I understand the aspect of cognitive development, I just need to understand the educational applications of the theory.
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One of the most common controversies I find related to Piaget has to do with when to teach algebra. Many claim that algebraic thinking needs to be introduced earlier, and in fact it has been moved steadily down from high school to middle school and even elementary school. Yet others claim, based on Piaget's stages of learning, that some children learn to think abstractly at different ages and not all of them are capable of learning algebra in early adolescence.
The simple answer is that we need to understand what stage of cognitive development our students are at, and create our teaching strategies accordingly. Based on Piaget's theories you should not be using truly abstract ideas until adolescence.
Examples would be that you teach a young child through purely sensory experiences. You give them objects to touch, taste, and smell. You offer them different textures and colors in their toys. As a child gets a little older you use language to coordinate with the sensory input. You give textures names, you give colors names, you give tastes names, as well as the objects around them. Young school age children can begin to grasp concepts outside their immediate outlook but still need concrete reinforcement. You teach math concepts with manipulatives. You teach new vocabulary with pictures. You also take into account the child's limited view of the world, vivid fantasy life, and lack of understanding of time. As children grow you can make more connections to the real rather than fantasy world to teach concepts and they begin to understand and apply the abstract.
One key to remember is that all children develop at a different rate and that depending on what age your students are you may have students at different levels of cognitive mastery and therefore you must differentiate your instruction to meet the needs of these learners at their cognitive level.
Piaget's approach to education challenges behaviouralist notions of knowledge, learning or facts being "pumped in" to a child by the teacher. Piaget argued that learning does not happen so automatically, and that it is, as #2 identifies, a process and not a product. It is perhaps highly ironic that education nowadays involves so many targets and standards that children are expected to attain at specific ages. We seem to have forgotten the wisdom of such thinkers as Piaget.
Piaget's theories can be applied in many different educational realms. For example, his basic premise that involved stages of development gives heavy weight to the individual maturation of a child. This has educational implications in that teachers have to understand that "mastery" occurs at different paces for the different learners in the classroom. There is no one "end" in which everyone has "learned." Rather, it takes place organically, and teachers have to understand that this maturation and growth must be built into the curriculum and the planning of instruction. I think that another relevant idea of Piaget comes in his view of education, in general:
Knowledge is a process, not a product; it is dynamic, never static, self-regulatory rather than imposed from without. Even as mollusks taken from the lake of Neuchâtel and placed in an aquarium change very little after five or six generations, the human organism has a built-in blueprint that determines the course of cognitive evolution, a course not unlike that of the evolution of scientific thought.
The idea of knowledge being a process and not solely a product is profound for education. In a modern setting where so much is connected to high stakes standardized assessment and standards based educational reform, Piaget's words about process and product, the dynamic nature of understanding, and the evolution of the mind haunt the modern educator. The applications of this idea would be to construct a realm whereby stakeholders are able to embrace an approach to learning that is rooted in the child's development and their own sense of understanding. This is probably where the realm of authentic assessment and student driven approaches to learning would be something Piaget embraces.
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