Montessori Mathematics and its Essential Support to Piaget’s Theory of Development of Logico-Mathematico thought
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Mathematics is every where around us. For most of us, students, sees mathematics as a subject that requires to be passed in order to graduate and fulfill our dreams in life. Mathematics plays important roles I the field of science, may it be hard or soft science.
Without numbers, math, how could we ever determine the price of a certain good? As for the store owner, how could he be going to ask price to the customers if he does not give any price of his commodities?
In a deeper sense, math is treated as a language that simplifies the long sentences in the article into a one equation. The geometry, calculus, trigonometry and much more are being used in the academe in order to give the students of a mental view of how long is the Great Wall of China and how tall is the Mount Everest.
But how thus a child would treat mathematics? Do they easily understand the number that their teachers wanted to recite in front of the class? How do they feel when we talk about “Mathematics”? What do they think, imagine and the like when we asked them about the first thing that comes out of their mind when they hear the word mathematics.
Students from the different academic level have different perspectives of mathematics from one another. There are many factors that contribute to the learning process of a student and from those, the understanding of a certain thing [in this paper we are refereeing to mathematics], underlies.
In this paper, we are going to discuss mathematics in the context of a child’s life. Then, after doing so, we relate this into how children construct and recognize patterns of ideas in a logical and sequential manner.
Moreover, there will be a part of this paper wherein we will talk about the similarity between logico-mathematico thought and Montessori’s concept of the mathematical mind.
Mathematics in the Context of a child’s Daily Life
Since birth a child is already encountering mathematics in his daily experiences. There are times the parents of the child play with their child by counting the child fingers or by asking the child to count the marble that he has. Learning during this stage is accompanied by playing in order to get the attention of the child and make him to be more interested to learn things around him.
There are also times wherein the child uses numbers [math] in saying the quantity of his toys or marbles to other children. In this situation, the child uses mathematics in describing things that are common on our sight.
Even during the bonding moments of the parent and child, there are times that the parent teaches the child in answering the questions that involves numbers [counting] like the number of age that the child has and the number of the child’s siblings. Inside the classroom, a child enables to solve mathematical problems through the aid of the mathematical tools and other concrete models that can be easily understand and explained to the children.
Mathematics, as we all know, is full of abstraction that needs deeper mental perception that children does not have during this period in their lives. Therefore, there exists a need for the use of “manipulative materials” in order to educate the children regarding mathematical concepts and principles (Post, 1981, p. 1).
After learning the tools and importance of mathematics on the child daily living, the latter uses the said tools and ideas to solve real life problems. The first thing that comes into the child’s mind when a child encounters a problem is that they ask first their selves, why and how did that happened? After doing so, they will now remember the things their teachers and parents taught and come up with a solution from the best that they could to solve the problem.
The use of the mathematical models only serves as a guide in their pursuit of answers but the factor that contributes most in the child’s formulation of solution is their experiences in the house, during their younger days, and in schools. It is where in this period the idea of logico-mathematico comes in.
Logico-mathematico states that we develop our cognitive capabilities through self-motivated action in the world. This only means that we acquire of capabilities and develop our personalities out of the experiences that we had in the past years.
Moreover, logico-mathematico can be treated as an experience that that supposedly puts characteristics to the objects, order and any other relationship that can be associated with regards to the object concerned (Durak, 2005, p. 4-7).
Since mathematics is considered to be a self contained system, therefore, there is a need for the child to understand first the difference of mathematics in the physical world and in the social world. Meaning, they must understand that the syntax of mathematics urging is very different from the syntax of daily language (White, 2004, p. 1) before using all the mathematical ideas and concepts into daily purpose of living.
Social-arbitrary knowledge is defined as the knowledge build up by the humanity which includes the information regarding rules, laws, morals values and the like of the society that should be take into consideration every time the child makes his/her actions (http://www.wheaton.edu).
Just like the Logico-mathematico kind of thinking, Montessori’s mathematics also encourages the children to discover on their own the answer on every problem that they may encounter.
Advocates of Montessori mathematics believes that children enjoys the exploration of math and geometry concepts if they are related to the real life situations they are experiencing (Olaf, 2004, p. 1).
According to the article of Olaf, they observed that children tends to enjoy making up problems for each other [classmates] after encouraging them to make problems, especially the story problems, related to their lives and the subjects they are studying.
In order to help a child in figuring the answer to the problems and easily understand the various abstract ideas in mathematics, teachers of Montessori mathematics uses tools or devices in order to give the students a clearer picture of the mathematical concepts and ideas they are going to discuss. Some of these tools are the concrete rods, number cards, spindle boxes, Seguin boards, trinomial cubes, board sets for different mathematical operations and a lot more.
According to the advocates of Montessori mathematics, that activity that includes the use of the said mathematical learning devices attracts the students to learn more of the subject area they are teaching.
For instance, the use of the concrete rods, it helps the child to better understand numerals and serves as a good introduction tool to fractions. The said above concrete materials also improves the use of the child’s senses in the learning process (http://www.montessoriland.com).
Based from Piaget’s cognitive development theory, child’s development is based on the level of maturity the child has. The more mature the student, the deeper the capability of the child in understanding various concepts in the learning process, in this paper we are talking about mathematics. The cognitive development theory of Piaget is categorized into four development period.
First is the Sensori-Motor stage wherein the development of the child relies on the child’s ability to discover the relationship of its environment to his self (http://evolution.massey.ac).
The second stage is the Preoperational stage wherein the child becomes egocentric that makes him difficultly appreciate the life from any other viewpoint (Lin, 2002, p.1).
The third stage is the Concrete Operational stage wherein the child starts to think logically and organize his/her way of thinking and the egocentric thinking vanishes into the personality of the child (http://social.jrank.org). It is also during in this stage wherein the child thinks sequentially.
The last stage of Piaget’s cognitive development theory is the formal operation in which the ability to think and perceive abstract ideas. During this stage the emergence of skills like logical thinking, deductive reasoning and systematic planning starts to experience by the individual (Wagner, 2007, p. 1).
Mathematics is just one of the many things that an individual must be able to understand in order to be guided in his/her journey in life. It is really hard to earn knowledge on this area but through continuous efforts and development on the manner of approach it would be as ‘friendly’ as it may seem.
The different processes and ways of making mathematics interesting still has to be improved to cope up with the kind of living we have now in order to prepare those children to the “harmful” environment that we have.
From generation to generation, the behavior of the children varies and there is no clear trend about this, in this regard the advocates of Montessori mathematics must be able to adjust on what ever the changes may be on the kind of approach they are going to implement to teach the young generation with the right knowledge about our society.
At this point, I just want to give emphasis on the importance of finding or developing of new ways in making the learning of a child more effectively. The provision of concrete tools and devices really helps a lot in the success of the learning’s of the children. If there would be improvements on this area, then it would be a great help.
Learning is a continuous process. It starts from the day we are born up to our last breath here on earth. The future of the students depends not only on how they will find their way to the solutions but also to the efforts of our elders for they has the responsibility to guide us and provide means on how will those student achieve their goals in life.
http://www.wheaton.edu/intr/Moreau/courses/563/cogdev.pdf Wheaton.edu, Cognitive Development: Overview of Piaget, p.1, 2005
http://www.montessoriland.com/site/1270021/page/457130 Montessoriland.com, Mathematics Products, p.1, 2007
http://evolution.massey.ac.nz/assign2/MH/webpage.htm Evolution.massey.ac, Jean Piaget’s Stage Theory, p.1-7, 2007
http://social.jrank.org/pages/157/Concrete-Operational-Thinking.html Social.jrank.org, Concrete Operational Thinking, p.1, 2007
http://tomweston.net/PiagetDurak.pdf Durak, A.J., Piaget and Marxist Theory, p.4-7, 2005
http://coe.sdsu.edu/eet/Articles/piaget/index.htm Lin, Shu-wan, Piaget’s Developmental Stages, p.1, 2002
http://www.michaelolaf.net/1CW612math.html Olaf, Michael, Montessori Philosophy, AGE 6-12+ YEARS GEOMETRY, MATH, MEASUREMENT, p.1, 2004
http://education.umn.edu/rationalnumberproject/81_4.html Post, Thomas, The Role of Manipulative Materials in the Learning of Mathematical Concepts, p.1, 1981
http://psychology.about.com/od/findex/g/formalops.htm Wagner, Kendra Van, Formal Operation Stage, p.1, 2007
http://www.emis.de/proceedings/PME28/RR/RR031_Mitchelmore.pdf Mitchelmore, Michael and White, Paul, Abstraction in Mathematics and Mathematics Learning, p.1, 2004