Asumsi Dasar Karakteristik Matematika, Subyek Didik dan Belajar Matematika Sebagai Dasar Pengembangan Kurikulum Matematika Berbasis Kompetensi Di SMP

By : Dr. Marsigit, M. A.
Reviewed by: Resti Safitri (09301241012)

Student of Mathematics Education 2009 in Yogyakarta State University
(http://resti-safitri.blogspot.com/)

Pengalaman dan kegiatan belajar merupakan kegiatan yang perlu dilakukan siswa dalam rangka mencapai kompetensi dasar dan materi pembelajaran. Ditinjau dari kompetensi yang ingin dicapai, pengalaman belajar dapat menghafal, menggunakan, dan menemukan. Dilihat dari

sisi materi maka pengalaman belajar dapat berkaitan dengan diperolehnya fakta, konsep dan prinsip. Pengalaman belajar dapat diperoleh baik di dalam maupun di luar kelas. Pengalaman belajar siswa perlu didukung dengan ketersediaan sumber bahan, baik berupa objek langsung maupun objek tak langsung yang bersifat kontekstual. Dengan demikian maka strategi pembelajaran yang dikembangkan dapat bersifat : Menekankan pada pemecahan masalah, belajar diberbagai konteks kehidupan sehari-hari,  mendorong siswa sebagai active learners, menghargai

keunikan diri siswa dan memperhatikan keanekaragaman perbedaan siswa, belajar melalui

cooperative learning, dan mengembangkan asesmen dalam sistem pengujiannya.

Kurikulum Berbasis Kompetensi dirancang agar di dalam proses belajar matematika siswa mampu melakukan kegiatan penelusuran pola dan hubungan; mengembangkan kreativitas

dengan imajinasi, intuisi dan penemuannya; melakukan kegiatan pemecahan masalah; serta mengomunikasikan pemikiran matematisnya kepada orang lain. Untuk mencapai kemampuan tersebut dikembangkan proses belajar matematika yang memperhatikan konteks dan penerapannya dalam kehidupan sehari-hari.

Untuk semua jenjang pendidikan, materi pembelajaran matematika meliputi (Ebbutt dan

Straker, 1995):

1.      Fakta : meliputi informasi, nama, istilah dan konvensi.

2.      Pengertian : meliputi membangun struktur pengertian, peranan struktur pengertian, konservasi, himpunan, hubungan pola,urutan, model, operasi, dan algoritma.

3.      Keterampilan penalaran : meliputi memahami pengertian, berfikir logis, memahami contoh negatif, berpikir deduksi, berpikir sistematis, berpikir konsisten, menarik kesimpulan, menentukan metode, membuat alasan, dan menentukan strategi.

4.      Keterampilan algoritmik, meliputi : mengikuti langkah yang dibuat orang lain, membuat langkah secara informal, menentukan langkah, menggunakan langkah, menjelaskan langkah, mendefinisikan langkah sehingga dapat dipahami orang lain, membandingkan berbagai langkah, dan menyesuaikan langkah.

5.      Keterampilan menyelesaikan masalah matematika  meliputi: memahami pokok persoalan, mendiskusikan alternatif pemecahannya, memecah persoalan utama menjadi bagian-bagian kecil, menyederhanakan persoalan.

Pengembangan pengalaman dan kegiatan belajar merupakan esensi dari kurikulum matematika berbasis kompetensi, dimana didalamnya terkandung telaah tentang karakteristik matematika, subyek didik dan belajar matematika.

HOW TO DEVELOPH TEACHING LEARNING PROCESS OF MATH IN SENIOR HIGH SCHOOL OR JUNIOR HIGH SCHOOL

By Resti Safitri

 (09301241012/Pmatsub’09)

A.           INTRODUCTION

Learning mathematics in general is still dominated by teacher-centered learning paradigm, which is often referred to as learning (direct teaching). Teachers actively transfer knowledge to students, while students receive lessons with passive. Mathematics is taught as a form that is so, not as a process. As a result, creative ideas students can not develop, train less reasoning power and are not used to seeing other alternatives that may be used in solving a problem. Students are only able to remember and memorize formulas or mathematical concepts without understanding their meaning.

Meanwhile, not a few students who view mathematics as a subject is boring, creepy and even scary, so that low student motivation to learn mathematics and many students try to avoid math. Many students find it difficult to understand mathematics because mathematics is abstract, while our minds are accustomed to thinking about concrete objects. Teachers are not accustomed to using learning methods that enable students and enables students to relate math to real life, learning methods with a contextual approach, constructivist approach, problem-based learning, and so on. Teachers are accustomed to using mechanistic and strukturalistik learning model, namely the teacher explained, gives formulas and examples, then students are given problems to work on. As a result many students are still having difficulty learning mathematics.

The development of science and technology have changed the paradigm of very fast system and method of learning. Students as learners are required to master the learning material as measured by competency. On the other side of the paradigm shift teaching system also appears on the transfer of knowledge is initially more emphasis on the teaching process (teaching), based on the content (content base), is abstract and only for certain groups and in the process of teaching tend to be passive, Currently education began to shift in the learning process (learning), based on the problem (base case), is contextual and is not limited only to certain groups so that students are required to more actively study and develop learning materials by optimizing other sources.

One alternative that is a model of learning with guided discovery approach, because this model  can be developed in addition to cognitive abilities of students, also can enhance students' abilities in terms of communicate the mathematics and social skills. In this paper will discuss aspects - aspects of learning for learning mathematics can be run effectively and the results are as expected.

B.            CONTENT

1.             Lesson plan

A lesson plan is a teacher's detailed description of the course of instruction for one class. A daily lesson plan is developed by a teacher to guide class instruction. Details will vary depending on the preference of the teacher, subject being covered, and the need and/or curiosity of children. There may be requirements mandated by the school system regarding the plan.

In essence, the preparation of lesson plans aimed at designing the learning experience of students to achieve learning goals. There is no mindset (algorithms) for constructing a specific lesson plans, because the draft is supposed to be rich in innovation in accordance with the specifications of teaching materials and students' learning environment (natural resources and local culture, community needs and developments in science and technology).

In general, the characteristics of lesson plan is good is as follows:

a)         Contains teaching and learning activities to be implemented by teachers who will be a learning experience for students.

b)        Learning steps systematically arranged so that learning objectives can be rice.

c)         Structured learning steps as detailed as possible, so that when lesson plan is used by other teachers (eg, the three subject teachers are not present), easily understood and not cause a double interpretations.

While there are many formats for a lesson plan, most lesson plans contain some or all of these elements, typically in this order:

a)       Title of the lesson

b)      Time required to complete the lesson.

c)       List of required materials.

d)      List of objectives, which may be behavioral objectives (what the student can do at lesson completion) or knowledge objectives (what the student knows at lesson completion).

e)       The set (or lead-in, or bridge-in) that focuses students on the lesson's skills or concepts. These include showing pictures or models, asking leading questions, or reviewing previous lessons.

f)       An instructional component that describes the sequence of events that make up the lesson, including the teacher's instructional input and guided practice the students use to try new skills or work with new ideas.

g)      Independent practice that allows students to extend skills or knowledge on their own.

h)      A summary, where the teacher wraps up the discussion and answers questions.

i)        An evaluation component, a test for mastery of the instructed skills or concepts, such as a set of questions to answer or a set of instructions to follow.

j)        Analysis component the teacher uses to reflect on the lesson itself ,such as what worked, what needs improving.

k)      A continuity component reviews and reflects on content from the previous lesson.

A well-developed lesson plan reflects interests and needs of students. It incorporates best practices for the educational field. The lesson plan correlates with the teacher's philosophy of education, which is what the teacher feels is the purpose of educating the students. Unit plans follow much the same format as a lesson plan, but cover an entire unit of work, which may span several days or weeks. Modern constructivist teaching styles may not require individual lesson plans. The unit plan may include specific objectives and timelines, but lesson plans can be more fluid as they adapt to student needs and learning styles.

The instructor must decide whether class assignments are whole-class, small groups, workshops, independent work, peer learning, or contractual:

1. Whole-class : the teacher lectures to the class as a whole and has the class collectively participate in classroom discussions.
2. Small groups: students work on assignments in groups of three or four.
3. Workshops: students perform various tasks simultaneously. Workshop activities must be tailored to the lesson plan.
4. Independent work: students complete assignments individually.
5. Peer learning: students work together, face to face, so they can learn from one another.
6. Contractual work: teacher and student establish an agreement that the student must perform a certain amount of work by a deadline.

2.             Student worksheet

Student Worksheet is one type of teaching aids, and some even categorize the types of math learning aids. In general, student worksheet is a learning device as a complement or a means of supporting the implementation of Learning Implementation Plan. Student worksheets in the form of sheets of paper in the form of information and the questions (the questions) that must be answered by learners. Student worksheet is very well used to promote the involvement of learners in learning both the methods used in the application of guided and to provide practice development. In the process of learning mathematics, student worksheet aim to find a concept or principle and application of concepts or principles.

Student worksheet is a stimulus or the guidance of teachers in learning to be presented in writing so that the writing needs to consider criteria for graphical media as a visual medium to attract the attention of learners. At least student worksheet as a media card. While the student worksheet must consider the contents of the message elements of the writing of graphic media, the hierarchy of matter (math) and the selection of the questions as an efficient and effective stimulus.

Intended use of student worksheet in teaching and learning are as follows:

a)      Providing the knowledge, attitudes and skills that need to be owned by learners.

b)      Checking the level of understanding learners of material that has been presented.

c)      Develop and implement a difficult subject matter delivered orally.

The benefits gained by the use of in student worksheet learning process is as follows:

a)      Enabling learners in the learning process.

b)       Assist learners in developing the concept.

c)      Train learners in discovering and developing process skills.

d)     As guidance teachers and learners in carrying out the learning process.

e)       Help learners acquire a record of the material learned through the learning activities.

f)       Helping learners to add information about the concepts learned through systematic learning activities.

Student worksheet compile steps are as follows:

a)         Curriculum analysis to determine the material that requires teaching materials student worksheet.

b)        Constructing the map needs student worksheet.

c)         Determining the titles studentworksheet.

d)        Writing studentworkheet.

                                                         i.            Formulation derived from the basic competencies student worksheet special guidebook syllabus development.

                                                       ii.            Determine assessment tool.

                                                      iii.          Organization of the material.

3.                  Small group discussion

Small group discussion is a process that involves small groups in an informal face to face interaction with the aim of sharing experiences or information, solve problems or make decisions. Group discussion is one strategy that allows students to master a concept, it can enhance students' creativity, foster the ability to interact, as well as language skills.

Teaching small groups and individuals is the usual classical form of teaching that allows teachers in the same time face some small groups who learn in groups and some of those students who work or study on an individual basis. The format of teaching is characterized by the presence of more intimate interpersonal relationships and healthy between teachers and students, the opportunity for students to learn according to their abilities, interests, ways, and his speed, the assistance of teachers, student involvement in designing learning activities, as well as the opportunity for teachers to play different roles in learning activities.

Each teacher can create a format for organizing students' learning activities, small groups and individuals in accordance with the objectives, subject (matter), the needs of students, as well as time and facilities available. Skills taught small groups and individuals need to master teachers because their application can meet the learning needs of different students. In addition, small group learning and individual giving the possibility of a healthy interpersonal relationships between teachers and students, the process of mutual learning among students with each other, facilitate the acquisition of the teacher in monitoring student learning, can enhance students' motivation to learn, can foster the spirit of mutual help , as well as allowing teachers to devote its attention to specific student learning so that it can find an appropriate learning approach for students.

The advantages of group discussion for students are:

a)      Students can share information in exploring new ideas or solve a problem .

b)      Improve understanding of key issues .

c)      Develop the ability to think and communicate .

d)     Increasing involvement in planning and decision making.

e)      Built up a healthy spirit of cooperation as well as a cohesive group and responsible.

4.                  Various interaction

Variations stimulus was a teacher in the context of activities of teaching and learning interaction process intended to overcome the boredom of students so that, in the teaching-learning situation, students are always showing diligence, enthusiasm, and full participation.

Benefit of various interaction are:

a.       To create and enhance the students' attention to aspects of teaching and learning relevant.

b.      To provide an opportunity for the development of talent want to know and investigate the students about new things. 

c.        To foster positive behavior toward teachers and schools in many ways more vibrant teaching and learning environment better.

Variation in teaching and learning activities are intended as a change in the teaching process, which can be grouped into three groups or components, namely:

a.       Variations in the way teachers teach, variation in the way teachers teach include: the use of sound variations (voice teacher), Focusing on students (focusing), silence or silence teachers (teacher silence), made contact and motion perspective (eye contact and movement), movement body expression, and the change of position and movement teachers in the classroom teachers (teachers movement).

b.      Variations in patterns of interaction and student activities. The pattern of teacher interaction with students in teaching and learning activities  very diverse, ranging from activities that are dominated by the teacher until the child's own activities are carried out. The use of variations in patterns of interaction are not intended to cause boredom, tedium, and to bring it to class for student success in achieving its objectives.

5.                  Various teaching method

A teaching method comprises the principles and methods used for instruction. Commonly used teaching methods may include class participation, demonstration, recitation, memorization, or combinations of these. The choice of an appropriate teaching method depends largely on the information or skill that is being taught, and it may also be influenced by the aptitude and enthusiasm of the students.

For effective teaching to take place, a good method must be adopted by a teacher. A teacher has many options when choosing a style by which to teach. The teacher may write lesson plans of their own, borrow plans from other teachers, or search online or within books for lesson plans. When deciding what teaching method to use, a teacher needs to consider students' background knowledge, environment, and learning goals. Teachers are aware that students learn in different ways, but almost all children will respond well to praise. Students have different ways of absorbing information and of demonstrating their knowledge. Teachers often use techniques which cater to multiple learning styles to help students retain information and strengthen understanding. A variety of strategies and methods are used to ensure that all students have equal opportunities to learn. A lesson plan may be carried out in several ways: Questioning, explaining, modeling, collaborating, and demonstrating.

Explaining form is similar to lecturing. Lecturing is teaching by giving a discourse on a specific subject that is open to the public, usually given in the classroom. This can also be associated with modeling. Modeling is used as a visual aid to learning. Students can visualize an object or problem, then use reasoning and hypothesizing to determine an answer.

 Demonstrations are done to provide an opportunity to learn new exploration and visual learning tasks from a different perspective. A teacher may use experimentation to demonstrate ideas in a science class. A demonstration may be used in the circumstance of proving conclusively a fact, as by reasoning or showing evidence.

Having students work in groups is another way a teacher can direct a lesson. Collaborating allows students to talk with each other and listen to all points of view in the discussion. It helps students think in a less personally biased way. When this lesson plan is carried out, the teacher may be trying to assess the lesson by looking at the student's: ability to work as a team, leadership skills, or presentation abilities. It is one of the direct instructional methods. Collaborating is great in that it allows to actively participate in the learning process. These students who learn best this way by being able to relate to the lesson in that they are physically taking part of it in some way. Group projects and discussions are a great way to welcome this type of learning.

6.                  Student Reflection

Reflection is thinking for an extended period by linking recent experiences to earlier ones in order to promote a more complex and interrelated mental schema. The thinking involves looking for commonalities, differences, and interrelations beyond their superficial elements. The goal is to develop higher order thinking skills.

Reflection is a proses of  reviewing in experience  describe, analyze, evaluat and inform it. The student exchange abaut their learning experience to another student reflection is to take ownership of student’s learning, that is student center  and motivation student to study harder at their learning.

7.                  Cognitif Schema

Cognitive Theory, developed by Jean Piaget, a Swiss psychologist who lived from 1896-1980. This theory concerns the emergence and acquired schemata-schemes of how one perceives its environment-in developmental stages, when a person obtains a new way of mentally representing information.

Application of cognitive learning theory in learning, teachers must understand that students are not as easy an adult in his thinking process, pre-school age children and early elementary school learning to use concrete objects, active students is overlooked, a teacher organization of the material by using a pattern or logic particular from simple to complex, teachers create meaningful learning, individual differences students to achieve student success.

Piaget describes the educational implications of cognitive theory is:

a.       Focusing attention to the ways of thinking or mental processes, not just the result. Teachers must understand the process used so that children arrive at these results. Experience - learning experience developed in accordance with the stage of cognitive function and if teachers are attentive to students' approach used to arrive at certain conclusions, then it can be said of teachers are in a position to give the experience in question.

b.      Stresses the role of students in their own initiative and active involvement in learning activities. In the classroom, Piaget emphasized that the teaching of knowledge so (ready made knowledge) of the children are encouraged to determine their own knowledge through spontaneous interaction with the environment.

c.       Understand the existence of individual differences in terms of the progress of the development. Piaget's theory assumes that all students grow and pass through the same developmental sequence, but the growth took place at different speeds. Therefore, teachers should make efforts to regulate activities in the class consisting of individuals - individuals into form groups - small groups of students rather than the activity in the form of a classical.

d.      Prioritize the role of students to interact with each other. According to Piaget, the exchange of ideas - ideas can not be avoided for the development of reasoning. Although the reasoning can not be taught directly, its development can be simulated.

8.                  Student conclution

Student conclution is one good way to present a material to student. To make the student conclusion efective, the teacher should be:

a.                   Make a plan what will do in class.

b.                  The teacher still involve in the small group during discussion, so the teacher know what  the student’s need.

c.                   After presentation over, the teacher review who the student present and also give a chanche to the student make conclusion.

d.                  The student should be present their result to compare with other group.

9.                  Apperception

Apperception is what the teacher and student doing to get prepare to do the learning or leson. Actually apperception belongs to student, the teacher just help
the student to construct their understanding abaut the topic. Student construct the students construct their own way of solving problems. on a video that never aired during last week's lecture, we can see that the teacher asked the students about solving a problem, then students are asked to answer it.

Students answer , according to the capability it has. Then the teacher shows the answers of some of the students and presented in front of the class. with shows the answers from other students, students can see the variation of solving the problem. With these students can think of many ways to solve those problems, not only with one step. In this case only a mentor teacher and the student who actively construct their own completion of the matter.

10.              Assessment

Assessment of learning is a process or a formal effort to collect information relating to the crucial variables of learning as an ingredient in decision making by teachers to improve processes and student learning outcomes. Important variables are defined at include lack of knowledge, understanding, skills and attitudes of students in teacher learning obtained with different methods and procedures in both formal and informal.

The use of appropriate types of assessment that will determine your success in accessing information relating to the learning process. Selection of assessment methods should be based on information targets to be achieved. The information in question is the learning outcomes achieved by students. There are five categories of learning outcome goals that deserve to be the basis in determining the types of assessment that will be used by teachers. The five learning outcomes are:

a.      Knowledge Outcomes, is a mastery of students' knowledge on the substance of a subject.

b.      Reasoning Outcomes, which shows students' ability to use knowledge in the conduct of reason (reason) and solve a problem.

c.       Skill Outcomes, the ability to demonstrate specific achievement-related skills based on the mastery of knowledge.

d.       Product Outcomes, the ability to make a particular product that is based on the mastery of knowledge.

e.    Affective Outcomes, achievement of certain attitudes as a result of studying and applying knowledge.

The main purpose of assessment use in learning (classroom assessment) is to help teachers and students in making decisions propesional to improve learning.  Assessment aims to, among other:

a.                  diagnose students' strengths and weaknesses in learning.

b.                  Monitor student progress.

c.                   Determine the levels of student ability.

d.                  Determine the effectiveness of learning.

e.                   Influencing public perceptions about the effectiveness of learning.

f.                   Evaluate the performance of classroom teachers.

g.                   Clarify the purpose of teacher-designed learning

11.              Various media and teaching aids

 Media actually will be helpful in realizing the goal of education despite the many flaws in it . It is expected kreatifity teachers in choosing which media is more suitable to be applied in the classroom. In this case to consider is the material to be delivered, the situation pre-class facilities and infrastructure.

Although in its use of technology and other types of media are needed to help teachers and students in learning activities, but there are some advantages and disadvantages in its use. Among the advantages or usefulness of learning media, such as:

a)             Presentation pesanagar not be verbalistis (in the form of words, written or oral mere).

b)             Solve frontier of space, time and the senses, such as:

                                                   i.            Objek that is too large is replaced by reality, images,movies or model small.

                                                 ii.             Obyek assisted with micro projectors, film frames, film or images.

                                               iii.            That motion is too slow or too fast can be helped by TAME lapse or high speed photography.

                                               iv.            Events that occurred the past can be played back through the recording film, video, film frames, picture or verbally. Obyek that are too complex (machines) can be presented with models, diagrams, etc.

                                                 v.            Concept  that are too broad (the mountain air fire, earthquakes, climate, etc.) can be visualize in the form of movies, movie frame, pictures, etc..

With appropriate use of educational media and varied nature of the passive learner can be overcome. In this case the medium of learning is useful for:

a)                   Improve excitement of learning.

b)                  abilitymore direct interaction between the students with the environment and the fact.

c)                   Ability students learn on their own according to ability and interests of each.

There are some disadvantages with respect to the movement of advance of other visual teaching materials overemphasized visual  themselves with no other activities related to the design, development, production, evaluation, and management of visual materials. Besides the visual material is also seen as merely a tool for teachers in the learning process so that the integration of teaching materials and tools are ignored. Audio visual flaws: too much emphasis on the mastery of the material of the development process and still looked upon as an audio-visual aids teachers in the learning process.

C.                CONCLUTION

Mathematics is a very big role in the life to come, but today's mathematics courses remains a scourge for most students. Causes of mathematics remains a problem for students is the lack of interest among students studying mathematics, students still think that mathematics is difficult and unpleasant, and teachers are also less variable in the present matter of mathematics, students often do not even think that mathematics is a monster.

Who need to get the attention of teachers of mathematics at this time of paradigm shift from teaching to learning. Learning paradigms are not enough students studying with the teacher's instructions in transferring knowledge to students, but students need to construct a mathematical study. Constructivist math teachers are teachers who construct learning theory by reflecting on how teachers use and form of mathematics. In a constructive learning, to consider the following matters:

a.              Students are actively involved in learning, students learn the material with a significant mathematical work and thinking, and students learn how to learn it.

b.             New information must be linked with other information so it blends with the information that has been owned by students to the understanding of the information (materials) complex occurred.

c.              Oreintasi learning is an investigation and discovery which is basically solving the problem.

Implications of the theory of constructivism on learning of them:

a.              Every teacher will have experienced that a matter has been dealt with clearly-explained but there are still some students who do not understand or do not understand the material being taught at all. This suggests that a teacher can teach a subject to the sisiwa well, but all or part of his students do not learn at all. Strenuous efforts of a teacher in teaching should not be followed with good results in their students. Because, only with a terrible effort the students will truly understand the material taught.

b.             The task of each teacher in facilitating the students, so that knowledge is built or constructed material students sendirisan not implanted by the teacher. The sisiwa must be able to actively assimilate and accommodate new experiences into cognitive framework.

c.              To teach well, teachers must understand the mental models used by the students to know their world and reasoning that develop by student to support the models were.

d.             Students need to construct their own understanding for each concept so that teachers in the teaching materials rather than "lecturing", explained or similar efforts to transfer knowledge to students but to create situations for students that help them make the mental constructions are needed.

e.              The curriculum is designed so that condition that allows the knowledge and skills can be constructed by learners.

f.              Problem-solving exercises are often done through a study group to analyze problems in everyday life.

g.             Learners are expected to be active and be able to find a way of learning that suits him. Teachers only as a facilitator, mediator, and friends who make the situation conducive to the construction of the self knowledge learners.