SECCIÓN VI MODELOS DE FORMULARIOS
CONDICIONES ESPECÍFICAS 4.1 VIGENCIA DE LA OFERTA
Jerome Burner worked on the process of thought in general. Later he
applied this to the process of learning mathematics. He devised experiments to help him observe how mathematical thinking in children
develop. The investigation concerned the individual strategies by which
a child tries to discover a given logical relationship.
The procedure in most of the experiments was to present a number of cards to the child. Each card has its diagrams of triangle, circle or square separately or a combination of these. Each card was red or green or blue.
So there were three variable- number, shape and colour-each with three
values. A concept such as red triangles was thought of by the experimenter and the subject chose cards to which the experimenter answered either Yes or No: if the card was red and had triangles on it.
and No if not. Subjects were asked to find the concept, which the experimenter had in mind in the least number of trials. Sometimes more
variables were used, sometimes the numbers of choice were restricted.
From this single procedure, Bruner was able to claim that learning in
general depended on four factors.
(i) the structure of the concept that is to be learnt:
(ii) the nature of the learner’s intuition:
(iii) the desire of the learner to learn:
(iv) the readiness for learning- (biological readiness ).
Thus Bruner considered adequacy of both the subject matter and the
learner himself necessary for the leaning of Mathematics. By this hemeant that the learner must be intuitively ready to learn and the materials to be learnt must be presented in a form (or structure ) that
matches the learners “readiness stage” This led to his controversial, but yet popular, assertion that” Any concept can be taught effectively in some honest form to any child at any age provided such a concept is
introduced at the child’s language level”.
This sort of reasoning let him to attempt a classification of these levels or stages. The following are the three stages through which Brunner says
a child goes through in cognitive learning.
i.
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The Enactive Stage: At this stage the child thinks only in terms
of action. This stage is characterised by the mode of representing past events through motor responses. The child enjoys touching and manipulating objects as teaching proceeds. Specifically no
serious learning occurs at this stage. Topics can however be
introduced to a child at this stage using concrete materials. The
child’s methods of solving problems are limited because he
cannot “act the solution” – he cannot solve problems.
ii.
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The Iconic Stage: This is the stage of manipulation of images.
Here he builds up mental images of things already experienced.
Generally, such images are composite being forEDU from a
number of experiences of similar situations. Learning at this stage is usually in the form or in terms of seeing and picturing in the
mind any object which transforms learning. The child uses thinking thereby making transfer of learning considerably easy.
Bruner emphasised that before any image is forEDU to represent a sequence of acts, certain amount of motor skills and practice have to take place.
The Symbolic Stage: At this stage child possesses the ability to
evaluate learning. Logic, language and mathematical symbols are used to discuss what has been learnt. Acquired experiences are translated into symbolic form The three stages can be illustrated
this way, using the concept of addition of positive whole numbers. Consider the problem: 3+2=5. First the child must work
with block, marbles, counter or other real objects. Take the three first, take another two mix them up and then count the mixture
(or union) At the second stage he will be able to work with
worksheets containing pictures of objects(images). Instead of the physical objects he is now able to recognise their image and can
solve something like it while not necessarily requiring the production of the ducks physically. At the final stage he can solve
the real’ problem 3 + 2 =5 using symbols 3 and 2 and 5. Bruner’s three stages, in brief of the sequence
Action
Image Word
and correspond approximately to Piaget,s sensor motor
Perceptual Abstract
modes of cognitive functioning. Also Burner, like Piaget, believe that all
mathematics could be learnt by discovery approach provided that searchis started early enough in the life of the child by presenting to him
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concrete materials relevant to the concept we want him to learn at a
higher stage. For example properties of a triangle could be taught by making sure children play with triangular shape object at the pre-school(Kindergarten ) stage, draw enough image of triangles at the at the primary school and by the junior secondary school they would have
been sufficiently equipped with various terms to discover for themselves
some, if not all the properties of a triangle.
3.2.2 The Contributions of Robert Gagne
As a behaviourist psychologist. Gagne devoted his time to study the conditions of learning. He believes that learning occurs as a result of interaction between the learner and the environment. Learning is known to have taken place when we notice (observe) Gagne maintains that the stages described by Piaget are not necessarily the inevitable result of an
inborn “timetable” but are instead a consequence of children having learned sets of rules that are progressively more complex. How do children acquire these sets of rules? According to Gagne children are
“taught “ the rules by their physical and social environment.
Notice the differences in what Piaget and Burner are claiming on one
hand and what Gagne is saying. If we follow Piagets (and Burner) assertion we will assume that children will develop complex concepts.
Understanding and problem solving skills when they are ready. That is
when their nervous systems have matured sufficiently and they had
enough experiences with simpler more elementary problems.
Mathematics teachers who see learning as a process of discovery are likely to borrow heavily from Piaget and Burner. Others who will see learning as produced primarily by children environment are likely to take their cues from Gagne. The following illustrates Gagne contribution
to the learning of mathematics.
We have already mentioned that he emphasised the idea of pre-requisite knowledge in learning mathematics. That is the idea that one cannot master complex concepts without mastering the fundamental concepts
necessary for such complex concepts. For instance. a child can not successfully add fractions without the knowledge of finding common
denominator of fraction. That is,
A child cannot do 1 + 2 unless he has been
3 5
Lead to learn that 1 = 5 and 2 = 6
3 15 5 15 therefore 1 + 2 = 2 + 6 = 11
3 5 15 15 15
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EDU 808 Mathematics Curriculum and Instructions in Secondary Schools