5.3 Operation
5.3.3 Logging and scanning
The last two of the Piagetian Stage are designated as Concrete
Operational and Formal Operational. These are the stages that span theage range from 8 to 15 (16) an age range that includes your mathematics
students.
3.3.1 Characteristics of the Concrete Operational Stage
Operations are mental actions that are reversible. The first indicator of the stage `is the ability of the child to conserve number i.e. to realize that
the number of objects is unchanged regardless of their arrangement.
However, the same child may assume that the length of a pipe cleaner is changed when it bent or that the amount of clay in a round ball is altered
if the clay is made into a long thin pipe. In other words, ability to conserve numbers, length, mass, volume etc. are not all achieved at the
same time even though, all the schemes require the ability to consider
several perceptions simultaneous and the ability to reverse a mental
action.
By the end of the concrete operational stage, the preadolescent is able conserve all of the preceeding relationships. If you are a physics minor, you all realize that volume can be a fairly complex concept, because it includes the ideas of interior volume, liquid volume, and displacement
volume. Underlying all the above major advances of the stage of concrete operations are the ability of the child to classify and the ability
of the child to work with relationships which order makes a difference.
The ability to classify is perhaps the most powerful of our thinking tools. In and out of school, we are asked to learn hierarchies of classification systems addition, much learning depends upon the ability
to perform multiple classification i.e. the task we give a child when we
ask him to select an isosceles right triangle from a collection of shapes.
Piaget’s data show that these abilities are developed gradually throughout other concrete operation stage as a result of interaction with
sufficient experiences that require multiple classification. Ordering relationships are inseparable parts of learning to classify. For example,
the full meaning of statements such as “6 is greater than 4 which is greater than 3” is not grasped by the pre-conceptual operation youngster.
But the concrete operational child correctly conclude that “6 is therefore,
greater than 3”and later, is able to reverse these relationships and
handies “less than” statements.
However, all of the above abilities are developed during the stage of
concrete operation by interacting with “Concrete” content. What is
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essential about the “concrete content” is that the experiences be real to
the child and that such experiences reflect in as tangible a way as possible the concept or rule being developed. An enormous concrete
base of activities, verbal responses, visual responses, and visual experiences is required for initial learning of an idea which are the
instructional modes suitable for teaching concrete operational students?
3.3.2 Characteristics of Formal Operational Stage
The mental structures that develop by the end of the concrete operational stage are still available for use by the adolescent or
preadolescent. They must be used to solve many real-life problems.
In some cultures the concrete operational structures are the most
complex that can be identified.
At the stage of formal operations, the adolescent can deal with the
“form” of the situation and need not resort entirely to the concrete aspects of the problem. Piaget has identified four characteristics of the
former operational stage, all of which depend upon one/another. They are(1) the treatment of the real as a subset of the possible, (2) hypothetic deductive reasoning, (3) Combinatorial analysis and (4) prepositional
thinking.
At the formal level, reality is considered as a subset of the possible, with
the result that hypotheses may proceed from non observed and non-
experienced phenomena. This characteristics of the formal stage, i.e. the ability to imagine the possible as containing the real, the formal thinker from the restrictions of his or her sense. Further formal operations are characterized by prepositional thinking. The elements manipulated by the formal thinker are logical propositions, statements containing rawdata rather than just the raw data itself.
This is where you use conjunction, disjunction, implication, negation
and equivalence. This type of thinking is what Piaget called second
degree thinking operation, that result in statements.
The four characteristics of formal operations, outline the manner in which the intellectually mature adolescent thinks. Presented with a new
situation, the adolescent begins by classifying and ordering the concrete elements of the situation. The results of these concrete operations are
divested of their intimate ties with reality and become simply propositions that the adolescence may combine in various ways. Using
combinatorial analysis, the student regards the totality of combinations
as hypotheses that need to be verified and rejected or accepted.
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EDU 808 Mathematics Curriculum and Instructions in Secondary Schools
3.3.3 Implications of Piaget’s Theories for Mathematics Instructions
Stage Independent Theory
There are some clear signals to the teacher in the functioning of
adaptation
First, if assimilation- accommodation is to occur the gap between the new experience and past knowledge cannot be too large. If you analyse the nature of the content and search for prerequisites, some prior needs
will be identified.
Then you can informally diagnose through home work assignments or a
shared question/answer period or any number of teaching methods.
If adaptation is such an individual matter and the resulting knowledge is more heterogeneous than homogeneous, then diagnosis must be able to
identify differences in individual understanding.
Diagnostic questions or tasks can be constructed to assess recall,
comprehension or the ability to use concepts in a novel way.
The Stage-Dependent Theory
The fact the cognitive structures are nested and that one never loses the mental abilities of earlier stages is significant; if a task is truly one that requires formal operations, the student cannot operate on it meaningfully unless he or she has had sufficient concrete experience on which to draw. The teacher must analyse the content and its concrete prerequisites. Must diagnose the students intuitive background and must
plan instruction to close the gap between the intuitive and the abstract.
4.0 CONCLUSION
Although it may be difficult or undesirable to attempt to hurry the stages of development, just described, teachers are important in providing appropriate readiness activities and in asking appropriate questions.
Otherwise, the children may be delayed in achieving the various stages
of development. Two of the following four basic factors affecting mental development-experience and social transmission are strongly
affected by teachers.
These four factor affecting mental development as the child proceeds through the stages of development just described are organic growth,
(maturation), experience, social interaction or transmission, and
equilibration.
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