The Piagetian concept of learning mathematics is basically an exposure to an overwhelmingly vast amount of concrete materials, in the pre-operational and operational stages of the child’s development. These developmental stages occur predominantly in the elementary level. It is therefore important for teachers in the grade school to provide and expose their pupils to varied concrete and manipulative devices. These would help the children experience the learning of the concept as well as make them “visualize” mathematics. Materials which children can handle and worked with are called manipulatives. These materials are manipulated by the learners to be able to come up with possible solutions or possible understanding of the abstract concepts in mathematics. Jean Piaget has considered the operational stage as the best period where concrete materials are being bombarded to the learners. The idea it to give the children enough exposure to similar characteristics or properties such that they can abstract what is common. For example, a child may not be able to define the numeral “three”. The teacher may let him work with different objects like 3 green guavas, 3 yellow pencils, 3 blue books and so on. From here, the child can abstract the concept of “threeness” despite his not being able to define it. Here, the concretization of the abstract concept is being manifested through the use of manipulatives.
Mathematics Manipulatives for Teaching
Mathematics per se is very abstract discipline and one has to imagine two worlds or two levels of existence. These worlds include the “physical real world” and the “idea world”. Physical situations give rise to idealized reflections in the idea world. Jerome Bruner, one of the cognitive psychologists, considered the concretization of ideas as the “enactive” level of learning. In this level, the child “plays” with the concrete materials. The teacher may provide structured activities with these materials, such that these materials can provide a transition to the “iconic” level or the semi-concrete stage of learning. The drawings, pictures, films and similar types of materials are categorized in the “iconic” level. These are generally experienced by the learners in the print media that they see. That is why, textbooks and workbooks are limited in terms of their being “enactive”. Adequate exposure to concrete and semi-concrete materials, can provide a smooth transition to the “symbolic” level of learning. This is considered by Piaget as the formal operation stage of development, wherein the learners are now doing the abstract operation. The logical thought process of the learner would then be ready to abstract the concepts they have learned through the concrete and semi-concrete materials. Piaget considered that the evolution of intelligence involves a continuous organization and reorganization of one’s perception of and reaction to the environment. This involves “assimilation” which is the fitting of new situations into existing psychological framework and “accommodation” which is the modification of behavior by developing new cognitive structure (Post, 1986).
The rational behind the use of manipulatives is to facilitate efficient teaching and effective learning. It is believed that learning is facilitated when the learner can experience what they are learning. From experiencing the learning activity one uses most of his sense, thereby enhancing more retention and permanence in the process of learning. The inherent motivational factor of the manipulative provides the learners incentive to work and sustain the desire to finish the activity or task to be done. While the learners work on the manipulatives, they can discover or investigate how the concepts are developed. This could trigger in the learner the motivation to discover more concepts or solutions because they get the actual “feel” of the materials.
Teachers desirous of breaking the monotony of the usual “chalk-talk” presentation of a mathematics lesson, may shift to providing manipulatives to the learners. Teachers may have their learners construct the appropriate manipulative so that they can exhibit their understanding if the concept. An ingenious and creative teacher can make his classroom a mathematics laboratory where children can work creatively and independently. Here the children can also interact with one another as well as work cooperatively in the process of investigation and discovery. In this manner, mathematics classes may become a venue for innovative learning… a learning children will always cherish in their lifetime.
 Jerome Bruner