One of the main researchers who has connection with the constructivist movement was David Ausubel. In 1968, he put forward the case that the most important thing for teachers to know at the outset of the teaching is what each student knows already. However, he held a different approach to how the teaching material should be presented in the classroom or by self-study than Bruner. He argued that students need guidance if they are to learn effectively and he advocated the direct instruction learning approach. Ausubel (1968) focussed on both the presentational methods of teaching and the acquisition of knowledge. He made a major contribution to learning by studying and describing the conditions that lead to ‘meaningful learning’. He attempted to find ‘the laws of meaningful classroom learning’.
According to Ausubel, meaningful learning presupposes:
• “That the learning material itself can be non arbitrarily (plausibly, sensibly, and non randomly) and substantively (non verbatimly) related to any appropriate cognitive structure (possesses “logical meaning”).
• That the particular learner’s cognitive structure contains relevant anchoring idea(s) to which the new material can be related.
• The interaction between potentially new meanings and relevant ideas in the learner’s cognitive structure gives rise to actual or psychological meanings. Because each learner’s cognitive structure is unique, all acquired new meanings are perforce themselves unique.”
Ausubel et.al, (1978)
The meaningful learning processes exist when the new concept can be linked to the pre-existing concept in the learners’ cognitive structure (for example, already existing relevant aspect of knowledge of an image, an already meaningful symbol, a known concept or a proposition). The new concept interacts in a non arbitrary (in the sense of plausibly, sensibly and non randomly), and substantive (non verbatimly) basis with established ideas in cognitive structure. Thus, meaning derives directly from associations that exist among ideas, events, or objects. As the new knowledge is subsumed into the existing knowledge, it interacts and modifies it and the entire new matrix now becomes more elaborate and new linkages form between concepts. Obviously, this theory can only become reality if the teacher finds out what the learner already knows. Orton (2004) argued that, if an attempt is made to force children to assimilate and accommodate new mathematical ideas that cannot link to knowledge which is already in an existing knowledge structure, then the ideas can only be learned by rote. In contrast with meaningful learning, rote learning results in arbitrary literal assimilation of new knowledge into cognitive structure. It occurs when no relevant concepts are accessible in the learner’s cognitive structure.
‘Rote-meaningful’ learning is a continuum, which relies on the learner and differs from one learner to another. The feature of the cognitive structure of the learner interconnects in a diverse degree from topic to topic in the ‘rotemeaningful’ continuum. The learner’s existing knowledge and the way that new knowledge is linked to existing knowledge involves subsumption. Ausubel postulates that cognitive structure is hierarchically organised which means the less inclusive sub-concepts and details of specific data are organised, under the more inclusive concepts. Therefore, good expository teaching should be given to the learner to ensure that a new concept is linked to relevant existing knowledge. According to Ausubel (1968) the advance organiser is “an advanced introduction of relevant subsuming concepts (organisers) which can facilitate the learning and retention of unfamiliar but meaningful verbal material”. The idea of advance organizer was introduced by Ausubel in the following two cases:
a) When the student does not process proper subsumers, e.g. when the material is totally new and the learner does not have relevant information to which they can relate the new material.
b) When the student obtains appropriate subsuming information, but the information is insufficiently developed and is not likely to be identified and linked to the new information.
The existing components of the knowledge structure to which new learning needed to be correlated; subsequently they become recognized also as ‘anchoring’ ideas or concepts. At this point Ausubel’s idea of subsumers is similar to Bruner’s view of readiness. So, if the subsumers are there the student is efficiently ready. Orton (2004) stated that meaningful learning implies an understanding of constraints. He also added that any theory of learning mathematics should take into account the hierarchical nature of the subject, and there should not be many occasions when new knowledge cannot be linked to existing knowledge. For instance, it is not possible to learn about integers and about rational numbers unless the natural numbers are understood meaningfully .
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