The three prominent researchers of cognitive theories that is related to the learning and teaching of mathematics are Robert Gagne, Jerome Bruner and David Ausubel, they all put forward their ideas initially in the 1960s , at that time, all three were established in their carriers and recognized as authorities in their own right. All the three attempted to define cognitive theories of instruction; this coincided with periods of tremendous growth in scientific knowledge and expansion of, what was now in these Western Countries Universal Secondary Education
Bruner advocated discovery learning and gained very wide acceptance at least in schools.
The three stages of Bruner’s theory of intellectual development are;
i. enactive where a person learns about the world through actions or objects
ii. Iconic where learning occurs through using models and pictures
iii. Symbolic which describes the capacity to think in abstract terms. The underlying principle for teaching and learning at least mathematics is that a combination of concrete, pictorial, then symbolic activities will leads to more effective learning. Mathematics teachers should apply the burner’s theory in the following progression: start with a concrete experience then move to pictures and finally use symbolic representation. Students should equally be encouraged by their teachers to use discovery learning techniques since anything discovered by the students themselves tends to be retained for a longer period of time.
David Ausubel’s notable contribution to the subject matter was the notion of the “advanced organizer” the advance organizer simply means a device or a mental learning aid to help students “get a grip “ on the new information. Put in more difficult language, according to David Ausubel, the advance organizer is a means of preparing the learners cognitive structure for the learning experience about to take place. It is a device to activate the relevant schema or conceptual pattern so that new information would be more readily ‘subsumed’ into the learners existing cognitive structure or mental depiction. The implication of Ausubel’s theory of advance organizer in teaching and learning of mathematics is that the teacher should always ignite and sustain the attention and interest of his students while delivering his lessons
In the other hand, Robert Gagne was concerned with the problem of determining just what skills and knowledge are required for someone to be an effective performer at a given task. He suggested that a task would be best learnt by following specific sequence of nine events namely: gaining attention, informing the learner of the objectives, stimulating recall of prerequisite learning, presenting new materials, providing guidance, eliciting performances, providing feedback about correctness, assessing performance, and enhance retention and recall. These notions of task analysis and the importance of the correct sequencing of instruction are followed by most mathematics teachers when designing their progrmames.
Again the concept of Gagne’s knowledge hierarchy leads to the assumption that it is important to present all the necessary lower level facts before proceeding to teach at higher level, related to this is the concept that people can reason with higher – level concepts if they have learned all the prerequisites lower -level information
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