The Teaching Of Mathematics

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The aim of this paper is to outline the learning theories of three psychologists and how they are applicable to the teaching of mathematics. The teaching of mathematics has become of great concern since the passing grades are falling dramatically. In all honesty, the teaching of mathematics was never top priority as literacy was the major priority.

It has now come to the fore, as children are not turning out the expected results and thereby measures have to be put in place to equip teachers in order to have different and varied ways to in cooperate children with different learning disabilities.


Jean Piaget was a Swiss biologist, philosopher, and psychologist best known for his work in the area of developmental psychology. Like Sigmund Freud and Erik Erikson, Piaget divided cognitive growth and development into fixed stages. But Piaget's particular focus was on the intellectual or cognitive development of children and on the way in which their mind's processed and progressed in knowledge. Piaget's central thesis was that children (1) develop self-centric theories about their environment, and about objects or persons in that environment, and they grow; and (2) that children base these theories on their own personal experiences interacting with persons and objects in their environment; (3) that the child used "schemas" to master and gain information about the environment; and (4) that the sophistication of a child's cognitive structures increased as the child grew and developed, as did the child's "schemas". Schemas, which are the child's tool bag of actions and responses to make things happen, start with rudimentary interactions such as grabbing and mouthing objects and eventually progress to highly sophisticated skills such as scientific observation. Piaget divided the child's path of development into four stages which began with birth and culminated in the teen years. These stages are: Sensorimotor stage (0-2 yrs), Preoperational stage (2-7 yrs), Concrete operations (7-11 yrs), and Formal operations (from 11-15 and up). A chief tenet of Piaget's theory is that these stages do not vary in order, cannot be skipped, and should not be rushed (http:/​/​, September 18, 2012)

Santrock (2001) states that for Piaget, two processes are responsible for how children use and adapt their schemas (a concept or framework that exists in an individual's mind to organize and interpret information): assimilation and accommodation. Assimilation is the incorporation of new knowledge into existing knowledge. Accommodation is the adjustment to new information by children (p. 49).

These stages he generalized as:

Sensorimotor stage (birth - 2 years old) -- Child interacts with environment through physical actions (sucking, pushing, grabbing, shaking, etc.) These interactions build the child's cognitive structures about the world and how it functions or responds. Object permanence is discovered (things still exist while out of view).

Preoperational stage (ages 2-7) -- Child is not yet able to form abstract conceptions, must have hands-on experiences and visual representations in order to form basic conclusions. Typically, experiences must occur repeatedly before the child grasps the cause and effect connection.

Concrete operations (ages 7-11) -- Child is developing considerable knowledge base from physical experiences. Child begins to draw on this knowledge base to make more sophisticated explanations and predictions. Begins to do some abstract problem solving such as mental math, etc. Still understands best when educational material refers to real life situations.

Formal operations (beginning at ages 11-15) -- Child's knowledge base and cognitive structures are much more similar to those of an adult. Ability for abstract thought increases markedly (Santrock, 2001, p. 49-50).


Vygotsky (1896-1939) a Russian also believes like Piaget that children actively construct their knowledge. According to Santrock 2001, there are three claims that capture the heart of Vygotsky:

The child's cognitive skills can be understood only when they are developmentally analyzed and interpreted. Taking this approach means that in order to understand any aspect of the child's cognitive functioning, one must examine ots origins and transformations from earlier to later forms.

Cognitive skills are mediated by words, language and forms of discourse, which serve as psychological tools for facilitating and transforming mental activities. Language, he states is the most important of these tools that will help the child plan activities and solve problems.

Cognitive skills have their origins in social relations and are embedded in a sociocultural backdrop (p. 60). He believes that the development of memory, attention and reasoning involves learning to use the inventions of society, such as language, mathematical systems and memory strategies.

Within these claims, Vygotsky has give unique ideas about the relation between learning and development. These ideas qualify his view that cognitive functioning has social origins. One such idea is the Zone of Proximal Development. This is the range of task that are too difficult for children to master alone but that can be learned with guidance and assistance from adults or more-skilled students (p. 60).

McLeod (2010) states that "Vygotsky views interaction with peers as an effective way of developing skills and strategies. He suggests that teachers use cooperative learning exercises where less competent children develop with help from more skillful peers - within the zone of proximal development." He further states that, "Vygotsky believed that when a student is at the ZPD for a particular task, providing the appropriate assistance (scaffolding) will give the student enough of a "boost" to achieve the task. Once the student, with the benefit of scaffolding, masters the task, the scaffolding can then be removed and the student will then be able to complete the task again on his own."


Jerome S. Bruner (1915- ) is one of the best known and influential psychologists of the twentieth century. He was one of the key figures in the so called 'cognitive revolution' - but it is the field of education that his influence has been especially felt (Smith, 2002).

According to Smith (2002), "in the 1960s Jerome Bruner developed a theory of cognitive growth. His approach (in contrast to Piaget) looked to environmental and experiential factors. Bruner suggested that intellectual ability developed in stages through step-by-step changes in how the mind is used. Bruner's thinking became increasingly influenced by writers like Lev Vygotsky and he began to be critical of the intrapersonal focus he had taken, and the lack of attention paid to social and political context."

In his research on the cognitive development of children, Jerome Bruner proposed three modes of representation: "Modes of representation are the way in which information or knowledge are stored and encoded in memory" (McLeod, 2008).

Enactive representation (action-based)

Iconic representation (image-based)

Symbolic representation (language-based)


(0 - 1 years)

This appears first. It involves encoding action based information and storing it in our memory. For example, in the form of movement as a muscle memory, a baby might remember the action of shaking a rattle.

The child represents past events through motor responses, i.e. an infant will "shake a rattle" which has just been removed or dropped, as if the movements themselves are expected to produce the accustomed sound. And this is not just limited to children.

Many adults can perform a variety of motor tasks (typing, sewing a shirt, operating a lawn mower) that they would find difficult to describe in iconic (picture) or symbolic (word) form.


(1 - 6 years)

This is where information is stored visually in the form of images (a mental picture in the mind's eye). For some, this is conscious; others say they don't experience it. This may explain why, when we are learning a new subject, it is often helpful to have diagrams or illustrations to accompany verbal information.


(7 years onwards)

This develops last. This is where information is stored in the form of a code or symbol, such as language. This is the most adaptable form of representation, for actions & images have a fixed relation to that which they represent. Dog is a symbolic representation of a single class.

Symbols are flexible in that they can be manipulated, ordered, classified etc., so the user isn't constrained by actions or images. In the symbolic stage, knowledge is stored primarily as words, mathematical symbols, or in other symbol systems.

Bruner's constructivist theory suggests it is effective when faced with new material to follow a progression from enactive to iconic to symbolic representation; this holds true even for adult learners. A true instructional designer, Bruner's work also suggests that a learner even of a very young age is capable of learning any material so long as the instruction is organized appropriately, in sharp contrast to the beliefs of Piaget and other stage theorists (McLeod, 2008).



Children who are in the upper grades (6, 7, 8, 9 etc) are in the process of moving from concrete to the abstract stage of their cognitive processes. Therefore different methods need to be used in order to help students to understand the concept. For example, the use of fraction circles to help student understand how to add, subtract, multiply and divide fractions. Students are allowed to use these until they are proficient in algorithms. All in all every lesson should incorporate hands on experiences in which the students discover the rules for themselves


Vygotsky emphasizes the social contexts of learning and that knowledge is mutually built and constructed. The principles of using the students' zone of proximal development and scaffolding can be used. Show them how to scaffold. In a classroom, the teacher should set up a lesson plan to include some guided practice (where the teacher assists the students in doing a new activity or skill) and some independent practice (where the students practice the skill on their own after learning about it).

In a math class, for example, you might scaffold a lesson by presenting information on multiplying fractions and showing a few examples where you multiply fractions on the chalk board. Then you put a few more fraction-multiplication problems on the board and ask the students to help you solve the problems by talking them through the process as a group. Finally, you give the students a few more problems where they have to multiply fractions on their own. In this way, you took the students from not knowing anything about multiplying fractions to knowing how to do it on their own; you brought them through the zone of proximal development (Cook).


While Bruner has influenced education greatly, it has been most noticable in mathematical education. The theory is useful in teaching mathematics which is primarily conceptual, as it begins with a concrete representation and progresses to a more abstract one. Initially, the use of manipulatives in the enactive stage is a great ways to "hook" students, who may not be particularly interested in the topic.

Furthermore, Bruner's theory allows teachers to be able to engage all students in the learning process regardless of their cognitive level of the concept at the moment. While more advanced students may have a more well-developed symbolic system and can successfully be taught at the symbolic level, other students may need other representations of problems to grasp the material (Brahier, 2008).