Developing Students Conceptual Understanding Of Mathematics Education Essay

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The Lev Vygotsky learning theory states that understanding of a concept by a student during instruction is based on the learners social activity whereas the Piaget's learning theory states that for teaching learning process to be effective, then the individual learner must be allowed a room to interact with the equipment, model or the specimen in the surrounding so as to be able to understand and eventually invent or reproduce. Conventionally the Vygotskian learning theory is utilized in support of valuable and efficient learner-teacher interactions. It is for that reason that the Vygotsky theory of learning is known widely as social theory. On the other hand, Piagetian learning theory is customarily applied to prop up efficient and effectual learner-resource interaction.

3.1 Piaget learning theory:

Jean Piaget was chiefly engrossed in the way information and knowledge developed and grew in human beings particularly children as they grow. This interest informed the development of his famous and popular learning theory. The conception of cognitive arrangement is principal to his theory of learning. According to Piaget, cognitive organization alters throughout the processes of adjustment which include: assimilation, classification, accommodation, class inclusion, conservation, decentration, egocentrism, operation, schema and stage. Here, assimilation entails the elucidation of actions in terms of cognitive structure available whilst accommodation is defined as the changing of the cognitive structure so as to make good judgment of the milieu. He referred to classification as the ability to group together on the grounds of common characteristics whereas class inclusion meant a more understanding than the simple classification. With regard to conservation, he said that conservation is the realization that objects or sets of objects stay even when they are rearranged or caused to appear different. Decentration is defined as that ability to reallocate position from one categorization to the other while egocentrism is known as the conviction that you are the heart of universe in addition to the fact that all things revolves about you; the corresponding inability to see the world as others do and adjust to it. Operation is that process of working out a task in your head for instance, little kids (those in sensorimotor stage and the ones in pre-operational stage) have to operate or act, and attempt stuffs out in the actual and real world, to do things (like counting of fingers): elder children and even adults can handle extra task in their heads. The schema is the mental presentation of a group of perceptions, concepts, actions and/or ideas that go together. Finally, stage is a developmental duration in which a child is able to understand certain items, but not others. Cognitive growth is made up of a continuous endeavor to acclimatize to the surroundings in terms of both the assimilation and accommodation, processes which go together. In this wisdom, the learning theory of Piaget is quite akin in natural world to other constructivist view of learning. Piaget duly considered the development of cognitive structures as an isolation of genetic system. With regard to Piaget's two main theoretical concepts: accommodation and assimilation, knowledge is an equilibrium process which belongs to both the biological interactions as well as cognitive ones and take shape in four imperative stages.

Stages of Cognitive Development:

Sensori-motor- Distinguishes self from other objects. Identifies self as an agent of deed (Birth-2 yrs)  and begins to operate with intent. For example a child pulls a cord to put handy in activity or shakes a jangle to create a sound.  Here the child does achieve object permanence. It realizes too that things persist to subsist even whilst no longer there to the general sense (pace Bishop Berkeley).

 Pre-operational-a child learns to use speech and to symbolize matter by descriptions and

  (2-7 years) words. Thoughts are still egocentric. The child has complexity taking the opinion of others. It classifies substance by a solitary mark: e.g. groups jointly all the ruby blocks in spite of of form or all the cube blocks without regard to colour. 

Concrete operational- the child can think logically and coherently about events and

(7-11 years)  substance. Can classify objects taking account of several features.

Formal operational- the child can logically think out abstract ideas. Can think

(11 years and above) hypothetically.

3.1.1 The theory of Piaget in the class:

Piaget explored intensively and extensively the implications of his supposition to nearly all aspects of cognition, intelligence as well as moral development. Many of the practical conducted by Piaget were paying attention to the advancement of mathematical and logical concepts. The Piagetian theory has been utilized broadly to instruction performance and curriculum drawing in basic learning (e.g., Bybee & Sund, 1982; Wadsworth, 1978). The ideas of Piaget have incredibly been influential on others.

Piaget's learning theory is grounded on four fundamental development stages above described that children undergo so as to learn. In every stage, the education procedure is quite different and a little further complex. Piaget deeply held that children should play, practice, and conduct experiment and raison d'être in order to study. He had conviction that human being could not be given knowledge that they understand o the spot. Humans contain to put up their personal awareness and this is done via experimentation. Experience permits children to generate schema; mental models and afterward the schema may be distorted through assimilation, accommodation and classification (Barbers, August 27, 2008). Assimilation is regarded as the course of taking in; that of absorbing a few event or experience to some schema. On the other hand, accommodation according to Bee is that time one change the schema as a consequence of fresh information and stability is at what time one strives in favor of and arrives at equilibrium of the knowledge collected. As the child conduct experiment with novel things, they are capable to code it into intelligence by mentally classifying it according to knowledge they had earlier processed. In a classroom situation, a teacher is supposed to base or design instructional deliverance, classroom control and management as well as assessments and tests on these schemes (mental presentations) that already the learner is familiar with for objective achievement.

Founded on the Piaget's learning theory, the curriculum and instructional design and delivery in to say a fifth or sixth ranking classroom is to be typically kinesthetic (Barbers, 2008). Children in mathematics class in particular should constantly be given practicing assignments which are hands on and regularly be checked by the teacher. They should be having something they physically can do like counting bottle tops etc. and even experiment with. Learners are expected to hear, see and ultimately handle the task at hand. For instance, in the project that teaches about density of substances, learners should have glass of water and perhaps an egg availed to them. They are expected to predict between the egg float and water which one will float in a bid determine which between the two is denser.

3.1.2 Teaching Mathematics toward Piaget theory

Using Piaget's learning theory results in particular recommendations for a specified developmental stage of cognitive skills. For instance, children in the stage of sensorimotor; the mathematics teachers is tasked to try by all means to offer an affluent, motivating and interesting atmosphere with ample models and other learning materials to interact and possibly play with. However, those in the real operational stage; the learning activities should be planned such that they entail tribulations such as classification, arranging in order, locating, and conservation using tangible objects.


The main ideas employable here are loyal and bound to the four developmental stages postulated by Piaget: Children resolve to give varied explanations of realia at diverse stages of cognitive growth. Cognitive development is facilitated via providing actions or situations with the aim of engaging learners in addition to require adjustment; that is assimilation and accommodation. Learning resources and activities ought to engage the suitable stage of psychomotor or mental operations meant for a child of a particular age. Such a step should be observed and incorporated by the teacher when scheming and planning for a lesson. This measure evades asking the learners to execute tasks which are far ahead of their current cognitive capacity and capability. Use teaching techniques and methods that enthusiastically incorporate learners and present challenges to learners.

3.2 Vygotksy theory:

Vygotksy theory also known as the Social Development Theory proposes that social interaction must precede growth; perception and that cognition is the finished item for consumption of socialization and social conduct.

3.2.1 Zone of proximal development:

In the Vygotskian point of view of learning, the famous Zone of Proximal Development (ZDP) is proposed to be one's latent competence for development via the help of a knowledgeable (teacher) person (Vygotsky, 1962/1934). The ZPD is one of the Vygotsky's three fundamental themes. The ZPD is actually the space between a learner's ability to carry out a task under adult supervision and/or in the company of peer partnership and the learner's ability in solving the hitch alone. According to Vygotsky's conviction and believe, learning take place in this very zone (Crawford, 1996). The implication of the Vygostkian standpoint is that it deletes the conventional borders linking the learner entity endeavor and social exchanges as the learner comes to 'know'.

In fact, to Vygotsky's viewpoint, human beings do use paraphernalia that grow from traditions, such as language and symbols, to arbitrate their communal environments. To begin with kids expand these tools to provide exclusively as social functions, the manner to express desires. Vygotsky convincingly trust that the incorporation and internalization of these apparatus led to advanced judgment skills (acquisition of knowledge).

3.2.2 Teaching mathematics toward Vygotsky theory:

Vygotsky's learning theory does promote education contexts whereby learners participate and take an active role in the process of learning, just like Piaget's proposal. Indeed here is the point of convergent between Piaget's and Vygotsky's theories. The roles of both the instructor and the learner are for that matter shifted, as a teacher must work in partnership with his or her learners so as to aid in facilitation of the meaningful construction of knowledge in learners. Learning process as a result becomes a reciprocal knowledge for the learner and the instructor.

The Vygotsky's sociocultural theory of learning was duly analyzed by Davydov who made use of the theory in elementary mathematics curriculum. Davydov hinted that mathematics needs at the elementary level a logical, pedagogical and psychological analysis. He says that for logical analysis, one is able to obtain basic mathematics concepts just for conceptual development which Vygotsky (1929) called the child's "natural arithmetical endowment". For psychological analysis Davydov said you discover the child's capacity to grasp fundamental mathematical concepts whereas pedagogical analysis creates instructional procedures able to develop links between the learners' background knowledge and the scientific concepts. It involved "teaching experiment" planning and implementation (Davydov, 1975; 6). The instruction programme of Davydov attained this by winning kids in succession of tricks that needed contrast of quantities, or the creation of quantitative associations of both equality and inequality. Such actions were in the children's capability, and they may well use daily comparatives like thicker, shorter, or even heavier to express the dealings. Vygotsky's famous zone of growth was free for learners by screening that all these diverse comparisons relating dissimilar criteria, might be represented via the generalities such as ">" "<" or "=". The abstractions were encompassed by personal denotation, though, owing to imaginative instruction strategies like use of leading questions; the teacher deliberately staging mistakes; clashing and amalgamation of verbal and visual depiction of arithmetic concepts. The visual representations are intangible and contain a corresponding figure to the numerical representation A < B. instructors acted as though kids' figurative view of quantities were comparable to broad algebraic formulae, whilst the children were still attached to a good degree a more tangible and realistic perceptive of the representations. Even so, within the circumstance of instructional discourse, instructor is able to shift the children gradually in the direction of greater abstraction and generalization (Newman and Cole, 1989). The events of instruction that Davydov discovers in his study on elementary mathematics curriculum, demonstrate that teachers are socializing children into the social dialogue of mathematics.

3.3 How these theories can be applied in the class to reduce the misconceptions and error in mathematics:

The absolute truth as above demonstrated is that both the Piaget's and the Vygotsky's theories of learning share applicable similarities especially with regard to teaching and learning of mathematics and science concepts. Like Piaget suggests that during learning the child should lead and be permitted to interact with materials; an opinion largely shared by Vygotsky. Their studies have greatly influenced and contributed to teaching and learning process in terms of curriculum development. Vygotsky sociolcultural learning theory laid firm emphasis on the communicative function of symbol systems. That the symbols give explanation for ways in which pupils derive sense of symbols. They are more likely to remember what they see and touch than if learning was based on sheer verbalism. The learner will also be in a position to distinguish apparently similar concepts. According to Piaget's theory, learner interacts with the resources. The significance of the setting (classroom) and tools available for learners use play a critical role in pupils development (Southerland et al, 2001).