5.3 Operation
5.3.2 Virtual channel definition
Jean Piaget was a French-Swiss psychologist who was originally trained as a biologist. For more than fifty years he studied and analysed the growth and development of children’s thinking. His school in Geneva is noted for the study of psychological problems underlying the learning of
mathematics. His work has the greatest significance for teachers of
mathematics especially at the primary level.
This applies to his study of children’s general intellectual development and specifically to the development of mathematical concepts. You will
now study this in more detail.
Piaget view cognitive development in terms of well-defined sequential stages in which a child’s ability to succeed is determined partly by his biological readiness for the stage and partly by his experiences with activity and problems in earlier stages. The age 0-2 years is knows as the Sensorimotor stage where the child relates to its environmental through
its senses only.
Towards the end of the second year of life, children ‘have rudimentary understanding of space and are aware that objects have an existence
apart from their imEDUiate experience of them.
The pre-operational stage, (2-7 years) which generally cover the cognitive development of children during the pre-school
(KINDERGARTEN) years, is marked by the ability to deal with reality
in symbolic ways.
The thought processes of children in this stage are, however, limited by
Centering (inability to consider more than one characteristics of an
EDU 808 Mathematics Curriculum and Instructions in Secondary Schools
object at a time). Children at this stage also have difficulty with
reversibility (The ability to think back to the causes of events).
Because of these deficiencies, they cannot conserve (retain) important
characteristics of objects and event, and cannot engage in logical thinking in any concrete sense. The child is said not to posses the
concept of conservation of number, volume, quality or space.
Piaget demonstrated the lack of conservation in two experiments The following is an account of the experimental procedure:
The child was presented with two rows of five plastic squares each row arranged (as show below) in one-one correspondence. All the squares
are equal in size.
Child’s
Experiementer’s
The experimenter then says to the child. The first row is yours and the
second row in mine.
Question: Have we both the same number of squares Child’s response: Yes (Both the same)
The experimenter later arranges the squares as shown below by spreading out one row (the child’s).
Child’s
Experimenter’s
He then repeats the statement and question to the child. The child responded by saying he (the child) has more-that is they are no longer
both the same.
The child at the pre-operational stage is influenced by the perceptual features of the stimulus (i.e plastic squares). He lacks the ability to see
that movement has not altered the plastic squares. Conservation of numbers is a fundamental requirement in the understanding of numbers,
for without it a child cannot match sets by pairing to establish equivalent
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sets. For instance, for the child who is learning a number sentence such as:
3 + 2 = 5
000 + 00 = 00000
and forms a union of disjoint set of three counters and another set two
counters, the physical movement of the counters into a new pattern may mean: they change in their number characteristics; and though they mayrecite 3 plus 2 equals 5, he lacks understanding.
The implication for the mathematics teacher is that it is a waste of time
(and probably harmful to children) to try to tell children things that cannot be experienced through their senses, that is, through seeing, feelings as well as hearing. Abstract mathematical ideas should therefore
not be introduced at this stage. Children at this stage must be permitted to manipulate objects and symbols, so as to be able to appreciate reality.
Mathematically oriented recreational facilities (e.g games, play blocks, counters, marbles etc) are important tools for learning mathematics at
this stage.
The period of concrete-operational stage (7-12 yrs) is particularly important to the primary school teacher because most primary school
children are in this stage of development. This stage marks the beginning of what is known as logico-mathematical aspect of experience. It is illustrated by the case of the child who learn that counting a set of objects leads to the same result whether he counts from
front to back, back to front, or whatever configuration in which the objects are arranged. Also logico-mathematical experience underlies the physical act of grouping and classifying in what is known as the algebra
of sets.
Piaget studies the concrete operational stage using the concept of conservation of invariance, which is a basic characteristics in this stage.
For example a child is shown two identical glasses containing the same
amount of water as illustrated in Figure 2.5(a).
Fig (a) (b)
The water in one glass is then poured into a taller glass with smaller diameter as shown in Figure 2.5(b).
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EDU 808 Mathematics Curriculum and Instructions in Secondary Schools
If the child understands the amount of water is still the same no matter
the nature of the glass container and rejects what perception tells him (one look like it contains more water), he is using logic and has arrived at the concrete operational thought level for his concept. The amount of water is conserved and remains invariant (unchanged) after the transferinto another container. This is referred to as the concept of conservation of invariance. The child realizes that the process can be
reversed- that if the liquid is poured back into the initial container, the
amount should remain the same.
Another example is the process of adding one number to another number. In thought, this would be: if I want to add 4 to 3, I would take a
collection of three marbles, physically and four more and then determine
the result. But in practice I would represent the process internally.
Reversibility would imply that if I recognize that 3+4 = 7 then I would
recognize that 7-4 = 3.
The psychological condition for a reversible operation is that of
conservation, i.e + A is reversed by –A.
The concrete-operational stage is therefore important for mathematics learning because many of the operations a child is able to carry out at
this stage are mathematical in nature. For example the operation of classification, ordering, construction of the idea of nature, spatial and
temporary operation. This includes all the fundamental operations of elementary logic of classes and relations of elementary mathematics,
geometry and even physics.
There is however one limitation at this stage. Children have difficulty
from hypothetical assumptions. Care must be taken in the type of
materials that is included in their mathematical curriculum.
Children can reason abstractly if they are not affected by the limitations of the concrete-operational stage. They can proceed to the fourth stage of Piaget’s process of cognitive developments. This is known as the
formal-operational stage, from about 12 years. Only one-fourth of adolescents and one-third of adults, however, ever fully functions at the
formal operational level as shown by results of research.
At this level, the child now reasons or hypothesizes with symbols or ideas rather than needing objects in the physical world as a basis for his
thinking. He can use the procedures of the logician or scientist, a hypothetic-deduction procedure that no longer ties his thoughts to existing reality. He has attained new mental structures and constructed
new operations.
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