product defines process/ outcomes drive the curriculum (teaching, learning and

assessment)

3. High expectations of success

- all students can succeed (in reaching the exit outcomes)

- every student should develop his/her full potential (student- centered approach)

- expect high level of achievement
- regular feedback on student's performance/progress/competence

4. Expanded learning opportunities

- provide challenging, stimulating and enriching (or remedial) learning

experiences/environments/resources

- use a variety of teaching, learning and assessment strategies/methods/techniques

- cater for individual needs and differences (or learning styles)
- cater for self-directed, self paced and self accessed in learning

-

MATHEMATICAL PROCESSES/ACTIVITIES AND ITS RELATION TO TEACHING AND LEARNING

The processes of mathematics are the thinking activities associated with doing mathematics:-

Basic activities :

 1. counting - dealing with quantities/numbers  and it grows becomes arithmetic, algebra (generalization of  
     arithmetic), number theory, statistics, probablity , decision science, acturial science, risk management
     so on...

2. measuring - dealing with length, area, volume,etc  and it grows becomes geometry, trigonometry, so on...

3. finding relationships between variables- dealing with variables and it grows becomes algebra,
    calculus, so on...


More advanced activities :

1. Mathematical modelling
   - the process of developing/designing/
     constructing/formulating a mathematical model from real
     world/everyday life/natural phenomena/word problems (heat,light,
     energy, waves etc) to become mathematical models (in the form of equations, functions, graphs, tables etc)

    eg  Newton's law motion, F=ma
         Boyle's law PV=c
         Plans and elevations, earth as sphere, bearing etc

2. Searching for patterns

   Maths is the study of patterns of various kinds in
  - numbers (patterns of numbers- odd, even,prime
    numbers, arithmetic, geometric progressions etc),

   - shapes (patterns of shapes- transformations -
     translation, reflection and rotation,
    geometrical patterns etc )

   -relationships ( form of functions and
    graphs- sine,cosine curves ; probability as a pattern of
    chances etc)

Other related activities include classifying/categorization, symbolizing, abstracting, defining,assuming
conjecturing, applying, collecting data, reasoning (inductively and deductively), proving, number sense
estimation, data handling, recognising and representing relationship mathematically etc..
(ie the process of mathematical thinking)

The products of mathematics are the results of mathematical thinking and activities include definition of concepts (defined concepts), postulates (eg  a line can be drawn between any two points), methods for solving problems, rules, formulas, theorems, mathematical models etc
(ie the product of mathematical thought)

Brief historical and philosophical background of mathematics its relation to the teaching and learning mathematics

Introduction :

History:

Mathematics has been regarded as the backbone of human civilization.
History of mathematics is the history of civilization.
Mathematics is the mirror of civilization.
Mathematics has led to the development of various subjects, vocations and technology.
Mathematics is the gate and key to all sciences.
All things are mathematical

Philosophy:

There two schools of thought :

1. Mathematics is discovered ( pure mathematics/abstract mathematics)

Mathematics is a dynamic (in the making) subject (vs static subject) whether it was discovered (ie mathematicians like scientists who form theories/laws/principles from simple ideas/examples, intuition, making conjectures/hunches, through questioning and observations .
eg discovered mathematical concepts such as numbers, relations, functions, ratio, proportion, rate, etc.
 
 OR

Mathematics is invented/created (applied mathematics)
(ie mathematicians like technologists who apply mathematics in solving real life problems) eg invent/create  symbols, formulas, methods, heuristics, algorithms, models, strategies etc.

Based on two schools of thought in the philosophy of mathematics, discuss its implications to the teaching and learning maths in the classroom.