SECONDARY SCHOOL INTEGRATED CURRICULUM (KBSM - MATHEMATICS)

Integrated curriculum model : Why integrate?

1. The main goal of KBSM (1989) is to develop potentials of the students in a holistic and integrated manner (ie intellectually, spiritually, emotionally and physically balanced)

within the context of National Philosophy of Education.

2. Mathematics should be seen as an integrated whole (ie connections or relationships of mathematical topics) not as a discrete or fragmented topics. (eg relationships between fraction, ratio, proportion and percentage)

3. Mathematics as an integral part of human experiences, activities and problem solving, emerging from everyday life, interaction with other disciplines especially with science and technology.

How to integrate?

Integration of mathematical contents through three main components ( ie thematic approach):

1. Numbers such as counting and calculating ( ie whole numbers, fractions, decimals, percentages multiples and factors etc

2. Shapes - familiarizing with topics such as angles ,lines, polygons, circle,solid , trigonometry, earth etc.

3. Relations- understanding rules, laws and relationships in the topics such as set, functions and graphs, matrix, statistics, probability etc.

How to implement in teaching and learning?

1. Teaching and learning as an integrated approach ie integration of mathematics as problem solving, mathematics as communication, mathematics as reasoning and mathematical connections.

2. Balance between understanding of concepts and mastering of skills.

3. Apply mathematical skills in real problem solving situations.

4. Inculcate historical elements into mathematics teaching and learning.

5. Emphasize on mathematical thinking strategies.

6. Using inquiry- discovery method

Discuss the factors that may lead to the problems in the implementation of the current KBSM mathematics curriculum in schools.

METACOGNITION AND MATHS EDUCATION

Metacogniton means thinking about thinking ie thinking about one's own thinking process or

knowledge about our own cognitive processes.

It is the ability to monitor one's current level of understanding (self-monitoring) and determine when it is not adequate, be aware of the improvements needed(self-awareness) or correct/modify the faulty thinking involved(self-corrective actions).

Metacognition involves self reflection is an important part of an active learning especially in solving mathematical problems which supports the development of critical and creative thinking skills.

Main operations of metacognition (for example in solving mathematical problem by using Polya Model - understand the problem, device/select a plan/strategy, carry out the plan and looking back ) are planning, executing/implementing, monitoring and evaluating.

Ideally, students should be able to assess their thinking before(do I know what to do?- given data, what to find? , during (am I on the correct track? - using right formula/method?- should I consider alternatives?) and after a problem solving process ( Is the answer correct/reasonable?- check the solution).

In other words it is a self-regulated/controlled or self-directed of learning which promotes independent learners among the students. Metacognition is just as any skill, it improves with practice.

Metacognitive skills can be develop through cooperative learning/group work where students can think aloud( describe precisely their thinking), sharing their thought via description and/or visual representation with other students in the group.

Examine a mathematics textbook (choose one problem in form 4 mathematics), design teaching and learning activities which incorporate metacognitive skills in that lesson.

HOW KNOWLEDGE GROWS ( eg. SCIENCE)

Scientific knowledge is the product of continuous process of inquiry which begins and grows through the acquisition of scientific skills by the scientists:

Scientific skills can be divided into 2 types: Science process skills (SPS) and manipulative skills (MS). SPS involve cognitive processes that are related to thinking skills(critical and creative thinking). MS involve psychomotor skills related to handling of instruments in carrying out experiments.

Some of the main science process skills are as follows: observing, classifying, measuring and using numbers, making inference, predicting, communicating, using space-time, interpreting, defining operationally, manipulating variables, making hypotheses and experimenting.

In each SPS there may be one or more thinking skills involved.

Making observation (eg movement of an object)

- characterizing (eg force, mass, velocity)

- relating (eg F=ma)

Careful observations of nature are main sources of scientific knowledge

Classifying

- looking at similarities and differences (eg animals and plants)

- compare and contrast (eg types of animals)

Measuring and using numbers

- looking for pattern

- sequencing

Making Inference (draws early conclusion)

- relating (one variable with another variable)

- looking at similarities and differences

- analyzing

Predicting ( eg weather, and wave)

- relating

- visualizing (mental representation)

- searching for patterns

Communicating (verbal, mathematical, symbolic etc)

- involve all thinking skills

Using space - time (eg study of shape, motion and change; relativity theory)

- sequencing

- arranging

- relating

Interpreting (eg function and graph )

- summarizing

- generalizing

- meaning

Defining operationally (eg force, momentum, heat etc)

- making analogy

- visualizing

- analyzing

Controlling variables (eg pressure and volume)

- characterizing

- looking at similarities and differences

- relating

- analyzing

Making hypothesis

- relating

- predicting

Experimenting - involve all thinking skills

Some of the main manipulative process skills are as follows: using equipment/ apparatus, handling apparatus safely, keeping safely all scientific equipments and apparatus, careful handling of live specimens (in biology), mixing solutions, pouring solutions etc (in chemistry) etc.

Discuss in what ways the understanding on how knowledge of grows will effect the role of science teachers in the classroom?

COGNITIVE PSYCHOLOGY AND MATHS EDUCATION

• Learning processess are complex, behaviorist psychology cannot explain all learning.

• Theoretical inadequacy/limitations to explain all that our students learn and do.

• Learning is not only limited to observable behaviour that always simple and straightforward but it is more of thought processes (or cognition).

• Learning as internal process (cognitive procesess) that cannot be observed directly; changes in behaviour is a reflection of internal change (the cognitive structure) in the student’s mind.

• Major focus is on knowing (understanding,memory, thinking, perception,meaning,decision making, sense making,problem solving, concepts or ideas etc)

Main representative theorists:

• Piaget ( cognitive development theory)
•   Bruner (discovery learning)
•   Ausubel (meaningful learning)
• Gagne’ (classification of human learning)

With cognitive approach, the emphasis in teaching and learning has shifted from product to process (ie process of mathematical thinking , process of problem solving etc)

KEMAHIRAN BERFIKIR KRITIS DAN KREATIF

Contoh pemikiran kritis :

1. mencirikan- menyatakan ciri/sifat sesuatu konsep (eg segitiga, fungsi,luas dsb)

2. membanding dan membezakan- menentukan persamaan dan perbezaan (eg hubungan dan fungsi, pembolehubah bebas dan bersandar, membezakan fakta dgn pendapat)

3. mengumpul dan mengelaskan- mengumpul, menyusun, membuat urutan, mengkategori mengikut ciri /kriteria tertentu (eg no genap, perdana, jujukan aritmetik dan geometri)

4. menganalisis- mencerakinkan maklumat (eg menganalisis maklumat dlm masalah berayat)

5. mentaksir dan menilai- mentaksir maklumat, menilai ketepatan, kemusabahan sesuatu maklumat (data, hujah,andaian, definisi, pendapat,cadangan, pernyataan,sumber)

Contoh pemikiran kreatif:

1. menjana idea baru- mengemukan, mencetus idea, mencadang alternatif( eg kaedah baru penyelasaian masalah)

2. membuat inferens-membuat kesimpulan awal drp data, maklumat, pemerhatian dan ujikaji yg perlu dibuktikan kebenarannya.

3. menghubungkait- mencari perkaitan diantara pembolehubah, menentukan hubungan suatu konsep dgn konsep lain.

4. mensintesis- menggabungkan, menyepadukan idea idea, merekabentuk, menggubal model

5. membuat gambaran mental- menvisualisasi/menggambarkan/membayangkan dlm minda/mengkonsepsualisasi sesuatu idea/situasi/memperluaskan daya imaginasi (eg konsep had, infiniti, kadar)

6. merekacipta - menghasilkan idea asli, membuat inovasi/penambahbaikan/pengubahsuaian kaedah tanpa terikat kpd kaedah yg lazim.

COOPERATIVE LEARNING IN MATHEMATICS

Cooperative learning began with the strong belief that learning is most effective if students are actively involved in sharing ideas and work cooperatively to complete an academic tasks.

There five essential elements of cooperative learning:

1. positive interdependence- the success of one learner is dependent on the success of the other learners.

2. promotive interaction- individual can achieve promotive interaction by helping each other, exchange resources, challenging each other's conclusions, providing feedback, encouraging and striving for mutual benefits.

3. individual accountability- teachers should assess the amount of effort that each member is contributing by random randomly calling students to present their group's work.

4. Interpersonal and small-group skills- teachers should provide opportunities for group members to know each other, accept and support each other, communicate accurately and resolve differences constructively.

5. Group processing- teachers should provide opportunities for the class to assess group progress. Group processing enables group to focus on good working relationship, facilitates the learning of cooperative skills and ensures that members receive feedback.

Cooperative learning represents a shift in educational paradigm from teacher-centered approach to a more student-centered learning in small group.

The main problems and challenges which arise in incorporating cooperative learning :

1. fear of the loss of content coverage- CL often take longer than conventional methods, it may waste a lot time to complete the taks.

2. students lack the skills to work in group- teachers are often concern with students' participation in group activities.

3. do not trust students in acquiring knowledge by themselves- teachers have the knowledge and expertise but they have to tell their students what and how to learn.

4. lack of familiarity of some teachers with CL methods- intensive in-service course should be implemented to overcome the problem.

5. teacher need to prepare extra materials for CL class use- require a lot of extra work, time and new burden for teachers.