A Very Important Application of Piaget's Theory - Research Paper Example

Montessori mathematics as support for Piagets theory The contention of this paper is that Montessori mathematics provides essential support for Piagets theory of development of logico-mathematical thought. The support for Piagets theory is indicated in how Montessori education implements adoption and organization of information, classification, ordering, and use of conversation. The support is also expressed in Montessoris notion of reversibility or irreversibility in acquiring knowledge in mathematics. Fundamental support for Piagets theory is also expressed in the sequencing of Montessori teaching, particularly in how the four basic mathematical operations are taught in Montessori schools. Support for Piagets theory is also expressed in the Montessori emphasis for exploration so children can learn at their pace. At the core of Piagets theory is an assertion that "children construct, or create, logic and number concepts from within rather than learn them by internalization from the environment" (Kamii 2009 citing Piaget 1971, Piaget and Szeminska 1965, Inhelder and Piaget 1964, and Kamii 2000). Piaget distinguished three kinds of knowledge: physical knowledge, social knowledge, and logico-mathematical knowledge (Kamii 1996: 99). Piaget taught that the logico-mathematical knowledge is only partly acquired from objects because, for instance, the similarity between two blocks of different colours is not observable but is deduced by an individual through putting things in relationships with the relationships earlier discovered (Kamii 1996: 100). In other words, for Piaget, individuals or children use a logico-mathematical framework within their minds to acquire knowledge (Kamii 1996: 100). According to Kamii (1996: 100-101), through what Piaget described as logico-mathematical framework, a learner or student or child acquires knowledge through: 1. Classifying or categorizing/comparing 2. Logically deducing or acquiring knowledge through cognition from within himself/herself The two are clearly applied in Montessori schools as learning proceeds from the concrete to the abstract as shown in figure 1. Figure 1. Montessori Curriculum: from concrete to abstract Source: Tim Seldin, Chairperson, International Montessori Council (Concurrently, Mr. Seldin is also President of The Montessori Foundation) Applying Piagets teaching, Kamii concluded that children create their own arithmetic or mathematics in acquiring logico-mathematical knowledge using their ability to think and, thus, the goal of math education is to assist learners or children invent procedures for solving mathematical problems as well as in constructing "a network of numerical relationships" (Kamii 1996, 101). As pointed out by Piaget, mathematical knowledge is different from physical knowledge because the former is not observable while the latter is (Kamii 1996: 102). Following Piaget, Kamii said that "there is no such thing as addition fact" because sums are internalized or constructed from within (1996: 102). In illustrating Piaget thinking, Kamii said that one example of a network of relationship is in the situation when a child learns that 5 + 6 = 11 because 5 + 6 = 10 +1 (1996: 102). Other examples on how children invent computational procedures for solving mathematical problems (Kamii 1996:105-106) are as follows: 1. To solve a subtraction mathematical problem 53-24, there can be several options: a) treat it as 50-20=30, 3-4=1 less than zero, and 30-1=29; b) view the mathematical operation instead as 50-20=30, 30-4=26, and 26+3=29; and c) confront the problem as 50-20=30, 30+3=33, and 33-4=29. 2. To address a mathematical problem in multiplication like 125x4, some of the options are: a) handle it as 4x100=400, 4x20=80, 4x5=20, and 400+80+20=500; or b) consider it as 4x100=400, 4x25=100, and 400+100=500. 3. Finally, children can handle division this way: 74/5 will be solved by the operation 5+5+5+… until a numeral close to 74 is reached and a fraction is employed for the remainder. According to Kamii, the Piaget method does not use textbooks because textbooks employ "repetition and reinforcement from the outside" (Kamii 1996: 102). Instead, an educational approach that follows Piagets theory uses "numerical reasoning in daily living, group games, and problem-solving". As we will show later, the Montessori method uses experiences, group games, and problem-solving in pedagogy or teaching children. Acquiring knowledge through the rigours of living has been the fundamental way that a child has been acquiring knowledge and, for Kamii, it follows that the teacher schooled in the ways of Piaget should use the same basic procedure in the pedagogy (Kamii 1996: 102). The use of a problem-solving approach is promoted because the Piaget approach emphasizes a "constructivist program" wherein exchange of views or conversation is important (Kamii 1996: 105). Dialogues and discussions are extremely important in the Montessori method, especially as critical outlook is promoted. In the "constructivist program", the teacher does not reinforce right or correct wrong ones because that can stop all thinking (Kamii 1996: 105). Teachers schooled in the ways of Piaget encourage his or her class to express agreements or disagreements immediately (Kamii 1996: 105). If no one among the teachers students is able to give the correct answer then this indicates that the question is beyond the comprehension of the teachers students and should not have been asked at all in the first place (Kamii 1996: 105). Nevertheless, according to Kamii, if the students debate long enough, the students will eventually arrive at the correct answers. International Montessori Council Chairperson and concurrent President of the Montessori Foundation Tim Seldin described Montessori education as follows (2): There is rare reliance on texts and workbooks because "skills and concepts are abstract, and texts simply dont bring them to life." Instructions do not emphasize drills and memorization. Students acquire knowledge through experience, investigation, and research. All classes teach critical thinking. Students are taught to become unafraid of mistakes because mistakes are normal in the learning process. There is focus on the development of student self-discipline, purpose, and motivation. There is an emphasis on the connections among the various subjects. Consistent with the aforementioned, the Montessori Foundation description of the Montessori way of teaching mathematics is as follows: "Hands-on materials" to promote a mental image of mathematical concepts and mathematical operations. All operations in mathematics are made clear and concrete and "children internalize a clear image of how mathematical processes work". Although progression from concrete to the abstract is observed, related subjects are linked as relevant to the students experience. The approach makes possible the introduction of concepts in algebra, geometry, logic, statistics, and arithmetic early enough. In addition, students are introduced to new experiences which make possible the introduction of new concepts, learning, and skills. Students are also introduced to new problem-solving situations as one such type of new experiences. Students who are able to grasp relatively simple concepts are introduced to relatively complicated concepts. An important approach being followed by the Montessori way of teaching mathematics is to connect mathematics with the students "direct experience". A continuing goal of Montessori mathematics education is to ensure a "concrete representation of abstraction" to allow the child to develop within his or her mind a mental picture of concepts. The Montessori Foundation emphasized that the Montessori way of teaching mathematics is based not only on the works of Maria Montessori but also on the works of Jean Piaget. Thus, even as the Montessori teaching methodology starts from experiential, the experience is linked to several disciplines as shown by Figure 2 even if the sequence is from the concrete to the abstract. Figure 2. Integration in the Montessori Curriculum Source: Tim Seldin, Chairperson, International Montessori Council The Montessori emphasis on experiential knowledge highlights a very important application of Piagets theory. In Piagets view, knowledge that comes from experience is "irreversible" as "…everything experiential is irreversible" (Calloway 215 citing the works of Jean Piaget and Piaget and Inhelder). This means that knowledge acquired through experience becomes deeply internalized by the learner and forgetting the acquired knowledge becomes difficult or very unlikely. Meanwhile, following Piaget and Montessori principles, Pound emphasized that effective teaching of mathematics among young children requires that the teaching be done in a playful manner, in a way that the teacher has enthusiasm and in a way that the learning, teaching, or knowledge is connected with the everyday experiences of children (105). Pound stressed that if the mathematics teacher is bored, then his or her students will also be bored. At the same time, however, Pound emphasized that rich learning experience should not be equated to mere games, songs, and rhymes because they are not replacements for real life experience (106). In other words, Pound emphasized the need to connect the teaching of mathematics with real life experiences and to introduce students to new real life and problem-solving experiences as part of an approach in teaching mathematics in the Montessori way. In effect, Pounds remark echoed what Maria Montessori recommended on page 255 of her book, The Absorbent Mind. On the page, Maria Montessori pointed out that the task of an educator is to "teach the teacher where he or she intervened needlessly" (255). Work Cited Callaway, Webster. Jean Piaget: A Most Outrageous Deception. New York: Nova Science Publishers, 2001. Haylock, Derek and Anne Cockburn. Understanding mathematics in the lower primary years: A guide for teachers of children 3-8. London: Paul Chapman Publishing (A Sage Publications Company), 2003. Inhelder, Barbel, and Jean Piaget. The Early Growth of Logic in the Child. New York: Harper & Row, 1964. Kamii, Constance, Piagets Theory and the Teaching of Arithmetic. Prospects 26.1 (March 1996), 99-111. Kamii, Constance. Young Children Reinvent Arithmetic. 2nd ed. New York: Teachers College Press, 2000. Kamii, Constance. "Teachers Need More Knowledge of How Children Learn Mathematics". Virginia, USA: National Council of Teachers of Mathematics, 2009. 23 December 2009 from . Montessori Foundation. "Montessori and the Study of Mathematics". The Montessori Foundation, 2007. 23 December 2009 from . Montessori, Maria. "The New Teacher". Chapter 17. The Absorbent Mind. Adyar, Madras, India: Theosophical Publishing House, 1949. Piaget, Jean. Biology and Knowledge. Chicago: University of Chicago Press, 1971. Piaget, Jean. The Psychology of Intelligence. Trans. Malcolm Piercy and D.E. Berlyne. London and New York: Routledge Classics, 2003. Trans. of La Psychologie de lintelligence. 1947. Piaget, Jean, and Alina Szeminska. The Childs Conception of Number. New York: W. W. Norton & Co., 1965. Pound, Linda. Thinking and learning about mathematics in the early years. New York: Routledge, 2008. Seldin, Tim. Montessori 101: Some Basic Information that Every Montessori Parent Should Know. International Montessori Council: The Montessori Foundation, 2006. Read More