# Mathematical theories

Theories of mathematics learning (Piaget, Bruner, Vygotsky). Compare and contrast between them with suitable examples and explain how children develop mathematical knowledge and skills.

Theories of mathematics learning

Mathematics is learnt using various theories such as behaviorism, constructivism, social-culturalism and embodied mathematics.

Behaviorism theory

Behaviorism theory explains concepts in mathematics using the environment of the learner without affecting the internalities of the learner. It lays its basis on the response of the stimulus skinner model that applies science methods in study of human learning (Cottrill, 2003). The theory controls learning by affecting situational variables, behavioral and consequences of behaviors. Behaviorism theory is the most common methods used in learning and one that led to usage of quantitative and empirical learning.

Constructivism theory

Constructivism theory argues that learners’ knowledge is obtained from setting of his current environment. Piaget also developed evolution theory to describe phenocopy from establishing genome equilibrium. He explains that students undergo cognitive epigenesis whereby students find equilibrium through assimilation and accommodations (Ojose, 2008). The theory explains that students use reflective abstraction to move from one level to a higher level.

Piagent learning uses a stage theory that describes a child’s mental development. Children are seen to think differently in their different stages of development. Stages are pre-operational, concrete operational and formal operational thinking. During adolescence stage, children are prepared and ready to get to formal operational way of thinking (Ojose, 2008). Moreover, piagnet explains that student’s progress from one understanding level to another by reflective abstraction. Intraoperational stage involves a student focus on transformational object separate from other actions and objects. Interoperational thoughts come about when students begin relating with actions using reflective abstraction. Tran’s operational stage involves a reflection on the interrelations and their transformation into objects in larger systems.

Socio-culturalism theory

Socio-culturalism theory explains that the intelligence of students comes from normal socio interactions with the world. Moreover, speech, social interaction, and co-operative activity are contributors in the world. Language used by students helps in building cognitive tools that are controlled from by the conscious mind. The teacher participates by establishing the relationship between the given sign and its meaning. In this stage zone of proximal development is the distance between developmental level of a student (working on problems) and her potential development (working with an adult). The teacher becomes the holder of the tool and consciously controls concepts on behalf of the child until he gains ability to digest knowledge from external sources (Cottrill, 2003).

Piaget and Vygotsky have similarity with their ideas describing that children have active roles in their learning processes. Vygotsky idea is constructive in that the child requires to internalize knowledge gained from outside. Confrey exaggerates Piaget’s view of students understanding arguing that it should be done in isolation. Vygotsy differs with Piaget view of speech role. Vygotsy explains that social interaction has the possibility to restrict diversity in class.

Piaget explains that children involve in egocentric speech and utterances that are not directed to others and expresses in ways for listeners to understand. The speech plays little role in cognitive development. In addition, the speech gets social as the child matures. However, vygotsky argued that languages and thoughts just emerge and the transition from paralinguistic reasoning to verbal reasoning is nonsocial utterances of a child. Further development of children their private speeches end up becoming inner speeches.

Embodied mathematics theory

Lakoff and Nunez explained conceptual metaphor has a vital role in mathematical ideas and views as well as cognitive unconscious. It goes beyond findings used in cognitive science to explain the number of mathematical concepts that come into learners mind through metaphors and blending.

Frameworks explaining how children develop mathematical knowledge and skills

An epistemological framework with Action-Process-Object-Schema explains that concepts in mathematics move from action (intraoperational) to process (interoperational) through reflective abstraction (interiorization). The later process is formed into an object (transoperational). Constructed objects may be de-encapsualted to the process when in need.     Coordination of processes and actions forms schemas, which are thematized to objects (Cottrill, 2003). The framework describes mental constructions made by a student while understanding a concept (genetic decompositions). It involves establishment of instructional treatments that make students create constructions in genetic decomposition. Instructional treatments involve using computer programming languages that are mathematical, learning strategies that are cooperative and lecturing alternatives.

Other frameworks describes mental constructions as either structural) objects) or operational (processes). Reflective abstractions are described to move from processes to objects including interiorization, condensation, and reification. It expresses objects as resembling a dualism other than dichotomous relation. Moreover, images of students about concepts are figurative or metaphoric.

Vinners concept of image and Schemas differ in that vinner describes mental images as coalescences of pictures in the mind and arranged to categories that match with mathematical vocabularies. Students may also develop mathematical knowledge and skills through notational system, which represents mental structures into their physical world (Cottrill, 2003). Educators design educational systems that are functional and elegant with computers to describe the world as being social, mental, and material.

Learning takes in the relationship of the student interpreting and representing mathematics sharing their meanings with others. Students understand how to solve problems under conditions with adequate resources of basic knowledge of mathematics, techniques. In addition heuristics or techniques should be available to enable solve problems and the ability to control and select available resources in an appropriate manner and belief systems that help students endure during problem solving process.

References

Cottrill, J. (2003). An overview of theories of learning in mathematics education research.

Ojose, B. (2008). Aplying Piaget’s theory of cognitive development to mathematics instruction. The mathematics educator, 1(18), 26-30.