Unit 2.1
Developmental progression in the
learning of mathematical concepts
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Jean Piaget : Cognitive Theory
• Laid foundation for Cognitive Psychology
• Focus not only on the external behaviour of the
learner, rather importance is given to internal
behaviour of the learner
• Our thinking process change radically, though
slowly, from birth to maturity because we
constantly strive to make sense of the world.
• Believed that learning is a function of certain
process
• They
are,
organization,
adaptation,
equilibration, and operation
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1. Organization
• Intelligence was not random, but as set of
organised cognitive structures
• People are born with a tendency to organize
their thinking processes into psychological
structures
• These psychological structures are our systems
for understanding and interacting with the
world
• These structures are called schema – mental
systems or categories of perception and
experience
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Organization……………….
• A schema is a concept or framework that exists in
an i di idual’s mind to organise and interpret
information
• Schema can range from simple to complex
• Schemas are the basis building blocks of thinking
• Eg. Different voices from environment, sucking
through straw
• Organization is the ongoing process of arranging
information and experience into mental system
or categories (schemas)
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2. Adaptation
• the adjustment to a new environment is
adaptation
• People inherit the tendency to adapt to their
environment
• Two basic process are involved in adaptation:
a) Assimilation
b) Accommodation
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a) Assimilation
• Fitting the new information into existing
schemas.
• It is a process of incorporating new objects
and experiences into the existing schemas.
• It is applied to every new object and in every
new situation
• It is trying to understand something new by
fitting it into what we already know.
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b) Accommodation
• Altering existing schemas or creating new ones
in response to new information
• It occurs when a person must change existing
schemas to respond to a new situation.
• If data cannot be made fit any existing schemas,
then more appropriate structures must be
developed.
• It is the modification of the individuals internal
cognitive structures.
• Accommodation accompanies assimilation
• Here the child remains active, and explores,
questions, experiments, etc.
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Adaptation ……………………
• People adapt their environments by using
existing schemas whenever these schemas
work (assimilation) and by modifying and
adding to their schemas when something new
is needed (Accommodation)
• By assimilating new to the old and by
accommodating the old to the new, the person
learns.
• This process of adaptation continues thought
life.
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4. Operation
• A mental activity that transforms or
manipulates
• eg. Adding, multiplying etc.
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3. Equilibration
 The act of searching for a balance
 It is the search for mental balance between cognitive
schemas and information from the environment
 If we apply a particular schema to an event or
situation and the schemas works, then the
equilibrium exists.
 If the scheme does not produce a satisfying result,
then disequilibrium starts and we become
uncomfortable.
 This motives us to keep searching for solution
through assimilation and accommodation and thus
our thinking changes and moves ahead.
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Object permanence is the understanding
that objects continue to exist even when they
cannot be observed (seen, heard, touched, smelled
or sensed in any way).
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Sensorimotor stage
• The hild’s thi ki g i ol es seei g, o i g,
touching, tasting, etc.
• Develops object permanence
• Begins to make use of imitation, memory and
thought
• Begins to recognize that objects do not cease
to exist when they are hidden
• Moves from reflex actions to goal directed
activity.
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•
•
•
•
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Learning Mathematics at sensorimotor Stage
They have the ability to link numbers to objects
(one dog, two cats, three toys).
To develop mathematical capability of a child in this
stage, the hild’s ability might be enhanced.
At this stage they have some understanding of the
concept of numbers and counting.
We should provide solid mathematical foundation
by providing activities that incorporate counting
It enhances hildre ’s conceptual development of
number.
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Do not understand the concept of conservation, the
principle that quantity remains the same even we change
the shape
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Pre-operational Stage
• Children has not mastered mental operation
but moving towards mastery
• Ability to use symbols, words, gestures, signs,
images, etc.
• difficulty with reversible thinking (thinking
backward)
• Difficulty with conservation (the principle that
some characteristics of an object remain the
same despite changes in appearance)
• Children face difficulty in considering more
than one aspect at a time, i.e. decentring,
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• They lack basic logical operational skills.
• Pre-operational children are egocentric
They think the world is created for them
I a ility to see the orld through other’s
perspective
Should be gone by age 5
• They indulge in collective monologue
• Animism- belief that inanimate things are
alive.
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Learning Mathematics at preoperational stage
• Child links together unrelated events, see
objects as possessing life.
• Does not understand another point of view
and cannot reverse operations
• For eg. A child at this stage who understands
that adding 2 to 5 yields seven . But the
reverse operation of taking two from seven is
not possible.
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Learning Mathematics at preoperational stage
• Child should engage with problem-solving
tasks that incorporate available materials such
as blocks, sand, and water.
• While the child is working with a problem, the
teacher should elicit conversation from the
child.
• The verbalization of the child, as well as his
actions on the materials gives a basis that
permits the teacher to infer the mechanisms
of the hild’s mental thought process.
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Learning Mathematics at preoperational stage
• Childre ’s perceptions are generally restricted
to one aspect or dimension of an object.
• Eg. Concept of conservation. The child use
only one dimension, height as the basis for
the judgment of another dimension, volume.
• Teachers should employ effective questioning
about characterizing objects.
• For eg. A teacher could ask students to group
the shapes according to similar characteristics.
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•Concrete operational stage
• Children at primary level
• Understand the concept of conservation
• Can think logically, use analogies, and perform
mathematical transformation (5+9 is the same
as 9+5)
• Is also know as reversibility which promotes
logical thinking
• Abstract thinking is not possible
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Learning Mathematics at
Concrete Operational Stage
• Characterized by Seven types of conservation :
Number, Length, liquid, Mass, Weight, area, volume.
• Conservation of number is mastered by age 6
• Conservation of length and weight is mastered
by age 8 or 9.
• Perform operations like identity: if something
is added or taken away the material remains
same
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Learning Mathematics at
Concrete Operational Stage
• Develop transitivity
• Eg. if A is bigger than B, and B is bigger than C
then what is the relation between A & C?
• The major change of this period is that the
development proceeds from pre-logical
thought to logical solutions and concrete
problems.
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Learning Mathematics at
Concrete Operational Stage
• They can consider two or three dimensions
simultaneously (Decentration)
• For eg. In the liquid experiment, if the child notices the lowered level of the
liquid, he also notices the dish is wider , seeing both the dimensions at the
same time.
• Seriation and classification are the two logical
operations that develop during this stage – both are
essential for understanding the number concept.
• Seriation= it is the ability to order objects according to
increasing or decreasing length, weight, or volume.
• Classification = grouping objects on the basis of
common characteristics
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Learning Mathematics at
Concrete Operational Stage
• Hands-on experiences and multiple ways of
representing a mathematical solution can be
the ways of fostering development of this stage.
• The importance of hands-on experience cannot
be over emphasized at this stage.
• These activities helps the students to make
abstract ideas concrete.
• Hands-on mathematical ideas and concepts are
useful for solving problems.
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Learning Mathematics at
Concrete Operational Stage
• Concrete experiences are needed to explore
concepts such as place values and arithmetic
operations.
• Materials like: pattern blocks, cubes, tiles,
geoboards, tangrams, dice, etc….
• Teachers can use convenient materials in
activities such as paper folding and cutting
• Through the use of these materials students
acquire experience that help to lay foundation
for advanced mathematical thinking.
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Learning Mathematics at
Concrete Operational Stage
• Teachers should help the students to make
connections between the mathematical concepts
and the activity.
• Children may not automatically make connections
between the work they do with manipulative
materials and the corresponding abstract
mathematics.
• Children tend to think that the manipulations they
do with models are one method for finding a
solution and pencil- and paper math is entirely
separate.
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Education, Mysore
Learning Mathematics at
Concrete Operational Stage
• For eg.
• How a four by six inch rectangle built with
wooden tiles relates to 4 multiplied by 6 or
four groups of six?
• Teacher could help students make connections
by showing how the rectangles can be
separated into four rows of six tiles each and
by demonstrating how rectangle is another
representation of four groups of six.
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Learning Mathematics at
Concrete Operational Stage
• Creating opportunities for students to present
mathematical solutions by multiple ways (eg.
Symbol, graphs, tables) is one tool for
cognitive development in this stage.
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Formal Operational Stage
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Thought process becomes quite systematic
Mental manipulation- number of variables
Imagination develops
Develop experimental spirit
Grater abstraction and metacognition
Ability to judge truth of logical relationships
Reflective thinking
Debate – for and against a topic
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Learning Mathematics at
Formal Operational Stage
• The child is capable of forming hypotheses
and deducing possible consequences
• Can allow the child to construct his own
Mathematics
• Child begins to develop abstract thought
patterns; here reasoning is executed using
pure symbols without the necessity of
observant (perceptive) data
• For eg. The learner can solve x+3x = 12 without having to refer to a
concrete situation like “ Geetha has two chocolates and her friend has
thrice as many. Together they have 12. So how many chocolates Geetha
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has?’
Education, Mysore
Learning Mathematics at
Formal Operational Stage
• Reasoning skills in this stage include:
clarification,
inference,
evaluation
and
application
1. Clarification: It requires students to identify and
analyse elements of a problem, allowing them to
interpret the information needed in solving a
problem.
 Teachers can help students enhance their
mathematical understanding by encouraging
students to take out relevant information from
a problem statement
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Learning Mathematics at
Formal Operational Stage
2. Inference : students at this stage are
developmentally ready to make inductive and
deductive inferences in mathematics
3. Evaluation : Evaluation involves using criteria
to judge the adequacy of the problem
4. Application : Students connecting
mathematical concepts to real-life situation
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Learning Mathematics at
Formal Operational Stage
• The teacher can provide critical direction by
emphasizing methods of reasoning
• Thus the child can discover concepts through
investigation.
• While the teacher studies the hild’s work to
better understand his thinking , the child
should be encouraged to self- check,
approximate, reflect and reason
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Learning Mathematics at
Formal Operational Stage
• All students in a class are not necessarily
operating at the same level.
• Teacher should try to find out their students
cognitive levels to adjust their teaching
accordingly.
• In general, the knowledge of Piaget’s stages
helps the teacher understand the cognitive
development of the child as the teacher plans
stage appropriate activities to keep studnets
active.
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