The stage of concrete operations: 7 to 11 years
An operation is a rule-following transformation which we can make using
our mental representations. In other words it is a type of logical thought
process. Examples of operations include arithmetical operations such as
adding, subtracting, multiplying and dividing. The ability to conserve
requires mental operations including reversibility – the understanding that a
transformation can be un-done, and compensation – the recognition that one
change – e.g. flattening a ball of clay will make it thinner but larger in area.
Another example of an operation is converting a statement into its opposite,
as illustrated by the child who said, ‘When I grow down I want to be small’.
During the stage of concrete operations, children can perform mental
operations with real situations only. Concrete means real and tangible.
An example is of a child at a family meal who looks at the other people
around the table and asks, ‘Are you the oldest one dad?’, and having this
confirmed comments, ‘That means you’ll die first’. Here the child is
applying a simple rule: first assuming that everybody lives for the same
length of time, and then deducing that they will die in the same order that
they were born. The assumption is faulty, but the logical operation is correct.
The father, with the advantage of greater knowledge, merely replies, ‘I hope
so son’.
This ability to perform operations means that children at this stage succeed
in the conservation tests of the previous stage.
They can also manage seriation tasks. This means they can put concrete
objects (or their representations) in order. For example if you told a child
that Malcolm is taller than his brother Simon, who is taller than their sister
Leonie, they could tell you which child was shortest.
Children in this stage can also perform transitivity operations – these
involve making new deductions from information given. For example,
given the information about the three children above, they could correctly
answer the question, ‘Is Malcolm shorter or taller than Leonie?’
They can also demonstrate the ability of class inclusion. Piaget’s class
inclusion test is an example of this. Piaget and Szeminska (1941) showed
children twenty wooden beads. Eighteen were brown and two were white.
Each child was asked several questions:
1. Are all the beads wooden?
2. Are there more brown beads or white beads?
3. Are there more brown beads or more wooden beads?
Pre-operational children usually answer the first two questions correctly.
However they usually answered question three by saying that there were
more brown beads. Children in the stage of concrete operations usually
answered all three questions correctly. Notice that in question two, the child
has to think about two separate classes of beads – the brown ones and the
white ones. But in question three, the two classes are not separate – they
overlap. The class of brown beads is included in the class of wooden beads
(this is class inclusion). The wrong answer is given, perhaps because the
child assumes that if brown beads are one class, and they are the majority,
they must have more members than any other possible class.
The stage of formal operations: 11 years and beyond
In this stage, adolescence can apply operations not only to concrete
situations, but also to abstract concepts. One example of a formal operation
is substitution used in algebra. The individual is able to use a symbol – such
as ‘A’ to stand for an unknown variable. This enables them to perform other
arithmetical operations on ‘A’. For example, if told that 3A=21, they can
discover that A=7.
During the stage of formal operations, adolescents can acquire and use
abstract concepts. These are concepts which refer not to real objects or
situations, but to inventive or hypothetical ones. For example, many
psychological concepts are abstract. One example is the concept of
conservation. Conservation is an idea used to make sense of how children
perform and tests like the three beakers test. You cannot see or touch
conservation – you can only infer that it is occurring when a child gives the
right answer. Other abstract concepts include: repression, energy, bias,
validity and justice.
Adolescents also begin to think hypothetically. This means they can suggest
what might be the result if something were to happen. For example, being
able to say what might happen if all the polar ice melted.
The stage of formal operations also features greater flexibility in thinking.
For example, adolescents are able to try one way of solving a problem, and
then if it is unsuccessful, they discard it and think of another way. They
have a range of strategies instead of being limited to one approach.
Piaget devised several tests of formal operational thought. One of the
simplest was the ‘third eye problem’. Children were asked where they
would put an extra eye, if they were able to have one, and why. Schaffer
(1988) reported that when asked this question, 9 year olds all suggested that
the third eye should be on the forehead. However, 11 year olds were more
inventive, some suggesting that a third eye placed on the hand would be
useful for seeing round corners.
Another task Piaget used was to give participants some string and some
weights which could be attached the string, and to use these to find out what
factor or factors determine how fast the pendulum swings. Participants can
vary the length of the pendulum string, and vary the weight. They can
measure the pendulum speed by counting the number of swings per minute.
To find the correct answer, the participant has to grasp the idea of the
experimental method – that is to vary one variable at a time (e.g. trying
different lengths with the same weight). A participant who tries different
lengths with different weights is likely to end up with the wrong answer.
Stage
Characteristics
Ability to deal with
situations is limited to:
1. having sensations and
Substages 1-3
producing actions
(including reflex
responses)
Sensorimotor
2. the ‘here and now’
stage
Intentional actions emerge
Trial and error behaviour
Substages 4-6 Object concepts develops
Simple pretend play
Language acquisition
Symbolic thought develops
Preconceptual Egocentrism
period
Animism
Centration
Preoperational
Judgements based on
stage
appearance not logical
Intuitive
thought
period
Less egocentric
Unable to conserve
Can perform operations
with real situations
including:
Stage of concrete operations Conservation
Seriation
Transitivity
Class inclusion
Can perform operations on
unreal, abstract situations
Stage of formal operations
Abstract concepts acquired
Hypothetical thinking
Flexibility in thinking
Typical Age
0-8 months
8-24 months
2-4 years
4-7 years
7-11 years
12+ years