For quite some time now, the teaching and learning of Mathematics has been a subject of study by researchers, theorists and others alike. The reasons for this keen interest are not only far-fetched but also very glaring. In the first instance, Mathematics is the backbone of any technological development. Furthermore, it is a core subject, which is offered by all students. The purpose of this section is to review and give an x-ray of how PBL fits into the theories of learning proposed by Piaget and Vygotsky. It also reviews the theory of constructivism in relation to the PBL.
2.9.1 Piaget’s theory
Jean Piaget, a Swiss psychologist and one of the most prominent developmental psychology researchers during the 20th century, had an early career in science and later became interested in the development of children. His research methodology is described as quasi-clinical, primarily one-to-one interviews and direct observation in classrooms. He also studied epistemology (the study of how knowledge is acquired), and regarded the child’s incorrect responses to be as important as the correct ones (Ashlock, Johnson, Wilson & Jones 1979).
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Among Piaget’s major contributions was his theory that children pass through distinct stages of mental and emotional development. These stages; sensori-motor, pre-operations, concrete operations, and formal operations represent distinctive differences in the qualitative thinking abilities (Ashlock et al, 1979). Piaget discovered from his investigations of knowledge growth that he could learn a great deal about knowledge and its development from careful observation of those who were just beginning to develop and organise their intelligence (Shulman, 1987).
Education in Piaget’s view merely refines the child’s cognitive skills that have already emerged. Piaget also views the teacher as a facilitator and guide, not a director, who provides support for children to explore their world and discover knowledge (Santrock, 2005). More so, Piaget opposed teaching methods that treat children as passive receptacles (Byrnes, 2003). This view is one of the tenets of PBL in which the teacher becomes a facilitator rather than being a dispenser of repository knowledge and learners are given the opportunity to explore the world around them and make meaningful contributions to learning thereby making learning learner-centred. Piaget introduced the concept of reflective abstraction to describe the construction of logico-mathematical structures by an individual during the course of cognitive development (Tall, 1991). In PBL, leaners are free to interact with one another and the learning materials to foster the development of problem solving skills and a minimum dose of abstract thinking through reflection is involved. In Piaget’s theory reflective abstraction has no absolute beginning but is present at the very earliest stages in the coordination of sensori-motor structure. More so, reflective abstractions continue up through higher Mathematics to the extent that the entire history of the development of Mathematics from antiquity to the present day may be considered as an example of the process of reflective abstraction (Tall, 1991).
The educational implication from Piaget’s work and its use in the PBL classroom is that children learn best from concrete activities. If implemented in schools, the use of concrete objects significantly alters the role of the teacher and the nature of the learning environment. The teacher thereafter becomes less of an expositor and more of a facilitator that promotes and guides children’s learning rather than teach everything directly (Santrock, 2005). This is one of the hallmarks of PBL. Piaget emphasised the important role that student-to-student interaction plays in both the rate and the quality at which intelligence develops. In the PBL
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classroom, learners are given the opportunity to interact with one another and this could serve as a springboard for exchanging, discussing, and evaluating one’s own ideas and the ideas of others thereby making leraners to be more critical of self and others. Piaget posited that the opportunity to exchange, discuss, and evaluate one’s own ideas and the ideas of others promotes in children a more critical and realistic view of self and others (“decent ration”). The educational implication of Piaget’s view in PBL Mathematics classroom is that, students learn best by making discoveries, reflecting on them, and discussing them, rather than blindly imitating the Mathematics teacher or doing things by rote which blocks meaningful learning. 2.9.2 Vygotsky’s Theory
Another developmental theory that focuses on children’s cognition is Vygotsky’s theory. Like Piaget, Vygotsky emphasised that children actively construct their knowledge and understanding. In Piaget’s theory, children develop ways of thinking and understanding by their actions and interactions with the physical world. In Vygotsky’s theory, children are more often described as social beings than in Piaget’s theory. Children develop their ways of thinking and understanding primarily through social interaction. Their cognitive development depends on the tools provided by society, and their minds are shaped by the cultural context in which they live (Santrock, 2005). In the PBL classroom, learners engage in social interaction and discourse and Mathematics as an object of learning is made more meaningful when learners are given minimum level of support and guidance. Zone of Proximal Development (ZPD) is Vygotsky’s term for the range of tasks that are too difficult for the child to master alone but that can be learned with guidance and assistance of adults or more children that are skilled. He also defined ZPD as the place where new external ideas are accessible to the learner with those ideas already developed. Thus, the lower limit of the ZPD is the level of skill reached by the child working independently. It is also referred to as spontaneous concepts, that is, ideas developed within. The upper limit is the level of assistance of an able instructor that is called scientific concepts, that is, ideas external to the learner. Closely linked to the idea of the ZPD is the concept of scaffolding. Scaffolding means changing the level of support (Van de Walle, 2007). In the PBL classroom, scaffolding is a major ingredient to facilitate meaningful learning in which the teacher guages the level of support being offered to the learners. The level of support given is gauged by the level of difficulty of the problem. If the problem is too simple, the zone is so small that the problem is
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not a problem any more and the scaffolding may be unnecessary thus very little learning has occurred and this may lead to leaners’ disinterest. On the other hand, if the problem is too cognitively challenging the zone is simply too big for the learner to bridge even with the help of their peers’ and teacher scaffolding. The learner looses interest and motivation, thus terminating the learning opportunity.
The Vygosky theory sees learners as a social being while the Piaget theory focuses on leaners as a cognitive being and the integration of these two theories form the bedrock of the theory of constructivism.
2.9.3. The Theory of Constructivism
Constructivism is a theory about how we learn. It also suggests that children must be active participants in the development of their own understanding. If the assertion is true, it follows that it is how all learning takes place regardless of how we teach (Van de Walle, 2007). From a constructivist perspective, some principles to follow when teaching Mathematics include: (i) making Mathematics realistic and interesting, (ii) considering students prior knowledge, (iii) making the Mathematics curriculum socially interactive (Middleton & Goepfert, 1996). Constructivist teaching, however emphasise that children have to build their own scientific knowledge and understanding. At each step in science learning, they need to interpret new knowledge in the context of what they already understand. Rather than putting formed knowledge into children’s minds, in the constructivist approach, teachers help children construct scientifically valid interpretations of the world and guide them in altering their scientific misconceptions (Martins, Sexton, Franklin, & Gerlovich, 2005).
Some constructivist researchers such as Von Glassesfeld (1990), Sahu (1983) and Kaput (1992) have investigated the processes by which students modify their cognitive representations as they create external representations and use conventional symbols to express their thinking. Given that Mathematics educators almost universally accept that learning is a constructive process, it is doubtful if any take the representational view literally and believe that learning is a process of immaculate perception (Cobb, Yackel, & Wood, 1992). However, as Ernest (1991) observed, the term constructivism itself coverspanoply of theoretical positions. Some of these appear to be eclectic positions with researchers attempting to combine the notion of learning as active construction with aspects of the
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representational view. According to Cobb et al (1992) learning is described as a process in which students actively construct mathematical knowledge as they strive to make sense of their worlds. On the other hand, learning can in practice be treated as a process of apprehending or recognising mathematical relationships presented in instructional representations. These two characterisations of mathematical learning reflect differences in the emphasis given to the students and to the teachers’ inter-presentations of instructional representations. The view of learning as active construction implies that students build on and modify their current ways of knowing mathematical concepts. In the PBL classroom, learners are given the opprotunuty to construct their own knowledge of Mathematics through schematization of the learning process in which previously learned knowledge serves as precursors and anchors to the new knowledge. In constructivism collaboration is emphasised (Adler, 1997) and this form the basis of PBL. PBL classroom allows learners to collaboratively engage in decision making regarding the solution to a problem at hand with learners not loosing their autonomy and control. In constuctivism knowledge gained is relatively permanent (Adler, 1997) and PBL relies on the heuristics of problem solving in developing and consolidating knowledge in learners. When students make public conjectures and reason with others about Mathematics, ideas and knowledge are developed collaboratively, revealing Mathematics as constructed by human beings within an intellectual community (Ball, 2000).