Educational theories developed by cognitive psychologists provide educators with methods that can be used in teaching mathematics. Four of the most famous cognitive psychologists who developed such theories are Piaget, Vygotsky, Bruner, and Dienes. Their theories greatly influenced education throughout the century on how teachers approach the teaching and learning of mathematics (Smith, 2006). These theories gave birth to the theory of constructivism, a theory “that views the child as creating knowledge by acting on experience gained from the world and then finding meaning in it” (Smith, 2006).
Through the interaction with concrete materials such as math symbols and story problems, children can understand mathematics more. Jean Piaget (1896-1980) developed the theory of Genetic Epistemology. The theory centralizes on the concept of cognitive structure. According to Piaget “cognitive structures are patterns of physical or mental action that underlie specific acts of intelligence and correspond to stages of child development” (Kearsley, 2007b). The theory further states that that there are four primary cognitive structures (or development stages).
The first stage is the sensorimotor stage, which takes place in the first two years of the child’s life. In this stage, the child learns about its environment through motor actions such as seeing, hearing and touching. The second stage, the preoperational stage, happens between the ages 3 to 7 years. According to Piaget, in this stage, the child learns language quickly and is able to use symbols as representation of real objects. It is not until the third stage, the concrete operational stage, that the child is able to learn math tasks.
In this stage, which takes place between the ages 8 to 11, the child learns to think systematically and quantitatively. The child learns to become logical but still depends upon concrete referents. The final stage of cognitive development according to Piaget is called the formal operations stage, which takes place between the ages 12 to 15 years. In this stage, the child retains its logical and systematic thinking but is now able to apply the processes they learned to more abstract problems (“Jean Piaget’s stage theory”). To develop this theory, Piaget studied his own children.
He found out that learning as a product of maturity and that certain tasks are preemptive on the child’s learning. Although Piaget’s theory has always been under scrutiny, it still influences much of today’s classroom. According to Smith (2006), the strengths of Piaget’s theory “include a focus on the child’s thinking, or the process, not just the answer; self-initiated, active involvement in a rich environment; avoiding pushing the child to be adultlike, and viewing the role of the teacher as a guide or resource person” (Smith, 2006) (p 14).
Lev Semenovich Vygotsky (1896-1934), a famous Russian psychologist, developed the Social Development Theory. This theory proposes that social interaction influences learning. In his book Mind in Society, Vygotsky (1978) states that Every function in the child’s cultural development appears twice: first, on the social level, and later, on the individual level; first, between people (interpsychological) and then inside the childe (intrapsychological). This applies equally to voluntary attention, to logical memory, and to the formation of concepts.
All the higher functions originate as actual relationships between individuals. (Vygotsky, 1978) Vygotsky views learning differently from that of Piaget. While Piaget believes that cognitive development is a product to be obtained, Vygostky views learning as a process that should be analyzed. Vygotsky calls the phenomena of social development as the zone of proximal development. This phenomenon differentiates the child’s actual development level when they solve problems independently and the potential development level when they solve problems under the guidance of adults or peers more capable (Vygotsky, 1978).
That is, a task that the student cannot perform alone can be accomplished through the help of others who are more capable to perform the task. Once the students hear other people’s thoughts, they now make use of private speech — they talk to themselves to guide their thinking. Vygostky views that children need guidance in the early stages of learning so that they can understand a concept. As the process carries on, this guidance is gradually trimmed down so that the student will learn to master the skill and later on do the task independently.
Teachers may ask the student to discuss how he got the answer for a specific problem while others listen. Although typical classroom tests only evaluate what the child already knows (Smith, 2006), these may still help teachers determine the level of discussion he will have to undertake. There are other assessments which can elicit the higher zone of proximal development (Smith, 2006). These assessments help the teacher determine the level of understanding the student has on a particular topic.