Feb 12, 2013

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.

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.


  1. Wahyu Berti Rahmantiwi
    PPs Pendidikan Matematika Kelas C 2016

    Menurut teori Piaget perkembangan intlektual siswa berkembang melalui tahap definisi yang baik dimana anak-anak dapat mengembangkan konsep melalui interaksi dengan lingkungan. Berbeda dengan teori belajar Vygotsky yang menekankan peranan orang dewasa dan anak-anal lain dalam memudahkan perkembangan anak, teori Piaget justru memandang pembelajaran yang dilakukan melalui penemuan individual.piaget memandnag kumpulan konsep dapat digunakan ketika siswa berinteraksi dengan lingkungannya. Hal ini sejalan dengan pembelajaran matematika yang bersifat abstrak dengan tujuan untuk memudahkan maka dibawa dalam kehidupan sehari-hari.

  2. Kunny Kunhertanti
    PPs Pendidikan Matematika kelas C 2016

    Dala hal ini piaget mengungkapkan bahwa anak-anak menciptakan pengetahuan mereka tetang dunia. Namun, dia juga berpendapat bahwa dalam penciptaan dan pengembangan pengetahuan mereka, anak-anak dibatasi oleh struktur konseptual absolut, terutama matematika dan logika; Dengan demikian, Piaget menerima pandangan absolut tentang pengetahuan, terutama matematika (ibid, hal.185). Kesalahan pun sering dilakukan oleh anak-anak maupun remaja, namun Guru dapat membantu anak mengatasi kesalahan dalam pemikirannya dengan memberinya pengalaman yang mengekspos mereka sebagai kesalahan dan menunjukkan jalan menuju koreksi kesalahan.

  3. Elli Susilawati
    Pmat D pps16

    Siswa adalah subjek utama dalam kegiatan penemuan pengetahuan. Mereka menyusun dan membangun pengetahuan melalui berbagai pengalaman yang memungkinkan terbentuknya pengetahuan. Mereka harus menjalani sendiri berbagai pengalaman yang pada akhirnya memberikan percikan pemikiran (insight) tentang pengetahuan-pengetahuan tertentu. Hal terpenting dalam pembelajaran adalah siswa perlu menguasai bagaimana caranya belajar (Novak & Gowin, 1984).

  4. Elli Susilawati
    Pmat D pps16

    Siswa adalah subjek utama dalam kegiatan penemuan pengetahuan. Mereka menyusun dan membangun pengetahuan melalui berbagai pengalaman yang memungkinkan terbentuknya pengetahuan. Mereka harus menjalani sendiri berbagai pengalaman yang pada akhirnya memberikan percikan pemikiran (insight) tentang pengetahuan-pengetahuan tertentu. Hal terpenting dalam pembelajaran adalah siswa perlu menguasai bagaimana caranya belajar (Novak & Gowin, 1984).

    PPS-MAT D 2016
    Beberapa implikasi teori Piaget dalam pembelajaran: Memaklumi akan adanya perbedaan invidual dalam hal kemajuan perkembangan. Teori Piaget mengasumsikan bahwa seluruh siswa tumbuh melewati urutan perkembangan yang sama, namun pertumbuhan itu berlangsung pada kecepatan yang berbeda. Ditambah cara berfikir anak kurang logis dibanding dengan orang dewasa, maka guru harus mengerti cara berfikir anak, bukan sebaliknya anak yang beradaptasi dengan guru, pendidikan disini bertujuan untuk mengembangkan pemikiran anak, artinya ketika anak-anak mencoba memecahkan masalah, penalaran merekalah yang lebih penting daripada jawabannya. Oleh sebab itu guru penting sekali agar tidak menghukum anak-anak untuk jawaban yang salah, tetapi sebaliknya menanyakan bagaimana anak itu memberi jawaban yang salah, dan diberi pengertian tentang kebenarannya atau mengambil langkah-langkah yang tepat untuk untuk menanggulanginya.

  6. Gamarina Isti R
    Pendidkan Matematika Kelas B (Pascasarjana)

    Menurut Piaget perkembangan intelektual anak melalui tahapan yang jelas. Anak dapat menciptakan pengetahuannya sendiri dan menganalisisnya.
    Sedangkan peran guru dapat dengan membantu anak mengatasi kesalahan dalam pemikirannya dengan memberinya pengalaman yang mengekspos mereka sebagai kesalahan dan mengarahkan jalan menuju koreksi kesalahan. Anak dapat diberikan banyak situasi untuk mengatasi kesalahannya. Oleh karena itu sebagai pendidik kita harus mengetahui tahapan yang sedang dialami oleh anak, karena hal itu dapat menjadi langkah penyesuaian pelaksanaan pembelajaran agar pengetahuan yang akan dicapainya sesuai dengan kemampuan yang sedang dialaminya.

  7. Angga Kristiyajati
    Pps UNY P.Mat A 2017

    Terima kasih Banyak Pak Prof. Marsigit.

    Sepemahaman kami, pada teori piaget terdapat tiga konsep dasart yaitu skema, proses adaptasi pada tiap-tiap tahap dan tahapan pengembangan. Ketiga tahapan tersebut akan berjalan dengan optimal ataupun tidak tergantung pada beberapa faktor yaitu kemampuan guru dalam memfasilitasi siswa (mengajar), kemampuan atau kompetensi siswa, waktu dan tempat kapan suatu materi disampaikan kepada siswa, dan yang utama adalah kemampuan guru dalam interpretasi dan menerapkan metode yang tepat dalam pembelajaran.

  8. Luthfi Nur Azizah
    PPs PM A

    Menurut Piaget dalam Slavin (2006) perkembangan kognitif setiap anak melalui 4 tahap. Setiap tahap memunculkan kemampuan yang baru dalam pengolahan informasi. Tahap pertama yaitu sensori motor, yaitu pemahaman dibangun dengan pengalaman indera dan tindakan-tindakannya. Kemudian dilanjutkan tahap pra-operasional, dimana dunia direpresentasikan dengan kata dan gambar. Selanjutnya yaitu tahap operasional konkret, seorang anak akan memulai untuk berpikir secara rasional, mampu mengklasifikasikan objek, serta memahami suatu percakapan. Tahap terakhir yaitu operasional formal, yaitu anak akan memiliki kemampuan berpikir abstrak dan refleksif. Dengan mengetahui perkembangan kognitif tersebut, sebagai guru hendaknya bisa menempatkan pengetahuan mana yang akan digunakan sebagai awal pembelajaran dan bagaimana keterurutan pembelajaran agar sesuai dengan tahap perkembangan kognitif siswanya. Dengan begitu, siswa tidak akan terbebani secara kognitif selama kegiatan pembelajaran berlangsung.

  9. Dimas Candra Saputra, S.Pd.
    PPs PMA 2017

    Assalamualaikum Prof,
    Banyak teori Piaget yang dapat diimplikasikan dalam pembelajaran matematika. Diantaranya adalah bahwa pelaksanaan pembelajaran matematika disesuaikan dengan tahap perkembangan siswa.Selain itu, proses belajar menurut Piaget hanya dapat terjadi melalui interaksi dengan objek di sekitarnya. Siswa diberikan kesempatan yang luas untuk membangun pengetahuannya sendiri atau dalam diskusi kelompok. Maka objek-objek yang dimaksud dalam proses interaksi tersebut disesuaikan dengan tahapan perkembangan siswa. Misalnya, sumber belajar yang diberikan pada siswa di SD akan berbeda dengan siswa di SMP.