u n it 2 .1 d e velop menta l p rog res sion in th e le ar ... · pdf fileu n it 2 .1 d e...

Unit 2.1

Developmental progression in the learning of mathematical concepts

5/2/2016 1 Dr. Priya Mathew St. Joseph's College of Education, Mysore

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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Dr. Priya Mathew St. Joseph's College of

Education, Mysore

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

5/2/2016 3 Dr. Priya Mathew St. Joseph's College of

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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.

5/2/2016 14 Dr. Priya Mathew St. Joseph's College of Education, Mysore


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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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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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• 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.

Learning Mathematics at preoperational stage


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• 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. 5/2/2016 30


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• 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. 5/2/2016 31


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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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• 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

Learning Mathematics at

Concrete Operational Stage


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• 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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• 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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• 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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• 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. 5/2/2016 38


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• 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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• 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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• 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

• 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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• 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

has?’ 5/2/2016 45 Dr. Priya Mathew St. Joseph's College of

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• 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 5/2/2016 46 Dr. Priya Mathew St. Joseph's College of

Education, Mysore



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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• 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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• 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. 5/2/2016 49 Dr. Priya Mathew St. Joseph's College of

Education, Mysore


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