# Theory History

**Bruner’s Stages of Representation:-** Jerome Bruner is a psychologist who focused much of his research on the cognitive development of children and how it relates to education. While he has made many contributions to the field of psychology, his greatest contributions have been in the educational field. At the time of his research, behaviorism was the primarily practiced theory in America’s classrooms (Brahier, 2009, p. 49).

Initially, Bruner was interested in how the mind organized and categorized information. Because his early career focused on cognitive psychology, Piaget’s theories played a large role in his initial studies. Over time, however, as he began to specialize more on learning, Vygotsky and his ideas on the Zone of Proximal Development and scaffolding came to be increasingly influential to Bruner’s research (Smith, 2002). Each of Bruner’s stages of representation builds off of the knowledge and information learned in the previous stage, or in other words, the stage before acts as scaffolding for the next stage. The theory has come to play a huge role in mathematics education, particularly with the encouraged use of manipulatives. Eventually, Bruner’s stages of representation came to play a role in the development of the constructivist theory of learning as well (Culatta, 2012).

## Theory Definition

Jerome Bruner theorized that learning occurs by going through three stages of representation. Each stage is a “way in which information or knowledge are stored and encoded in memory” (Mcleod, 2008). The stages are more-or-less sequential, although they are not necessarily age-related like Piaget-based theories. Going through the stages is essential to truly understand the concept, as it helps the learner understand why.

**Bruner’s Stages of Representation**

1. __enactive (action-based)__

Sometimes called the concrete stage, this first stage involves a tangible hands-on method of learning. Bruner believed that “learning begins with an action – touching, feeling, and manipulating” (Brahier, 2009, p. 52). In mathematics education, *manipulatives* are the concrete objects with which the actions are performed. Common examples of manipulatives used in this stage in math education are algebra tiles, paper, coins, etc. – anything tangible.

2. *iconic* (image-based)

Sometimes called the pictoral stage, this second stage involves images or other visuals to represent the concrete situation enacted in the first stage. One way of doing this is to simply draw images of the objects on paper or to picture them in one’s head. Other ways could be through the use of shapes, diagrams, and graphs.

__enactive (action-based)__

*iconic*(image-based)

### 3. __symbolic (language-based)__Sometimes called the abstract stage, the last stage takes the images from the second stage and represents them using words and symbols. The use of words and symbols “allows a student to organize information in the mind by relating concepts together” (Brahier, 2009, p. 53). The words and symbols are abstractions, they do not necessarily have a direct connection to the information. For example, a number is a symbol used to describe how many of something there are, but the number in itself has little meaning without the understanding of it means for there to be that number of something. Other examples would be variables such as x or y, or mathematical symbols such as +, -, /, etc. Finally, language and words are another way to abstractly represent the idea. In the context of math, this could be the use of words such as addition, infinite, the number three, etc.

## Application for Teachers of Mathematics

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, 2009, p. 54). In addition, by having all students go through each of the stages, it builds a foundation for which the student can fall back on if they forget or as they encounter increasingly difficult problems. For these reasons, it is essential that the teacher go through each of the stages with the whole class; however the time spent on each stage can and will vary depending on the student, topic, etc.

## The Theory & Mathematical Academic Language

Another important part of the theory’s application, is the academic language. The development and use of an academic language is crucial for successfully learning the concept. This primarily takes place in transitioning from the iconic stage to the abstract, language-based, symbolic stage. “Since language is our primary means of symbolizing the world, [Bruner] attaches great importance to language in determining cognitive development” (Mcleod, 2002). The correct academic language needs to be taught and used in the symbolic stage in order for the student to demonstrate that they can not only come up with the correct answer but that they understand the problem and process for getting it. In this context, the academic language involves not only vocabulary and mathematical terms but also mathematical symbols such as +, -, (, ), /, etc.