PREPARING RESEARCH PROPOSAL IN EDUCATION

A research proposal provides a road-map to the study and normally consists of five parts (based on UTM format)

1. Title of research (working title)
2. Introduction
3. Review of the literature
4. Methodology of research
5. Expexted findings and summary

How many pages (10-30 or 40-70 or 70-100 pages or more ?)

When to present ? 2nd sem (master) or 3 sem (PhD) or ??) - full time or part - time

Introduction : 

- providing the background (issues/arguments/phenomena) and the context of the problem to be dealt with (including objectives, significance and scope of research)
- formulating/conceptualized the statement of the problem
(problem well-defined is half-solved!!!)

Review of the literature :

- expand upon the context and background of the study and further define the problem/a key concept (concepts)
- review relevant research, unanswered questions (gap), untried method, findings of others you are challenging or extending
- develop theoretical framework of the study
- provide a basis for rationale of the study and the formulation of hypotheses.

Methodology of research :

- describing the methods, design and procedures of the study
- quantitative or qualitative or mixed methods ( instrumentation, sampling, data collection and analysis)
- Gantt chart/plan/schedule of tasks in the research

Expected findings and summary :

- brief descriptions of the findings 


   

kemahiran berfikir aras tinggi dlm matematik - ketibaan makna

DRP highlight teaching and learning activities in the field of education the knowledge whatsoever about what to do when there is a meaning to arrival (the arrival of meaning) about something (eg a concept) kedlm soul / conscience man.

Similarly in mathematics that involves various concepts DRP easy KPD hard and complex. somebody told students understand the meaning kedlm when they reached him.

Apa buktinya?
Various data can work with them antaranya pelajar dapat menerangkan dgn jelas makna sesuatu konsep itu dan dpt menggunakan konsep dan kemahiran yg berkaitannya dlm proses penyelesaian masalah.
Discuss dgn contoh contoh yg sesuai mengenai pernyataan diatas.

PPS 2873 - Current Issues in Mathematics Education

Synopsis :

This course aims at exploring critically issues and trends  related to four main aspects namely curriculum, teaching and learning, assessment and research in mathematics education from both local and international perspectives. It focuses on the concepts and philosophies underlying the implementation of curriculum, teaching and learning and assessment in mathematics education. Students are expected to do a lot of independent and critical reading from local and international mathematics education research journals.

Assessment methods :

1. Review of selected research articles and presentation (individual) - 30%
2. Project - analysis of secondary school mathematics textbook (group of 2/3) - 30 %
3. Final exam (comprehensive) - 40 %

PROBLEM POSING, PROBLEM SOLVING AND PEDAGOGY

Assumption (epistemologically) -  mathematics as result of human problem posing and solving
                                                    -  mathematics as a mental construction (creation, invention or
                                                       discovery)  

In term of learning, social constructivism identifies all learners of mathematics as creators of mathematics involving problem posing and solving.

Therefore as a consequences of problem posing and solving pedagogy :

1. School maths for all should be centrally concerned with mathematical problem posing and solving
    (reduce content- oriented mathematics curriculum)

2. Inquiry, investigation, problem posing or formulation should occupy a central place in the school
    maths curriculum and precedes problem solving

3. The pedagogy (teaching, learning and assessment) should be process and inquiry(or investigation)
    focused (vs product)

4. learner-centered view of investigation as a learner directed activities (new questions posed, new
    situations are generated and explored- promotes active learning)

5.  Increase learner autonomy and self- regulation ( develop reflective and meta-cognitive skills)


Mathematical problem posing (formulation, investigation etc) is divergent (creative thinking and higher order thinking) as the process of mathematical problem solving (critical thinking and higher order thinking - eg by using Polya method)) is convergent.








                                             

PROBLEM-BASED LEARNING(PBL) IN MATHEMATICS

Paradigm shift from traditional teaching model (content-based, teacher directed and student as knowledge recipient) to problem- based learning model( problem motivated, teacher as facilitator and student as problem solver).

Main characteristic of PBL approach is that,
the problem (real-world problem : unstructured and authentic (vs simulated) is the starting point of learning. (discuss its implications to the current maths curriculum).

eg (solve the following problems)

1. The costs for two different kinds of heating systems for a three- bedroom home are given below

solar system - cost to install rm 29700 and operating cost/year is rm 150
electric system - cost to install rm 5000 and operating cost/year is rm 1100

After how many years will total costs for solar heating and electric heating be the same?
What will be the total costs for both systems at that time?

2. Two ordinary six-sided dice are rolled, what is the probability of getting a sum of 8?
3.  Working together, Ahmad and Ali can complete a job in 4 hours. Working alone, Ahmad requires 
     6 hrs more than Ali to do the job. How many hrs does it take Ali to do the job if he works alone?

Benefits of PBL:

1. creating meaningful learning(content/topic/concept) through inquiry (emphasis on critical, logical  creative thinking, deep reasoning and metacognition))

2. encourage the development broad -based mathematical problem solving strategies (heuristics)
 rather than content learning in a limited sense.

3. development of self-directed/regulated/independent learners - students assume major responsibility
    for the acquisition of knowledge.




 

Broad- based mathematical problem solving strategies

...in USA (cont..)

NCTM (National Council of Teachers of Mathematics) - focus on concept development and problem solving.

By learning and acquiring a variety of broad-based ( general ) mathematical problem-solving
strategies (heuristics) students are equip to be a better problem solvers across the topics in mathematics or transferring those skills to a variety of problems.  Some of the general problem solving strategies are :

1. Characterize the problem : What is given? What is needed?What is missing? etc
2. Have you seen this before? : or different form ?
3. Look for pattern : eg Gauss recognized a pattern 1+2...+100 = ?
    1+100=2+99=...101 (50 pairs)
     50 @ 101 = 5050
4. Simplification/reduction :  can the problem be broken up into smaller or manageable
    sub-problems?
5. Work backwards : when trying to prove a theorem, it may begin from the conclusion and back track logically
6. Modeling/simulation : a mathematical model may be developed that simplify some complicated process/phenomena in the real word (representing/translating into a mathematical forms eg table, diagram, chart, graph, equation, relationship, function, inequalities, matrices, etc)
7. Logical reasoning/arguments - inductive and deductive reasoning,
8. guess and check/improve - develop a sense of estimation
9. make and test conjectures
10. formulate/pose problems from situations within and outside mathematics


     

TEACHING MATHEMATICAL PROBLEM SOLVING

In USA :

1. Philosophy of mathematics education for the 21 st century :
   The goal of teaching mathematics is to help all students develop mathematical power
   ie to produce effective problem solvers and powerful mathematical thinkers.

2. Mathematics must be seen as an integrated whole ( not as a separate and unrelated topics),
    as a part of human experience, emerging from everyday experience, interaction with science
    and technology and other fields.
   

3. Spend more time on developing broad- based mathematical problem solving skills
    ( general problem solving techniques ie heuristics) and less time on perfecting routine
     computations.

4. Teaching mathematics as problem solving

  - problem solving as a means as well as a goal of instruction
  - apply problem solving skills to solve problems in new contexts with emphasis on multi-steps
    and non-routine problems.
  - recognize and formulate (posing) problems from real word situations/phenomena
  - mathematics is problem-centered and application- based
  - subject to be investigated, discovered, explored and created

5. Problem solving is seen as the most important means to develop powerful mathematical thinkers.

TEACHING AND LEARNING APPROACHES IN MATHEMATICS

There are two main approaches in teaching and learning mathematics based on the psychological theories in education:

 Behaviorist approach :

1. drill -practice (practice makes perfect)
2. mastery of skills (lower order thinking skills- knowledge, comprehension and application)
3. performance- based  (how to do) - suitable for routine/familiar problems
4. focus on algorithm (procedures/steps of calculation)
5. mistakes and errors should be avoided/minimized
6. teacher- centered (focus on teaching)

Cognitive approach :

1. construction of meaning (searching for meaning)
2. conceptual understanding (higher order thinking skills - analysis, synthesis and evaluation)
3. thinking- based (emphasis on why) - suitable for non- routine/ unfamiliar problems
4. focus on heuristic (general methods of solving problems) - Polya's Model
5. mistakes and errors is good indicators of misconceptions and difficulties
6. student- centered (focus on learning)

Discuss with suitable examples on how to use behaviorist and cognitive approaches in teaching and learning mathematics in the classroom. Why teachers need to master both approaches?




 

PERSPECTIVES OF MATHEMATICS ANT ITS RELATION TO TEACHING AND LEARNING

Perspectives of mathematics :

Mathematics as a dynamic and continually expanding field of human creation, involving the process of inquiry, thinking, reasoning (with its intellectual rigour), discovery and invention as well as a cultural product of various civilizations.

1. Mathematics is a practical and problem- driven and problem solving knowledge (ie mainly arise from practical or real- life situations).

2. Mathematics is a science of numbers, shape and space and relationships.

3. Mathematics as a language

4. Mathematics as a way of thinking


Assignment 1 :

Discuss in what ways do the perspectives of mathematics influence the curriculum design in mathematics at the secondary school level ?

Assignment 1

Assignment 1 (20%) with presentation

Problem solving is a fundamental process and an integral part of mathematics.
Critically discuss with suitable examples the epistemology (origin/sources/discovery/grows/development/potential/validity/limitation (if any) of one mathematical concept/topic/branch in the contexts of problem solving (refer to at least one mathematics textbook at the primary or secondary school).

Sinopsis MPS 1813

MPS 1813 -  PENYELESAIAN MASALAH DALAM PENDIDIKAN MATEMATIK
                      (PROBLEM SOLVING IN MATHEMATICS EDUCATION)

                      Key concepts/questions :

                      1. Mathematics - brief history and philosophy of mathematics related to problem 
                          identification, formulation and solution.

                     2.  Mathematics as problem solving - models (strategies, methods, heuristics
                          and techniques).

                     3.  Mathematics education (teaching, learning and assessment) from the perspective of
                          problem solving.

                     4.  Research on problem solving in mathematics education.

                     5.  Issues and trends of problem solving in mathematics education.

                     



                    

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.

TECHNOLOGICAL LITERACY

What do you understand by technology?

Many definitions ?- means different thing to different people

Epistemologically which knowledge comes first, how it began ? science, mathematics or technology? (discuss)

Science - study of nature and fenomena, how the world/universe works (satisfying people's curiosity)

Mathematics - study of numbers, shapes, patterns and relations (as a language of science)

Technology - study how to solve practical problems and serves human needs

In the broadest sense, technology extends our abilities to change the world around us : to cut, shape, or put together materials; to move things from one place to another, to reach farther with our hands, voices, and senses.

We invent technologies to change the world to suit us better which may relate to survival needs such as food, shelter, defence etc.

Developments in science and mathematics often stimulate innovations in technology by offering new kinds methods to be solved (discuss examples).

In particular, the progress of science and mathematics ( intensity of discoveries of scientific and mathematical ideas) has contributed to advancement of technologies (agriculture, manufacturing, medical,transportation, construction, ICT etc - technological inventions) for thousands of years and still continues to do so.

Any invention is likely to lead to other inventions. Discuss.