Kualitas Kedua dalam RPP pada Skema Pencapaian Kompetensi 'is never ending effort'

Oleh: Euis Kurniawati, SPd

Direkomendasikan oleh Dr Marsigit

Pada perkuliahan terakhir hari Rabu (31 Desember 2008) sampailah kami pada suatu kesimpulan akhir tentang materi kuliah Perencanaan Pembelajaran Matematika (PPM). Ternyata apa yang kami harapkan selama ini dari kuliah PPM yaitu berupa 'produk' RPP (Rencana Pelaksanaan Pembelajaran atau Lesson Plan) yang kami bayangkan, yang baku, baik dan benar (adakah?) sesuai kebutuhan di sekolah, yang siap pakai (siap difoto copy oleh rekan-rekan di sekolah/daerah) terhempaskan sedalam-dalamnya ke dasar jurang pemikiran kami yang masih dalam kategori 'mengkhawatirkan'.

RPP yang kami harapkan tersebut adalah RPP Formal, RPP kualitas Pertama. RPP yang hanya menampakkan alur proses pembelajaran, yang bisa diperoleh dari contoh yang sudah ada, bahkan kami bisa saja mencarinya di internet.

Senyatanya, RPP yang beliau (Dr. Marsigit M.A.) maksudkan, yang beliau harapkan agar kami pahami, adalah RPP dalam ranah kualitas Kedua.

Dengan segala keterbatasan saya dalam memahami dan memaknai RPP dalam kualitas Kedua, saya mencoba untuk menuangkannya sebagai suatu wujud permulaan pemahaman yang saya miliki, tentu saja jauh dari sempurna.

Apa yang dimaksud dengan RPP?

RPP (Rencana Pelaksanaan Pembelajaran) adalah seperangkat persiapan yang dilakukan oleh guru meliputi skenario sebelum, selama, dan setelah kegiatan pembelajaran berlangsung. Sedangkan menurut Dr. Marsigit M.A., RPP pada dasarnya merefleksikan aktivitas yang terjalin antara siswa dan guru.

Perbedaan nyata antara kegiatan pembelajaran di Indonesia dengan negara lain adalah terletak pada 'persiapan'. Contohnya persiapan (RPP) yang digunakan di negara Jepang berisi tentang hal-hal yang akan dikerjakan oleh siswa. Sedangkan RPP yang digunakan di Indonesia berisi hal-hal yang akan dikerjakan oleh guru. RPP yang biasanya disusun oleh guru di Indonesia, kebanyakan berupa RPP legal formal yang dipergunakan untuk kepentingan karir atau dinas semata belum menyentuh hakikat RPP yang sebenarnya (Nature Leson Plan).

RPP dapat ditinjau dari berbagai skema, diantaranya: Struktur Pembelajaran (Pendahuluan, Kegiatan Inti, dan Penutup), Skema Pencapaian Kompetensi (Will, Attitude, Knowledge, Skill, dan Experience), Skema Interaksi (Klasikal, Kelompok dan Individua), Skema Variasi Metode (Induksi-Deduksi) Skema Variasi Media atau alat bantu pembelajaran (LKS dan Alat Peraga) dan Skema Variasi Sumber Belajar (Buku Teks, Internet atau Blog dan ICT).

Kualitas Kedua dalam RPP

Kualitas Kedua dalam RPP pada Skema Pencapaian Kompetensi (Will, Attitude, Knowledge, Skill, dan Experience) dapat dipandang sebagai suatu cara dalam memaknai RPP berdasarkan hakikat kompetensi yang ingin dicapai dan ditampilkan dalam RPP.

1. Will: Kemauan, Kehendak (Senang, gembira)

Bagaimana upaya yang dilakukan guru dalam merancang suatu RPP yang dapat mengeksplorasi dan membangkitkan kemauan atau kehendak siswa untuk belajar dan terlibat aktif dalam kegiatan pembelajaran secara menyenangkan dan dengan perasaan gembira, jauh dari rasa was-was, cemas, bahkan takut. Bagaimana upaya guru dalam membawa dunia matematika ke dunia anak (siswa). Contohnya: menghargai bahasa ibu dalam pembelajaran matematika (selawe = 25; selangkung = 50).

2. Attitude: Sikap, Pendirian (Sabar, tepat waktu)

Bagaimana upaya yang dilakukan guru dalam merancang suatu RPP yang dapat menghidupkan 'ruh' pembelajaran matematika dalam diri anak yaitu berupa keuletan, ketekunan, sifat pantang menyerah, kemampuan mengkomunikasikan ide dan gagasan, kemampuan berpikir kritis dan logis, serta kemampuan mencari solusi dari suatu permasalahan. Sehingga melahirkan suatu sikap atau pendirian yang positif bagi siswa misalnya berupa sifat sabar, tekun, serta tepat waktu (disiplin).

3. Knowledge: Pengetahuan (Mengetahui)

Bagaimana upaya guru dalam merancang suatu RPP yang dapat menumbuhkan kesadaran pada diri siswa agar mereka 'mengetahui' eksistensi mereka dalam dunia matematika, pembelajaran matematika dan pendidikan matematika. Bagaimana upaya guru membangkitkan kesadaran siswa akan pentingnya membangun pengetahuan (khususnya matematika) mereka sendiri dengan potensi, cara, dan keunikan mereka masing-masing.

4. Skill: Keterampilan (Terampil)

Bagaimana upaya guru dalam merancang suatu RPP yang dapat mengasah potensi siswa dalam hal keterampilan mengemukakan ide/gagasan; keterampilan memilih solusi dari permasalahan matematika; keterampilan menyanggah, menjawab, membuktikan suatu hipotesa; keterampilan menggunakan dan menyusun alat peraga; dan sebagainya. Inti dari semuanya itu adalah agar siswa terampil menggunakan potensi keunikan mereka untuk menghadapi dan mencari solusi (secara cerdas) bagi permasalahan dalam pembelajaran matematika. Sehingga matematika tidak lagi menjadi mata pelajaran yang menakutkan dan membosankan bagi siswa.

5. Experience: Pengalaman (Kebermaknaan)

Bukankah pepatah mengatakan bahwa pengalaman adalah guru yang terbaik? Maka sangatlah diperlukakan upaya guru dalam merancang suatu RPP yang dapat menjadikan kegiatan pembelajaran matematika sebagai suatu pengalaman penting, berkesan dan berharga bagi siswa. Pengalaman akan dirasakan penting, berkesan dan berharga bagi seseorang jika ada makna yang mendalam dari pengalaman tersebut. Kebermaknaan dalam belajar akan dicapai mana kala siswa merasa terlibat dan berperan aktif secara menyenangkan dan penuh perasaan gembira. Kebermaknaan dapat tercapai dalam kegiatan pembelajaran yang memberi ruang dan kesempatan sebesar-besarnya bagi siswa untuk mengkonstruksi pengetahuan dengan cara dan potensi unik mereka masing-masing, dengan bimbingan guru (tanpa mendominasi) yang mampu melayani kebutuhan para siswanya yang heterogen.

Catatan:

Tulisan ini belum memuaskan bagi saya. Mungkin karena RPP dalam kualitas Kedua merupakan pengetahuan baru bagi saya yang juga sekaligus memancing rasa penasaran saya untuk dapat merancangnya. InsyaAlloh ke depannya akan diupayakan untuk dapat terealisasi. Saya harus lebih banyak belajar dan terus berusaha, never ending effort (meminjam istilah Pak Marsigit).

Elegi Seorang Guru Menggapai Batas

Oleh: Marsigit

Mulialah hati, pikiran dan tindakan pedagogik guru, karena tiadalah seorang guru bermaksud memberikan keburukan bagi siswanya. Totalitasnya dia mengidamkan kebaikan dan keberhasilan bahkan keberhasilan tertinggi jika memang mungkin bagi siswanya. Maka hati, pikiran dan jiwanya menyatu menjadi motivasi yang kuat bahkan mungkin SANGAT KUAT untuk mewujudkan tindakan pedagogik : MEMBIMBING, MENGAWASI, MEMBEKALI, MENASEHATI, MEWAJIBKAN dan kalau perlu MENGHUKUM siswanya DEMI KEPENTINGAN SISWA.
Ketika rakhmat Nya menghampiri guru sedemikian hingga guru dengan sengaja atau tak sengaja menembus BATAS jiwanya sehingga memperoleh kesempatan MELIHAT DIRINYA dari tempat nun jauh secara mandiri maupun berbantuan orang atau guru atau nara sumber, maka adalah suatu tempat dimana BATAS ITU akan digapainya. Dalam batas itulah guru menemukan FATAMORGANA.
Atas fatamorgana itu maka terdengarlah nyanyian AKAR dan RUMPUT yang sayup-sayup sampai kira-kira bait-baitnya begini:

-Jangan jangan motivasimu yang sangat KUAT telah MELEMAHKAN INISIATIF nya, apakah itu batas yang kau cari?
-Jangan-jangan maksudmu MEMBIMBING telah menjadikannya TERGANTUNG DAN TAK BERDAYA, apakah itu batas yang kau cari?
-Jangan-jangan maksudmu MENGAWASI telah menjadikannya mereka merasa menjadi makhluk yang memang selalu perlu diawasi dan dengan demikian identik dengan KEBURUKAN, apakah itu batas yang kau cari?
-Jangan-jangan maksudmu MEMBEKALI telah menjadikanmu SOMBONG DAN TAKABUR serta menjadikannya RENDAH DIRI DAN TAK BERDAYA, apakah itu batas yang kau cari?
-Jangan-jangan maksudmu MENASEHATI adalah AMBISI DAN EGOMU yang menyebabkan mereka hidup TIDAK BERGAIRAH, apakah itu batas yang kau cari?
-Jangan-jangan maksudmu MEWAJIBKAN adalah pertanda HILANGNYA NURANIMU sekaligus HILANGNYA NURANI MEREKA, apakah itu batas yang kau cari?
-Jangan-jangan maksudmu MENGHUKUM adalah SEMPITNYA HIDUPMU dan juga SEMPITNYA HIDUP MEREKA, apakah itu batas yang kau cari?
-Jangan-jangan maksudmu DEMI KEPENTINGAN SISWA adalah keadaan HAMPA TAK BERMAKNA atau bahkan MENYESATKAN, apakah itu batas yang kau cari?

Orang tua berambut putih datang dan berkata:
"ITULAH SEBENAR-BENAR BATAS YANG KAU CARI, DAN BATASMU TIDAK LAIN TIDAK BUKAN ADALAH BATAS MEREKA. ANTARA DIRIMU DAN MEREKA ITULAH SEBENAR-BENAR ILMUMU TENTANG DIRIMU DAN TENTANG DIRI MEREKA.
JIKA KAMU BELUM MENGETAHUI BATASMU JANGAN HARAP RAHMAT ITU DATANG MENGHAMPIRIMU KEMBALI, TETAPI JIKA TOH KAMU TELAH MENGETAHUI BATASMU MAKA ADALAH KODRAT NYA BAHWA KAMU MASIH HARUS BERJUANG MENGGAPAI RAHMATNYA.
NAMUN MAAF SAYA TELAH SALAH UCAP, KARENA SEBENAR-BENARNYA YANG ADA, ADALAH TIADALAH ORANG PERNAH DAN DAPAT MENGGAPAI BATASNYA. UNTUK ITU MARILAH KITA TAWADU' DENGAN KODRAT DAN IRADATNYA. TETAPI JANGAN SALAH PAHAM BAHWA IKHTIARMU MENGGAPAI BATASMU ADALAH JUGA KODRATNYA.
AMIN YA ROBBIL ALAMIN"

The Implication of Vygotsky’s Work to Mathematics Education

By Marsigit
Proper organisation of the learning is the key factor in the pedagogic processes described by Vygotsky in which the teacher holds for the responsibility of the child's learning. This implies careful diagnostic assessment of the child's existing category system and appropriate sequencing of learning experiences to move the child from that point towards the next defined curricular goal (Galloway and Edwards, 1991). The process of generalization indicates the abstraction of rules and the beginnings of the development of internal consciousness and higher cognitive functioning (Evans, 1986); through this process the curriculum is changed and developed to meet the needs of the pupils more fully. By concentrating on the analysis of the overall process of education, Vygotsky sees teachers occupying a didactic role. He defined intelligence as the capacity to learn from instruction (Sutherland, 1992); it implies that the teacher should guide her pupils in paying attention, concentration and learning effectively; the teacher should scaffold a pupil to competence in any skill. Vygotsky places the teacher firmly alongside the child in a process of jointly constructing meaning and so emphasises the importance of language and communication in the construction of an understanding of the world (Galloway and Edwards, 1991).The teacher's role then is to make the classroom as rich an interactive learning community as she or he can and through language to lead children into new zones of proximal development (Gipps, 1994); and he suggested that instruction is most effective when it is addressed to the child's zone of proximal development (Blenkin & Kelly, 1984). Internalization of the learning is demonstrated through the ability to transfer the learning to new situation (Evans, 1986). Vygotsky proposed that every specific state of a pupil's development is characterized by an actual development level and a level of potential development (Hoyles, 1987); the pupil is not able to exploit the possibilities at the latter level on her own, but can do so with educational support; thus, teaching should provide 'scaffolding' for voyaging into the next level of intended learning. Hoyles (1987) concluded that the ideas of providing 'scaffolding' leads on to think about this model of teaching which does not necessarily lead to conflict between the learner's autonomy and pedagogic guidance.
It is important to note here that Vygotsky at one time acknowledge the operation of societal or social institutional forces; Vygotsky and Mead studied social processes in small group interaction in terms of interpersonal dynamics and communication. As emphasized by Vygotsky (1978), the social context affects development at both the institutional and material level, as well as the interpersonal level. In development, children adapt their cognitive and social skills to the particular demands of their culture through practice in particular activities; children learn to use physical and conceptual tools provided by the culture to handle the problems of importance in routine activities (ibid, p. 328). Study after study has documented the absence in classrooms of the fundamental tool for the teaching of children: assistance provided by more capable others that is responsive to goal-directed activities (Tharp and Gallimore, 1988). To provide assistance in the ZPD, the teacher must be in close touch, sensitive and accurate in assisting. There should be opportunities for assisted performance, for using of small groups and for the maintenance of a positive classroom atmosphere that will increase independent task involvement of students, new material and technology with which students can interact independent of the learner (ibid, p. 58).
The explicit implication from above propositions for the teaching of primary mathematics is that the children need to actively engage with mathematics, posing as well as solving problems, discussing the mathematics embedded in their own lives and environment as well as broader social context (Ernest, 1991). The appropriate of teaching, as he suggested, may include a number of components : genuine discussion, both student-student and student-teacher, since learning is the social construction of meaning; cooperative groupwork, project-work and problem solving for confidence, engagement and mastery; autonomous projects, exploration, problem posing and investigative work, for creativity, student self-direction and engagement through personal relevance; learner questioning of course contents, pedagogy and the modes of assessment used, for critical thinking; and, socially relevant materials, projects and topics, including race, gender and mathematics, for social engagements and empowerment. Related to the resources of teaching, Ernest (1991) suggested that due to the learning should be active, varied, socially engaged and self-regulating, the theory of resources has three main components : (1) the provision of a wide variety of practical resources to facilitate the varied and active teaching approaches; (2) the provision of authentic material, such as newspaper, official statistics, and so on for socially relevant and socially engaged study and investigation; and (3) the facilitation of student self-regulated control and access to learning resources.
When cognitive change is considered as much a social as an individual process, new question arise about when and how to track or measure change (Newman, et al., 1989). This is about the role of assessment in the process of instructional interaction. In the 'dynamic assessment', derives from a particular interpretation of Vygotsky's zone of proximal development (ZPD), the ZPD provides a very interesting alternative to the traditional standardized test (Newman, et al., 1989). For Vygotsky, assessment which focuses only on a child's actual level of attainment or development is incomplete and gives only a partial picture. Instead of giving the children a task and measuring how well they do or how badly they fail, one can give the children the task and observe how much and what kind of help they need in order to complete the task successfully; in this approach the child is not assessed alone; rather, the social system of the teacher and child is dynamically assessed to determined how far along it had progressed. Assessment tasks and outcomes should be open to pupil discussion, scrutiny and negation where appropriate, and student choice for topic for investigation and project-work (Ernest, 1991). Further, he suggested that the content of assessment tasks, such as projects and examination questions, should include socially embedded mathematical issues, requiring critical thinking about the social role of mathematics.
Within the ZPD, and suggest that clarification and communication of purpose, aims, and expectations are central to strategy for self-assessment; the variation in assistance to the child that Tharp and Gallimore describe permeate this account of development activities as assessment itself is treated as a performance. He found that, by interviewing the children in the six classes aged between 5 and 9, pupils self-assessment provide the basis for development activities with the clarification of purposes, aims, and expectation through the use of long-term aims and short-term target. Tharp and Gallimore's model provides a framework for developing the ways in which children can be encouraged to assess their own progress; the clarification and evaluation of targets become a zone in which each child's performance is assisted by their teacher (ibid, p.236); as they become involved in their own assessment they gradually take over the task and complement the wide range of skills and talents with each child begins school. So the purpose of mathematics education should be enable students to realize, understand, judge, utilize and sometimes also perform the application of mathematics in society, in particular to situations which are of significance to their private, social and professional lives (Niss, 1983, in Ernest, 1991). Accordingly, the curriculum should be based on project to help the pupil's self-development and self-reliance; the life situation of the learner is the starting point of educational planning; knowledge acquisition is part of the projects; and social change is the ultimate aim of the curriculum (Ernest, 1991).
References:
Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.
Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Tharp, R. and Gallimore, R., 1988, 'A theory of teaching as assisted performance' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.

The Implication Of Piaget's Work to Mathematics Education

By Marsigit
Especially attractive to workers in mathematics education were Piaget's conceptions that children's intellectual development progresses through well-defined stages, that children develop their concepts through interaction with the environment, and that for most of the primary years most children are in the stage of concrete operations. Further, she stated that in mathematics education, it was a natural consequence of belief in Piaget's theory about the central role of interaction with objects that, when they learn mathematics, children should be expected to work practically, alone with their apparatus, and to work out mathematical concepts for themselves. Piaget proposes that children create their knowledge of the world; however, he also argued that in the creation and unfolding of their knowledge, children are constrained by absolute conceptual structures, especially those of mathematics and logic; thus, Piaget accepts an absolutist view of knowledge, especially mathematics (ibid, p.185). For primary children (7 to 12 years) in which Piaget called they are in the stage of concrete operation, mental action occurs in a structure with its counteraction - adding goes with its reverse operation subtracting, combining with separating, identity with negation (Becker, 1975); thinking shows many characteristics of mature logic, but it is restricted to dealing with the real; an eight-year old, for example, has no trouble ordering a set of sticks according to height, but might fail to solve the problem.
In order to draw out the explicit implications of Piaget's work for mathematics teaching in primary school for the children who are at concrete-operational stage, it is useful to divide this stage into three stages : early concrete-operational (7-9 years), middle concrete-operational (10-12 years) and late concrete-operational (13-15 years). The children at early concrete-operational and middle concrete-operational will be discuss in the following. The child at early concrete-operational stage is confined to operations upon immediately observable physical phenomena; therefore, he states that the implications of the teaching mathematics may be translated as : (1) both the elements and operations of ordinary arithmetics must be related directly to physically available elements and operations, (2) there should be no more than two elements connected by one operation even with the restriction and the result must be actually closed to avoid the problem of any doubt about the uniqueness of the result, (3) the only notion of inverse is physical, (4) there is no basis for seeking a consistency in relationships with a system of elements selected two at a time and connected by an operation.
The child at middle concrete-operational tends to work with qualitative correspondences, e.g. the closer, the bigger; and thus is still reality bound and not capable of setting up a reliable system based on measurement. For these reason he outlined that its implications for teaching mathematics in the primary school are that : (1) Children begin to work with operations as such but only where uniqueness of result is guaranteed by their experience both with the operations and the elements operated upon; this in effect means two operations closed in sequence with small numbers or one familiar operation using numbers beyond his verified range, e.g. the child can cope with items involving the following types of combinations, (3+8+5) and (475+234); (2) The developing notion of the inverse of an operation tends to be qualitative; children regard substracting as destroying an effect of addition without specifically value of 'y' in y+4=7, they regard 'y' as a unique number to which '4' has been added, substracting '4' happens to destroy the effect of the original addition; (3) a basis exists for the development of a notion of consistency as being a necessary condition for a system of operations but the child tend to recognize the need without being able to give a logical reason for it. Preadolescent child makes typical errors of thinking that are characteristics of his stage of mental growth; the teacher should try to understand these error; and, besides knowing what errors the child usually makes, the teacher should also try to find out why he makes them (Adler, 1968). For these reason, further he suggests that an answer or an action that seems illogical from the teacher point of view on the basis of teacher's extensive experience may seem perfectly logical from the child's point of view on the basis of his limited experience.The teacher can help the child overcome the errors in his thinking by providing him with experiences that expose them as errors and point the way to the correction of the errors.
The child in the pre-operational stage tends to fix his attention on one variable to the neglect of others; to help him overcome this error, provide him with many situations (Adler, 1968). Due to the fact that a child's thinking is more flexible when it is based on reversible operations, the teacher should teach them pairs of inverse operations in arithmetic together, and teach that subtraction and addition nullify each other, and multiplication and division nullify each other (ibid, p.58). As Piaget summed up in Copeland (1979), 'numerical addition and subtraction become operations only when they can be composed in the reversible construction which is the additive group of integers, apart from which there can be nothing but unstable intuition'. Adler (1968) also suggested that physical action is one of the basis of learning; to learn effectively, the child must be a participant in events; to develop his concepts of numbers and space, for example, he needs to touch things, move them, turn them, put together or take them apart. Children should have many experiences in sorting common classroom materials, working with concrete shapes and sizes and colors, and discussing all sorts of relationship; this activities provides a basis for determining in a clearly defined way what progress children are making in their ability to realize, as well as copy, basic spatial distinction; and, most children will be ready at first-grade level to learn the basic shapes.
Since there is a lag between perception and the formation of a mental image, the teacher needs to reinforce the developing mental image with frequent use of perceptual data, for example, let him see the addition once more as a succession of motions on the number line when the child falters in the addition of integers (Adler, 1968). What the mind 'represents' may be and often is different from what is 'seen' or 'felt' by small children (Copeland, 1979); the teachers need to know the stages through which children go in developing the ability to consider geometric ideas. In order that the students are ready to learn a new concept, the teacher should examine the mastering of student's prerequisites concept; Piaget's theories suggested that learning was based on intellectual development and occured when the child had available the cognitive structures necessary for assimilating new information (Leder, 1992). In his teaching, the teacher needs to look at the way pupils go about their work and not just at the products; he also needs to listen to pupil's ideas and try to understand their reasoning and discuss the problems so that pupils reveal their ways of thinking. These activities are actually in the framework of teacher's method of assessing students' thinking.
Piaget developed his 'clinical method' as a way of exploring the development of children's understanding, and employed observation along with interview as a means of accessing children's views of the world (Conner, 1991). He is the pioneers who advocated using observations of children in real situation; the observation, that is more than just looking, serves a useful assessment purpose involves : looking at the way pupils go about their work and not just at the products, listening to pupils' ideas and trying to understand their reasoning, and discussing problems so that pupils reveal their ways of thinking (ibid, pp. 50-51). The observation were used to support the hypothesis that the children, at a certain stage, were discriminating between 'means' and 'ends' (Becker, et al.,1975). In this interview, the answers of students at various ages are then analyzed to see how properties of 'mental structures' change with age (ibid, p. 218). The justification of whether the Piaget's idea of assessment is practical or not in the classroom practice depends much on : the philosophy of mathematics education in which we start to do so, the characteristics of his paradigm of cognitive development or student's competence, the capability of the teacher ; and, in general, this is dependent on its interpretation.
References:
Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.
Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Tharp, R. and Gallimore, R., 1988, 'A theory of teaching as assisted performance' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.

Vygotsky's Work and Its Relevance to Mathematics Education

By. Marsigit
One of the most fundamental assumptions that guided Vygotsky's attempt to reformulate psychology was that in order to understand the individual, one must first understand the social relations in which the individual exists (Wertsch, 1985); Vygotsky argued that the social dimension of consciousness is primary in time and in fact, the individual dimension of consciousness is derivative and secondary. Thus, to explain the psychological, we must look not only at individual but also at external world in which that individual life has developed (Tharp and Gallimore, 1988). The first key feature of Vygotsky's theory is that of internalization. The process by which the social becomes the psychological is called internalization (Tharp and Gallimore, 1988); the individual's plane of consciousness is formed in structures that are transmitted to the individual by others in speech, social interaction, and the processes of cooperative activity; thus, individual consciousness arises from the actions and speech of others. Wertsch (1985) listed that Vygotsky's account of internalization is grounded in four major points : (1) internalization is a process wherein an internal plane of consciousness is formed; (2) the external reality at issue is a social interactional one; (3) the specific mechanism at issue is the mastery of external sign forms; and (4) the internal plane of consciousness takes on a 'quasi-social' nature because of its origins. In the beginning of the transformation to the intramental plane, the child need not understand the activity as the adult understands, need not be aware of its reason or of its articulation with other activities (Tharp and Gallimore, 1988); all that is needed is performance, through assisting interaction; through this process, the child acquires the plane of consciousness of the natal society and is socialized, acculturated, made human.
The second key feature of Vygotsky's theory is that of the zone of proximal development; this refers to the gap that exists for children between what they can do alone and what they can do with help from someone more knowledgeable or skilled than themselves (Gipps, 1994). Vygotsky introduced the notion of zone of proximal development in an effort to deal with two practical problems in educational psychology (Wertsch, 1985): the assessment of children's intellectual abilities and the evaluation of instructional practices. He argued that it is just as crucial, if not more so, to measure the level of potential development as it is to measure the level of actual development; existing practices were such that 'in determining the mental age of a child with the help of tests we almost always are concerned with the actual level of development' (ibid, p. 68). Vygotsky argued that the zone of proximal development is a useful construct concerns processes of instruction (ibid, p. 70); instruction and development do not directly coincide, but represent two processes that exist in very complex interrelationships.
Assisted performance defines what a child can do with help, with the support of the environment, of others, and of the self (Tharp and Gallimore, 1988); the transition from assisted performance to unassisted performance is not abrupt. They present problems through the zone of proximal development (ZPD) in a model of four stages; (1) the stage where performance is assisted by more capable others (Stage I); (2) the stage where performance is assisted by the self (Stage II); (3) the stage where performance is developed, automatized, and 'fossilized' (Stage III); and (4) the stage where de-automatization of performance leads to recursion back through the ZPD (Stage IV). Tharp and Gallimore (1988) outlined the propositions of the problems in the Stage I as follows : (1) before children can function as independent agents, they must rely on adults or more capable peers for outside regulation of task performance; (2) the amount and kind of outside regulation a child requires depend on the child's age and the nature of the task; (3) the child may have a very limited understanding of the situation, the task, or goal to be achieved; at this level, the parent, teacher, or more capable peer offers directions or modeling, and the child's response is acquiescent or imitative; (4) only gradually does the child come to understand the way in which the parts of an activity relate to one another or to understand the meaning of the performance; (5) ordinarily, the understandings develops through conversation during the task performance; (6) the child can be assisted by questions, feedback, and further cognitive structuring that is such assistance of performance has been described as scaffolding, by Wood, Bruner, and Ross (1976); (7) the various means of assisting performance are indeed qualitatively different; (8) a child's initial goal might be to sustain a pleasant interaction or to have access to some attractive puzzle items, or there might be some other motive that adults cannot apprehend; (9) the adult may shift to a subordinate or superordinate goal in response to ongoing assessment of the child's performance; (10) the task of Stage I is accomplished when the responsibility for tailoring the assistance, tailoring the transfer, and performing the task itself has been effectively handed over to the learner; this achievement is gradual, with progress occuring in fits and starts.
For the Stage II, Tharp and Gallimore (1988), outlined the propositions: (1) the child carries out a task without assitance from others; however, this does not mean that the performance is fully developed or automatized; (2) the relationships among language, thought, and action in general undergo profound rearrangements - ontogenetically, in the years from infancy through middle childhood; (3) control is passed from the adult to the child speaker, but the control function remains with the overt verbalization; the transfer from external to internal control is accomplished by transfer of the manipulation of the sign from others to the self; (4) the phenomenon of self-directed speech reflects a development of the most profound significance; self-control may be seen as a recurrent and efficacious method that bridges between help by others and fully automated, fully developed capacities; (5) for children older than 6 years, semantic meaning efficiently mediates performance; (6) for children, a major function of self-directed speech is self-guidance; this remains true throughout lifelong learning. For the Stage III, Tharp and Gallimore (1988), outlined the propositions: (1) once all evidence of self-regulation has vanished, the child has emerged from the ZPD into the development stage for the task; (2) the task execution is smooth and integrated; it has been internalized and automatized; (3) assistance from the adult or the self is no longer needed; indeed assistance would now be disruptive; (4) it is at this stage that self-consciousness itself is detrimental to the smooth integration of all task components; (5) this is a stage beyond self-control and beyond social control; (6) performance here is no longer developing; it is already developed. For the Stage IV, Tharp and Gallimore (1988), outlined the propositions: (1) there will be a mix of other-regulation, self-regulation, and automatized processes; (2) once children master cognitive strategies, they are not obliged to rely only on internal mediation; (3) enhancement, improvement, and maintenance of performance provide a recurrent cycle of self-assistance to other-assistance; (4) de-automatization and recursion occur so regularly that they constitute a Stage IV of the normal development process; after de-automatization, if the capacity is to be restored, then the developmental process must become recursive.
References:
Wertsch, J.V.,1985, Vygotsky and The Social Formation of Mind,London : Harvard University Press.
Tharp, R. and Gallimore, R., 1988, 'A theory of teaching as assisted performance' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.

Piaget's Work and its Relevance to Mathematics Education

By Marsigit
Piaget's theory of intellectual development focuses on two central aspects of the progressive view of childhood; first, on the centrality of children's experience, especially physical interaction with the world; second, on the unfolding logic of children's thought, which differs from that of the adult (Ernest, 1991). Piaget proposed four major stages of intellectual development: (1). the sensori-motor stage (birth to 1 1/2 to 2 year), (2). the pre-operational stage (2 to 7 years), (3). the concrete operational stage (7 to 12 years), (4). the formal operational stage (12 to 15 years and up). The characteristic in which a remarkably smooth succession of stages, until the moment when the acquired behaviour presents seems to be recognizes as 'intelligence' (Piaget and Inhelder, 1969); there is a continuous progression from spontaneous movements and reflexes to acquired habits and from the latter to intelligence. They further stated that this mechanism is one of association, a cumulative process by which conditionings are added to reflexes and many other acquisitions to the conditioning themselves. They then regarded that every acquisition, from the simplest to the most complex, is a response to external stimuli, that is a response whose associative character expresses a complete control of development by external connections. They described that this mechanism consists in assimilation, that reality data are treated or modified in such a way as to become incorporated into structure of subject.
In moving from the sensory-motor stage to operational thought, several things must occur during the preoperational period (Becker, et al., 1975); there must be a speeding in thought or actions, there must be an expansion of the contents and scope of what can be thought; and there must be concern not only with the results of action but also with understanding the processes by which a result is achieved. Piaget and Inhelder (1969) outlined that there are three levels in the transition from action to operation; at the ages of two or three there is a sensory-motor level of direct action upon reality; after seven or eight there is the level of the operations in which concern transformations of reality by means of internalized actions that are grouped into coherent and reversible systems; and between these two level there is another level obviously represents an advance over direct action in which the actions are internalized by means of the semiotic function and characterized by new and serious obstacles. Toward the end of the preoperational stage, the basis for logico-mathematical thinking has been laid in the use of language, but the child is still far from reaching operational thought (Becker, et al., 1975).
During the years between two and seven the child learns much about the physical world; some of this is spontaneous, while other is deliberately taught by parents and teachers; despite the many intellectual feats of the period, children do not reason in a logical or a fully mathematical way. Children's thinking in the pre-operational period is characterized by what Piaget called moral realism as well as animism and egocentrism (ibid, p.102). Animism is the failure to adopt one stance towards inanimate objects and another towards oneself; moral realism is the consequence of viewing morality in one sense only; egocentrism is the consequence of the child's taking only one perspective; and the child achieves the next stage of intellectual development when at last he can consider a situation from several different aspects - in other words, he can de-centre (ibid, p.103). After many experiments, Piaget and his colleagues concluded that there is a sequence of development for each of the conservations; each experience requires that the child must judge whether the two things are still the same or are different when the entities is transformed in appearance by being changed in shape or transferred to another receptacle (Sutherland, 1992). It has been shown that children in the period of concrete operation can perform the mental operation of reversibility and can attend to several aspects of a situation at once (ibid, p.109).
For Piaget, an operation is a mental action (Becker, et al., 1975) that usually occurs in a structure with its counteraction - adding goes with its reverse operation subtracting, combining with separating, identity with negation; an operation is said to be concrete if it can be used only with concrete referent rather than hypothetical referents. The first obstacle to operations (Piaget and Inhelder, 1969), then, is the problem of mentally representing what has already been absorbed on the level of action. In the concrete operational stage, thinking shows many characteristics of mature logic, but it is restricted to dealing with the 'real' (Becker, et al., 1975). The second obstacle to this stage is that on the level of representation (Piaget and Inhelder, 1969); achieving this systematic mental representation involves constructive processes analogous to those which take place during infancy; the transition from an initial state in which everything is centered on the child's own body and actions to a decentered state. The third obstacle is related to the complexity of the using of language and the semiotic function involving more than one participant.
Formal operations involve thinking in terms of the formal propositions of symbolic logic and mathematics or in terms of principles of physics (Becker, 1975); one can deal with the hypothetical and one can deal with operations on operations. Piaget studied the development of logical thinking in adolescence and reflective abstraction, that very human capacity to be aware of one's own thoughts and strategies. Piaget assert that the basis of all learning is the child's own activity as he interacts with his physical and social environment; the child's mental activity is organized into structures and related to each other and grouped together in the pattern of behaviour (Adler, 1968). Piaget also asserts that mental activity is a process of adaptation to the environment which consists of two opposed but inseparable processes, assimilation and accommodation (ibid, p. 46). The child does not interact with his physical environment as an isolated individual but as part of a social group; as he progresses from infancy to maturity, his characteristic ways of acting and thinking are changed several times as new mental structure emerge out of the old ones modified by accumulated accommodations (ibid, p.46).
Piaget found that there is a time lag between the development of a child's ability to perceive a thing and the development of this ability to form a mental image of that thing when it is not perceptually present (Adler, 1968). The development of the child's concepts of space, topological notions, such as proximity, separation, order, enclosure, and continuity, arise first; projective and Euclidean notions arise later; and his grasp of order relation and cardinal number grow hand in hand in the concept of numbers (ibid, p.51). Piaget also asserts that a child progresses through the four major stages of mental growth is fixed; but, his rate of progress is not fixed; and, the transition from one stage to the next can be hastened by enriched experience and good teaching (ibid, p.53). Based on all the above propositions, some of their implications for the mathematics teaching in the primary school can be asserted. Piaget maintained that internal organization determines how people respond to external stimuli and that this determines man's unique 'model of functioning' which is invariant or unchangeable (Turner, 1984); a person attempts to make sense the environmental stimulus by using his existing structure or by assimilating or accommodating it.
The structure and their component schemes were said to change over time through the process of equilibration; if a subject finds that her present schemes are inadequate to cope with a new situation which has arisen in the environment so that she cannot assimilate the new information, she will be drawn, cognitively, into disequilibrium (ibid, p.8). Given these fundamental postulates of Piaget's theory : internal organization, invariant functions, variant structures, equilibration and organism/environment interaction; what then are the implications for mathematics education in the primary mathematics school ?
References:
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Adler, I., 1968, Mathematics and Mental Growth, London : Dennis Dobson.
Becker, W., et al., 1975, Teaching 2: Cognitive Learning and Instruction, Chicago : Science Research Associates.

Bagaimana Melakukan Diskusi Matematika dengan Siswa?

Oleh: Marsigit
Jika seorang guru ingin melakukan diskusi matematika dengan siswa maka pastikanlah apakah diskusi akan dilakukan secara klasikal, pada kelompok atau secara individual. Berdasarkan pengalaman penulis, maka secara umum dapat diberikan semacam pedoman bagaimana seorang guru seyogyanya melakukan diskusi:
1. dengarkanlah apa yang mereka katakan
2. usahakan suasana bersifat mendukung diskusi
3. usahakan suasana berpikir merdeka
4. usahakan agar siswa mempunyai inisiatif
5. usahakan agar siswa mempunyai rasa percaya diri
6. gunakan kata-kata yang mudah dipahami oleh siswa
7. usahakan hubungan yang baik antara guru dan siswa
8. berusaha mengetahui ide/gagasan siswa
9. menampung ungkapan siswa sebagai informasi
10. utarakan balikan secara bertahap
11. gunakan ilustrasi untuk mempermudah komunikasi
12. menghargai setiap ide yang disampaikan oleh siswa
13. usahakan agar diskusi bersifat efaktif dan bermanfaat.
14. memahami bahwa siswa memerlukan penjelasan
15. fokus pada pertanyaan siswa/penanya (jangan Jawa: disambi)
16. memikirkan hal-hal yang terkait dengan pertanyaan.
17. memikirkan solusi jawaban
18. bersifat jujur dan terusterang
19. berpikir positif (tidak prejudice) terhadap si penanya
20. tidak mempunyai kepentingan ganda misalnya mengajukan pertanyaan yang sulit untuk tujuan menghukum
21. gunakan referensi-referensi untuk mencari solusi.
22. kembangkan sikap menghargai dan empati
23. hindari sikap menang sendiri/otoriter
24. letakkan persoalan pada jejaring konsep yang lebih luas
25. usahakan agar siswa senang melakukannya kembali
26. jika memungkinkan libatkan siswa yang lainnya
27. menyadari manfaat jawaban terbuka
28. menyadari bahwa pertanyaan siswa adalah awal dari pengetahuannya
29. memperhatikan alokasi waktu dan konteks sekitar

Pada pembelajaran tradisional guru terpaksa sibuk mengontrol siswanya

Oleh: Marsigit

Pada pembelajaran tradisional, guru terpaksa melakukan berbagai kegiatan kontrol agar siswa bersikap kooperatif dan memperhatikan guru. Hal ini disebabkan karena guru belum mampu mengembangkan skema pembelajaran untuk melayani berbagai macam kebutuhan akademik siswa. Berikut contoh cuplikan berbagai aterensi yang ditemukan penulis yang menggambarkan bagaimana guru melakukan kontrol terhadap siswa:

Aterensi 1:
Wahyu ngantuk ya ?
Nah tadi sudah makan belum ?
Sudah.
Banyak nonton T.V. ya ?
Nah berapa coba ?

Aterensi 2:
Bagaimana cara penyelesaiannya ?
Penyelesaiannya bagaimana ?
Bagaimana Anto ?
YANG LAIN DIAM.

Aterensi 3:
Sudah apa belum ?
Nah kalau sudah diteliti dulu dari nomor satu sampai nomor sepuluh
AYO PRIHANTO KAMU SUDAH SELESAI ?
SUDAH DITELITI ?
AYO DITELITI LAGI
COBA SAMPAI BETUL
TIDAK HANYA DILIHAT LHO YA PRIHANTO

Aterensi 4:
Ayo Candra duduknya bagaimana ?
Hayo yang tertib Rona, ayo Rona duduknya yang bagus.
Nanti supaya betul semua

Aterensi 5:
Nol dapat dikurang lima ?
Sebetulna dapat tapi ini kurang ya to ?
Jadi barang tidak ada dikurangi lima, maka ...
SETERUSNYA KITA BAGAIMANA ?
Pinjam
Pinjam pada...?

The Role of Cognitive Development Theory for Mathematics Education

By. Marsigit
Theories about how children think and learn have been put forward and debated by philosophers, educators and psychologists for centuries; however, the contemporary thinking about education, learning and teaching is not 'brand new'; certain theories have been absorbed and transformed over time or translated into modern terms; and, some of them become prominent and influential (Wood, 1988).There is no question at all to the fact that anything related to the term 'cognitive development' is greatly embeded to the work of two greatest figures of developmental psychology in twentieth century, Jean Piaget and Lev Vygotsky. Piaget's influence on the primary mathematics curriculum and on research developmental psychology has been immense; Vygotsky's work has been gaining in influence over the past ten years. Traditionally, primary education has looked to child development and psychology for theoretical guidance and underpining (Gipps, 1994); Piaget's positive contribution, however, was both to start a theoretical debate about young children's intellectual development and to encourage the close observation of children; Vygotsky, the Russian psychologist, has given us a number of crucial insights into how children learn, of which to have particular consequences for classroom.
Observing child's behaviours when she interacts with surrounding objects or people, may be the starting point to discuss about the mechanisms of her cognitive development. In the interactions she may look at the object, take hold of it, listen to the sound or talk to the people; more than just these, she may also categorize, memorize or even make the plan for a certain activity. Such behaviour is taken for granted, much is automatic, yet for it happen at all requires the utilization of complex cognitive processes (Turner, 1984). By perceiving or attending to the visual and auditory surroundings, she may keep these in her mind. Her recognition of the functions of the objects, for example that the chair has the function for sitting, is related to the using of her memories and her developing the concepts of a 'chair'. Cognitive processes underlie the ability to solve problems, to reason and to learn. Implicitly, the above proposition lead that the term of 'cognitive development' is associated with the development of the processes and the content to which these processes are applied. Behaviourists characterize internal processes by associating them with the 'stimulus-response'. The reason why a person gives a particular response to a particular stimulus was thought to be either because the two were associated in some way, that is, the response was 'conditioned', or because the appearance of this response had been rewarded previously (Turner, 1984). Information-processing approach assumes that a person who perceives stimuli, stores it, retrieves it, and uses it (ibid, 5); information is transformed in various ways at certain stages in its processing.
Piaget (1969) admitted that any explanation of the child's development must take into consideration two dimensions: an ontogenetic dimension and a social dimension (in the sense of the transmission of the successive work of generations). Piaget used a biological metaphor and characterized mathematical learning as a process of conceptual reorganization. At the heart of Piaget's theory is the idea of structure; cognitive development, and in particular the emergence of operational thought, is characterized in term of the emergence of new logical or logico-mathematical structures. Further, Light states that Piaget's theory has a functional aspect, concerned with intelligence as adaptation, with assimilation, accommodation and equilibration; his main contribution and influence lay in his structural account of cognition.
Central to Piaget's view of the child is the assumption that the child actively constructs his own ways of thinking through his interactions with the environment (ibid, p.216). Piaget used observations of his own children to formulate some aspects of the development of intelligence. Absolutists view mathematical truth as absolute and certain; and, progressive absolutists view that value is attached to the role of the individual in coming the truth (Ernest, 1991). They see that humankind is seen to be progressing, and drawing nearer to the perfect truths of mathematics and mathematics is perceived in humanistic and personal terms and as a language (ibid, p.182). Piaget provides 'a license for calling virtually anything a child does education (McNamara, 1994); moreover, an analysis of the development of the progressive movement in the UK suggests that it was only after child-centered methods were established in some schools that educationists turned to psychologists such as Piaget to provide a theoretical justification for classroom practice. The other foundation for a number studies in Psychology, in which Piaget played a prominent part, seems to be influenced greatly by Durkheim's assumption, as Luria cited that the basic processes of the mind are not manifestations of the spirit's inner life or the result of natural evolution, but rather originated in society. And Vygotsky stressed on using socio-cultural as the process by which children appropriate their intellectual inheritance.
Reference:
Ernest, P.,1991, The Philosophy of Mathematics Education, London : The Falmer Press.
Gipps, C., 1994, 'What we know about effective primary teaching' in Bourne, J., 1994, Thinking Through Primary Practice, London : Routledge.
McNamara, D., 1994, Classroom pedagogy and primary practice, London : Routledge.
Piaget, J. and Inhelder, B., 1969, The psychology of the child, London : Routledge & Kegan Paul.
Turner, J.,1984, Cognitive Development and Education, London: Methuen.
Vygotsky, L.S, 1966, 'Genesis of the higher mental functions' in Light, P. et al. , 1991, Learning to Think : Child Development in Social Context 2, London : Routledge.
Wood, D., 1988, How Children Think & Learn, Oxford: Basil Blackwell.