UNIT 4: CHILDREN’S CONCEPTUALISATION OF
MATHEMATICS
UNIT STRUCTURE
4.1
Learning Objective
4.2
Introduction
4.3
Piaget’s Theory of Mathematics Learning
4.4
Vygotsky’s Theory of Learning
4.5
Sociological Background on Mathematical Knowledge
4.6
Role of Language in a Mathematical Classroom
4.7
Let us Sum up
4.8
Further Reading
4.9
Answers to Check Your Progress
4.10
Model Questions
4.1
LEARNING OBJECTIVE
After going through this unit, you will be able to –
describe the Piaget’s theory of mathematics learning
explain the Vygotsky’s theory of learning
discuss the effect of sociocultural background of children on their
mathematical knowledge and
identify the role of language of communication in a mathematical
classroom.
4.2
INTRODUCTION
Children’s conceptual knowledge develops by dealing with the
concept in different contexts and at different levels. If a child’s understanding
of a concept is thorough, he or she is in a better position to devise strategies
to solve problems based on it. For example, a child who understands what
multiplication means may use multiplication tables for solving a multiplication
problem, or s/he may add one of the numbers repeatedly to solve it, or s/he
may break up a number and apply distributing.
Jean Piaget of Switzerland and L.S. Vygotsky of Russia are two of
110
Teaching of Mathematics for Primary School Child
Unit 4
Children’s Conceptualisation of Mathematics
the pioneers who studied the way how children learn. They observed
children, built their hypothesis of how they learn and then tested their
hypothesis. In the process, they built up a whole theory of constructivism.
In this unit we are going to discuss the theories of learning by Piaget
and Vygotsky and the implication of these theories of learning in mathematics
education.
Along with that we will also discuss the effect of sociological
background of children on mathematical knowledge and the role of language
in a mathematical classroom.
4.3
PIAGET’S
LEARNING
THEORY
OF
MATHEMATICS
Jean Piaget (1896-1980), the famous child psychologist of
Switzerland, after undergoing lots of experiments, had suggested that the
Cognitive development in children takes place through four separate
stages in a fixed order. These stages are:
Sensory-motor Stage (from birth to 2 years)
Pre-operational Stage (about 2 to 7 years)
Concrete Operational Stage (about 7 to 11 years) and
Formal Operational Stage (about 11 to 14 years)
LET US KNOW
To understand the nature of learning in children, the
teacher must have a thorough knowledge of the
cognitive development in children so as to make
education in the elementary stage more effective.
Children learn continuously from their surrounding with the help of
their sense organs – that is by seeing, hearing, touching and through
other sensory experiences. The ability of learning in this way is termed
as cognition. The mental or intellectual development that takes place
in children, as a result of cognition, is termed as cognitive or mental
or intellectual development.
Teaching of Mathematics for Primary School Child
111
Unit 4
Children’s Conceptualisation of Mathematics
We know that the physical growth of children takes place in stages
following some fixed order. For example, before learning to run or
jump, a child learns to sit, stand and walk respectively. In a similar
way, the cognitive development of children also takes place through
separate stages in a fixed order. This order is universal in nature i.e.
it follows the same order in case of all human children. So, the mental
development of children keeps on taking place from one stage to
another in an order.
According to Piaget, children of a certain age are able to gain or
develop a predetermined amount of cognition only. Therefore, they should
not be forced to learn beyond their cognitive level.
Now let us discuss the stages of cognitive development put
forwarded by Jean Piaget and how each stage of development is related to
the learning of mathematics.
Jean Piaget (1896-1980)
Piaget was a Swiss
psychologist
and
philosopher famous for
his
epistemological
studies with children.
Sensory-motor stage (from birth to about two years):
During this time of pre-childhood stage, the infant behaves in a
various ways such as holding an object by hand and then bringing it to
mouth for sucking, listening and looking towards the direction of the sound.
In this way by using their sense organs and through various physical and
mental activities, the mental and muscular development of children keeps
on developing. They learn to pronounce words and gradually the language
development also begins to take place from this stage onwards.
The children at this stage need concrete experiences to understand
concepts and ideas. Also at this stage, children are egocentric and can only
see the world from their own perspective.
If we specifically address how the Sensorimotor Stage applies to
mathematics, we can see how learning is developing. First, at the age of
around eight months children begin to have an understanding of object
permanence in which they are able to find objects that have been taken out
of their view. Children also begin to link numbers to objects (e.g., one dog,
two cats, three pigs, four cows). The concept of one-to-one correspondence
also starts developing in the children at this stage. One-to-one
correspondence is the ability to match numbers to objects and objects to
112
Teaching of Mathematics for Primary School Child
Children’s Conceptualisation of Mathematics
Unit 4
objects. Children begin to match one object to one person or one toy to one
person.
Parents of children in this stage of development should lay a solid
mathematical foundation by providing activities that incorporate counting
and thus enhance children’s understanding about the concept of number.
For example, parents can help children count their fingers, toys, and
chocolates. Questions such as “Who has more?” or “Are there enough?”
could be a part of the daily lives of children as young as two or three years
of age.
Pre-operational Stage (about 2 to 7 years):
The characteristics of this stage include an increase in language
ability, symbolic thought, egocentric perspective, and limited logic. This
stage also marks the beginning of solving more mathematically based
problems like addition and subtraction.
Children at this stage are restricted to only one aspect or dimension
of an object. Therefore they are also limited in their rational and logical thinking,
and thus limited in their mathematical abilities. Also, since children are
restricted to one dimensional thinking, they are influenced by the visual
representation of things, i.e. if two things look different; the child is likely to
conclude they are different while really the visual representation could simply
be a different perspective. Let us consider the following examples:
By arranging some marbles in the following way, if we ask a
child at pre-operational stage, “Number of marbles in both the
rows is same or not?”
OOOOO
O O O O O
The child would answer that the there are more number of
marbles in the bottom row.
Similarly by showing two containers with same size but different
shapes if we ask the child, “Are both the container of same
size?”
Teaching of Mathematics for Primary School Child
113
Unit 4
Children’s Conceptualisation of Mathematics
The answer of the child would be that the first container is bigger
in size than the second one. This means the child is unable
concentrate both on the height and breadth at the same time.
Since a child in the preoperational stage is still limited to the concrete
world, therefore, the teachers must make use of concrete materials such
as blocks, paper cuttings, etc. while teaching the young students. The
concept of geometric shapes may also be introduced at this stage of
development by asking the students to group the shapes according to similar
characteristics. This may be followed by questions, such as, “How did you
decide where each object belonged?” “Are there other ways to group these
together?” Engaging in discussion or interactions with the children may
engender the children’s discovery of the variety of ways to group objects,
thus helping the children think about the quantities in novel ways.
Concrete Operational Stage (from 7 to 11 years):
The concrete operational stage lasts from the age of seven to twelve
years of age. This stage is characterized by remarkable cognitive growth,
when children’s development of language and acquisition of basic skills
accelerate dramatically. The children at this stage develop the ability to think
in more systematic and logical manner. They start classifying based on
several features and characteristics rather than solely focusing on the visual
representation. Let us consider the following example:
Children at concrete operational stage are likely to think twice before
deciding whether the quantity of water present in both these two containers
are of same size or not. Different shapes of the containers bring about a
change in the thinking process of the child. So, it can be seen that the child
can now think in a logical manner that the change in appearance of an
object does not alter either its quantity or its number.
114
Teaching of Mathematics for Primary School Child
Children’s Conceptualisation of Mathematics
Unit 4
Since children can now classify based on several features, they are
able to consider 2 or 3 dimensions. While children were previously limited
to their own point of view, now they can take into account others perspectives.
They can also begin to understand the ideas of seriation and classification
more thoroughly, both of which are essential for understanding number
concepts. Seriation is the ability to order objects according to increasing or
decreasing length, weight, or volume. On the other hand, classification
involves grouping objects on the basis of a common characteristic (for
example, from among various types of objects they can now separate those
objects which are made of plastic materials), can concentrate on different
aspects of the same object (for example, shape, colour, weight, length,
breadth etc.). The ability to present solutions in multiple ways also develops
at this stage. Inorder to develop the ability of presenting multiple solutions in
a child , discussions in a classroom can be very helpful. This experience of
sharing solutions could open the child upto the idea that there is not always
one right way to solve every problem; there could be multiple ways that are
equally correct solutions. The child in this developmental stage will also
have the skills to make basic observations and routine measurements which
increases the ways they classify objects/situations. The concept of one-toone correspondence also develops further at this stage.
Formal Operational Stage(about 11 to 14 years)
The last stage of development that Piaget identifies is the Formal
Operational Stage, which children enter roughly between the ages of eleven
to sixteen years old and continues throughout adulthood. At this stage, the
thought process of the child becomes quite systematic and reasonably well
integrated. Children do not need the concrete experiences which they
required to understand mathematics in the previous stages. They also begin
to understand abstract concepts which lead to much more complicated
mathematical thinking, for example, estimating the area under a curve, which
is not based on a concrete experience, comparing negative and positive
integers using a one-to-one correspondence, etc.
So we have seen that Jean Piaget’s theory gives us an idea on how
a concept develops and outlines how a child progresses through this
developmental process of learning.
Teaching of Mathematics for Primary School Child
115
Unit 4
Children’s Conceptualisation of Mathematics
CHECK YOUR PROGRESS
Q 1: Fill in the gaps:
i. According to Piaget, the cognitive
development of children takes place through .....................
separate stages in a ....................... order.
ii. The children should not be forced to learn beyond their
....................... .
Q 2: State whether the following statements are true or false.
i. Ability of counting numbers develops in the children at
the Sensory-motor stage of learning.
ii. Children begin to have an understanding of object permanence
at the age of around two months.
iii. At the pre-operational stage, children can recognize only one
aspect or one dimension of an object.
iv. Children begin to understand abstract concepts at the concrete
operational stage.
4.4
VYGOTSKY’S THEORY OF LEARNING
The Russian psychologist, Vygotsky was the first child psychologist
who had stressed on the importance of the social setting on the learning
process of children. For example, when the children come to school, most
of them know some number, names; they can compare sizes and shapes
of some objects. But because of their social backgrounds,some may even
recognize numerals by the time they come to school. Others coming from
different backgrounds, like a vegetable seller, may not recognize numerals
Vygotsky, Lev
but would have experience of dealing with complicated operations on
Semyonovich (1896-
numbers.
1934). Russian
Psychologist
Vygotsky also pointed out the idea that the potential for cognitive
development is limited to a certain time span, which he named the “Zone
of Proximal Development” (ZPD). ZPD is the difference between what a
learner can do without help and what he or she can do with help from the
elders. In simple language, ZPD refers to the student’s capacity for learning.
116
Teaching of Mathematics for Primary School Child
Children’s Conceptualisation of Mathematics
Unit 4
Full development during ZPD depends upon full social interaction.
The range of skill attained by a child increases more with adult guidance or
peer collaboration than what can be attained alone. It is seen that by the
help of adults, children can do and understand more than they can on their
own. Since adults can play important roles in the learning processes of a
child, therefore, the social setting or the family background in which a child
is born plays an important role in the cognitive level of the child.
The teachers should make effort to know their students’ social
background and their prior knowledge and experiences. This will help you to
help your students to achieve their potential as learners by providing learning
and consolidation tasks that are within your students’ ZPD. Students in any
classroom differ in many ways. When teachers develop instructional plans
that acknowledge their differences, students can learn in ways that are
suitable for and meaningful to them.
Moreover, the teacher should make the environment of the classroom
comfortable for the children, so that they can feel secure and wanted. This
will help to build their self confidence. They would work and act in the same
way as they are used to at home or in the playground with a sense of freedom.
This would help to nurture their intellectual development and their learning
will become more interesting, instead of being monotonous. In such a
classroom situation, the teacher allows the children to do what they want to
do; make them practice or even test them which at the same time goes on
guiding and supporting them. The school should not be a place where
children feel that they are being constantly tested or put under scrutiny
Teaching of Mathematics for Primary School Child
117
Unit 4
Children’s Conceptualisation of Mathematics
because of which they may develop phobia or make mistakes. Thus the
teacher can help to change the school into a place where children can
explore and learn with pleasure and confidence.
CHECK YOUR PROGRESS
Q 3: Fill in the gaps:
i. Vygotsky had stressed on the importance of
the ........................ on the learning process of children.
ii. According to Vygotsky, the potential for cognitive development is
limited to a certain time span, which he named as the ....................
iii. The range of skill attained by a child increases more with
........................... than what can be attained alone.
4.5
SOCIOLOGICAL
BACKGROUND
MATHEMATICAL KNOWLEDGE
ON
Mathematical knowledge can be broadly divided as conceptual and
procedural knowledge. A teacher needs to develop and interrelate both of
them them in the classroom. Each child is part of a social environment and
has interactions with the physical world around her/him. This shapes the
mind of the child. As any class room contains children from various
backgrounds and with a variety of abilities, therefore, the kind of interaction
in the classroom varies from child to child and is unpredictable.
The socio-cultural background has a lot of effects on a person. Let
us take an example- suppose s/he has grown up in Australia instead of in
her/his own home town some where here in the north eastern region; then
definitely the person would have different teachers, friends, relatives etc. S/
He will become more interested in the area of subject matter according to
the socio-cultural setup of Australia rather than that of her/his native place.
Social setting, therefore, has a great influence on the learning process of a
person.
It is found that the children before starting formal schooling possess
some pre-conceptual knowledge related to mathematics according to their
social setup. For example, the children belonging to a tea garden area of
118
Teaching of Mathematics for Primary School Child
Children’s Conceptualisation of Mathematics
Unit 4
the north eastern region know well as to what quantity of dried tea leaves
can be obtained from the related quantity of fresh tea leaves. Practically
that knowledge helps them a lot to understand and apply the same in
problems related to weights and measures under metric system. Similarly,
some children coming from some other and different backgrounds may
have some basic concepts of number, shapes, size etc. in their own way
learning there from their background even before coming to school.
Sociocultural background, therefore, plays a significant role in a
child’s learning of mathematics.
4.6
ROLE OF LANGUAGE IN A MATHEMATICAL
CLASSROOM
Language is needed as a pre-requisite for conveying mathematical
notions to children in the mathematics classroom. Language itself is
something that the children try to master before they begin to learn other
subjects. But in learning mathematics, children have to understand language
as well as mathematics, which is by no means easy for a child. When a
child is not able to understand a particular mathematical concept, it may
just be due to some confusion created by the language used for explaining
the mathematical concept in the classroom by the teacher.
Thus, the language of mathematics and mathematics as a language,
both play an important role on the children’s learning process. Some
examples are cited below, which may arouse confusion in child’s
understanding of mathematical concepts because of language interference–
A teacher of class –II, while explaining concept of place value to her
children by citing the example of 11, may say, “one and one make eleven”.
This can create confusion in the minds of the children because they have
already learnt that one and one actually makes two.
From this example it can be seen that the correct use of language is
vital for making a child understand a concept clearly. Wrong use of language
is apt to make the child confused. Sometimes children coming from certain
backgrounds may not be familiar with some words that are used in the text
books and by the teachers. For example, not knowing the meaning of words
Teaching of Mathematics for Primary School Child
119
Unit 4
Children’s Conceptualisation of Mathematics
such as ‘same’, ‘different’, ‘few’, ‘as many as’, ‘equal to’, ‘each’ etc. can
obstruct their understanding of mathematics.
Another source of confusion is when different words express the
same mathematical concept. For example, ‘equals’, ‘makes’ and ‘is the same
as’ are all represented by the sign ‘=’.
Even for older children language can be a barrier to their
understanding of mathematics. What language is used to convey certain
mathematical ideas may decide how the children will receive and understand
those ideas.
At another level, children can be confused by the grammatical
complexity and length of a sentence. For example, let us take this sentence:
“What number between 25 and 30 cannot be divided exactly by 2 or 3?”
This sentence is definitely a complex one. To make it easier to understand
for the child it can be rewritten as “Look for a number between 25 and 30.
You cannot divide this number exactly by 2 or 3. What is the number?”
Thus, from the above examples it becomes clear that the role of
language in a mathematical classroom is an important factor which
influences a child’s mathematics learning.
Hence, children at all levels must be given many opportunities to
talk about mathematical ideas. They should be given a suitable environment
to learn how to use mathematical words or phrases orally, before they are
expected to represent mathematics symbolically.
CHECK YOUR PROGRESS
Q 4: Does social background affect the concept
of mathematical knowledge of the children?
...............................................................................
...................................................................................................
Q 5: Is language necessary in a mathematical classroom?
...................................................................................................
...................................................................................................
120
Teaching of Mathematics for Primary School Child
Children’s Conceptualisation of Mathematics
4.7
Unit 4
LET US SUM UP
We have discussed the Piaget’s and Vygotsky’s theories of learning.
According to Piaget, the Cognitive development in children takes place
through four separate stages in a fixed order. He also said that children
in a particular age contain a predetermined amount of cognition only
and therefore, they should not be forced to learn beyond their cognitive
level
Vygotsky was the first psychologist to stress the importance of social
setting for child’s learning. According to him sociocultural background
affects the children’s mathematical knowledge. Each child is a part of
a social environment and as such her/his mind is shaped by it.
The role of language in a mathematical classroom is also vital for
children to acquire conceptual understanding of mathematics. Wrong
use of language may confuse the mind of the child affecting her/his
understanding of the concepts.
4.8
1)
FURTHER READING
Zevenbergen, R., Dle, S., Wright, J,R., (2005). Teaching of Mathematics in Primary Schools. Australia: Allen & Unwin.
4.9
ANSWERS TO CHECK YOUR
PROGRESS
Ans to Q No 1: (i) Four, Fixed
(ii) Cognitive level
Teaching of Mathematics for Primary School Child
121
Unit 4
Children’s Conceptualisation of Mathematics
to Q No 2: (i) True
(ii) False
(iii) True
(iv) False
Ans to Q No 3: (i) Social Setting
(ii) Zone of Proximal Development (ZPD)
(iii) Adult guidance or peer collaboration
Ans to Q No 4: Yes
Ans to Q No 5: Yes
4.10 MODEL QUESTIONS
A. Very Short Questions
Q 1: Mention the four stages of cognitive development suggested by Jean
Piaget.
Q 2: At what stage of development children can solve simple addition and
subtraction problems in mathematics?
Q 3: Write some of the characteristics of Concrete operational stage of
development.
Q 4: Write down the full form of ZPD.
Q 5: Children coming from different socio-cultural background also differ
in their learning capacities.True/False?
Q 6: Language used by a teacher in teaching mathematics plays an
important role in children’s learning of the subject. True/False?
B. Short Questions (Answer in about 150 words)
Q 1: Compare Piaget’s Pre-operational Stage and Concrete operational
stages of developments.
Q 2: Explain the idea of ZPD as suggested by Vygotsky.
Q 3: How does socio-cultural background affect children’s learning
processes?
Q 4: How do children get confused by the grammatical complexity and length
of a sentence? Explain with an example.
122
Teaching of Mathematics for Primary School Child
Children’s Conceptualisation of Mathematics
Unit 4
C. Long Questions (Answer in about 300-500 words)
Q 1: Discuss the Piaget’s Theory of Mathematics Learning.
Q 2: Explain the Vygotsky’s Theory of Leaning.
Q 3: The language of mathematics and mathematics as a language, both
play an important role on the children’s learning process. What do you
mean by this statement? Explain with examples.
*** ***** ***
Teaching of Mathematics for Primary School Child
123
REFERENCES
Unit 1
National Council of Teachers of Mathematics, Principles and Standards
for School Mathematics, , Reston, VA: Author.
Polya, G., How to solve it. (1957) Garden City, NY: Doubleday and Co.,
Inc.
Simon, M. and Blume, G., (1996), “Justification in the Mathematics
Classroom: A Study of Prospective Elementary Teachers”, Journal of
Mathematical Behavior, vol. 15, 3 - 31.
Wilcox, S., Lanier, P., Schram, P., and Lappan, G., (1992) Influencing
Beginning Teachers’ Practice in Mathematics Education: Confronting
Constraints of Knowledge, Beliefs, and Context, Research Report
No. 1992-1, East Lansing, MI: National Center for Research on Teacher
Education.
Unit 4
Loop, E.(2011). How to Apply Piaget’s Theory to Teaching Mathematics.
Retrieved
on
6-6-2012
from
http://www.ehow.com/
how_7741298_apply_piaget_theory_classroom.html.
McLeod, S.A. (2010). Zone of Proximal Development. Retrieved on 96-2012 from http://www.simplypsychology.org/Zone-of-ProximalDevelopment.html.
Reedal, K.E. (2010). Jean Piaget’s Cognitive Development Theory in
Mathematics Educatiom. Retrieved on 9-6-2012 from http://ripon.edu/
macs/summation.
Zevenbergen, R., Dle, S., Wright, J,R., (2005). Teaching of
Mathematics in Primary Schools. Australia: Allen & Unwin.
124
Teaching of Mathematics for Primary School Child