Another psychologist who contributed a significant idea in educational theory is Jerome Bruner (b. 1915), an American psychologist. His theory about learning has similarities with Piaget’s theory. While Piaget has four development stages, Bruner has three — the enactive mode, the iconic mode and the symbolic mode. In the enactive mode, the child only acts and reacts to what he sees. It is not until between the ages 2 to 3 that the child forms a variety of sensory image.

This is the iconic mode in which the child is already able to recall a visual, auditory, or tactile image of an object. In this mode, the child also learns to compare and contrast. The symbolic mode comes at around the age of 5 or 6. This can be of the form of an oral language, picture story drawing, or number writing. The child learns to think abstractly in this mode. Bruner believes that although each of these skills is dominant in one of each stage, it is still present and accessible throughout the learning process (Hollyman, n. d. ).

Bruner’s three modes are often used in today’s math instruction: physically doing math with manipulatives; doing mental math by thinking in terms of memories of visual, auditory or kinesthetic clues; and finally being able to use number symbols with meaning. Bruner also developed the Constructivist Theory. According to this theory, “learning is an active process in which learners construct new ideas or concepts based upon their current/past knowledge. The learner selects and transforms information, constructs hypotheses, and makes decisions, relying on a cognitive structure to do so.

” (Kearsley, 2007a). According to Bruner, the role of the instructor is to encourage students to learn principles by themselves. The environment should be interactive enough. The instructor should consider the student’s current state of understanding in rendering information (Kearsley, 2007a). Dienes (1967) proposes that there are five levels of mathematical thinking. In the first stage, which Dienes describes as free play, the child actively explores the environment but does not notice everything.

Some objects that may have common characteristics go unnoticed. In the next stage, generalization, the child begins to recognize patterns, regularities, and common attributes across different models. In the representation stage, the child is able to recognize and use symbols to represent an object. The next stage is symbolization. Although symbols are being used in the representation stage, the child is able to use the symbols to describe relationship such as words and formulas. The final stage is called formalization.

In this stage, the student is able to categorize, order and recognize the relationships and properties of all two- and three-dimensional figures as part of the structure of the discipline of mathematics. As with Bruner, Dienes believes that skills acquired from the previous stages is present throughout the child’s development process (Smith, 2006). Mathematics teaches problem solving in order to train young learners in creative, persistent thinking. Problem solving includes matching, classifying, ordering, patterning, and thinking about numbers.

However, today, the math curriculum fails to teach the connections between the symbols and their practical uses. Manipulation of numbers has become only a means to pass a course. Students now fail to see the relation of math to everyday life. Teens and young adults avoid taking math as much as possible. In other words, to students math doesn’t make any sense, except to mathematicians. For over two decades, the National Council of Teachers of Mathematics called for reform recommending the problem solving should be the focus of school mathematics (Edwards, 1980).

According to Smith (2006), problem solving should be focused on in order to understand mathematics better. Specifically, problem solving can help students Build new mathematical knowledge through problem solving; Solve problems that arise in mathematics and in other contexts; Apply and adapt a variety of appropriate strategies to solve problems; Monitor and reflect on the process of mathematical problem solving. * (p. 52) Parents and educators alike agree that problem solving should be the focus of the school mathematics curriculum.

All aspects of the curriculum, from science to social studies, stress logical, flexible thinking and math is one subject that can students with this since, in math, children explore many kinds of reasoning. In learning math, they discover how it explains the real world. Math not only helps people handle everyday tasks. As Smith (2006) stresses, math “is a way of thinking. ” Smith (2006) further stresses the two essential ingredients of problem solving — an interesting, challenging problem, and a positive atmosphere. In the succeeding sections, we will discuss these two elements in more detail.