Constructivism as an Educational theory in Mathematics
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Introduction Learning is an inevitable process for human beings and even animals. A young one is born with little or no knowledge of the surrounding, both within and without. In this case, things start acquiring a given shape through the learning process. Learning becomes the preoccupation of human beings as their growth and development takes course. In the former times, learning was accomplished in an informal setting. As development and modernization took shape, dedication was formalized and confined to the classroom. Classroom teaching methods has been revolutionized with development of various theories and philosophies for the same. Among them is constructivism theory. This theory has been applied in many subjects including mathematics. This paper explores constructivism as an educational theory in mathematics.
Constructivism as an Educational theory Constructivism theory is coined around a philosophy of self experience of the learner. This theory is coined on the basis of how a student connects information or instructions give in class. Jean Piaget, who came up with this theory suggest that learning is accomplished through a connection of the experience of the learner with new information. In most cases, the new information or ideas leant in the classroom or anywhere is not really taken in as intended. Instead, a student processes information given based on the prior knowledge he or she has. The new information given is processed only if it has some connection with prior experience of the learner. This means that the student learn through discovering the surrounding by self experience, research, and discovery. This theory seeks to revolutionize information processing method of the student. In the traditional learning situation, the student memorizes information given by the teacher and later regurgitates the same during examinations time thereby rendering him or her ineffective in practical application of the same. However, constructivism teaching system enables the student to retain and at the same time apply the information leant in classroom. This makes the student to be active both mentally and physically. The goal of constructivism in a mathematics class is to make the students think more about a given formula and explain it in their own way of understanding. In this case, the teacher does not lecture students but rather creates an environment for them to make mental constructions about a given problem (Leikin & Zaslavsky, 1999).
History of Constructivism The concept of constructivism is an ancient philosophy that dates back to the period between 470 and320 BC. The ancient philosophers, including Aristotle, Plato, and Socrates employed constructivism in their philosophies by alleging that it was the way of formulating knowledge. However, they merely mentioned it meaning that it remained a mentioned concept that was not applied. In 19710, Vico explored constructivism and made its application formal. He explained that the truth was equivalent to a creation, meaning that something spoken ought to have a physical backing to be considered as valid. Vico’s notion on constructivism was later expanded in the 20th century by Dewey and Ceccato. According to Leikin and Zaslavsky (1999), Dewey and Ceccato believed that knowledge could not be pushed beyond the experience of the learner. Later, Jean Piaget made a big contribution to constructivism model of learning. In fact, the theory of constructivism in practice today is based on his ideas. He came up with a learning model for cognitive constructivism. Piaget did an experiment in a cognitive class whereby he discovered three alternatives for students in the event of learning. Piaget discovered that in cases in which information taken in does not coincide with the information familiar to the student, the student is disoriented. As a result, he or she may choose not to believe the outcome, loose interest in the outcome, or alter the outcome to suit his or her predictions and understanding. The latter option was explored by Piaget who termed it meaningful learning or simply cognitive restructuring. This therefore became the goal of learning. A new and meaningful idea acquired by a student undergoes cognitive restructuring to reach the meaning that coincides with the physical world (Nilson, 2010). The movement from concrete to an abstract idea forms the basis of Piaget’s learning process. Students construct their own comprehension of a body of knowledge at their convenient tempo and make use of words to convey their thoughts on the same. The concept of constructivism underwent further development in 1960s. Various scholars, including Goldin, Vygotsky, Lerman, and Ernest developed social constructivism to build on the same. Social constructivism was based on the idea of interaction. There was a belief that interactions boost construction understanding. The proponents of this theory further believed that thoughts are in the form of language and mathematics was not a stationary body of facts but rather an evolving way of thinking that was constructed through interactions. This means that constructivism can be defined basing on the amalgamation of Piaget’s concepts and social constructivism tenets. This means that students construct meaning of a given mathematical concept based on what is familiar to them by experience or interacting with one another (Pon & Gallery, 2001). The development of constructivism continued with the emergency of radical constructivism and constructionism. Radical constructivism was postulated by Von Glaserfed who alleged that knowledge can be constructed to a radical limit. According to Von Glaerfed’s idea, knowledge is subject to the experience of the student. This means that one can explore knowledge to a level related to his or her experience. Constructionism, on the other hand, has been incorporated in constructivism as an emphasis on the need to involve students in the blueprint and creation of tasks that have personal significance to them. Constructivism is in continued expansion and projection (Pon & Gallery, 2001).
Comparison of Constructivism and Traditional Teaching and Learning Traditional teaching in this context means focus of the information whereby the teacher gives out information based on the structures or curriculum set in place. The information is logical and follows a certain scheme. In this case, learning entails the acquisition of precise mental representation of rigid mathematical structures and relationships that does not depend on the collective activity of the leaner. For instance, in a mathematics class, a teacher brings a new concept whereby students take note on the same. The teacher thereafter gives a solved practical example. Thereafter, students are allowed to labor on their own. However, the practice is different in constructivism a class. These differences mainly manifest in characteristic behavior on the same. In traditional lesson, it is the teacher who identifies the topic or issue to teach whereas in the constructivism lesson, the student is left to identify the topic based on prior interaction with the same. The issue or topic identified is seen as relevant to the student in constructivism because it is from the student himself, though it is less likely to be relevant in the context of the teacher. However, the student might not see the relevance in the topic introduced by the teacher for the case of traditional teaching. In traditional teaching and learning, the teacher asks students questions to determine their height of comprehension. However, in constructivism class, students ask questions as a result of being curious in their grasp of the concept (Nilson, 2010). In a traditional class, it is the duty of the teacher to provide resources to implement learning process. The resources identified include notes, experts, and any physical materials for the same. On the other hand, a student in constructivism class is on the fore front in material acquisition. He or she identifies the materials required for the same and looks for means of accessing them. At the same time, the student plans on how to investigate a given concept in constructivism class. A traditional class does not extend the learning process outside the classroom except in few occasions involving take-away homework. For the case of constructivism class, students extend the learning process outside the classroom. Given that they associate learning with practical situation, they extend the learning process outside the classroom whereby they compare theory with a real-world situation. At the same time, mathematical concepts and skills in a constructivism class emerges out automatically because they are sought for by students. However, there are no new concepts in a traditional mathematics class because formulas and concepts are determined. At the same time, there are no new concepts and skills applied to a given situation in a traditional class. Instead, predesigned formulas and concepts are used. For the case of constructivism class, there are new concepts and skills applied to a given situation. Generally, a student in a constructivism class is more active than a student in a traditional class.
Constructivism in a Classroom Constructivism class in this case is how it would appear in a classroom. This can be done considering the role of the teacher, the classroom environment, and the curriculum.
The Role of a Teacher The teacher provides a favorable environment for engaging students into constructive thinking. To gauge the level of understanding, the teacher may request students to give details in their own understanding of the previous topic. If the lesson is the first one, the teacher starts by asking students to think of an activity within a mathematical topic in consideration. For instance, if it is to do with algebra, the students may be asked to explain the real life situation of the case. At the same time, the teacher engages students by telling them to think at an ordinary level. The teacher ensures that everyone is involved by taking views from every student in class. The teacher thereafter analyses every constructed view of each student and challenges the student at some point. Because all views are diverse, and some are likely to be wrong, the teacher embraces the view of each student and encourages him or her to be involved more. The teacher thereafter directs the lessons to the outside environment and at the same time taking control of the classroom.
The Classroom Environment The setting in the classroom is set to suit a constructive environment for students. Considerations are made for flexibility and small-sized groups. As a result, a classroom is arranged to accommodate groups of fours. At the same time, the sits are arranged to allow movement of the students in between. Materials needed for the lessons are strategically placed in the classroom for students to access them easily.
The Lessons The lesson is given to the class inform of a problem. Students are challenged to work on the problem in small groups and later regroup to compare the solutions found. The lessons are given in theories for the student to provide practical answers.
Examples of Theories to Practice There are various situations teachers can engage students to gain practical from theory. An example of mathematical lesson to engage students in practical from theory is statistics. In this case, the teacher can provide handouts containing statistics questions, such as finding mean, mode, median, and standard deviation among many. The teacher thereafter provides newspapers to every group and challenges them to find articles related to statistics and use those articles to answer questions in the handouts. The students can be encouraged to use resources such as the teacher, textbooks, calculators, fellow students to present their findings.
Another example may include engaging students into practical trigonometric equations. For instance, calculating the equation of one side of a triangle or finding the equation of the intercept of the slope in practice. At the same time, computer graphics may be used for representing information in statistics and algebra.
Suggestions for Implementing Constructivism in Secondary School Classroom Implementation of constructivism in a classroom should be done considering several steps and principles. In the first place, the teacher should create problems pertinent to students. If the problems posed are relevant to the students, they will approach it sensing the relevance of the problem to their lives. For instance, classroom of students whose relatives are from the Middle East may find the subject of reconciliation and peace relevant to their lives. The teacher should thereafter structure learning process around vital ideas. This means making a step from a narrow point to a broader point. According to Nilson (2010), vital points are the building blocks of a whole. The students should be knowledgeable in locating the vital points. For instance, for case of statistics, students are supposed to learn vital concepts in the same. In the process, big ideas are applied to various subjects studied in a classroom. The students’ point of view should be monitored closely and guarded because they are the windows of reasoning. In a traditional class, students are forced to confine to the teacher’s thinking. As a result, teachers challenge their ideas to coincide with their thinking. For the case of constructivism, students are given the opportunity to explain the reasons behind their answers. This means that lessons should be tailored in relation to the requirements of the student. Also, the curriculum should be tailored to meet the suppositions and development of students. Students normally show interest to concepts that appeals to them. The role of the teacher in this case is to observer students and tailors the lesson to accommodate them. In this way, students will be able to pursue their area of interest. During the process, the teacher should monitor the progress of students (Concept to Classroom, 2012).

References
Concept to Classroom. (2012). How do I apply constructivism in my classroom? Retrieved Oct 08, 2012, from Concept to Classroom: http://www.thirteen.org/edonline/concept2class/constructivism/exploration.html
Leikin, R., & Zaslavsky, O. (1999). Cooperative learning in mathematics. Mathematics TeacherJournal, 92 (3), 240-246.
Nilson, B. (2010). Teaching at Its Best: A Research-Based Resource for College Instructors. San Francisco: John Wiley and Sons.
Pon, N., & Gallery, E. (2001). Constructivism in the Secondary Mathematics Classroom. Retrieved Oct 08, 2012, from Dagmar Walker: http://people.ucalgary.ca/~egallery/volume3/pon.html
Wood, D. (1998). How Children Think and Learn. Oxford : Blackwell Publishers Ltd.