BES-143 Pedagogy of Mathematics

TEACHING-LEARNING OF MATHEMATICS

UNIT 5

Approaches and Strategies for Learning Mathematics 5

UNIT 6

Organizing Teaching-Learning Experiences 27

UNIT 7

Learning Resources and ICT for Mathematics 52

Teaching-Learning

UNIT 8

Assessment in Mathematics 75

UNIT 9

Professional Development of Mathematics Teacher 101

Block

2

Indira Gandhi

National Open University

School of Education

BES-143

Pedagogy of

Mathematics

EXPERT COMMITTEE

Prof. I. K. Bansal (Chairperson)

Former Head, Department of Elementary

Education, NCERT, New Delhi

Prof. Shridhar Vashistha

Former Vice-Chancellor

Lal Bahadur Shastri Sanskrit

Vidhyapeeth, New Delhi

Prof. Parvin Sinclair

Former Director, NCERT

School of Sciences

IGNOU, New Delhi

Prof. Aejaz Mashih

Faculty of Education

Jamia Millia Islamia, New Delhi

Prof. Pratyush Kumar Mandal

DESSH, NCERT, New Delhi

Prof. Anju Sehgal Gupta

School of Humanities

IGNOU, New Delhi

Prof. N. K. Dash (Director)

School of Education

IGNOU, New Delhi

Prof. M. C. Sharma

(Programme Coordinator- B.Ed.)

School of Education

IGNOU, New Delhi

Dr. Gaurav Singh

(Programme Co-coordinator-B.Ed.)

School of Education

IGNOU, New Delhi

SPECIAL INVITEES (FACULTY OF SOE)

Prof. D. Venkateswarlu

Prof. Amitav Mishra

Ms. Poonam Bhushan

Dr. Eisha Kannadi

Dr. M. V. Lakshmi Reddy

Dr. Bharti Dogra

Dr. Vandana Singh

Dr. Elizabeth Kuruvilla

Dr. Niradhar Dey

Course Coordinators : Prof. M.C. Sharma, SOE, IGNOU

Dr. Anjuli Suhane, SOE, IGNOU

Course Contribution

Unit 5 and 8

Dr. Adhya Shakti Rai

Associate Professor, Dr. Shakuntala Mishra

National Rehabilitation University, Lucknow

Dr. Anjuli Suhane

Assistant Professor, SOE, IGNOU, New Delhi

Unit 6

Dr. Anjuli Suhane


Unit 7

Sh. Ajith Kumar


Unit 9

Dr. Sarah Basu

COURSE TEAM

Language Editing

Prof. Amitav Mishra

Assistant Professor

SOE, IGNOU, New Delhi

Format Editing

Dr. Anjuli Suhane

Assistant Professor


Proof Reading

Dr. Anjuli Suhane


April, 2017

Indira Gandhi National Open University, 2017

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NCERT, New Delhi

Course : BES-143 Pedagogy of Mathematics

BLOCK 1: UNDERSTANDING THE DISCIPLINE OF MATHEMATICS

Unit 1 Nature and Scope of Mathematics

Unit 2 Aims and Objectives of Teaching -Learning Mathematics

Unit 3 How Children Learn Mathematics

Unit 4 Mathematics in School Curriculum

BLOCK 2: TEACHING -LEARNING OF MATHEMATICS

Unit 5 Approaches and Strategies for Learning Mathematics

Unit 6 Organizing Teaching-Learning Experiences

Unit 7 Learning Resources and ICT for Mathematics Teaching-

Learning

Unit 8 Assessment in Mathematics

Unit 9 Professional Development of Mathematics Teacher

BLOCK 3: CONTENT BASED METHODOLOGY-I

Unit 10 Number System, Number Theory, Exponents and Logarithms

Unit 11 Polynomials: Basic Concepts and Factoring

Unit 12 Linear Equations, Inequations and Quadratic Equations

Unit 13 Sets, Relations, Functions and Graphs

BLOCK 4 : CONTENT BASED METHODOLOGY-II

Unit 14 Statistics and Probability

Unit 15 Parallel Lines, Parallelograms and Triangles

Unit 16 Trigonometry and its Application

Unit 17 Mensuration and Coordinate Geometry

4

BLOCK 2???

Block Introduction

The course

5

Approaches and

Strategies for Learning

Mathematics

UNIT 5 APPROACHES AND STRATEGIES

FOR LEARNING MATHEMATICS

Structure

5.1 Introduction

5.2 Objectives

5.3 Pedagogical Shift: From Behaviorist to Constructivist

5.4 Constructivist Approach for Teaching-Leaning Mathematics

5.5 Strategies for Teaching-Learning Mathematics

5.5.1 Inductive- Deductive

5.5.2 Analytic -Synthetic

5.5.3 Problem Solving

5.6 Techniques for Transacting Mathematics Curriculum

5.6.1 Drill and Practice

5.6.2 Play Way

5.6.3 Home Work

5.6.4 Assignments

5.7 Let Us Sum Up

5.8 Unit End Exercises

5.9 Answers to Check Your Progress

5.10 References and Suggested Readings

5.1 INTRODUCTION

Mathematics is one of the few subjects that have the practical, cultural and

disciplinary value. Mathematics has the potential to range across all the three

values, but due to inappropriate teaching-learning process, its potential is not

being utilized to its optimum level. Unfortunately, the current focus of

Mathematics education is mostly on memorizing formulae to solve specific

problems and applying them to examination questions. To make Mathematics

an instrument of all the three values, it must be taught in an engaging,

interesting and interacting manner. According to National Curriculum

Framework-2005, the main goal of Mathematics education in school is the

mathematisation of the child’s thought process. Mathematics relies on logic,

reasoning, problem solving, creativity and mathematical way of thinking.

These skills can be useful in many other subjects.

In this unit, we will discuss various approaches, strategies and techniques of

teaching-learning of Mathematics. We will also discuss the shift with paradigm

of learning Mathematics from traditional behaviorist approach of rote learning

and drill to constructive approach, where learner constructs his/her own

knowledge.


Teaching -learning of

Mathematics

6

5.2 OBJECTIVES

After going through the Unit, you will be able to:

• analyze the pedagogical shift in teaching Mathematics from behaviorist to

constructivist;

• illustrate the constructivist approach for teaching-learning Mathematics;

• explain about various strategies for teaching-learning Mathematics;

• differentiate among various strategies for teaching-learning Mathematics;

• identify and apply different strategies of teaching;

• help learners to apply problem-solving skills in solving mathematical

problems; and

• apply various techniques for transacting Mathematics.

5.3 PEDAGOGICAL SHIFT: FROM

BEHAVIORIST TO CONSTRUCTIVIST

Mathematics has always been praised for its usefulness and significance in life.

It plays a key role in deciding how individuals deal with various problems of

life. But, at the same time learner find it hard to understand how functions,

equations or geometric shapes, which can help them in everyday life. They

consider it as a difficult subject due to its nature of being abstract which may

be one of the reasons why Mathematics is not popular amongst many learners.

Even today, as in the past, many learners still struggle with Mathematics. In

recent years, there has been a debate over the different approaches by which

Mathematics is being taught in the schools. In the past, Mathematics was

taught by the traditional methods with direct instruction and rote-memorization

of facts and procedures. The recent Mathematics initiatives shift, teaching and

learning away from a traditional on learning rules for manipulating symbols to

active engagement of learners in learning Mathematics. In other words,

teaching-learning Mathematics has started shifting from behaviorist approach

to constructive approach. Constructivism's success may be due, in part to the

frustrations that educators experienced with behaviorist educational practices.

Beginning in the 1960s, behaviorism swept from the arena of psychology into

education with an air of authority that was startling. Schooling became

structured around the premise that if teachers provided the correct stimuli,

then learners would not only learn, but also their learning could be measured

through observations of learner behaviors. (Jones & Brader-Araje, 2002).

Skinner and Watson were the two major proponents of behaviorism. According

to behaviorists, all behavior is the result of an individual’s responses to external

stimuli i.e. the external environment contributes to learning. It emphasizes on

the effects of external conditions such as rewards and punishments in

determining the learning of learners. It focuses mainly on objectively

observable behaviors and consequently, does not count mental activities. In

contrast to the beliefs of behaviorists, the constructivists viewed learning as a

search for meaning.

Piaget and Vygotsky were strong proponents of constructivism who opined that

knowledge is constructed by the learner. The Skinner, a behaviorist,

constructed a teaching machine in 1958. Skinner’s teaching machine was a

rote-and-drill machine where a chunk of information was presented before the

7

Approaches and


Mathematics

individual in the form of programmed instruction. Constructivist model

believes that learning occurs as an internal cognitive activity wherein learners

construct knowledge (models) from their classroom experience. Constructivists

believe that children develop their knowledge through active participation in

their learning. However, Behaviorists believed that meaning exists in the world

separate from personal experience. In behaviorist model all instructional goals

are framed in specific, behavioral, and observable terms. Like the traditional

approach, the instructor is in the centre of the presentation and interaction and

the role of the learner is to absorb the presentation and material whereas

constructivist suggests that learning activities must have the characteristics of

active engagement, inquiry, problem solving, and collaboration with others’

real life. They consider that the teacher is a guide and facilitator, who

encourages learners to take part in discussions and formulate their own ideas,

opinions, and reach the conclusions.

Behaviorist supports deductive approach and Constructivist supports inductive

approach of teaching. Therefore, Constructivists focus on a different aspect of

education than Behaviorists. Constructivist sees how learners learn on their

own when learners are presented with stimuli and Behaviorists focus more on

how learners respond to positive and negative reinforcement provided by

teacher.

Check Your Progress

Note: a) Space is given below to write your answer.

b) Compare your answer with the one given at the end of this Unit.

1. Who were the major proponents of behaviorism?

………………………………………………………………………………

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2. What is the overall philosophy of constructivism?

………………………………………………………………………………

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3. Write down the any three differences between constructivist and

behaviorist approaches.

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Mathematics

8

4. Fill in the blanks.

I. For …………………………….. learners are not blank-slates.

II. ……………………….………………supports inductive approach and

………………supports deductive approach of teaching.

5.4 CONSTRUCTIVIST APPROACH FOR

TEACHING LEARNING MATHEMATICS

As we discussed that constructivism assumes that knowledge cannot be

transmitted to learner but is constructed by him/her. It is constructed by the

learner on the basis of experiences. In the process of knowledge generation

new experiences, talking to others and reflective thinking could be helpful.

Many times learning includes change in existing conception. There are three

ways for meaningful learning to take place which are as follows:

1) Addition to the existing knowledge,

2) Small modification to the existing knowledge,

3) Major changes in the existing knowledge.

Conceptual change does not take place easily. For conceptual change three

conditions are necessary, which are as follow:

1) Learner must encounter a situation which he/she is not able to understand

using existing knowledge, thereby producing dissatisfaction in the

learner.

2) Learner must come across some knowledge, which is intelligible to

him/her and seems plausible.

3) The new knowledge help learner to understand some new situations

which were beyond his/her reach earlier.

There are many shades of constructivism. It is not unique monolith philosophy.

As we discussed, Piaget the first constructivist laid emphasis on action by the

learner on the object which results in accommodation and assimilation.

Vygotsky another important constructivist is proponent of socio-cultural

perspective. For him zone of proximal development, scaffolding and peer

learning are three important considerations. Zone of Proximal Development

(ZPD) is that stage of development of a child where he cannot solve a problem

of his/her own but a sight hint and help by teacher or some other able person,

is sufficient to enable him to solve the problem. Scaffolding is the support

provided by an expert to a novice in initial stage of learning. Slowly as the

progress is made by the learner, support is withdrawn gradually peer learning

could take place in three ways peer tutoring, cooperative learning and peer

collaboration.

As constructive teacher, you should have a clear idea about learners’ previous

knowledge. You can use appropriate strategies to assess the previous

knowledge since this will be very important for designing suitable activities for

working in their ZPD.

Whenever learners find difficulties within the zone, it is the duty of the teacher

to provide assistance or support in the process. This can either be done by

the teacher or with the help of a more competent peer. This process of

9

Approaches and


Mathematics

assisting is technically known as scaffolding. Thus a constructivist teacher is

required to create opportunities for peer scaffolding and teacher-directed

scaffolding in order to stimulate knowledge construction.

The important task of a teacher is to design appropriate activities so that the

learners can work on it and construct expected knowledge with confidence

and a feeling of success. This of course needs ingenuity and creativity.

Creation of a learning environment which is stimulating, interactive, and

enlightening for the learners is the most challenging task of a constructivist

teacher. Is it possible to create a stimulating environment? Of course you can,

then how? Consider the following example where a teacher wants to help the

learners to comprehend the identity (a+b)² = a²+2ab+b² by using constructivist

approach.

As an introductory activity teacher can give the following task with the aim to

assess and strengthen the prerequisites needed for the proposed learning.

Task: The learners are asked to draw squares of different sizes on given graph

paper individually.

42

• Subsequently the teacher initiates the whole class discussion by providing

points for discussion:

• “Count the number of smaller squares inside the bigger square you

constructed”

• “Can you establish any relationship between the number so obtained and

the size of the side?”

• “Can you relate this with any other concept you learnt earlier?”

32

22

72



Mathematics

10

Learners may recognize that the area of the whole square is the sum of the

areas of all the four parts. They may find that areas of the square parts are 3²

and 4² respectively, and those of the rectangles are 3×4 and 4×3 respectively.

Then learner may conclude that (3+4)2 = 3

2 +4

2 + 2×3×4 = smaller square +

bigger square + one rectangle + other rectangle

So , (a+b)² = a²+ 2ab+b²

In this way teacher can stimulate the learners to participate actively in the

teaching learning process. The feelings of success as well as enjoyment will

naturally motivate the learners.

The task should provide insight to the learners to reach the relationship

between the side and area of a square.

Similarly by providing ample opportunities for learners to engage in dialogue,

both with the teacher and with one another; teacher can create an interactive

classroom. The experience of success during different stages will automatically

enlighten the learners both scholastically and co scholastically.

While organising different activities give opportunities to the learners to

initiate discussions, to ask questions, to work independently etc.

From the conceptual meaning of constructivism as learning theory you could

observe that knowledge construction is not the product of successive pouring of

information through teacher talk, but a natural consequence of personal

experience, inquiry, reflection and insight. Therefore, thoughtful and open-

ended questions revealing learners’ prior knowledge and experiences are

asked by the teachers in constructivist classrooms.

Activity For Practice:

1. You studied the concept of constructivism and its importance in the

classroom teaching learning process. Make a list of principles to be

considered by you while structuring classroom learning activities.

32

42

32 3 × 4

42

11

Approaches and


Mathematics

2. Select any topic from your choice from class IX Mathematics textbook.

And design activities to transact that topic through constructivist

approach.

3. You are teaching the concept of formation of simple linear equation.

Design one interesting example which connects to life situation of the

learner to teach formation of simple linear equation.

Check Your Progress



5. Who is the originator of Constructivist philosophy?

………………………………………………………………………………

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………………………………………………………………………………

………………………………………………………………………………

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6. What do you understand by constructivist pedagogy? Explain its benefits in

teaching learning in Mathematics?

………………………………………………………………………………

………………………………………………………………………………

………………………………………………………………………………

………………………………………………………………………………

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5.5 STRATEGIES FOR TEACHING-LEARNING

MATHEMATICS

We have discussed constructivist approach of teaching learning of

Mathematics. There are some strategies of teaching Mathematics like

inductive – deductive, analysis-synthesis , problem solving, discovery,

activity etc. which help the learner in constructing their knowledge. The

purpose of these strategies is to make teaching- learning more interactive as

well as effective. You can select a particular strategy based on the needs of

learners as well as its relevance to the content. Some strategies of teaching-

learning Mathematics are as follows:



Mathematics

12

5.5.1 Inductive – Deductive

It is a combination of inductive and deductive approach. Let us first discuss

the Inductive approach.

a) Inductive Approach

Inductive approach is based on the process of induction i.e. reasoning from

specific facts to general principles. Therefore, it proceeds from particular to

general, from concrete to abstract. It is a method of constructing a formula with

the help of a sufficient number of concrete and specific examples. Learners

arrive at the formula or general rule through the examples of particular cases. It

is based on actual observation and experiments. Inductive approach is a much

more learner-centered approach. The learners are encouraged to devise the

formula on their own. This approach is psychological in nature. It develops

scientific attitude, comprehension ability and logical thinking among learners.

The teacher’s role is only to facilitate the use of appropriate questions.

Inductive approach is suitable in the following situation:

• Introduction of new topic

• Formulation of rules

• Derivation of formulas

• Generalization

Mathematically speaking, inductive reasoning might take this form:

Step1 - Shows that something is true for specific items. (Particular concept).

Step2 - Shows that if it is true for one and more, then it must be true for the

rest. (General concept).

Let us discuss some examples to make clear idea about it.

Example 1: Sum of two odd numbers is even.

Solution:

Teacher: Draw the following table in your notebook. Write two odd numbers

and then sum it.

Now observe the table and try to find out mathematical relationship between

the numbers.

Particular concept:

First odd number Second odd number Sum of these numbers

1 1 2

1 3 4

5 7 12

15 3 18

.. .. ..

.. .. ..

13

Approaches and


Mathematics

General concept: Learners may conclude that the sum of two odd numbers is

even.

Example 2: Sum of the angles of the triangles is equal to 180°.

Solution:

Teacher: Draw a few triangles.

Measure and sum up the angles in triangle A.

Measure and sum up the angles in triangle B.

Measure and sum up the angles in triangle C.

and so on

Draw the following table in your notebook and fill up the values of various

angles of triangles.

Now observe the table and try to find out relationship between angles of triangle.

Particular concept:

Triangle Measure of

angle 1

Measure of

angle 2

Measure of

angle 3

Sum of

angles of a

triangle

A 50 50 80 180

B 45 50 85 180

C 30 60 90 180

D .. .. .. …

E … … … …

F … … … …

General concept: Learners conclude that the sum of the angles of the triangles

is equal to 180° or two right angles.

b) Deductive Approach

Deductive approach is based on deduction. It is just the opposite of inductive

approach. It proceeds from abstract to concrete, from general rule to particular

or specific instances, and from formula to examples, from unknown to known.

This approach is relatively more teacher-centered. In this approach, rules are

initially given by teachers and then learners are asked to apply these rules to

solve more problems of similar nature. This approach is mainly used in

Algebra, Geometry and Trigonometry because different relations, laws and

formulae are used in these sub branches of Mathematics. It is more useful for

teaching Mathematics in higher classes. This method is useful for revision and

drill work. It enhances speed and efficiency.

Example 1: Learners are told that ‘the sum of angles in a triangle is 180°’.

Let ABC be a triangle.

Construct a line DE

Parallel to BC through

Here ÐABC = ÐBAD

ÐACB = ÐCAE

D EA

C



Mathematics

14

But ÐDAB + ÐBAC + ÐCAE = 180º

So ÐABC + ÐBAC + ÐACB = 180º

Example 2 : Find a5 × a

8 = ?

Solution:

General concept: First teacher told the formula that “am × a

n = a

m+n”

Then learners solve the problem.

Particular concept : a5 × a

8 = a

5+8 = a

13

Differences between Inductive and deductive approaches

SN. Inductive Approach Deductive Approach

1 Process of learning from specific facts

to general principles.

Process of learning and

reasoning from general principles to specific facts.

2 Certain complex and complicated

formulae cannot be generated so this

method is limited in range and is not suitable for all topics.

It is suitable for all topics.

3 It is time consuming and laborious

method

It is a short and time saving

method.

4 It is a scientific method. It does not impart any training

in scientific method

5 It does not burden the mind. Formula

becomes easy to remember.

It puts more emphasis on

memory.

6 It is a learner centric approach. Learners are only passive

listeners. It is more teacher-centric.

7 It gives new knowledge. It does not give any new

knowledge.

8 It is a method of discovery. It is a method of verification.

9 It is an upward process of thought and

leads to principles.

It is a downward process of

thought and leads to useful results.

Inductive – Deductive approach is a combination of both Inductive and

Deductive approaches. This method can be used in totality for realizing the

desired goals of mathematical learning. Through, Inductive approach, rules and

generalization are established and formulae are derived, while Deductive

approach is helpful in applying the deduced results. It also helps in improving

skills and efficiency in problem solving. No induction is possible without

deduction and no deduction results without induction.

5.5.2 Analytic - Synthetic

We have seen that in its early stages, most mathematics originates in ideas and

concepts in logico-deductive form. The ability to understand and workout a

rigorous deductive structure using logic or reasoning is of great importance.

15

Approaches and


Mathematics

Analyses and synthesis are approaches which use reasoning and arguments to

discover relationships. Let us discuss both the approaches separately.

a) Analytic Approach

Analysis is the process of breaking a complex topic or substance into smaller

parts in order to gain a better understanding of it. In this method, a problem is

analyzed into smaller/simpler problems. All the related facts are analyzed to

seek help in proceeding to the known conclusion. The purpose of breaking it

into smaller parts is to figure out the hidden aspects of the problem. So,

basically this approach moves from unknown to known. This method helps

learners in discovering the things himself. It is a psychological method based

on the principle of interest, which inculcates the spirit of inquiry and

investigation in the learners (Katozai, 2002 as quoted by Asif, Khan and

Zaman (2010 ). It facilitates comprehension and strengthens the urge to

discover new facts. It also provides opportunities to learners to tackle the

problem confidently and intelligently. But it is not applicable equally well for

all topics.

Example 1: If a2

+ b2

= 14ab prove that 2log (a + b) = 2 log 4 + log a + log b

Proof:

To prove this using analytic method, begin with the unknown.

The unknown is 2 log (a+b) = 2 log 4 + log a + log b

Now, 2log (a + b) = 2 log 4+ log a+ log b is true

If log (a + b)2 = log 4

2 + log a + log b is true

If log (a + b)2 = log 16 + log ab is true

If log (a + b)2 = log 16ab is true

If (a + b)2 = 16ab is true

if a2

+ b2

= 14ab which is known and true

Thus, if a2

+ b2= 14ab, it is proved that 2log (a + b) = 2 log 4 + log a + log b

Example 2: If a

b =

c

d, prove that

( ) ( )2 2ac 4b c 4bd

b d

- -=

Proof:

The unknown part is ( ) ( )

2 2ac 4b c 4bd

b d

- -= is true,

If a cd – 4b2d = bc

2 – 4b

2d is true,

If acd = bc2 is true,

If ad = bc is true

That is, if a

b =

c

dis true,

Which is known and true.

Thus if a

b =

c

d, prove that

( ) ( )2 2ac 4b c 4bd

b d

- -=



Mathematics

16

Synthetic Approach

Synthesis refers to a combination of two or more entities that together form

something new. In this method we move from known to unknown and from

hypothesis to conclusion. It is just the opposite of analytic method. It is an

approach in which we collect and combine various facts to find out the

unknown result. It presents the facts in a systematic way and can be applied to

majority of topics in teaching of Mathematics . According to (Katozai, 2002 as

quoted by Asif, Khan and Zaman : 2010 ) it is the process of putting together

known bits of information to reach the point where unknown formation

becomes obvious and true.

Example 1:

If a2 + b

2 = 14ab prove that 2log (a + b) = 2log4 + loga + logb

Proof:

To prove this using synthetic approach, begin from the known.

The known is a2

+ b2 = 14ab

Now, a2 + b

2 + 2ab =14ab + 2ab

So, it becomes (a + b)2 = 16ab

Taking log on both side,

log (a + b)2 = log 16ab

2log (a + b) = log16 + log ab

2log (a + b) = log 42 + log ab

2log (a + b) = 2 log 4 + log a + log b

So if a2

+ b2

=14ab, 2log (a + b) = 2 log 4 + log a + log b

Example 2:

If a

b =

c

d, prove that

( ) ( )2 2ac 4b c 4bd

b d

- -=

Proof:

The known part is a

b =

c

d

Subtract 4b/c from both sides

a 4b c 4b

b c d c- = -

( ) ( )2 2ac 4b c 4bd

bc cd

- -=

i.e ( ) ( )

2 2ac 4b c 4bd

b d

- -= which is unknown..

17

Approaches and


Mathematics

Differences between Analytic and Synthetic Methods

S

N

Analytic Method Synthetic Method

1 This process refers to breaking down a

bigger problem into smaller components.

This process combines many

small known components to derive something new.

2 It leads from unknown to known. In

other words it leads from conclusions to

hypothesis.

It leads from known to

unknown. In other words it

leads from hypothesis to conclusion.

3 This approach is lengthy and time

consuming.

It is short and time saving

method.

4 It is known as psychological method. It is known as logical method.

5 This method encourages thinking and

reasoning. This approach promotes meaningful learning.

It puts more emphasis on rote

learning. It promotes memory work.

6 This approach is informal in nature and

it is disorganized.

This approach is very formal

and systematic.

7 In this approach learners can recall and

reconstruct any step easily, if forgotten.

It is very difficult for the

learners to reconstruct the steps, if forgotten.

8 It is based on heuristic lines. This approach doesn’t cater to

heuristic approach.

Both these methods look like opposite to each other but they go together. They

support and complement each other. It can be said that analysis lead to

synthesis and synthesis makes analysis complete as well as clear. That’s why it

is desirable that teacher should use analytic method while teaching

Mathematics and motivate the learners to use synthetic approach for

presentation.

Activity for Practice:

4. Write a plan to conduct a topic through analysis – synthesis metod in your

class. Try it out in your class. Reflect on the process and write the merits

and demerits of the process.

5.5.3 Problem Solving Approach

Learners learn mathematical thinking most effectively through applying

concepts and skills in interesting and realistic contexts which are personally

meaningful to them. Thus, Mathematics is best taught by helping learners to

solve problems drawn from their own experience. Real-life problems are not

always closed, nor do they necessarily have only one solution. The solutions to

problems which are worth solving seldom involve only one item of

mathematical understanding or only one skill. Rather than learners

remembering the single correct method, problem - solving requires them to

search the information for clues and to make connections to the various pieces

of Mathematics and other knowledge, experiences and skills that they have



Mathematics

18

already learned. Such problems encourage thinking rather than mere recall.

What is the most common thing that learners do when they encounter a

Mathematical problem which they don’t know how to solve? Some give up

very quickly while some of them ask other learners or their teachers for help.

This doesn’t help the learner in the long run as each and every individual some

time or other will encounter problem, which they have not been solved earlier

and their problem solving ability will play an important role in these scenarios.

According to NCTM(2000) “Problem solving means engaging in a task for

which the solution is not known in advance ”.

Any mathematical situation can

be a problem for a learner if learner has not previously learned about how to

solve that. Once the learner learns how to solve a problem, it becomes an

exercise. Teaching through problem solving and teaching problem solving are

two different approaches. Teaching problem solving usually works on guess

and check, working backward etc. methods.

In teaching through problem solving, teacher will setup the context and explain

the problem. Now, learners work on the problem and the teacher monitors

their progress. After stipulated time each learner of the class shares his/her

ideas with the whole class and then they compare as to which idea is best for

solving that particular problem. In this way learner learns many new

mathematical ideas and procedures.

Problem

Learner 1 Learner 2 Learner 3 Learner 4

Solution1 Solution1 Solution1 Solution1

Comparison and

Discussion

Final Solution

19

Approaches and


Mathematics

Let us see an illustration where a teacher used problem solving strategy.

Illustration

Teacher : Look at the following series and tell me the next number:

2,4,8,16,32,……

Mani: 34

Teacher: How did you find?

Mani : Next number is multiple of 2 because all numbers in the series are multiples

of 2.

Teacher : According to your guess the series is 2, 4, 8, 16, 32, 34…… Then, 34

is correct answer. Whether it is correct? Let us check. Tell me all the multiples of 2.

Mani : 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24, etc.

Teacher : Tell me, whether all these numbers are coming in the series.

Mani : No madam

Teacher : So, your guess is not correct and hence 34 is not the right answer. Any

other guess (put the question to whole class)?

Kanjan : 64

Teacher : Very good, how did you find?

Kanjan : When observed the series, I found that every number is the double of

previous number. So, next number may be double of pervious number.

Teacher : Let us check your answer

2×2 = 4

4×2 = 8

8×2 = 16

16×2 = 32

32×2 = 64

Teacher : Your answer is correct. Do you agree?

Learners : Yes, madam (All learners with loud voice).

From above illustration, we have seen that teacher has given a problem to

learners then learners have analyzed the problem, searched the expected

mathematical relation, found the solution. At that time the teacher monitored

their progress. After stipulated time each learner of the class has shared his/her

ideas and find the best solution of the problem.


5) How would transact the topic ‘linear equations’ through problem solving

strategy.



Mathematics

20

Check Your Progress



7. Fill in the blanks.

i. Inductive approach proceeds from…………to…………..and

………to…………… .

ii. Synthesis refers to a …………………….of two or more entities.

iii. .…………………… approach moves from unknown to known.

8. Write any three differences between inductive and deductive approach of

teaching learning of Mathematics.

........................................................................................................................

........................................................................................................................

........................................................................................................................

........................................................................................................................

9. In what ways analysis and synthesis approaches are looks like opposite to

each other.

........................................................................................................................

........................................................................................................................

........................................................................................................................

........................................................................................................................

5.6 TECHNIQUES FOR TRANSACTING

MATHEMATICS CURRICULUM

There are various techniques of transacting Mathematics Curriculum in an

effective manner. Some of the techniques are as follows-

5.6.1 Drill and Practice

Drill is one of the most essential ways (or methods) of learning in mathematics.

The controlling purpose of all teaching activity is to reduce necessary learning

to habit. Gaining mastery requires acquisition of habits, hence drill/practice

plays an important role in acquiring mastery. By and large, practice lessons are

of three types. The first category of lessons for mastery is of basic subject

matter, e.g., multiplication tables, addition combinations, fractional equivalents

of decimals and percentages, factorization, construction in geometry, etc. These

include subject matter which must be thoroughly mastered so that speed and

accuracy is ensured on which future learning can be based.

The second category includes lessons for the mastery of procedures. In

Mathematics one has to adhere to a systematic arrangement of steps, follow

correct algorithms to scrutinize and check the correctness of each step, label

21

Approaches and


Mathematics

appropriately parts in a diagram, sort out data, translate problems into symbolic

form, practice short cuts, etc.

The third category consists of lessons which strive to develop the power of

thinking and reasoning, and increase the concentration and interest of the

learner. Such lessons include quizzes, puzzles and historical material which

does not form part of a regular lesson.

Although, a certain amount of formal drill is inevitable, preference should be

given to functional or meaningful drill. Meaningful drill implies prior

understanding of content and its appropriate application. This drill is

purposeful and is determined by need as will as by use. An effective drill

lesson should be organized keeping in view the following considerations:

1. Drill should follow learning and understanding of basics. It should not

encourage rote memorization without understanding the subject matter.

2. Drill should be varied. Some routine procedures make learning monotonous

and uninteresting.

3. Drill should be individualized and rewarding to each learner. Each learner

should see its purpose and utility.

4. Drill periods should be short and the learner’s achievement should be

frequently tested.

5. Drill should not be planned merely to keep learners “busy”. It should be

based upon thought-provoking situations to avoid the repetition of any

process mechanically.

6. Drill may also provide diagnostic information about learners.

5.6.2 Play Way

Play way technique is a child - centered informal method of teaching

which suits the interest of the child and improves its academic proficiency

effortlessly. This method helps to develop interest in Mathematics, motivates

learners to learn more, and reduces the abstract nature of the subject to some

extent.(Patel, nd). Play way can be an effective way of teaching Mathematics to

learner. Although only some concepts can be taught through games, the most

important benefit of games like mathematical quiz, puzzle, tricks, riddles,

guessing game, etc. is the oral practice of various mathematical concepts. Let

us see an example of guessing game where learners need to decode the

structure like the one given below.

Game: Guess the number

Think of a number. Multiply it by 5 and add 10. Tell me the number, I will tell

you your number.

Unknowingly in the process learners will discover the structure of the problem

in terms of operations of multiplication and addition and reversal of these

processes in terms of subtraction and then division.

For example, in this problem if the number given is 90, then original number

can be calculated by subtracting 10 and dividing by 5. The number which one

has thought will be 18.



Mathematics

22

Some of the examples of mathematical puzzles, tricks, riddles and guessing

games has been given in the unit 6 of this block. Refer and use it in day-to-day

teaching.


6. Design a game for a introducing the concept of ‘ratio’ for class VII learners.

Reflect on whether the task is a game or not.

5.6.3 Home Work

Homework refers to tasks assigned to learners by their teachers to be

completed outside the class. The purpose of home work is to encourage

learners to review, apply, practice and integrate what he/she has learnt in the

classroom. Siddhu (2006) quotes that school time is insufficient to exhaust

everything provided in the curriculum of Mathematics. Homework has to be

given regularly to provide for application and practice and to supplement

classroom teaching. It provides an opportunity for learners to make, discover,

and correct mistakes so that they can learn from them. It is intended to engage

learners in exploration of concept beyond the class time. It helps learners in

long lasting learning and preparing for the next class. The fundamental purpose

of homework to learners is the same as schooling in general.

Home work may consist of open-ended questions or closed-ended questions.

Open-ended questions are developed to prompt learners to apply concepts,

solve problems, and make mathematical connections. No routine or prescribed

methods are given for responding to open questions. Therefore, it makes

teachers able to observe the strategies, skills, logic, concepts, and connections

used by the learners to solve the questions. Open questions may be classified

as open-development, open-process, and open-ended questions.

Let us see some examples of various type of questions.

Type of question Examples

Open-ended development

question

Write a challenging story problem for the

equation n +3 = 7.

Open-ended process

question

The length of a field is 3 times to its width. Its

perimeter is 4800m. What are the dimensions

of the field? Show your work.

Open-ended question

The average mass of 15 fishes caught in a pond

is 2.5 kg. The mode is 3 kg. What are the

possible masses of the 15 fishes? Explain your

thinking.

Home work has some drawback also. It can destroy curiosity and love of

learning in the learner. Studies have shown that it takes time away from

independent study and extracurricular, family, and social activities important to

childhood development. So it is very necessary for the teacher to carefully

monitor the amount of home work so that learner can get enough time for

social activities. It is also necessary to make sure that homework has

appropriate level of difficulty so that learner can complete the homework

independently. A rule of thumb for homework might be that "all daily

23

Approaches and


Mathematics

homework assignments combined should take about as long to complete as 10

minutes multiplied by the learners' grade level" and "when required reading is

included as a type of homework, the 10-minute rule might be increased to 15

minutes" (Cooper, 2007, cited in Marzano & Pickering, 2007, p. 77).

Homework review in class is an important part of Mathematics teaching-

learning. It should be well graded. It should be assessed as a part of the overall

assessment of a learner. Home work should be duly checked and corrected. If it

is not checked, the learners may fall into the bad habit of evading it or copying

it.

5.6.4 Assignments

An assignment is a task or work allotment. In this technique, the learners are

provided with the responsibility for his/her own learning. The teacher acts as an

advisor and guide in case of any difficulty encountered. The method has

several advantages. It encourages initiative and independence, and provides

learners with the maximum amount of individual practice. Teacher should keep

in mind that assignment should always be a task which is within the capability

of the learner and has some interest for him. It is the major important part of

every learner life. The learners are encouraged to keep their completed

assignments for future references.

Characteristics of good assignment

• Assigned task must be clearly defined.

• Assigned work must have correlation with previous knowledge and

experiences.

• Assigned work must be stimulating and directing the learning experiences

and activities.

• It should be precise as well as have sufficient information to enable the

learners to complete the task.

• Newer topics for the assignments must be proposed with the earlier

learning experiences.

• Teacher must know what they want from the learners to gain from the

experience.

• Assigned task must be interesting to be completed within the stipulated

time by the learners.

• Library facilities and other reference resources are mandatory for

completion of assignments.

Check Your Progress



10) Fill in the blanks.

i. …………… may also provide diagnostic information about learner.

ii. School time is not sufficient therefore………….is required.

iii. . …………technique is child centered.



Mathematics

24

11) Discuss the importance of homework in Mathematics.

......................................................................................................................

......................................................................................................................

12) Write any four characteristics of good assignments.

......................................................................................................................

......................................................................................................................

......................................................................................................................

5.7 LET US SUM UP

The following concepts/issues are dealt in this unit:

� Teaching and learning Mathematics has started shifting from behaviorist

approach to constructive approach. Constructivism states that individual

construct their own knowledge and understanding of the world through

experiencing and reflection.

� Constructivist approach in Mathematics believes that learner can construct

knowledge by active participation rather than acquiring knowledge by

observing lectures delivered by teachers.

� Inductive approach is based on the process of induction i.e. reasoning from

specific facts to general principles. Therefore, it proceeds from particular to

general, and from concrete to abstract.

� Deductive method is based on deduction. It proceeds from abstract to

concrete, from general rule to particular or specific instances, and from

formula to examples, and from unknown to known.

� Analysis is the process of breaking a complex topic or substance into

smaller parts in order to gain a better understanding of it.

� Synthesis refers to a combination of two or more entities that together form

something new. In this method we move from known to unknown and from

hypothesis to conclusion.

� According to Problem solving approach, learners learn mathematical

thinking most effectively through applying concepts and skills in

interesting and realistic contexts which are personally meaningful to them.

� Drill method gives learners an opportunity to learn certain concepts quickly

and effectively.

� Play way method can be used for teaching Mathematics by activities such

as mathematical games, checkers, magic squares, puzzles and building

blocks.

� The purpose of home work is to encourage learners to review, apply,

practice and integrate what he/she has learnt in the classroom.

� An assignment is a task or work allotment. In this technique, the learners

are provided with the responsibility for his/her own learning. The teacher

acts as an advisor and guide in case of any difficulty encountered.

25

Approaches and


Mathematics

5.8 UNIT END EXERCISES

1) Analyze the behaviorist approach of learning. How does it differs from

constructivist approach? Discuss the significance of constructivist approach.

2) Which is the best method of teaching Mathematics according to your

view? Justify your preference with suitable examples and arguments.

3) Illustrate and discuss the inductive–deductive method of teaching

Mathematics

4) Describe the use of analytic and synthetic methods in the teaching of

Mathematics.

5) What do you understand by the problem solving method? How will you

employ it in teaching Mathematics?

6) What is place and value of drill and practices in the teaching of

Mathematics?

7) Illustrate the constructivist approach for teaching learning Mathematics.

8) How will you employ the techniques of assignment? How is it different

from home work

5.9 ANSWERS TO CHECK YOUR PROGRESS

1) Skinner and Watson were the major proponents of behaviorism.

2) The overall philosophy of constructivists holds that learners construct

their own understanding based on their unique experiences and furnish

meaning on the world.

3) The difference between behaviorists and constructivists are as follows:

• Behaviorism supports traditional teacher-centered approach whereas

for constructivists, learner is at the center of teaching learning

process.

• Behaviorists focus more on how learners respond to positive and

negative reinforcement provided by the teacher, whereas

Constructivists see how learners learn on their own when learners are

presented with stimuli.

• Behaviorist supports deductive approach and Constructivist supports

inductive approach of teaching.

4) (i) Constructivists (ii)Constructivist, Behaviorist

5) Jean Piget

6) Refer section 5.4

7) (i)from particular to general and from concrete to abstract (ii)combination

(iii)Analytic

8) Refer section 5.5.1


10) (i) Drill (ii) home work (iii) Play way




Mathematics

26


5.10 REFERENCES AND SUGGESTED READINGS

• Anthony, G and Walshaw, M (2009) , Characteristics of Effective

Teaching of Mathematics: A View from the West, Journal of Mathematics

Education, Dec 2009, Vol. 2, No. 2, pp.147-164

• Asif, M., Khan, M. M. and Zaman, K.(2010). Comparative Study of

Analytical and synthetic methods of Teaching Mathematics, Journal of

International AcademicResearch,Vol.10,No.3.available at www.uedpress.

org/ojs/index.php/jiar/article/download/10/25 (retrieved on 10.04.2016)

• Carbonell, L. (2004) Instructional Development Timeline. Retrieved

January 21, 2008, from http://www.my-ecoach.com/idtimeline/

learningtheory.html

• Chambers, P. (2010).Teaching Mathematics, Sage Publication, New Delhi.

• David, A. H., Maggie, M. K., & Louann, H. L. (2007). Teaching

Mathematics Meaningfully: Solutions for Reaching Struggling Learners,

Canada: Amazon Books.

• Gupta, H. N., and Shankaran, V. (Ed.), (1984). Content-Cum-Methodology

of Teaching Mathematics, New Delhi: NCERT.

• James, A. (2005). Teaching of Mathematics, New Delhi: Neelkamal

Publication.

• Jones, M.G. & Brader-Araje, L. (2002), The Impact of Constructivism on

Education: Language, Discourse, and Meaning, American Communication

Journal, Volume 5, Issue 3, Spring 2002

• Kumar, S. (2009). Teaching of Mathematics, New Delhi: Anmol

Publications.

• Marzano, R. and Pickering, D (2007). The Case For and Against

Homework Educational Leadership, Vol. 64, issue 6, p. 74-79 available at

http://www.ascd.org/publications/educational-leadership/mar07/vol64/num

06/The-Case-For-and-Against-Homework.aspx (retrieved on 03/04/2016)

• NCTM (2000). Principles and Standards for School Mathematics,

Washington, DC: NCTM; available at http://www.nctm.org/standards/

content.aspx?id=16909. (retrieved on 06.03.2016

• NCERT (2005). National Curriculum Framework, New Delhi: NCERT.

• NCTE (2009). National Curriculum Framework for Teacher Education,

NCTE, New Delhi.

• Mangal, S. K. (1993). Teaching of Mathematics, New Delhi: Arya Book

Depot.

• Patel, R.,(nd) Innovations in Teaching of Mathematics, available at

http://www.waymadedu.org/learnersupport/rachnamadam.pdf (retrieved on

12.04.2016 )

• Siddhu, K.S. (2006). Teaching of Mathematics, New Delhi: Sterling

Publishers.

• Van de Walle, J. A. (2004). Elementary and Middle School Mathematics:

Teaching Developmentally. New York: Pearson Education, Inc.

27

Organizing Teaching-

Learning ExperiencesUNIT 6 ORGANIZING TEACHING-

LEARNING EXPERIENCES

Structure

6.1 Introduction

6.2 Objectives

6.3 Planning and Designing Learning Experiences

6.3.1 Characteristics of Good Learning Experiences

6.3.2 Sequencing of Learning Experiences

6.4 Involving Learners in Teaching-Learning Process

6.5 Levels of Planning

6.5.1 Annual Planning

6.5.2 Unit Planning

6.5.2.1 Concept Mapping for Unit Planning

6.5.3 Lesson Planning

6.6 Planning Instruction for Children with Special Needs

6.7 Learning Mathematics through Puzzles, Riddles and Tricks

6.7.1 Tricks

6. 7.2 Puzzles

6.7.3 Riddles

6.8 Let Us Sum Up




6.1 INTRODUCTION

Planning is essential, not only in teaching, but also in every sphere of life. It

helps, not only to achieve the predefined objectives, but also gives right

direction to execute any work with reduced wastage of time. The teachers are

the pioneers of planning function as they plan instruction in order to realize the

objectives set by them. In the process of planning, the teacher arranges the

content or subject matter under series of topics and sub-topics. Once the

content is identified, you can decide on what you want the learners to

accomplish. Thereafter, you can set out your lesson plans for teaching specific

concepts.

In this unit, we will discuss how to plan learning experiences at various levels

and involve learners in Mathematics teaching-learning process. We will also be

familiar with various steps and process of unit planning and lesson planning.

Discussion will also take place how we can make Mathematics leaning

interesting by learning through Mathematics puzzles, tricks and riddles. As a

teacher of Mathematics you must have come across the challenges that children

with special needs face in Mathematics learning. This issue will also be dealt.


Mathematics

28

So, the discussion in this unit will help you to plan instruction for learners of

Mathematics.

6.2 OBJECTIVES

After going through the unit, you will be able to:

• describe major guidelines for planning an instructional programme;

• design learning experiences ;

• appreciate involvement of learners in teaching-learning process;

• develop a unit plan through concept maps;

• design a lesson plan based on 5-E approach;

• plan appropriate learning experiences for children with special needs; and

• transact mathematical concepts through quizzes, riddles, tricks etc.

6.3 PLANNING AND DESIGNING LEARNING

EXPERIENCES

Planning, designing and organization of learning experiences provide the

framework for effective teaching. In classroom teaching-learning, it is

necessary to plan activities that would provide directions to both teachers and

learners. Firstly, effective and proper planning involves making tentative

decisions regarding expectations for a given course and deciding how this can

be best accomplished, enriching teaching-learning experiences. Secondly,

organising these experiences involves formulation of objectives, strategies and

the variety of activities in accordance to the diverse interests of the learners.

Careful planning gives direction to cognitive abilities and clarification of

concepts invoking thoughtful pondering for making learning a joyful

experience. This may form the base of developing sound pedagogical

foundation among the learners, fostering successful application of knowledge

in day-to-day life.

6.3.1 Characteristics of Good Learning Experiences

The criteria of good learning experiences are:

i) Objective - based: Learning experiences should be appropriate in

relation to the behavioural changes expected by a changing the

objectives. Objectives are the end points to be achieved by the learner and

provide the base for designing learning experiences. Objectives are

formulated in relation to subject matter. Teacher can design learning

experience keeping in view the objectives of the subject matter to be

taught.

ii) Learner orientation: Learning experience is meant to provide desirable

behavioural changes in the learner through activities. To meet the

requirements of individual differences, the learning experiences should be

imaginative and easily adaptable to the interests and abilities of the

learners.

iii) Richness of experience- Learners differ in maturity, interests and

abilities. To plan effective and meaningful instruction for learners with

29


Learning Experiencesdiverse interests, experiences, abilities and rates of learning; the teacher

must visualize the range of learning experiences to capture skills of

different learners in their class. Creative work, community projects,

construction activities and experiments utilizing the skills and

mathematical abilities should be introduced that challenge the thinking of

the learners.

iv) Suitability to the mental level: Learning experiences facilitate the

teacher to achieve different objectives by creating appropriate learning

environment for the diverse learners. Learning experience is planned

according to the mental level of a particular group of learners.

v) Practicability: Teacher plans the learning experiences in such a way that

maximum learning takes place. The interaction provided through learning

experiences helps the learners to learn by promoting their thinking.

Designing practicable learning experiences should promote socio-

cognitive interaction between the teachers and learners in the class room.

vi) Evaluation: In order to check the realization of objectives of instruction,

teacher selects suitable mode to evaluate the learners. Learning

experiences are the means to help the learners to attain desired objectives.

Hence, every learning experience is viable for the periodic evaluation by

the teachers.

6.3.2 Sequencing of Learning Experiences

In an actual teaching plan, learning experiences are effective when they are

organized as a related whole. Generally in any particular lesson or unit, the

selection and sequencing of learning activities are done at three levels begin

with opening activities, moves to application and creative activities and ends

with concluding activities, that wrap up the unit. The sequence of organizing

activities provides learners’ opportunity to use thinking process in a sequence

that begins with observing, comparing, generalizing and inferring, and extends

thinking to include analyzing, and creating.

Level 1: The Preparatory or Readiness Stage

The opening activities or learning experiences for each main idea are

motivational to secure the interest, focus attention and build readiness of the

learner towards the topic .The learner should be given first hand experiences

with concrete objects or real life situations so that he / she observes, locates and

discovers some known mathematical ideas involved in the situation.

Level 2 : Exploratory or Development Stage

These learning experiences provide for experimentation, doing and thinking

analytically. These help in exhibiting connections or relationship between what

is known and what is to be concluded.

Level 3: Generalization Stage

The learner learns more about mathematical ideas as he/she observes, analyses

and discovers relationships and formulates generalization from a number of

problems or situations. These are the culminating activities which encourage

the learner to bring together different main ideas and generalize understanding.

At this stage, the learner’s mind has matured and he/she can express

mathematical ideas more meaningfully. Exhibits, reports, discussions,

individual notes etc., fall in this category of activities. Culminating activities


Mathematics

30

usually result in a presentation with emphasis on educational outcome. The

learning experiences may be planned for various purposes such as:

a) To explain a complex mathematical idea by making learners do or think in

parts;

b) To provide an opportunity to analyse and make generalizations;

c) To develop new vocabulary and use known vocabulary;

d) To coordinate the idea of mathematics and some other ideas like, social or

related to other branches of knowledge;

e) To provide experience in the process of problem-solving, i.e. planning a

situation, gathering information, solving the problem and verifying the

result;

f) To provide scope for application of mathematics ideas; and

g) To provide an insight into various methods of proof in Mathematics.

Let us see a class room situation where a teacher, Rama was trying to make

learners, of class 7th

, to represent variable quantities by letters and she designed

learning experiences in a sequence which are presented in the following table:

Level 1: The Preparatory or Readiness Stage : She told the class to write

their age in years in the table in the worksheet which was provided to the

learners in groups of two.

Level 2 : Exploratory or Development

Stage: Now Rama asked the learners to

discover the relationship between the ages

of two learners who formed the pair. She

helped the class in doing the task by telling

that they can represent the age of one learner

by the letter x and then represent the age of

the other learner in terms of x.

Level 3: Generalization Stage : She first called Rakesh and Rohan. Rohan

told her that he is one year older to Rakesh. She then asked Rohan to express

this relationship using a letter. Rakesh could do it. Rakesh told that if I

represent my age by x, then Rohan’s age will be x + 1.

Rama then asked the class that if she had assumed Rohan’s age as x, then

what will be the Rakesh’s age. Class struggled but finally came up with the

answer x – 1.

Then she told the whole class to express their age in terms of the age of their

peers.

Thus Rama tried to convey the learners how letters help when we wish to

express relationships.

At all the stages in school, learning of Mathematics, can be very interesting and

effective provided that there is the constructivist learning environment and

conditions for carrying out the activities.

Rakesh Rohan

2005 3 4

2007 5 6

2010 8 9

2014 12 13

2016 14 15

Figure 1

31


Learning Experiences

Check Your Progress



1) Why planning of is important in teaching-learning?

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

2) What are the characteristics of good learning experiences?

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

3) What is the importance of preparatory stage activities? How it is different

from activities organized at developmental stage?

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

6.4 INVOLVING LEARNERS IN TEACHING-

LEARNING PROCESS

We know that learner enjoy learning more while they are actively involved in

the process. Knowledge cannot be transmitted to learner but can be constructed

by the learners on the basis of their learning experiences. So, in the process of

knowledge generation, involvement of learners is very important. In teaching-

learning process, the role of teacher changes from ‘transmitter’ of knowledge

to ‘facilitator’ of knowledge construction.


1. Rama has designed learning experience to represent variable quantities

by letters. Design any other experience to represent variable quantities by

letters where all three stages are covered.


Mathematics

32

For development of mathematical concepts, you should ask those questions

which test learners’ ideas, provide feedback to learners, contrast learners’

ideas, make learners to use evidence to explain ideas, to apply their

conceptions to phenomenon, summarize results, and represent result

symbolically. Such activities help learners in the development of reasoning,

problem solving skills, logical and reflective thinking. To involve learners in

teaching -learning process you should:

• Ensure that each child is engaged in some learning activity according to

his/her interest and ability;

• Encourage learners to share, discuss, argue and compare their ideas;

• Provide help in the form of scaffolding, when asked for it;

• Provide opportunity to learners to express their preconception and to

identify them;

• Provide opportunity to elaborate their ideas and to discuss among

themselves;

• Inclusion of such activities that provide opportunity to challenge

misconceptions;

• Provide opportunity to learners to reorganise their ideas and opportunity to

teacher to introduce new ideas;

• Provide opportunity to apply new ideas in different situations.

As a teacher, you are responsible for creating an environment where

learners fully involve in process of mathematical thinking or knowledge generation; so that it can energise the construction of new mathematical

concepts. For example a teacher was developing her/his teaching of concept of

quadrilateral plan for teaching the class 7th learners. Her/his thought process

goes like this :

• What are the objectives of this lesson... yes really, I should help the

learners to develop the concept of a quadrilateral…. any more (?)….yes,

they should be able to differentiate other geometrical figures from a

quadrilateral….

• Of course, I should help them to identify different elements like sides,

angles etc of a quadrilateral.

• How can I help my learners in their knowledge construction process? .....

• I should start with an interesting introduction……

• How can I introduce it in a motivating way? ....

• Yes, since they have learnt already the concept of triangles and more over,

they are familiar with different objects. Hence if I can use these materials

while introducing the lesson differently, I can create interest among my

learners.

• With this thought process the teacher decided to design an introductory

activity. She thought of collecting different objects like paper, leaves,

bangles, coins, cut out of different geometrical shapes like triangles,

quadrilaterals, pentagons, etc. Then she planned class room process.

33


Learning Experiences• Should I start the class with a whole class discussion, or an individual

activity, or a group activity? …... if I am arranging a whole class discussion

sometimes I may not be able to create interest among all learners?

• individual activity ?

• Definitely no, since it may not be possible to arrange enough material……,

hence better option here is to start with a group activity after giving

appropriate directions.

This is another important area in which we should give stress. Teacher knows

exactly about her learners and she decided not to go for a whole class

discussion in the beginning and designed the plan accordingly.

• How can I organize this activity in the classroom? ...... I will divide the

learners in to four groups and each group will be given a set of geometrical

figures.

• What about the other objects? ..... Yes, definitely I can collect some

bangles, coins, papers etc and can include these also in the set……..

• I should prepare 4 different sets for this activity. What general instructions

I should give to the learners before the activity? .... Dear Learners, I am

going to give you a group task for that you will be grouped in to four. I will

give you a set of objects like geometrical figures and familiar objects. You

work in the group and classify those objects according to any criteria. You

have the freedom to classify them in to different groups according to any

criteria. After completing the activity, you prepare a chart in your notebook

based on the criteria and number of objects in each group like this. ( I can

draw the sample chart on the black board for your help).

Here teacher likes to enhance creativity of learners. If she directly asks

learners to classify the objects according to their shapes, her work can be

finished quickly, but by providing them an opportunity of alternative

classification, the creativity of the learners also can be enhanced.

• How to conclude this group activity? ….. I can ask any one from each

group to present the chart prepared by them followed by discussion in the

class. Through this whole class discussion I can probe the learners to see

the grouping done by different groups and sensitise the concept of four

sided closed figures as quadrilaterals.

In this way the teacher was able to design a plan for introducing the

concept of quadrilateral through different strategies like co-operative group

activity, inter group discussion, whole class discussion etc. by involving

learners.


2. Select any topic of your choice from secondary level mathematics

textbook and design learning experience where learners are actively

involved in knowledge construction.


Mathematics

34

6.5 LEVELS OF PLANNING

Planning enables you to think about your teaching in a systematic way before

you enter the classroom. In the teaching profession planning is often connected

with units and lessons. Planning of a unit/ lesson provides a useful framework

of carefully organized activities to accomplish expected goals of the prescribed

curriculum. The outcome of your planning is a coherent framework which

contains a logical sequence of tasks to be carried out for effective teaching and

learning. Planning in school begins with a plan for a year followed by unit and

lesson plan. Let us start with broader outline i.e. Annual Planning in the

following sub-section.

6.5.1 Annual Planning

At the beginning of the school year, you need to consider what you want the

learners in your class accomplish during the year. For this, you need to keep in

mind the background of your learners, and the expected goals of the prescribed

curriculum.

After identifying the goals, you need to sequence the way the mathematical

concepts could be taught. Since schools usually follow a system of dividing the

year into two terms; the next stage of planning would involve setting time

frame for teaching specific content areas in each of the two terms. This will

help to fit in all the Mathematics content that you think can be included; given

the background of the learners. So, what we need to ask ourselves at the

beginning of a year is :

• How much do most of the learners already know?

• How much do I expect them to accomplish by the end of the year?

• How should I sequence the concepts that I want to teach them?

• How should I distribute the total syllabus during the year?

Once the broad time frame is worked out, you need to identify the time

required for the teaching specific mathematics unit.

6.5.2 Unit Planning

Within the comprehensive scope of school curriculum, the individual teacher

needs to chalk out yearly course. Course is divided into various units. The

Check Your Progress



4) How can you involve learners in teaching learning of Mathematics?

……………………………………………………………………………………

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35


Learning Experiencesdivision of course into units helps to provide a framework for learning.

Disciplinary knowledge is organized into units; each unit consists of

interrelated facts, ideas, concepts, principles, and so on. Therefore, planning

the lesson construction should begin with detailed content analysis present in

the unit in order to appreciate knowledge components, their nature, cognitive

and social process behind that knowledge generation. Unit planning focuses

on-

� Content Analysis for the identification and development of key ideas/

concepts values/skills;

� Unit questions (Replacing objectives);

� Relevant pedagogical processes and strategies adapted for effective

learning;

� Learning resources required for classroom learning;

� Adaptations/Modifications (for children with special needs).

So, the unit planning is undertaken for the temporal distribution of the content,

identification of learning process, identification of the relevant pedagogical

process, resources and assessment strategies required for learning. The process

of unit planning directs teachers to reflect on what they want to accomplish in

each unit and in each class. Unit planning helps teachers to have control on

utilizing class time as productively as possible. It provides a sense of direction

and organization that helps to achieve significant academic gains within a

particular time period.

Content for the unit can be analyzed through concept mapping.

6.5.2.1 Concept Mapping for Unit Planning

Concept map can be used as an excellent planning device for planning

instruction. It helps in organizing and planning learning activities to enhance

learning experiences of the learners. Mapping the concepts may increase your

ability to provide meaningful learning experiences to learners by integrating

concepts. It facilitates to identify concepts and sub concepts that you want to

emphasize. Concept map facilitates the teacher to reconceptualize the course

content. It helps to understand relationship between facts and concepts through

cross-links, which leads to the development of lesson plan based on

constructivist pedagogy.

Concept map is a diagram that shows relationships among concepts. A concept

includes many attributes and prepositions. Concept maps consist of labels

(enclosed in circles or boxes) and relationships (a connecting line) linking two

concepts and sub concepts. The text on the line (linking words or phrases)

describes the relationship between these two concepts. ‘Prepositions’ contain

two or more concepts connected with each other by using linking words or

phrases to form a meaningful statement. A concept map may have many

‘prepositions’ which are taught or learnt independently.

Concept maps are developed by using paper-pencil or with the help of

‘Information and Communication Technologies (ICT)’. Steps for development

of a concept map is discussed in the Unit … course……….. of BES-143. So

please refer it . An example of concept map for a Unit “Lines and Angles” is

given in Fig 2.


Mathematics

36

Fig 2: Concept Map for the topic Lines and Angles

(Source:

After construction of concept map you can develop the unit plan. A format of a

unit plan has given below:

Format of a Unit Plan

Name of the Unit –Lines and Angles

Class- VII No of Sub Units- 2

No of periods required -7 Duration of period-40 minutes

S.No. Sub-Units No. of

Periods

1 Lines 03

2 Related Angles

• Supplementary Angles

• Complementary Angles

• Adjacent Angles

• Vertically apposite

04

37


Learning ExperiencesTable 1: Unit plan for Lines and Angles.

Concepts/Ideas/

Skills involved

Key

Questions

(Objectives)

Learning

Process

Learning

Strategies

Learning

Resources

Assessment

Strategies

Adaptation (for inclusion)

• Lines, line

segment, ray,

representation

• Intersecting

lines, parallel

lines

• Angles, vertex, arms, types of

angles,

measure of

angle, interior

and exterior of

angles

• Skill of using

ruler, compass,

• Do non-

parallel

lines form

angles?

• What are the

different

kinds of

angles ?

• Are supplemen

tary and

compleme

ntary

angles

pairs of

angles?

• Observation

• classification

• reasoning

• communication

• logical thinking

• Group work

• Presentation cooperative

learning

• Discussion,

• Inductive

• deductive

method

• Questioning

• Geogebra

software,

PPT

presentati

on,

• Paper

cuttings,

• Object related to

daily life

• Observation,

• drawings,

• reflection,

• portfolio,

• paper pencil test

• (Assess the drawings of lines

and angles for

correctness,

completeness,

and

misconception)

• Give the

figures of

lines, line

segment, and

ray, types of angles for

identification

• From the figure

learners name

different

angles

• Additional explanations,

illustrations

and practice

with objects and

animation/

geogebra

Having seen the unit plan preparation, let us now turn to lesson plan.


3. Select any unit of your choice from secondary class mathematics text book.

Develop a concept map and unit plan for selected topic.

6.5.3 Lesson Planning

In constructivist classroom, lesson plan focuses on creating learning situations

to meet the needs and interests of diverse learners. The entire focus of the

teacher is to plan meaningful learning opportunities that would facilitate all the

learners to construct the knowledge of the given textual content. Following are

the basic features of constructivist model of classroom teaching:

• Construction: Lesson plan in constructivist paradigm has to provide a

number of learning situations to help children construct their knowledge

based on their past experiences.

• Collaboration: Learning has a social character; therefore learning process

should provide sufficient space for collaboration on at work among

learners.

• Reflection: Reflection is vital in knowledge construction. Hence

construction of lesson plan has to provide deliberate space for reflection.


Mathematics

38

• Active: Constructive lesson plan has to carve out many situations to tap

active nature of the child.

• Evolution: Lesson plan has to take evolutionary character of knowledge

construction into account.

Different educationalists have suggested different models of constructivist

classroom learning which proposed some specific steps for constructivist

teaching. For constructivist teaching, the model developed by Roger Bybee is

widely used by practitioners. This model is best known as the “5 Es”. The 5Es

model provides a planned sequence of instruction that places learners at the

centre of their learning experiences, encouraging them to explore, construct

their own understanding of scientific concepts, and relate those understanding

to other concepts. Each of the 5-Es describes a phase of learning, and each

phase begins with the letter “E”: Engage Explore, Explain, Elaborate, and

Evaluate. An explanation of each phase of the 5E model follows:

Engage: Capture the learners attention, stimulate their thinking and help them

access prior knowledge. For this, the teacher has to create a situation to arouse

curiosity among learners.

Explore: Give time to learners to plan, investigate and collect information and

to think and reflect on the data to organize it, to make preliminary meaning.

For this, teachers have to facilitate learners with various activities such as

group work, discussion etc.

Explain: Encourage and involve learners is group work to share and analyze

the data they gathered.

Elaborate: Provide learners with on opportunity to extend and codify their

understanding of the concept and/or apply it to real-world situation. Learners

are to be encouraged and guided to apply, interpret, extend and enhance the

new concept (scaffolding). Teachers have to facilitate and guide learners in

expanding their understanding and developing interpretative /creative abilities

as well as reflective and critical thinking.

Evaluate: Evaluation of learners’ conceptual understanding and ability to

use skills begins at the stage of engage and continues throughout the model.

Outline of Lesson Plan

Name of Teacher trainee …………………………………………………….

Class : …………………………………………… Subject …………………

Topic : …………………………………………… Time Duration …………

Major Concepts (content Analysis): Teacher trainee has to analyze the given

content to identify the aspects of learning in terms of concepts, facts, theorems

and, conjectures.

Learning Objectives/Key Questions/Intentions: Teacher trainee has to

mention learning objectives or key questions for the given content.

Learning Process involved and required: Teacher trainee has to mention

skills that are involved and required for creating learning situations.

Learning Strategies: Teacher trainee has to identify learning strategies in

view of content analysis and inclusion of the same for effective learning.

39


Learning ExperiencesLearning Resources: Teacher trainee has to identify learning resources in

view of content analysis and inclusion of the same for effective learning.

Previous knowledge:

Introduction:

Teacher Activity Learner Activity

Engage

Engage

Presentation:

Learning

objective

Teacher and Learner Activities

1

2

Explore

Explain

Elaborate

Evaluation

Explore

Explain

Elaborate

Evaluation

Recapitulation:

Assignments:

A model lesson plan for your better understanding is given below.

Model Lesson Plan based on Constructivist 5 Es Approach

Name of Teacher Trainee ……..

Class : VII Subject: Mathematics

Unit: Playing with numbers

Topic : Prime and Composite Numbers

Major Concept (content Analysis): Prime numbers, composite numbers,

sieve of Eratosthenes, odd and even numbers

Prior knowledge: Factors, multiples, perfect numbers

Learning objectives: After completion of the topic learner will be able to:

• Understand the concept of prime numbers and composite numbers;

• Find factors of prime numbers and composite numbers;

• Understand sieve of Eratosthenes method;

• Analyze even and odd numbers.


Mathematics

40

Learning process involved: Classification, communication, interpretation,

reasoning, problem solving, argumentation etc.

Learning Strategies: Cooperative learning, role play, discussion etc.

Learning Resources: Construction paper, markers, cleared desktops, paper,

pencil, several small colorful pieces of poster board, marking tape etc.

Introduction:

Engage: Teacher writes numbers 1- 100 on the board. She decides to follow

the steps of sieve of Eratosthenes, and speaks them out loud as she goes.

Step 1 : Cross out 1 because it is not a prime number.

Step 2 : Encircle 2, cross out all the multiples of 2, other than 2 itself, i.e. 4, 6,

8 and so on.

Step 3 : You will find that the next uncrossed number is 3. Encircle 3 and cross

out all the multiples of 3, other than 3 itself.

Now, try to follow the similar steps for the next uncrossed numbers.

Presentation:

Explore: To form groups of two and ask learners to continue with the above-

mentioned procedure till all the numbers are either crossed out or encircled.

They may come up with a result like this:

1

2 3 4

5 6

7 8 9 10

11

12 13

14 15 16 17

18 19

20

21

22 23

24 25 26 27 28 29

30

31 32 33 34 35 36

37 38 39 40

41

42 43

44 45 46 47

48 49 50

51 52 53

54 55 56 57 58 59

60

61 62 63 64 65 66

67 68 69 70

71 72

73 74 75 76 77 78

79 80

81 82 83 84 85 86 87 88 89

90

91 92 93 94 95 96 97

98 99 100

41


Learning ExperiencesExplain: How are the crossed out numbers different from the encircled ones?

Is there a relation among all the encircled numbers and crossed ones?

The crossed out numbers are called composite numbers.

Learners may say that the crossed numbers ‘occur in other tables’ whereas the

encircled ones ‘occur only in 1’s and their own table’. They may say that all

the multiples of 2 are crossed, except 2 (no even number is prime except 2).

They may come to various conclusions regarding the concept of primes

numbers, composite numbers and statements in relation to odd-even numbers.

Elaborate: The learners will create their own construction squares. Each child

writes the numbers 101-120 on construction paper. The construction paper is

divided into small squares. The learners put cross sign on prime numbers .The

teacher displays a number on the white board, and ask if the number is prime or

composite. If it is composite, learner demonstrates prime factorization, again

using their construction squares. The teacher walks around the room to ensure

learners understanding.

Then, the learners put construction squares away. The teacher writes several

examples on the board for learners to complete with paper and pencil,

independently. She also assigns a few examples for homework.

The teacher checks for understanding by orally asking questions, listening and

observing learner response, throughout the lesson.

Evaluate: The teacher may ask the following questions to evaluate :

• Look at the table and tell me all the composite numbers less than 17.

• How many prime numbers exist between 10 and 50?

• How many even prime numbers exist between 50 and 100? (of course there

are none!)

• Find out all the prime number between 150 and 200 without using this

method.

Recapitulation: We have seen that numbers have patterns. You may notice

that the prime numbers exists in big numbers and small numbers. The smallest

prime number is two. The largest prime number is at infinity. Grouping is the

key to help determine if a number is prime or not. These groupings are also

known as factors.

Activity for Practice

4. Select a topic of your choice from secondary class mathematics text book.

Develop a lesson plan based on 5-E approach .


Mathematics

42

Check Your Progress



5) Why is unit planning important?

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6) How can concept mapping be useful in unit planning?

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7) What are various phases of 5 Es approach? Explain any one phase.

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8) List the important steps of a good lesson plan?

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43


Learning Experiences6.6 PLANNING INSTRUCTION FOR CHILDREN

WITH SPECIAL NEEDS

Nowadays, classrooms are becoming very diverse. A class consists of learners

with varied interests and abilities and also some children with visual

impairment, hearing impairment, learning disable. etc are likely to be present.

So, the traditional instructional strategy, fixed curriculum and fixed way of

assessment of learner performance would not work. Children with special

needs require additional support, adaptation in curriculum, teaching strategies

and assessment activities. So, at the time of designing learning experiences for

Mathematics, you must know about the learning needs of such children and

what difficulties face in Mathematics learning. According to that you can plan

your lesson. In the table 3.1, we are giving teacher’s action, learning resources

and adaption in assessment strategies for Mathematics teaching learning for an

inclusive classroom.

Table 3.1: Meeting the challenges of teaching Mathematics to children

with special needs.

Disability Area to be

compensated

Learning Experiences Learning

Resources

Adaptation in

Assessment

Visual

Impairment

Vision • Verbalize while writing

on black board

• Present subject content and information orally

• Reading aloud

• Assign work to learners in groups

• Collaborative strategies

• If possible, provide a copy of what you have

written on the board to

the learner

• Handover hand and hand under hand

method can be used

for identification of two and three

dimensional figures

• by using Braille

• by using mathematical Braille code

• Abacus

• Geo Board

• Special Geometry Kit

• Sphere wheel and Rubber Mat

• Taylor Frame*

• Audio Aids

• Tactile

• Haptic Device*

• Large print material

• Magnification Devices

• Screen Reader

• Talking material (Clock, calculator)

• Give a little extra

time

• Note taker

• Through oral presentation that

can be recorded

• Scribe/amanuen

sis

• Alternative Assessment

techniques


Mathematics

44

Hearing Impairment

Auditory and

Communica

tion

• Use of Sign Language

• Demonstration method

• When using visuals, allow time for the child

to view the board

• More opportunities to reading

• Lesson through Video

• Extra time to complete assignments

• Step-by-step directions

• Make use of

multimedia approaches for visual

representation of

course content.

• More than one mode of presentation like

manipulatives, verbal, pictorial, and symbolic

modes

• Use of visual

supplements

(projected materials,

whiteboard, charts,

vocabulary lists, lecture outlines)

• Captioning or scripts for announcements,

television, videos,

or movies

• Speech-to-text translation

captioning (i.e.,

computer on desk)

• Educational interpreter (ASL,

signed English, cued speech, oral)

• Buddy systemor notes, extra

explanations/directions

• Note taker

• Reduce quantity

of tests or test

items

• Use alternative tests

• Provide reading assistance with

tests

• Allow extra time

Intellectual

and Learning

Disability

Cognition

and Process • By using Multi sensory

approach

• Always use visual aids

and keep the focus on hands on and tactile

• Use real life contexts

• Frequent breaks

• List the steps/procedures for

multi-step problems

and algorithms

• Provide frequent checks for accuracy when

learners are doing class work

• Multimedia

Software

• Tape Recorder

• Videos

• Tactile

• Audio tapes

• Abacus

• Geo Board

• Screen Reader

• Talking material

(Clock, calculator)

• Extended

time

• Note taker

• Multiple testing

session

• respond on booklet

• Scribe/amanuensis

*Taylor Frame: The Taylor frame consisted of an aluminium frame and a set of metal pegs or

type with the patterns. The frame has rows of opening each set out as an eight pointed star.

The pegs could therefore be placed in the frame in one of the eight orientations, which could

be used to represent numbers, letters or signs. Maths can be composed in linear, vertical or in

algebraic notation.

**Haptic Device: Haptic devices are systems that can develop highly resolved two- or three-

dimensional space to give the user a physical feeling of the shape. Unfortunately, these

devices are very expensive, but would be the best way to represent text and non-text data.

45


Learning ExperiencesThe above ways of content presentation are definitely beneficial to all children

in your Mathematics classroom. When a teacher presents the subject content

through different ways by involving multi-sensory approach and also by taking

care of affective aspect; then it helps all types of learners including children

with disabilities in the classroom resulting in most effective teaching.

Check Your Progress



9) What types of adaptation are required in assessment process if child has

visual disability?

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

10) What types of learning resources can teacher use if child has intellectual

disability?

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

…………………………………………………………………………….

6.7 LEARNING MATHEMATICS THROUGH

PUZZLES, RIDDLES AND TRICKS

There are some issues related to Mathematics such as it is a difficult subject,

boring subject; it is a cold, dry and not interesting; and Mathematics is a

lifeless subject. How do these perceptions get rooted? As Mathematics

teachers, how can we change these perceptions? As teachers we can make

Mathematics interesting by using games, quizzes, riddles, tricks, etc. as

learning activities. Every child deserves to be successful and confident in

school life. Mathematical puzzle, riddle and tricks develop many life skills in

learners required to lead a successful and happy life. Children love games and

puzzles, so learning through games makes learning of Mathematics joyful and

interesting. Following are the benefits of recreational activities in Mathematics

teaching-learning situations:

• Encourage mathematical reasoning and logical thinking;

• Develop strategic thinking;

• Guide in concept development;

• Develop positive attitude towards Mathematics;


Mathematics

46

• Help to develop construction of ideas;

• Develop a variety of connections with the content that can form positive

memories of learning;

• Improve fundamental operations of Mathematics;

• Help teachers to cater to divers needs of children;

• Help in development of problem solving, critical thinking skills, decision

making skills etc.

Let us discuss in detail how to make Mathematics interesting by using puzzles,

riddles, tricks etc. in the following sub-sections.

6.7.1 Mathematical Tricks

We usually play games during our leisure time. A number of games are

developed in Mathematics, which apart from utilizing leisure time, create

interest, develop reasoning and logical thinking and positive attitude towards

Mathematics. Mathematical tricks look like magic. Let us see how a teacher

can show magic through mathematical trick in his class:

Mr. Ravi, a Mathematics teacher was teaching 9th

standard. He said the class

“Today I will show you a magic.” The entire class was very excited. He asked

one of the learners to come forward. All the learners wanted to go forward. Mr.

Ravi called Prashita to come forward.

Mr. Ravi started asking questions to Prashita:

• Keeping your age in mind, multiply the first digit of your age by 5.

• Add 3 to this number.

• Then double the obtained number.

• Add the second digit of your age to the obtained number.

• Tell me the number which you get.

• Now deduct 6 from the number that you got and you will have your age.

Prashita got her correct age which is 12. Now, all learners wanted to apply this

trick. So Mr. Ravi gave same instruction to the whole class. All children got

their correct age. They asked him how this magic happened. Mr. Ravi told that

this is not a magic, but only a trick. Then Mr. Ravi explained how this trick

worked.

How it works: Prashita is 12 years old. Let the first digit of her age be ‘x’ and

the second digit of her age be ‘y’. So, for this example, x is 1 and y is 2.So age

of the Pradhita is 10x + y.

Finally we got the right age.

Step 1 Multiply 1st digit of age by 5 5x

Step 2 Add 3 5x + 3

Step 3 Multiply by 2 2(5x + 3)

Step 4 Add the 2nd digit of age 2(5x + 3) + y

Step 5: Subtract 6 2(5x + 3) + y – 6

=10x+y

47


Learning ExperiencesLet us see one more example of trick that you can use in your class.

Example 1

Take any number; Double the number; Add 9 to this number ;Subtract 3 from

the above number; Divide it by 2; Next subtract your original number from the

obtained number.

Your answer is 3.

Trick : Let the number taken be x

Double the number and get 2x

Add 9 to the number and get 2x + 9

Subtracting 3 from the number and get 2x + 6

Divide the number by 2 and get x + 3

Subtract from the original number to obtain 3

That is why the answer is 3.

6.7.2 Puzzles

Puzzle is a statement apparently leading to a particular answer/result often

seeming to be impossible, whereas by means of deeper reasoning and analysis

is found a different answer altogether. Puzzles are called brain teasers. By

using puzzles, you can make Mathematics learning more interesting in the class

and develop reasoning and logical thinking in your learners. Here is an

example of puzzle, which can be used by a Mathematics teacher in his class to

make Mathematics interesting.

There were 3 friends. Once three of them climbed a tree and plucked some

mangoes. They saw the owner of the tree coming, so they put all the mangoes

in a basket and hide it under a tree. When the owner went off, one of the three

took out the basket and divided the mangoes in three equal parts and took one

part, the other man came and he also divided the remaining mangoes in three

equal parts but after dividing, he saw that one mango was left out so he took

one part and extra one mango. Lastly the third man came and took all left over.

It is found that three of them have got same number of mangoes. Find out what

was the total no of mangoes.

With the help of Algebra some of the learners can solve this puzzle. Solution is like that:

Let total no. of mangoes be x.

As stated in the puzzle, first one did three parts (x/3,x/3,x/3) and first man took

(x/3).

Remaining Part � ��

�

2nd

person divided remaining part in three parts but one mango was left.

So the division is like that: y+y+y+1.

2nd

person took y+1 number of mangoes.

Left over part was taken by 3rd

person that is 2y.

So �� 1 � 2

Or y=1 and x=6

So total no of manages is 6


Mathematics

48

6.7.3 Riddles

Questions with clever or surprising answers are popularly called riddles. A true

riddle always asks a question that can be answered reasonably. The riddle is

something different from puzzle. Puzzle is a mind-game, whereas riddles are

world’s oldest guessing games.A riddle is a statement or question or phrase

having a double or veiled meaning. All riddles are puzzles, but not all puzzles

are riddles. Here, are some examples of riddles that can be used to make

mathematics learning more interesting and meaningful.

Riddle1: Kapil and Manish were walking down the road with their pet dog

named Jackey. They started at the same time, in the same direction and to the

same point. Kapil walks with a speed of 4 km/h and Manish at 5 km/h. Jackey

ran towards Kapil passed through Manish and back again at a constant speed of

12 km/h. Jackey did not get slow on the turns. How far did Jackey travel in 1

hour?

Solution 1: Jackey travelled 12km in an hour. The reason is Jackey is running

with a speed of 12 km/h and the walking speed of Kapil and Manish have no

effect on the speed of Jackey.

Riddle2: What mathematical symbol can be placed between 6 and 8, to get a

number bigger than 6 and smaller to 8.

Solution 2 : A decimal point can be place between 6 and 8 i.e. 6.8

Riddle 3. A well is 20 meters deep. A frog climbs 6 meters during the day, but

falls back 4 meters during the night. Assuming that the frog starts from the

bottom of the well, on which day does frog get to the top?

Solution 3: Frog gets to the top on 8th day. The reason is frog climbed 2 meters

(6m – 4m) per day. In 7 days frog covers 14 meters and next day it climbs 6

meters and comes out of the well, so the total duration taken by the frog to

come out of wells is 8 days.


5. Design guessing games like mathematical riddle, puzzle, and tricks. And

use these in classroom teaching-learning to make mathematics more

interesting and meaningful.

Check Your Progress



11) What is the benefits of recreational activities in Mathematics teaching-

learning?

…………………………………………………………………………..

…………………………………………………………………………..

49


Learning Experiences12) How is mathematical riddle different from Mathematics puzzle?

…………………………………………………………………………..

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6.8 LET US SUM UP

In this Unit we have discussed that learning of learners depends upon suitable

and well organized learning experiences and knowledge cannot be transmitted

to learner but is constructed by the learner on the basis of his/her learning

experiences. So, involvement of learner is very important in teaching-learning

process. We have also discussed the levels of planning: annual planning, unit

planning and lesson planning. At the time of lesson planning, mathematics

teacher must know the learning needs and difficulties of the children with

special needs that they face in Mathematics learning. According to their

requirement, teacher must give additional support and adapt assessment

strategies. Teacher can also use Mathematics puzzles, riddles, tricks etc. to

make learning joyful and interesting.

6.9 UNIT END EXERCISE

1) Discuss the sequencing of organization of learning activities.

2) Observe a teacher teaching in a class and note down the following

• Major Concepts (content Analysis):

• Learning Objectives/Key Questions/Intentions

• Learning Process involved and required

• Learning Strategies

• Learning Resources

3) Prepare lesson plan on the following:

• Percent for class VIII.

• Linear equation in two variables for class IX…

• Pythagoras for theorem Class VII.




Mathematics

50

2) Objective based, Learner orientation ,Richness of experience, Suitability

to the mental level, Practicability, Evaluation

3) The opening activities or learning experiences for each of the main ideas

are motivational to secure the interest, focus attention and build readiness

of the learner for the topic .The learners should be given first hand

experiences with concrete objects or real life situations so that they

observe, locate and discover some known mathematical ideas involved in

the situation . At development stage, learning experiences provide for

experimentation, doing and thinking analytically. These help in exhibiting

connections or relationships between what is known and what is to be

known.



6) Concept mapping can be used as an excellent planning device for unit

planning . It helps in organizing and planning learning activities to

enhance learning experiences of the learners. Mapping the concepts may

increase your ability to provide meaningful learning experiences to

learners by integrating concepts. It facilitates identify concepts and sub

concepts that you want to emphasis. It facilitates the teacher

reconceptualise the course content. It helps to understand relationship

between facts and concepts through cross-links, which leads to the

development of lesson plan based on constructivist pedagogy.


8) Major highlights for lesson plan

� Major Concept (content Analysis)

� Learning objectives:

� Learning process involved

� Learning Strategies:

� Learning Resources:

� Introduction: Engage

� Presentation: Explore, Explain, Elaborate ,Evaluation

� Recapitulation

9) Give a little extra time, Note taker, through oral presentation that can be

recorded, Scribe/amanuensis, Alternative Assessment techniques

10) Multimedia Software, Tape Recorder, Videos, Tactile, Audio tapes,

Abacus ,Geo Board, Screen Reader, Talking material (Clock, calculator)


12) Questions with clever or surprising answers are popularly called riddles.

A true riddle always asks a question that can be answered reasonably. The

riddle is something different from puzzle. Puzzle is a mind game whereas

riddles are world’s oldest guessing games. A riddle is a statement or

question or phrase having a double or veiled meaning. All riddles are

puzzles, but not all puzzles are riddles.

51


Learning Experiences6.11 REFERENCES AND SUGGESTED READINGS

• Butter and Wrens (1950). The Teaching of Mathematics, McGraw Hill

Series in Education.

• Chaddha and Aggrawal (1998). The Teaching of Mathematics, Dhanpat

Rai & Sons, Ludhiana.

• IGNOU (2000). Teaching of Mathematics, ES- 342 Block1,IGNOU, New

Delhi.

• James, Anice (2005). Teaching of Mathematics, Neel Kamal Publication

Pvt ltd., Hyderabad.

• NCERT (2012). Pedagogy of Mathematics, NCERT, New Delh.

Mathematics Teaching


Mathematics

52

UNIT 7 LEARNING RESOURCES AND ICT

FOR MATHEMATICS TEACHING-

LERNING

Structure

7.1 Introduction

7.2 Objectives

7.3 Learning resources

7.3.1 Importance and Use of Learning Resources

7.3.2 Learning Resources from Immediate Environment

7.4 Mathematics Lab and Mathematics Corner

7.5 Mathematics Club and Forum

7.6 ICT Need, Importance and Use in Learning of Mathematics

7.6.1 Need and Importance of ICT

7.6.2 Use of ICT in Learning of Mathematics

7.7 Selection and Use of Appropriate Media

7.8 Let Us Sum Up



7.11 References and Suggested Reading

7.1 INTRODUCTION

The constructivist learning approach calls for the extensive use of various

learning resources as self-learning is emphasised in classrooms. Abundant use

of learning resources gains attention of learners in the learning processes and at

the same time it helps the teacher to sustain the involvement of learners in

learning. Apart from that, the learning turns into an enjoyable activity and all

round development of learners is assured both in cognitive and co-cognitive

aspects. Keeping the relevance of learning resources in mind, the widespread

use of learning resources is suggested both at elementary and secondary level.

In this unit, we would deliberate on the use of learning resources at school

level. Also the different types of resources that can be used in mathematics

classrooms will be discussed. Thus, the unit will help you in indentifying and

using various digital learning resources in your classroom.

7.2 OBJECTIVES

After going through the unit, you will be able to:

• describe learning resources and their importance in learning mathematics;

• identify various learning resources from immediate environment;

• develop activities for effective use of learning resources in mathematics

classrooms;

53

Learning Resources

and ICT for


–Learning

• explain the importance of math laboratory and math corners;

• identity the activities that can be undertaken by maths clubs and forums;

• describe importance and use of ICT in learning mathematics; and

• discuss the factors considered in the selection of appropriate media.

7.3 LEARNING RESOURCES

What are learning resources? Learning resources are texts, audio video

materials and digital aids that assist you in effective transaction of

curricular content. The major learning resource is the text book prepared by

central and state governmental educational agencies while a number of other

learning resources are also available. These may be manmade, improvised or

material available in the nature. You can also find learning resources in the

immediate environment, which will be discussed in sub section 7.3.2. It is also

to be noted here that, with the advancement of technology, a number of digital

learning recourses are also developed. Some of the common learning resources

are listed below:

Textbooks (Print and

Digital)

Work books

Activity Books

Flashcards

Posters

Educational games

Magazines and

Periodicals

Study Guides

Teacher Guides

Laboratories

Models

Reference Books

E-resources

Radio

Television

LCD projector

Computer

Internet Resources

Social networking Sites

Blogs

Wikis,

Discussion forum

Mobile learning

E-text

E-content

Virtual reality

OER

Second Life

Figure 7.1:Various Learning Resources

7.3.1 Importance and Use of Learning Resources

Learning resources serve many purposes. As you know, lecture method is the

easiest/common method. Many times, subject contents are not conveyed

appropriately to the learners through lecturing style. But, the creative

intervention of teachers can bring dynamism in the classrooms by employing

learning resources in lecture method. The same is true for other teaching

methods. Learning resources are imperative in teaching-learning situations due

to the following features. These resources:

� help learners to be involved fully in the learning process as learning

resources are powerful tool to gain and sustain motivation,

� facilitate learners to comprehend subject concepts effectively as they can

correlate the verbal instruction with real experience,

� assist learners to learn effectively and remember concepts for long,



Mathematics

54

� help learners to comprehend concept with clarity and bring vividness in

learning,

� help learners to concretize abstract concepts, and thereby enhances the

comprehension,

� reduce verbal communication on the part of teachers, and

� help learners to develop inquisitiveness, curiosity and interest in learning.

Apart from the above stated points, learning resources are important because of

the fact that learning is enhanced as learners experience hands on training and

real practice. At this point, make note of the following figure that also validates

the importance of learning resources.

Figure 7.2: How we learn

As a mathematics teacher, you may look for options that would enable you to

use learning resources effectively in your classroom. Let us discuss an

example.

Maya, a mathematics teacher was teaching the concept of ‘perimeter’ to her

learners. What she did was, instead of verbally introducing the concept, she

asked the learners to sit in groups. Thereafter, she asked them to complete the

assignment given on the computer (The class is conducted in the computer lab

and a group of five learners were assigned one computer). She was making use

of one of the web resources. This is a simple example. You may explore more.

Old Saying

What I hear I forget

What I see I remember

What I do I understand

We Learn

1% through Taste

1.5% through touch

3.5% through smell

11% through hearing

83% through sight

We Remember

10% of what we read

20% of what we hear

30% of what we see

50% of what we see & hear

80% of what we say

90% of what we say and do

55

Learning Resources

and ICT for


–Learning

(Source: http://www.mathopolis.com/questions Retrieved on 05/12/2016)


1) Prepare the list of learning resources and their usefulness in teaching

mathematics.

7.3.2 Learning Resources from Immediate Environment

The learning resources that we have discussed above are readily available or

are man made. But, for you as a teacher, it may not always be possible to

procure these resources for many reasons. In such cases, you can opt for natural

(immediate environment) learning resources. Such resources are available in

the classroom, own house or nature. Nature is the biggest reservoir of learning

resources. These resources are also known to be improvised aids or improvised

learning resources. Improvised learning resources are those resources that

are prepared from waste material or material available in the immediate environment. For example, if you want to teach 3D shapes, you can bring

empty match boxes, unutilized utensils, and so on. Even can prepare boxes

with unutilized cardboards, or thick paper cuttings, etc. Let us discuss an

example of using learning resources from our immediate environment.

Mrs. Radhika, a secondary school teacher of mathematics decide to teach

the concept of non intersecting lines, secant and tangent of a circle in

her class of Xth

standard. When a circle and line is given in a plane, there

are three possibilities: (i) the line will not touch the circle; (ii) the line

will touch at two points or the line may touch at a single point as shown

in figure 7.3. In the first case, we call the line segment PQ, a non-

intersecting line, the second a secant, and the third a tangent of the circle.

To teach these concepts, rather than drawing on the black board, she has

used learning resources from her immediate environment. It is

interesting to know what she used. Radhika brought a few sticks and

bangles, which were kept unutilized at her home. With those objects, she

was successful in organising the group activity and thereby helping the

learners to learn the afore mentioned concepts.



Mathematics

56

You may think of such learning resources as you plan your learning activities.

Fig 7.3: Non Intersecting Line, Secant and Tangent of a Circle

Check Your Progress

Note: a) Write your answers in the space given below.

b) Compare your answers with those given at the end of the Unit

1) Learning resources are important in mathematics teaching. Justify the

statement.

…………………………………………………………………………….

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2) List some of the learning sources found in nature, and explain, how you

would use the same in your classroom for mathematics teaching.

…………………………………………………………………………….

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7.4 MATHEMATICS LABORATORY AND

MATHEMATICS CORNER

Learning resources also include mathematics laboratory (math lab) and

mathematics corner, which slightly differ in their organization and material

possessed. Let us discuss the major differences among them “Mathematics

laboratory is a unique room or place, with relevant and up-to-date equipment,

known as instructional materials, designed for the teaching and learning of

mathematics and other scientific or research work, whereby a trained and

professionally qualified person (mathematics teacher) readily interacts with learners on specified set of instructions” (Adenegan,2003). Math lab is a place

where learners get opportunity to engage with mathematical objects, experiment

mathematical theories, solve mathematical puzzles and problems, play

mathematical games, experience hands on training, and so on. The material or

57

Learning Resources

and ICT for


–Learning

equipment that can be found in the mathematics laboratory includes, among

others, constructed (wooden/metal/plastic made) mathematical sets, charts and

pictures, computer(s), computer software, audio-visual instructional materials such

as projector, electronic starboard, radio, television set, tape recorder, video tape,

etc, solid shapes (real or model), bulletin board, three-dimensional aids, filmstrips,

tape photographs, portable board or whiteboard, abacus, cardboards, tape measure,

graphics, workbooks, graphs, flannel boards, flash cards, etc (Adenegan,2003).

Math lab consists of a number of materials and objects. Mathematics corner is a

miniature form of math lab. Math lab is highly organised, consists of several

objects/materials/instruments and requires specialized skills in developing them,

but math corners are simple and contains few mathematical objects and items. You

can setup a math corner at the corner of any other lab or on the corner of

classrooms. Usually, math corner is a place where learners find the

ordinary/common kinds of mathematical items and you can utilize these items

during the classroom interaction. In a way, math corners include math related

teaching-learning aids. It is to be noted that, the objects found in math labs can

also be found in math corners.

Figure 7.4: Mathematics Laboratory

(Source:http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics_Laboratory)

Now let us discuss the importance of math labs and corners. The math

labs/corners are important due to the following reasons:

� It helps learners to comprehend mathematical concepts effectively by

utilizing concrete objects and experiencing real situations.

� Learners can test and experience the theoretical knowledge and discover

different mathematical properties.

� It enhances the interest and motivation of learners to learn mathematics.

� Math labs provide objects and materials, which help learners to relate

concepts with their daily life activities and nature.

� Individual learning is promoted while exploiting math labs as learners

engage in exploration of mathematical contents in their own way.

� The cognitive development is supported and enhanced as learners exercise

both mind and body by engaging in learning activities.

� The teacher can demonstrate learning concepts by connecting with multiple

learning resources present in the math labs.



Mathematics

58

� It helps in the development of skill of enquiry and critical thinking.

� The principle of ‘learning by doing’ can be practiced by learners.

Now, let us see the objects that are generally found in math labs. It is your

obligation as a mathematics teacher to initiate steps to develop math labs in

your school. It is not necessary to have many items instead the basic objects

must be organised in the lab. While developing math labs, the following

objects/materials/equipments can be included in it.

Figure 7.5: Materials/Objects Found in Mathematics Library/Corner

Check Your Progress


b) Compare your answers with those given at the end of the Unit

3) What is a mathematics laboratory? How is it different from mathematics

corner?

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As a teacher, your concern should be how to use math labs and corners. Let us

discuss with an example.

Concrete Materials

beads, pebbles, sticks, ball

frames, seeds, balance,

weighs, measuring tapes, scissors, pins, abacus,

cardboard, board pins,

chart paper, graph paper

Pictures and Charts

Photographs of

mathematicians, history of

mathematicians, charts showing contributions of

mathematics, biographic

of mathematicians

Weighing and Measuring Instruments

as measuring tapes,

balances of different types, measuring jar and

graduated cylinder,

calculators

Drawing Instruments

compass, rulers,

protractors,stencils

Surveying Instruments

Angle mirror,Transit,Plane

table and

alidade,Clinometers,Sextant , Proportional dividers

Others/E-Reources

Models, Bulletin board,

Black board, Computers,E-

learning resources

59

Learning Resources

and ICT for


–Learning

Mr. Kishore was teaching the Herons formula to his ninth class learners. He

started the class as given below:

Kishore : How are you ?

Learners : Fine sir

Kishore : today, we are going to study a new concept.

After saying this, he took a photograph kept in the math corner. After

showing the photograph, he continued asking

Kishore : Do you know whose photograph is this?

Learners : No sir

Kishore : It’s ok. We will see who this mathematician is. Before that, I

will give you some triangles. Hope all of you know how to calculate the area

of triangle. Is it?

Learners : Yes, Sir.

Thereafter Kishore provides (The same is available in math corner and math

laboratory) different triangles to a group of 5 learners. Then he continues;

Kishore : Learners here is the task for you. You have to find the area of

the triangle given to your group.

Figure 7.5: Heron

This is a snapshot of the conversation of Kishore, where he was trying to teach

method of calculating the area of triangle using Herons formula. You might

have noticed that, to teach the concept, Kishore has used the photograph of

Heron and triangles of different area. This example is just a hint that shows

‘how a teacher can utilize the material/objects/equipments of the math

lab/corner’. You may think of such instances during teaching.


2. You have seen how Kishore has used math lab in teaching mathematics.

Suggest an activity that may be employed in math lab/corner as a learning

aid.



Mathematics

60

7.5 MATHEMATICS CLUB AND FORUM

Similar to math lab/corner, mathematics club and forum is also another

important learning resource. NCF (2005) suggested ‘mathematisation of

learner’s thought processes’ as one of the major goals of mathematics teaching.

How do we develop the skill of mathematisation among learners? You may

motivate learners to engage in math clubs and forums. Math clubs/forums are

to be viewed from two angles; a learning resource and as a place to engage

learners in extracurricular activities. Leaning resource in the sense that

mathematics teachers can utilize math club/forum to engage their learners to

discuss, debate and deliberate on various topics of mathematics. On the other

hand, different co-curricular activities such quizzes, study tours etc. can be

organised by mathematics clubs/forum.

Math club/forum is a group of individuals getting together to organise events,

discuss, debate on various topics pertaining to mathematics. The club arranges

various events such as birthdays of mathematicians, math days etc. Also, the

clubs and forums are engaged in organising discussions, debates, seminars,

study tours, etc. Ultimately, math clubs/forums help learners in developing

interest and motivation in mathematics learning. There are different ways of

involving learners in learning mathematics; math club/forums play a major

role. So as a math teacher it is your duty to initiate processes to develop math

clubs/forums. The math club/forums work under the guidance of the math

teacher.

Apart from this, math clubs/forums are important because of the following

reasons:

• Math clubs/forum help learners to engage in various activities related to

mathematics learning.

• Facilitate and arouse interest and motivation in learners to learn

mathematics.

• The leisure time can be properly utilized by involving in programmes

organised by math clubs/forums.

• Learners are exposed to various activities of math clubs/forums thus help

them to test theories learnt in their math classes.

• Provide opportunity to learners to initiate different programmes.

• Help learners to enhance skill of leadership, problem solving, joint

responsibility, hard work, etc.

• Math clubs/forums help learners to engage in activities where they can

discuss, contest and ponder over various themes of mathematics.

Let us see, how mathematics clubs/forums can be set up? What is the general

structure of such clubs? You might have seen various clubs/organisations in

your school and nearby areas. Such clubs organise events such as blood

donation camps, eye testing, cultural campaigns and so on. In such

organisation, we find office bearers and executive committees. In similar

fashion, math club/forums are set up in schools. For this, the initiation must

come from you as a mathematics teacher. So, it is pertinent to say that, you

have a bigger role in creating math clubs/forums.

To set up math club/forum, you can organise a meeting with students. In the

meeting, draft constitution of the club may be discussed and further course of

61

Learning Resources

and ICT for


–Learning

action may be initiated. The constitution can be prepared by you in consultation

with the head of the institution (Principal/Head Master). The points to be

included in constitution include; name of the club, aims and objectives of the

club, details of membership, etc. The club/forum should have head of the

institution as its patron and a mathematics teacher as convener. The office

bearers such as President, Vice-President, General Secretary, Joint Secretary,

and treasurer must be selected from the learners. After electing the office

bearers, the programmes to be organised may be discussed, and finalised. One

point to be noted here is that, it is not necessary to follow the format that we

have discussed instead you have full freedom to modify as per your need and

situation. The following activities can be undertaken by the math clubs/forums;

• Educational talks, lectures, key note addresses by renowned

mathematicians, teachers, math specialists, etc.

• Celebration of birth days of mathematicians and organization of other

important mathematical events, math days, etc.

• Discussions and debates on various topics and issues related to

mathematics.

• Quiz programmes.

• Conduction of math fairs, math olympiads, exhibitions, etc.

• Exhibition of mathematical models, aids, charts, etc.

• Seminars and workshops.

• Publication of magazines and periodicals on weekly/monthly/yearly basis.

Check Your Progress

Note: a) Write your answer in the space given below.

b) Compare your answer with those given at the end of the Unit.

4) Discuss a few activities that can be undertaken by mathematics

club/forums.

………………………………………………………………………………

………………………………………………………………………………

………………………………………………………………………………

………………………………………………………………………………

………………………………………………………………………………

Now, let us discuss the practical application of math clubs/forums.

Mr. George was engaged in teaching the concept of perimeter in class 9. After

teaching the concept, George directed the math club members to organise an

exhibition. George further suggested to the learners that the members of the

club should initiate steps to organise charts which contained different shapes

and ways of finding its perimeter explained in it. At the same time, the club

members could also develop models of the same. Learners agreed to it and club

office bearers requested all learners to prepare models and charts and the same

ware displayed in the exhibition.



Mathematics

62

As we have discussed in 7.4, the examples that show the practical applications

of math lab and math corner were to give you a hint. The practicalities of both

vary as per the nature of the concept, type of learners, classroom environment

and so on. You must be very cautious and creative in deciding effective

adoption of these learning resources.


3. The club activity discussed above resulted in development of charts and

models depicting the procedure for finding perimeter of various shapes.

Analyze a similar activity in your class and organise an exhibition.

7.6 ICT NEED, IMPORTANCE AND USE IN

LEARNING OF MATHEMATICS

ICT has become an inseparable component of teaching –learning process. What

do you mean by ICT? ICT stands for Information and Communication

Technology. ICT helps to store, process, disseminate, retrieve and transmit

information with the aid of technological medium. The UNESCO defines ICT

as “forms of technology that are used to transmit, process, store, create,

display, share or exchange information by electronic means. It includes, not

only traditional technologies like radio and television, but also modern ones

like cellular phones, computer network, hardware and software, satellite

systems and so on, as well as the various services and applications associated

with them, such as videoconferencing”. Thus ICT includes all technological

gadgets that help to store, transmit and communicate information.

What does ICT mean in educational context? Let us discuss with an example

from class room context. Imagine that a teacher wants to assess the progresse

of learning of his/her learners after teaching a particular concept, say for

example; volume of a cube . In such a situation, the teacher will teach the

concept and assess his/her learner using a computer made multiple choice test.

Thus it is evident that the teacher has made use of computer to asses his/her

learners in place of common paper pencil test. This is a way of utilizing ICT in

the educational context. Similarly there are multiple situations in the

educational process where you can employ ICT.

ICT in educational process is mainly employed in four ways, namely; teaching

learning, evaluation, administration and professional development. Let us

briefly discuss these aspects. Generally, teaching is primarily focused on

transaction of subject contents through lecture method, but with the emergence

of technology, many technological tools are employed for the same. For

example, virtual experiments, power point presentation, video conferencing,

internet, etc are used during the teaching –learning process. Thus, ICT is

widely adopted in teaching-learning processes. Similarly, in the case of

assessment and evaluation, multiple tools and software are used. For example,

online testing, computer tests, e –portfolios, etc., are used to assess learners’s

progress. ICT also finds application in administration and management. Storing

learners data in excel sheet, management information system (MIS) etc., are

some among them. ICT are used in professional development programmes.

63

Learning Resources

and ICT for


–Learning

Some of the latest technologies like, OERs, Massive Open Online Course

(MOOC), Free and Open Source Software (FOSS) assist various stakeholders

to professionally update and helps in career development. A snapshot of the

multiple roles of ICT in education is given below.

Figure 7.6: ICT in Education

Check Your Progress



5) Explain the applications of ICT in mathematics teaching-learning.

………………………………………………………………………………

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7.6.1 Need and Importance of ICT

You know that ICT has influenced teaching-learning, administration,

assessment and professional development of learners. Now, our discussion will

focus on its impact on mathematics teaching-learning. First of all we will

discuss the need and importance of ICT in the field of mathematics teaching-

learning. The following depicts the need and importance of ICT.

� The emergence of various learning resources has made the process of

learning easy for learners. Apart from that, teacher can succeed in

developing interest and motivation among learners with the aid of ICT

learning resources. The black boards, charts, models, etc. are the learning

Teaching and

Learning

• E-content

• OER

• E-learning

• Blog

• Wiki

• Mobile

learning

• Interactive

white board

Administartion

• Database

• MIS

• Record

Keeping

Assessment

• E-assesemnt

• E-test

• Online test

• Computer

tests

• E-portfolio

• Quiz tools

• E-rubrics

Professional

Development

• MOOCs

• SPOCs

• Discussion

Forum

• Online

Communities

• Online

Courses



Mathematics

64

resources of pre-digital era. In addition, teachers can also use digital

learning resources. Some of the digital learning resources are computer, e-

books, educational software’s, etc. Thus you may employ such digital

learning resources to make learning effective for learners.

� The shift in learning styles of learners proves the relevance of ICT in

teaching learning. It is common that, learners rely on traditional print text

books to comprehend subject knowledge. But, today’s learners are tech

savvy and prefer to use multiple digital devices for learning. Thus you

should supplement teaching with multiple ICT devices.

� Today constructivist approach of learning is practiced that help learners to

develop their own understanding of subjects based on their previous

experiences. In such a scenario, learners need to be supplied with multiple

sources (preferably digital in nature) as a supplement to build their own

knowledge and experiences of learning.

� Anywhere, any time learning is possible with the use of ICT. Learners get

opportunity to access information at their pace and time. As they search for

information, multimedia approach of education is encouraged. Thus,

learners’s weakness and strengths in learning can be easily identified and

remediated.

� ICT access helps learners to obtain latest information/knowledge in

different subjects.

� Multiple channels of communication are available that help learners to

interact, communicate and share information. Thus, flow of information

and knowledge is achievable that consumes less time.

� Learners can access various online repositories, online libraries, online

books, etc. Thus ICT provides opportunity for extra reading and rectifying

abstractness of concepts.

� ICT offers various devices and learning sources that support the learning

needs of learners with learning disabilities.

� ICT integrated education prepares learners to develop adequate skills and

all-round development.

� The efficiency and smartness of learning is enhanced with the use of ICT.

Learners learn better, comprehend knowledge with ease, retain the learned

contents and easily apply them in practical situations. It helps in

development of multiple skills both cognitive and physical.

� ICT helps teachers to present learning contents in multiple forms. The

teaching of complex concepts is made easy for learners with the aid of

ICTs. The theory of self and independent learning is promoted.

Activity for Practice 4:

4. Develop a blog focussing ICT’s importance in mathematics teaching

learning. Prepare a report of it.

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Learning Resources

and ICT for


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Check Your Progress



6) “ICT has much relevance in constructivist approach of learning”.

Comment on this statement.

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7.6.2 Use of ICT in Learning of Mathematics

In this section, we will discuss some of the ICT resources and method of

integrating them during teaching sessions. ICT includes multiple learning

resources and technologies starting from the radio to the most modern

augmented reality (virtual learning) like radio, television, LCD projector,

computer, internet communication, social networking, blogs, wikis, discussion

forum, mobile learning platforms, e-text, e-contents, virtual reality, OERs,

MOOCs, etc.

The National Policy on Information and Communication Technology (ICT) in School Education published in the year 2012 and National Mission

on Education through ICT (NMEICT)-2009, have advocated the adoption of

ICT at school and higher education level. The National Policy on ICT in

School Education (2012) recommended web-based digital repositories to host a

variety of digital content, appropriate to the needs of different levels of learners

and teachers. The National Repository of Open Education Resource

(NROER) is one among them. NROER is a collection of videos, audio files,

images, documents and interactive modules for all school subjects and grades

in multiple languages. Similarly, “e-Pathshala” (Web-site containing

approximately 364 eBooks, 137 videos and 100 audios this number is

increasing day by day) is another major initiative of e-learning for school

education.



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Figure 7.7: Major ICT initiatives of Govt of India (Source: http://mhrd.gov.in/)

One point to be stressed at this juncture is the creativity and thought process of

you as a teacher that would enable you to utilize technology in teaching.

Technology enabled learning is a major impact of ICT. There is a variety of

ways by which ICT can be utilized; it could be blog, wiki, e-content,

interactive white board and so on. Let us see how Naveen, a mathematics

teacher, of a government school, utilized ICT in teaching ‘ratio among the

volumes of right circular cone, hemisphere and right circular cylinder’. To

teach the same concept, Naveen had two options; either lecture method or

blended approach (using ICT). He went for the second option. What he did was

that after the theoretical explanation, he realized, learners are confused and

they found it difficult to comprehend the concept. At such a point, Naveen

utilized the OER repository of NCERT (Two screen shots of the OER are

shown below). This video is showing an activity, which help learners to

understand the ratio among the volumes of right circular cone, hemisphere and

right circular cylinder. Using this video presentation, Naveen could easily help

learners to gauge the described concept. Thus the use of NROER is an example

of utilizing ICT in teaching –learning.

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(Source: http://nroer.gov.in/55ab34ff81fccb4f1d806025/file/57d17e3816b51c090c38685a)

The pedagogy followed by Naveen is a mode of presenting the content using

ICT. So, you may bear in mind that, it is up to you as a teacher to decide the

ways of using ICTs in teaching learning. There are no stringent rules or styles

in using it. It depends on the imagination and creativity of the teachers.


5. Visit a few mathematics related websites and suggest learning activities

utilizing the visited websites.

6. Prepare a write up on various ICT initiatives of Govt. of India and their

usefulness in mathematics teaching

Check Your Progress



7) How will you use NROER repository for mathematics teaching?

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7.7 SELECTION AND USE OF APPROPRIATE

MEDIA

Do teachers use learning resource (media) at their own discretion and

convenience?

Is there any particular teaching resource that is applicable to all

classes/learners? Is it necessary to employ media with all teaching sessions?

These are some of the questions that you should be cautious about as you plan

for a media integrated teaching session. So you should definitely plan in

advance and prepare for handling a media integrated teaching session. In this

section, we will discuss some of the factors, that you should be aware of, for

integrating the media.

As you know, in the earlier sections we discussed various learning

resources/media that are useful for mathematics teaching. Thus, it is certain

that, teachers have the freedom to select any media from the basket of learning

resources/media. But how will a teacher select media? According to

Romiszowski (1997) the following factors influence the selection of media:

1) Task Factors: It refers to the nature of job in hand i.e. what are the

learning objectives? What are the behavioral changes that the teacher

wishes to develop in learners? What are the pedagogical approaches going

to be followed for transacting the curricular content? What time should be

devoted to the process? , etc.

2) Learner Factors: Learner factors include learners’s age level, motivational

characteristics, personality and individual differences, willingness for

learning, etc. Today, inclusion is emphasized in classrooms. In such

classrooms, learners with special needs are taught along with normal

learners. Thus, while selecting the media/learning resource for teaching,

care must be taken to meet the learning demands of both normal and

learners with special needs.

3) Economic/Availability Factors: It includes the cost of learning

resources/media, availability of media, working conditions of the media

and so on. As we know, a calculator is less costly compared to a computer.

So, if a mathematics teacher wish to teach concepts related to arithmetic

calculations, she/he may prefer simple calculators in place of computer.

This saves energy, time, complexities, etc. Similarly, situations that require

a camera, may utilize mobile cameras which are handy and mostly

available with teachers.

Now, let us discuss a practical example that interconnects all these three

factors. To teach the concept ‘bisector of a given angle’, the following

procedure may be followed. So, the task factor involved is ‘helping learners to

comprehend the process of drawing bisector to a given angle”. In this case, the

teacher anticipates that at the end of the class, learners would be able to draw

bisectors to any given angle. To teach the concept, the teacher has 40 minutes

and she/he planned a group activity. Why group activity? The number of

learners in the class was 40 and it was difficult for her to provide computer to

each learner. Thus, the learner factor involved here is the ‘number of learners

in the class’ and economic/availability factor is the distribution of computers as

computers are readily available in the computer laboratory. After deciding on

the medium, the teacher directs learners to complete the task mentioned in the

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Learning Resources

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self learning module. The module was set up in the computer before. This is an

example that shows how a mathematics teacher employed the three factors

discussed above.

Apart from the three factors discussed, you must also understand about the

concept of Technological Pedagogical Content Knowledge (known as

TPACK), a framework that help teachers to adopt technology in teaching

learning. In TPACK, ‘T’ stands for technology and refers to the knowledge of

teachers in technology that he /she wishes to employ in his/her classroom.

What are these technologies; for example virtual learning, web 2.0 & 3.0

applications, internet, audio clipping, video shots, e-contents, interactive

whiteboard, OERs, etc. ‘P’ is pedagogy that represents the knowledge of

teacher in pedagogical aspects of teaching. What are those pedagogical

aspects? For instance, the knowledge in various teaching methods, techniques,

styles of teaching, developmental stages of learners, etc. The letter ‘C’ denotes

the content knowledge. As you are aware, a teacher definitely should have

mastery over the subject. The content knowledge includes the knowledge in

terms, concepts, principles, theories, law, etc.

The TPACK framework is a guideline that every teacher can follow in

selecting the media. Before coming to that, let us explore a few more basic

aspects of TPACK frame work. In general, TPACK is the knowledge of an

individual in three components namely, technology, pedagogy and content.

Apart from that, TPACK also elucidates a few other components such as

Technological Knowledge (TK), Pedagogical Knowledge (PK), Content

Knowledge (CK), Technological Pedagogical Knowledge (TPK),

Technological Content Knowledge (TCK) and Pedagogical Content

Knowledge (PCK), as shown in figure given below. Thus, TPACK denotes the

interconnection of the individual components namely technology, pedagogy

and content. So, being a teacher, you must ensure that, while a technology is

selected for teaching a particular concept, these seven factors must be taken

care of.

Figure 7.8: The components of the TPACK framework (graphic from http://tpack.org)



Mathematics

70

Check Your Progress


b) Compare your answers with those given at the end of the Unit.

8) Discuss the factors to be considered in the selection of learning

resources/media for teaching.

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9) What is TPACK? Explain its relevance in the selection of technology for

teaching.

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Let us see an example for employing TPACK framework in teaching-learning

process. Data handling is one of the important concepts in mathematics. In data

handling learners study collection, organization, presentation and interpretation

of data. Thus, they start with the collection of data. Now, suppose you are

aiming at teaching collection of data. How will you choose an appropriate

technology? Let us interconnect the concept to be taught and TPACK frame

work. TPACK says, the teacher should have knowledge in technology,

pedagogy and content to integrate technology. In this case, then of course, the

teacher should have mastery over the content; collection of data. Now, come to

the second aspect of TAPCK; the pedagogy. So, what all pedagogical

approaches will be suitable in this context? You have studied various teaching

methods, techniques of teaching, models of teaching and so on. While selecting

the pedagogy, you should keep in mind various factors like; learners’s age,

maturity level, difficulty level of the topic and so on. Considering many such

factors, you can opt, Concept Attainment Model (CAM) as suitable since the

learners would be able to identify the concept of ‘collection of data’

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Learning Resources

and ICT for


–Learning

themselves. CAM will work out in groups. Thus, CAM and group activity is

the pedagogical part of TPACK framework.

The third aspect of TPACK is the selection of fitting technology. What do you

have in mind? Which technology will be apt here? Remember, you may ask

learners to search internet and collect data pertaining to temperature. But, will

that be suitable for learners of standard 7th

? It is better to select some other

technology. Also, the feasibility factor would be a hindrance, but, it is up to

you to select the technology suitable for your learners. Now, in this case let us

choose LCD projector, PPT presentation and internet. To prepare PPT

presentation, various data collected from internet would be used. Then, the PPT

will be shown to the whole class and learners will be directed to identify the

concept involved in it. The identification of the concepts will be attempted as a

group activity. Thus, the third aspect of TPACK framework includes LCD

projector, PPT presentation and internet. Some of the pictures used in for

teaching ‘collection of data’ are given in figure 7.9. The same pictures can be

used during PPT presentation.



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72

Figure 7.9

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Learning Resources

and ICT for


–Learning

While selecting the technology, pedagogy and content, you may also give due

weightage to other factors such as Technological Pedagogical Knowledge

(TPK), Technological content Knowledge (TCK), Pedagogical Content

Knowledge (PCK), and TPACK. If you find any mismatch on any of these

components, then that technology won’t be apt for teaching that particular

concept.


7. Apart from the factors discussed here, what other factors do you consider

important in selecting media/learning resources?

7.8 LET US SUM UP

The judicious integration of learning resources along with teaching methods

makes learning effective. Your duty as a teacher, is to select the appropriate

learning resources which would enable you to transact curricular contents and

thereby making learning enjoyable to learners. Thus the learning resources that

are readily available and also learning aids that could be procured from

immediate surroundings have been discussed. Mathematics laboratories and

mathematics corners also fall under the category of learning resources. The

importance of maths labs/math corners, ways of developing labs/corners,

materials to be kept in labs/corners is also discussed. In continuation to that,

the relevance of math clubs and forums is also discussed. We know that, ICT

plays a major role in the teaching –learning of mathematics. Therefore, the

need and importance of ICT, strategies in organising learning activities by

integrating ICTs are also discussed. The unit ends with the discussion on the

factors that are to be considered in the selection of learning resources.


1) Observe teaching sessions of any senior teachers and make a report on

learning resources that they use for teaching mathematics.

2) During the teaching practice sessions, set up a mathematics club in your

classroom and discuss the steps followed in creating it.

3) If you set up a mathematics lab, what modern ICT devices you would

prefer to include in it.

4) Discuss a learning activity for class X learners that involves ICT.

5) What factors would you consider while selecting learning resources/media

for teaching mathematics?

7.10 REFERENCES AND SUGGESTED READINGS

IGNOU (2008).AMT-01Teaching of Primary School Mathematics,AMT-

01,Block 1-5,SLM.New Delhi: IGNOU.

IGNOU (2010).LMT-01 Learning Mathematics ,LMT-01,Block 1-6,SLM.New

Delhi: IGNOU.

IGNOU (2012).BES-009 Teaching of Mathematics for the Primary School

Learner, Block 1-4,SLM.New Delhi: IGNOU.

NCERT (2005). National Curriculum Framework-2005. New Delhi: NCERT



Mathematics

74

NCERT (2006). Position paper: National Focus Group on Teaching of

Mathematics. New Delhi: NCERT.

NCTE (2009). National curriculum framework for Teacher Education. New

Delhi: National Council for Teacher Education.

Haylock, Derek. (2006).Mathematics Explained for Primary Teachers (third

edition).New Delhi: Sage Publications India Pvt.Ltd.

Sidhu Kulbir Singh (1994); The Teaching of Mathematics; New Delhi; (p17 –

p205) Sterling Publishers Private Limited.

Sunitha E; Rao; S.R. and Rao, D.B. (2006). Methods of Teaching Mathematics

New Delhi: Discovery Publishing House.

http://tc2.ca/uploads/PDFs/TIpsForTeachers/CT_elementary_math.pdf

retrieved on 05/12/2016

http://www.doublegist.com/teaching-resources-teaching-aids-enhance-

teaching-desired-social-behavioural/ retrieved on 05/12/2016

http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics_Laboratory


http://mathmagic-elements.blogspot.in/2011/04/mathematics-laboratory.html



1) The process of learning is made interesting, enjoyable and pleasurable

activity with the use of learning resources. The use of learning resources

also helps to fully involve learners in the learning process. Also refer

section 7.3.1

2) For example, various bottles and other similar objects can be used to teach

the concept of geometrical figures, 2D/3D shapes, etc.


4) Seminars, Quiz programmes, Debates, etc.

5) ICT can be used in teaching –learning, assessment of learners’s progress,

educational administration and professional development programmes.

Using PowerPoint presentations during teaching, storing learner data in

computers, etc are some of the practical applications of ICT.

6) The primary job a teacher in constructivist learning approach is that of a

facilitator of learning. In such an approach, learners themselves develop

knowledge by integrating their present knowledge with previous

experiences gained from either at home or his/her surroundings. Thus

teachers can aid learning by assigning learning tasks with help of ICTs. For

example, learners may be directed to watch educational videos (video may

be on seasonal change) and thereafter a group discussion can be organised.

This will help learners to develop their own knowledge about the concept

‘Time’.

7) Answer yourself.

8) Learner factors, Task factors and Economic factors


75

Assessment in

MathematicsUNIT 8 ASSESSMENT IN MATHEMATICS

Structure

8.1 Introduction

8.2 Objectives

8.3 Role of Assessment in Mathematics

8.4 Continuous and Comprehensive Evaluation in Mathematics

8.5 Preparation of Achievement Test

8.6 Tools and Techniques of Assessment of Learning Mathematics

8.6.1 Written Test

8.6.2 Observation

8.6.4 Anecdotal Record

8.6.4 Check List

8.6.5 Rating Scale

8.6.6 Rubrics

8.6.7 Assignment

8.6.8 Project

8.6.9 Portfolio

8.7 Assessment of Learning of Mathematics in Children with Special Needs

8.8 Let Us Sum Up




8.1 INTRODUCTION

Assessment is an integral part of teaching-learning process as it is a prime tool

for monitoring the progress and shaping learning. Now days, Mathematics is

being viewed not only as a traditional prerequisite subject for prospective

scientists, engineers, businessman etc, but, also as a fundamental aspect of

literacy for the twenty-first century. Keeping this in mind, about the

comprehensive view of Mathematics and its role in society, assessment should

aim at much more than just the test given at the end of course.

This Unit deals with several aspects of assessment in Mathematics. We will

begin this unit with discussion on the role of assessment in Mathematics and on

continuous and comprehensive evaluation in Mathematics. The Unit will also

discuss the procedures to be followed for construction of a good achievement

test and various tools and techniques for assessment in Mathematics learning.

Further, The Unit will also discuss the assessment of Mathematics learning of

children with special needs.


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76

8.2 OBJECTIVES

After the completion of the Unit, you will be able to

• describe the meaning and role of assessment;

• analyze the process of continuous and comprehensive evaluation in

Mathematics;

• prepare an achievement test for Mathematics;

• differentiate among various tools and techniques of assessment of learning

Mathematics;

• identify and use tools and techniques for assessment of learning

Mathematics; and

• modify the assessment to and techniques for children with special needs.

8.3 ROLE OF ASSESSMENT IN MATHEMATICS

The main aim of assessment is to collect information of learner’s achievement

and progress and provide direction for ongoing teaching and learning process.

Assessment can be done through both formal and informal activities.

Assessment in Mathematics refers to the process of identifying, gathering and

interpreting information about learners’ mathematical learning. Assessment is

the means, which deduces what learners know and what they do not. It

suggests teachers, learners, parents, and policymakers something about what

learners have learned and what more should be done in order to improve

performances in Mathematics. Assessment has a comprehensive meaning just

not limited to evaluation of student’s performances. Assessment can be used

for following purposes:

Assessment for learning: Assessment for learning occurs during the learning

process. Information obtained by this type of assessment is used by the

teachers to modify their teaching strategies, and learners use it to make changes

in their learning strategies. This approach of assessment helps teachers to

appraise the learners to monitor their learning; and guide the instruction at

process and provide feedback helpful to learners. It provides opportunities for

learners to develop an ability to evaluate themselves; make judgments about

their own performance and make necessary improvement.

Assessment as learning: Assessment as learning means an awareness of

learners regarding how they learn and use that awareness to make necessary

adaptations in their learning process. Therefore, they take an increased

responsibility for their learning. It involves setting of goal, monitoring the

progress and contemplating on results. It occurs throughout the learning

process.

Assessment of learning: Assessment of learning refers to a review process

which occurs at the end an learning unit. It provides measures of achievement

for the purpose of grading. It informs learners, teachers and parents, as well as

other stakeholders of the community about achievement at a certain point of

time to provide information regarding success.

Assessment in Mathematics must be planned keeping in mind its goals.

Assessment for and of learning, each has a role to play in supporting and

improving learner learning, and so, must be appropriately balanced.

77

Assessment in

MathematicsAssessment must be embedded in the learning process and interconnected with

curriculum and instruction.

8.4 CONTINUOUS AND COMPREHENSIVE

EVALUATION IN MATHEMATICS

The role of Mathematics in society has changed immensely. Evaluation process

in Mathematics has also been transformed to ensure consistency with the goals

of education. The whole pedagogy has been shifted from behaviorist approach

to constructivist approach. Since long the assessment experiences are mostly

the evaluation which is based on a behaviorist approach i.e. only discrete facts

and skills are evaluated, grading and ranking are provided. Current theories of

learning Mathematics suggest that learners are not passive learners simply

receiving information but actively constructing knowledge, too. These

changing views of Mathematics and the transformed role of teacher and the

learners have broadened the ways in which Mathematics is being taught.

Continuous and Comprehensive Evaluation (CCE); student's performance in

Check Your Progress



1) What is the meaning of assessment?

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2) In what ways assessment plays an important role in teaching-learning of

Mathematics?

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3) Fill in the blanks.

I. ……………..means an awareness of learners regarding how they

learn.

II. ……………… occurs at the end of the learning unit.


Mathematics

78

both scholastic and co-scholastic activities is assessed. CCE aims to reduce the

curricular workload on learners and to improve the overall abilities and skill of

learners by means of evaluation of learners’ performance in both types of

activities.

Continuous assessment of learners' work not only facilitates their learning of

Mathematics, but also enhances their confidence in application of learning in

Mathematics. This view changes the focus of assessment from summative

evaluation, where learners are evaluated at the end of unit and provided grades,

to the formative evaluation where learners are evaluated in the pursuit of

learning. So, the approach of evaluation solely for the purposes of grading and

ranking has been changed to approach of integrating evaluation with learning

activities that support learners' construction of knowledge. Evaluation must be

an integral part of the learning process rather than an interruption for it. The

Position Paper on Examination Reforms (2006), states that CCE should be

established to (i) reduce stress on children, (ii) make evaluation comprehensive

and regular, (iii) provide space for the teacher for creative teaching, (iv)

provide a tool for diagnosis and for producing learners with greater skills.

Continuous and Comprehensive evaluation includes both Formative as well as

summative evaluations. Therefore, the CCE enables the learners to be

evaluated throughout the term and at the end of term also suggestive scheme of

CCE is mathematics is given below:

Table 1: A Suggestive Blue Print of Continuous and Comprehensive

Evaluation in Mathematics

Continuous and

Comprehensive

Evaluation in

Mathematics

Components

of CCE

Components of

Formative and

Summative

Evaluation

Various types of

Activities/Questions

Formative

(throughout

the session)

(40 marks)

Two best out of

four Activities

( based on

activities, while

teaching a

Concept)

(20marks)

Group Activity ,

Assignments,

Group Game, Quiz,

Concept Mapping ,Project

Work,

Problem Solving,

Graphical

Representation,

Home Assignments,

Presentación (digital/

Graphs /diagrams etc.)

Laboratory

Activity-

Average of four

Activities

(10 marks)

Lab Ethics ,

Procedure of the

lab activity,

Recording ,

Verifying the solution of

the problem etc.

One

best out of two

Tests

(a small test

for a short duration after completing a

Multiple Choice,

Sequencing,

Fill in the blanks , Right /

Wrong,

Yes / No ,

Very Short Answer Type,

79

Assessment in

Mathematicsunit)

(10 marks)

Matrix type,

Short Answer Type, etc.

,

Summative (at the end of

the each

term)

(60 marks)

Two summative Evaluation conducted at the end of each term. ( 1

st term -20 marks &2

nd term -40

marks)

Therefore, Continuous and Comprehensive evaluation system has the capacity

to provide good quality Mathematics education to all the learners.

Check Your Progress



4) Tick the correct answer.

i) In CCE, evaluation is an integral part of the teaching-learning

process .(True/False)

ii) CCE is an outcome of Behaviorist approach. (True/False)

iii) Summative evaluation is done at the end of a term.(True/False)

5) How does continuous and comprehensive evaluation help the teacher?

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6) Write down any five activities used in evaluation of Mathematics.

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8.5 PREPARATION OF ACHIEVEMENT TEST

Achievement test is an instrument designed to measure the accomplishment of

the learners, in a specified area of learning, after a period of instruction. Hence,

this test developed for the purpose of testing the achievement of the learners’

can be given at the end of unit, term, semester, year, etc. These tests are

universally used by teachers mainly for the following purposes:

1) To measure whether the learners have achieved the objectives of the

planned instruction;


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80

2) To monitor learners' learning and to provide ongoing feedback to both

learners and teachers during the teaching-learning process;

3) To identify the learners' learning difficulties- whether persistent or

recurring; and

4) To assign grades.

Teachers help learners to enable them to develop some abilities, skills and

attitudes. After teaching, learners’ performance need to be evaluated

periodically. Teachers construct the tests to assess the achievement of learners.

Preparing an Achievement Test:

Let us consider the necessary steps in preparing an achievement test:

� Planning of the test

� Preparation of a blue print

� Preparation of test items

� Try out, preparation of scoring key and evaluation of the test

Step 1- Planning of the Test:

The first step for planning of the achievement test is to develop a design or

framework. For this, you have to:

A. Analyze the course content into different content units and decide the

weightage that is to be given to each in the test;

B. Decide the weightage to be given to different objectives being tested;

C. Decide the weightage to be given to different forms of questions to be

used in preparing a question paper;

D. Decide the weightage to be given to time and marks for different forms of

questions;

E. Decide the weightage to be given to the difficulty level in the test.

Let us see how we can prepare a good achievement test.

Weightage of the Content: The first part of the planning phase is to decide

about the weightage to be given to different units. You can include more units

in the annual examinations; but in quarterly or half yearly examinations, fewer

units should be included. So each unit would be given more marks in

comparison to the yearly examination. Let us take the example of Mathematics

subject for class IX. Let us take an example of a summative assessment with a

maximum of 25 marks and the duration of one hour. It incorporates to the

measurement of behaviors in the cognitive domain only. As an illustration,

term test we select, two units : Number system and Polynomials from class IX

mathematics, which have seven sub units and the question paper will be based

on these units. In the present context, the test covered the content of the

following units:

1) The Number line

2) Rational and Irrational Numbers

3) Decimal Expansion of real numbers

4) Operations on Real Number

81

Assessment in

Mathematics5) Degree of Polynomials

6) Remainder theorem of Polynomials

7) Factorization of Polynomials

The weightage given to each unit is presented in Table 2.

Table 2: Weightage given to different units

Content Marks Percentage

The Number line 15 60

Polynomials 10 40

Total 25 100

Weightage of Instructional Objectives: After deciding about the weightage to

be given to different units, you have to consider the learning objectives. Your

test is good only if it is able to evaluate the achievement of learning objectives

decided by you. You can allocate appropriate weightage to various objectives

like knowledge, understanding, application, skill, etc. For example, for the above

case , you may give weightage of 12%, 8%, 24%, 32%, 16% and 8% for

knowledge, understanding, application, analysis, synthesis, evaluation

respectively. The weightage given to different objectives is presented in Table 3.

Table 3: Weightage to instructional objectives

Objectives Marks Percentage

Knowledge 3 12

Understanding 2 08

Application 6 24

Analysis 8 32

Synthesis 4 16

Evaluation 2 08

Total 25 100

Weightage given to different forms of questions: The next step is to decide

about the weightage to be given to different forms of questions. Generally, in

an achievement test, a teacher has to include different types of items (essay,

short answer or objectives). The weightage given to different forms of

questions is presented in Table 4.

Table 4: Weightage given to different forms of questions

Forms of questions Weightage given

Essay Type

Short Answer Type

Objective Type

28

56

16

Total 100

The fourth step in the preparation of question paper is to give weightage to

marks and time for different forms of questions. The allotment of marks and

time to different forms of questions is presented in Table 5.


Mathematics

82

Table 5:Weightage given to marks for different forms of questions

Form of

questions

Marks

per

question

No. of

question

Marks Percentage

Objective type 0.5 14 07 28

Short answer

type

02 7 14 56

Long answer

type

4 1 4 16

Total 22 25 100

Estimation of Time: For teacher-made achievement tests, only the experience

of teachers should be enough for the estimate of time. You should try to

analyze and estimate the time for different types of questions. Here we have

taken hypothetically the total duration of 1 hr. For different forms of questions,

weightage given to time are presented in Table 6.

Table 6: Weightage given to time for different form of questions

Form of

questions

Time per

question (in

minutes)

Total no. of

questions

Total

Times(in

minutes)

Objective type 01 14 14

Short answer type 05 07 35

Long answer type 11 1 11

Total 22 60

The next step is to give weightage to difficulty levels of the items, which is

presented in Table 7.

Table 7: Weightage to difficulty level of questions

Difficulty Level Marks Percentage

Easy 5 20

Average 15 60

Difficult 5 20

Total 50 100

Step 2: Preparation of a Blue Print:

A blueprint is a three-dimensional chart showing different types of items with

marks for each topic/unit and each of the objectives. It shows the respective

weightage of marks for different objectives, and topics and various types of

items as prescribed by the school or in the syllabus or decided by the paper-

setter. These specifications have been discussed in the earlier steps of planning

of the blue-print.

Based on the above steps the final blueprint is developed. With the help of such

a table of specifications, you will be able to ensure the needed coverage of

units in the syllabus and assessment objectives. The final blueprint is presented

in Table 8.

83

Assessment in

MathematicsTable 8: Blueprint (Table of Specifications)

Objectives

Form of Q

Content

Knowledge Under-

standing Application Analysis Synthesis Evaluation

Grand

Total

O SA L O SA L O SA L O SA L O SA L O SA L

Unit 1 2

(4)

1

(2)

2

(4)

2

(1)

4

(1)

2

(1)

2

(1) 15

Unit 2 1

(2)

1

(2)

2

(1)

4

(2)

2

(1) 10

Total

Marks 3 0 0 2 0 0 2 4 0 0 4 4 0 4 0 0 2 0

25 Grand

Total 3 2 6 8 4 2

Note: Figures within the brackets indicate the number of questions and figures

outside the brackets indicate marks.

Entries made in this blueprint are only for illustration. You have to decide

about these while preparing the blueprint. However, it must confirm, to

weightage indicated in the design to the various objectives (12%, 8%, 24%,

32%,16%, and 8% ), content units (60% and 40%) and form of questions

(E=16%, S.A. = 56%, O.T. 28%) as reflected in this table of specification or

blueprint.

Step 3-Preparation of Test Items/Questions:

Test items form the very basis of testing. A test constructor should have good

knowledge of the subject. The test items should be clear, unambiguous and

according to the objectives. Different types of items - essay, short-answer and

objective types - should be prepared in sufficient numbers. Items of varying

difficulty should also be prepared. Experienced teachers are able to estimate

difficulty level by their judgment. Some items from question banks can be

taken up.

Step 4- Try Out, Preparation of Scoring Key and Evaluation of the Test :

After preparation of test items, a review is done on the basis of blueprint

requirements to assess the quality of items. It is time to be confirming the

validity, reliability and usability of the test. Try out helps us to identify

defective and ambiguous items, to determine the difficulty level of the test and

to determine the discriminating power of the items. Then only unambiguous

and objective based items are retained.

To maintain the objectivity and validity of test, you have to provide proper

instructions for marking. Objective type tests have key answers. Their answers

and corresponding marks should be given. Short answer questions are also

quite specific in nature and possible points or ideas in answers should be

mentioned with their corresponding marks. Essay type questions are lengthy

and need specificity for uniform marking. Important steps or points of answer

should be explicitly mentioned along with their corresponding marks. The

above guidelines for marking questions make our testing more reliable. These

achievement tests are used normally at the end of term/year as a part of


Mathematics

84

summative assessment. Care should be taken that summative assessment and

unit tests have adequate contribution to over all assessment of the learners.

Question wise analysis is given below:

Table 9: Question Wise Analysis

S.N Content Objective Form of

Question

Difficulty

Level

Marks

Section A

1.i Polynomial Understanding MCQ Average 0.5

1.ii Polynomial Analysis MCQ Difficult 0.5

1.iii Polynomial Understanding MCQ Average 0.5

1.iv Number system understanding MCQ Average 0.5

1.v Number system Knowledge MCQ Easy 0.5

1.vi Number system Knowledge MCQ Easy 0.5

1.vii Number system Analysis MCQ Difficult 0.5

1.viii Polynomial Application MCQ Average 0.5

1.ix Number system Application True/False Average 0.5

1.x Number system Application True/False Average 0.5

1.xi Number system Understanding True/False Average 0.5

1.xii Number system Understanding True/False Average 0.5

1.xiii Number system Application True/False Average 0.5

1.iv Number system Application True/False Average 0.5

Section B

2. Number system Evaluation Short Ans Difficult 2

3. Polynomial Understanding Short Ans Average 2

4. Number system Analysis Short Ans Difficult 2

5. Polynomial Knowledge Short Ans Easy 2

6. Polynomial Analysis Short Ans Difficult 2

7. Polynomial Understanding Short Ans Easy 2

8. Polynomial Understanding Short Ans Average 2

Section-C

9. Number

system/Polynomial

Synthesis/application Long Ans Average 4

85

Assessment in

MathematicsExample of an Achievement Test

Term of Examination: SA - I

Name of the School: XYZ

Class: IX

Time : 1hr Max. Marks: 25

Instruction: This test consists of three sections A, B and, C. All sections are

compulsory.

Section: A(0.5×14=7)

1) Choose the correct one:

(i) The remainder obtained on dividing p(x) = x3 + 1 by x + 1 is:

(a) 0 (b) (c) (d)

(ii) The value of k, for which the polynomial x3 – 3x

2 + 3x + k has 3 as

its zero is:

(a) 9 (b) –3 c) –9 (d) 12

(iii) If P(x) = cx + d, then zero of polynomial will be:

(a) – d/c (b) d/c (c) c/d (d) –c/d

(iv) Every Rational number is :

(a) natural number (b) an integer (c) a real number (d) a

whole number

(v) The smallest natural number

(a) 1 (b) 0 (c) –1 (d) none of the above

(vi) 1 is :

(a) a prime number (b) a composite number (c) both prime

and composite (d)neither prime nor composite

(vii) Which of the following is irrational?

(a) √4/9 (b) √12/ √13 (c) √7

(d) √81

(viii) If a+b+c= 0, then a3 + b

3+c

3 =

(a) 3abc (b) a2bc (c) ab

2c (d)2 abc

2) State true/false

(ix) Every whole number is a natural number.

(x) Every integer is a rational number.

(xi) Every point on the number line is of the form √x where x is a

natural number.

(xii) Every natural number is a whole number.

(xiii) Every irrational number is a real number.

(xiv) The decimal expansion of the number √2 is non-terminating non-recurring.


Mathematics

86

Section: B (2×7=14)

2) Let x be a rational and y be an irrational number. Is xy necessarily

irrational? Justify your answer by an example.

3) Factorize 36 x2

– 9 y2.

4) Justify that the square of irrational number is always rational.

5) Expand (4a – b + 2c)

2 and (3a

– 2b)

2.

6) Factorize 1 – 64a

3 – 12a + 48a

2.

7) If both (x + 1) and (x – 1) are factors of ax3

+ x2

– 2x + b find the value of

a and b.

8) Find the remainder when (y3 + y

2 – 2y + 5) is divided by( y – 5).

Section: C (4×1=4)

9) The polynomial x4

– 2x3

+ 3x2

– ax + 3a – 7, when divided by x + 1 leave

the remainder 19. Find the values of a. Also find the remainder when p(x)

is divided by x + 2.

Or

Locate √5 , √10 and √17 on the number line.

Check Your Progress



7) What are various steps for construction of an achievement test?

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

8) How does preparation of blue print help a teacher?

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..


1. Select a topic of your choice from class IX Mathematics textbook .Then

prepare a blue print and achievement test. While preparing the test, follow

the steps of constructing an achievement test.

87

Assessment in

Mathematics9) What should a teacher keep in mind at the test planning stage?

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

……………………………………………………………………………..

8.6 TOOLS AND TECHNIQUES OF ASSESSMENT

OF LEARNING MATHEMATICS

We have discussed how to develop an achievement test. Let us discuss tools

and techniques used to assess learned performance in Mathematics. Here are

some tools and techniques used in formative and summative assessment of

scholastic performance.

Table10: Tools and Techniques of Formative and Summative Assessment

Source: CBSE Manual (2010)

We will discuss some tools and techniques, which a teacher uses for assessing

in mathematics .

8.6.1 Written Test

In Mathematics, through the written test, we get a better indication of learners’

real achievement in learning by assessing their conceptual and procedural

understanding. It enables them to relate facts and principles to organize them

into a reasonable and insightful progression, which provides a fair justice to the

Formative Assessment (Flexible Timing)

Tools

Questions

Observation

Interview schedule

Checklist

Rating scale

Anecdotal records

Document analysis

Tests and inventories

Portfolio analysis

Rubrics

Techniques

Examination

Assignments

Quizzes

Collections

Projects

Debates

Elocution

Group discussions

Club activities

Demonstrations

Summative Assessment

(Written, End of Term)

Objective type

Short answer

Long answer


Mathematics

88

thoughts and ideas in a written expression. Written type test can be classified

into subjective/descriptive and objective type test.

A) Subjective /Descriptive Test: The subjective test in Mathematics usually

contains essay type and short answer type questions that is the questions

for which procedure of solutions is also evaluated. Through subjective

/descriptive test we can assesses learners problem solving ability.

B) Objective Type Questions: Now-a-days, objective type tests are

preferred because of their high validity and reliability. They have several

merits like maximum representation of the teaching aims and objectives,

broader coverage of the syllabus, efficiency, economic, time saving, easy

handling, etc. There is a minimum amount of fallacy in it. It can be

divided into:

a) Selection Type Questions:-True- false type, multiple choice

questions, matching questions and classification questions are the

sub-types of selection type questions.

e.g. A) 90º angle is called a Right Angle: true ( ) false ( )

b) Supply Type Items:- In this type of question, the learners have to

complete the answers by recalling and retaining its learning. This

type of learning can be attained by in depth study and memorization.

Supply type items can be a simple recall type where the learners

have to write a single, word, formula, number etc to complete the

answer.

e.g. The parameter of the circle is ----------------

8.6.2 Observation

Direct observation has been used as a way to assess mathematical skills since

the establishment of formal classrooms, hence Because Mathematics is a

subject that consists of step-by-step procedures, direct observation can be used

in conjunction with rubrics. By this technique, we can observe the interest,

skill, competency etc. It is a continuous process. Through observations,

teachers can assess children's abilities to communicate mathematically, apply

Mathematics concepts and skills, solve problems and work with others. A few

effective and efficient means for collecting observation information include the

following:

• Determining what skills or comprehensions are to be assessed.

• Carrying paper and a pen for recording observations.

• Using a checklist of desired behaviors and actions.

• Using a video camera to record observations

Observation schedule is used to collect information systematically and with

objectivity. Here is one example of using observation schedule for debate

competition.

89

Assessment in

MathematicsTable 11 : An example of an observation schedule for debate competition

S. No. Descriptors Score out of 5

1 Depth of knowledge of the content

2 Strength of the argument to conceive

3 Fluency, diction and pronunciation

4 Ability to contradict a given point of view

5 Respectful to the opponent

6 Ability to take criticism positively

7 Body language while arguing

*Source: CBSE Manual (2010)

8.6.3 Anecdotal Record

Anecdotal record is an observation method used frequently in the classrooms

in which the teacher summarizes a single developmental incident after the

event has occurred. A teacher records about what learners are learning, their

academic performance, learning behaviour, their achievements and social

interactions. Though it is an informal note but with its help, you can keep a

record of each and every learner of your class in a comprehensive manner.

Anecdotal notes should be used to record the day-to-day development of

students, as well as their specific behaviors like learner’s problem solving

ability, measurement ability, experimentation ability etc. These behaviuors/

observations need be recorded within two days of being observed to ensure

accuracy of information.

Here is one sample anecdotal record form.

Table 12: An Example of Anecdotal Record

Student: Anshika

Class: 9th

Observer: Mr.Mohit

Date: 7th July 2016, 11:45am

Setting: Classroom

Purpose: To observe Anshika understands of number line.

Observational question:

Is Anshika able to draw number line and show numbers on number line?

Observation details:

Anshika had developed the concept of natural number, whole numbers and

rational numbers. She listed five rational numbers between 1 and 2 and realized

that in fact there are infinitely many rational numbers between 1 and 2. She

concluded that, in general, there are infinitely many rational numbers between

any two given rational numbers. She drew the number line and showed the

number on the number line.


Mathematics

90

Analysis:

Anshika was able to apply the knowledge of number system. She demonstrated

confidence in drawing the numberline. Anshika was able to show any number

on number line.

8.6.4 Check List

Check list is an observational technique. It offers systematic ways of collecting

data about specific behavior, knowledge, performance and skills. Check lists

have two parts, in the first column statement and the latter is response yes/no

related to the statement. Let us see an example of checklist.

Table 13: Checklist for Problem Solving Skill

S.No. Aspect Yes No

1 Did the learner accept challenges in problem

solving willingly?

2 Did the learner apply knowledge learned from

previous learning tasks?

3 Did the learner reason and explain appropriately?

4 Did the learner see and or analyze relationships

and make connections?

5 Did the learner complete response the learner with

clear explanations?

6 Did the learner answer completely using correct

mathematical terms and symbols?

8.6.5 Rating Scale

Rating scales are extended form of checklists. In rating scales, w create

standards criteria for evaluating a performance and each standard has a definite

level of competence and we rate learners according to how well they perform

on each standard as they complete the task.. An example of rating scale for

experimental work in Geometry is given below:


2) Prepare a check list to assess reasoning ability of learners.

91

Assessment in

MathematicsTable 14 : Rating Scale for Experimental Work in Geometry

Student’s Name:____________ __________

Class:________________________________

Skills Observed Level of Mastery

Never

(1)

Sometimes

(2)

Generally

(3)

Mostly

(4)

Always

(5)

Learner identifies

geometric shapes

including circle, cone,

cube, cylinder,

pyramid, hexagon,

oval, parallelogram,

rectangle, square.

Learner has

understanding on

geometrical shapes.

Learner identifies

center, radius, and

diameter of a circle.

Learner classifies

shapes by the number

of sides.

Learner sorts and

identifies shapes by

attributes.

Learner construct

triangles including

scalene, isosceles, and

equilateral.

Learner construct

angles including acute,

right and obtuse.

Learner works with

precision and neatness.

Total Score

8.6.6 Rubric

A rubric is a scoring tool that divides the whole assigned work into component

parts with clear descriptions of each component at varying levels of mastery. It

provides a set of scoring guidelines that describe performance of learners. It

can be used for a wide array of assignments: papers, projects, oral presentations

etc. As learners demonstrate the performance, it is appropriate to assess the


3) Prepare a rating scale for problem solving skill of learners.


Mathematics

92

performance using a rubric. It is generally used to assess performance tasks and

open questions. Generally a typical rubric:

• contains a scale of points to be assigned: for example, 1 to 4.

• describes the characteristics of a response for each possible score.

An example of rubrics of mathematical project is given below for your better

understanding.

Table 15: Rubrics on Mathematics Project

Name of the Learner _____________________Class___________________

Section_________________________________Teacher_________________

Criteria 4 3 2 1

Submission

timelines

Project was received on

due time

Project was received 1

day late

Project was received 2

days late.

Project was received 3 or

more days late.

Completion All parts of the project

are completed

neatly and

correctly

All parts of the project

are completed

Some parts of the project

are completed

Few to no parts of the

project are completed.

Accuracy Each step of the project

was followed

and was

correct

One step of the project

was

incorrect

Several steps of the project

contained

error

Entire project was incorrect

Steps Every step of the problem

was

completed thoroughly

with work

shown

Most steps of the problem

were


with work

shown

Few steps of the problem

were


with work

shown

None of steps of the

problem were


with work

shown

Organization Learner completed

work in a

logical and sequential

manner that

is easy to follow

Learner completed

the work but

it is difficult to follow the

step used

Learner work is incomplete

but some

logical steps are shown

Learner work is incomplete

and no logic

is shown

Explanation Learner explained

how to solve and why the

chosen

methods

work

Learner explained

how to solve but could not

why the

chosen

methods work

Learner explained

only small part of work

Learner could not explain

any of the work

93

Assessment in

MathematicsDiagrams Learner created an

accurate

diagram graphs or

chart to help,

solve or to

show solutions

Learner created a

diagram,

graphs or chart that

contains

slight errors

Learner created a

diagram,

graphs or chart that

contains

many errors

Learner pdid not create

any diagram

graphs or chart

Knowledge of

terminology and

strategies

Demonstrates considerable

knowledge of mathematical

language and

strategies

Demonstrates thorough and

insightful knowledge of

mathematical

language and strategies

Demonstrates some


language and

strategies

Demonstrates limited


language and

strategies

Total Overall

Score____________

Comments:

Activity for Practice :

4) Prepare a rubric for assessing Mathematics lab work of learners.

8.6.7 Assignments

Assignments are used for both learning and evaluation. Evaluation of

assignment is an important aspect. When an assignment is given, it must be

based on the instructional objectives. The assignment should be evaluated

keeping in view of those objectives and the extent to which objectives have

been achieved. The assignment should be evaluated and grading should be

given. The assignment grade should also be included in the final assessment.

Let us see examples of assignments in mathematics.

Example 1: An assignments in statistics could be given to collect data from

school on class wise enrollment in session 2014-15, 2015-16 and 2016-17. And

represent them using bar graph and pie diagram. Draw conclusions on the basis

of graph and diagram.

Example 2:

Topic Assignment Specific Expectation from learners

Coordinate Geometry

History of

Co-ordinate

Geometry

Details of year wise significant

geometrical contribution

Create a picture gallery of these

Mathematicians and their contributions

What values they can learn from these

mathematicians?


Mathematics

94

8.6.8 Project

A project is a motivated problem, solution of which requires thought and

collection of data and its completion results in the production of something of

value to the learners.

Project enables learners to conduct real inquiry in an interdisciplinary manner.

It promotes problem-solving in Mathematics and connects it to real life

application.

Projects in mathematics provide opportunity to observe, collect data, analyse,

organize and interpret data and data and draw generalization.

A project could be individual or group project and could be presented in the

form of a document, report and/or a multimedia presentation.

Examples of Projects Level-VIII to X

• Geometry in Real Life: This project can enable the learners to apply

geometrical concepts, such as properties of triangles in real-life situations.

Learners can find the height of buildings, trees, etc.

• Project on BMI (Body Mass Index): In this project, learners can

investigate health conditions of a sample population by calculating B.M.I.

The detailed surveys, calculation, graphs, tables, etc. can be used to depict

the results of the project and also this project draws an interdisciplinary

linkage with biology.

The criteria of assessment of the project could be translated into a will defined

rubric.

Example

Criteria Level 4 Level 3 Level 2 Level 1

Content

Accurate,

precise, relevant and interesting

Accurate,

precise, but not so

interesting

Content has

some errors, relevant

but not so

interesting

Content

is accurate

and not

relevant

Creativity Very high High Moderate Low

Organisation Very well

organized and

sequenced

Well

organization

and

sequenced

Not so well

organized

Not well

organized

and content is

not sequenced

Originality The information

is

well researched

and original

The

information is

well

researched

The

information is

not original

The

information is

completely

copied

(Source: Pedagogy of Mathematics: Textbook for Two-year B.Ed. Course, NCERT,

2012, pp 248)

95

Assessment in

Mathematics

8.6.9 Portfolio

Portfolio is a collection of learner’s work. It can be designed to represent many

things in relation to children's Mathematics learning experiences. It compiles

academic work and other forms of educational evidence assembled for the

purpose of evaluating the curriculum quality, learning progress, academic

achievement, etc. It also helps in determining whether the learners have met

learning standards, helping the learners to reflect on their academic goals and

progress as learners. It provides a means for managing and evaluating multiple

assessments for each learner. It includes a variety of entries including test

reports, projects reports, essays, lab reports, assignments, problem solving

tasks, , a book review, photos, self-assessments, peer assessment, teacher

assessment, parents assessment etc. Following points must be remembered

while using portfolios:

• Provide learners the opportunity to provide input regarding the portfolio contents.

• Allow the learner to select some or all of the items. .

• The items chosen by learners must provide the insight into their real work,

their dispositions toward mathematics, and their mathematical

comprehension.

• The portfolio contents are developed over time, teachers must spend time to obtain information about the learning styles and patterns of the learner.

Check Your Progress:

Notes: a) Write your answer in the space given below.


10) What are the benefits of observation schedule?

……………………………………………………………………………

……………………………………………………………………………

……………………………………………………………………………

……………………………………………………………………………

11) How are anecdotal notes recorded?

……………………………………………………………………………

……………………………………………………………………………

……………………………………………………………………………

……………………………………………………………………………

12) What type of information should be the part of portfolio?

……………………………………………………………………………

……………………………………………………………………………

……………………………………………………………………………

……………………………………………………………………………


Mathematics

96

8.7 ASSESSMENT OF LEARNING OF

MATHEMATICS IN CHILDREN WITH

SPECIAL NEEDS

All the assessment tools may not be suitable for children with special needs as

their needs may differ from rest of the class. Adaptations and accommodations

are required as per the needs of the child and the assessment criteria should be

formulated as a teamwork following a discussion and consent of experts,

parents and the learners, while conducting assessment of learners with special

needs.

These learners may need adaptation in assessment process to find out their

current status or to make formative and summative evaluation.

Let us discuss some adaptations in assessment of different categories of

children with special needs.

For Learners with Low Vision: Low vision learners take more time to

complete homework or examination. These learners experience fatigue at the

end of the day which may affect the quality of work. Therefore, the following

considerations must be remembered:

• Allow oral exams or a scribe to write examination answers.

• If asking for examples in an exam, lessen the number of examples needed

to be given.

• Provide extra time to complete the exam.

• Administer the exam in more than one sitting.

• Reduce the number of questions to be answered.

• Question paper that consists of larger print can help low vision learners. A

few ways that can enhance the print are- Providing magnifiers; Simple,

bold and large text, Highlighting, Contrast between print and paper

background (example: black text, white paper)

• Provide extensions to assignment/Projects etc. deadlines.

For Learners with Visual Impairment( Blindness) : Much like with learners

with low vision, totally blind learners also face fatigue, and so, take extra time

to complete examination, projects or assignments. Therefore, some of same

considerations apply, but here are some more:

- Provide an alternate way of assessment (oral assessments, non-written

exams )

- Provide a scribe.

- Give the examination orally and record it with a print out or recorded

audio.

- Extra time for completion of task.

For Learners with Hearing Impairment: Some Considerations are as follows-

• Provide alternatives to oral viva’s questionnaire based assignments.

• Provide clear and simple feedback to the learners both verbally and in

writing.

97

Assessment in

Mathematics• Learners may require the use of a dictionary during exams.

• Keep examination instructions clear and short. Use simple language

• Avoid jargon unless it is crucial to the inherent requirements of the exam.

• Provide extra time in examination, particularly extra time for reading

questions. Some learners will prefer to have questions and instructions

‘signed’ to them.

• Arrange an alternative exam format (e.g. replacing short answer questions

with multiple choice questions) .

For Learners with Learning Disability: Some considerations are as follows:

• Allow extensions to assignment deadlines.

• Allow learners to submit an early draft of assignments to allow the

opportunity for feedback to the learner as a formative process.

• Ensure extra time in examination for reading and analyzing questions, and

for planning their answers. Assessment venue must be quiet and

distraction-free.

• Keep short written examination instructions and sentences within

examination questions. Questions using bullet points, lists or distinct parts

are more likely to be correctly interpreted.

Check Your Progress:



13) Why adaptation in assessment process is required in children with

special needs?

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

14) Suggest any five ways to assess performance in Mathematics of a

child with learning disability.

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….

………………………………………………………………………….


Mathematics

98

8.8 LET US SUM UP

Assessment is a prime tool for monitoring progress and shaping learning

during the teaching course and can be conducted through both formal and

informal activities. As for as assessment in Mathematics is concerned; it is the

process of identifying, gathering and interpreting information about learners’

mathematical learning. Continuous assessment of learners' work not only

facilitates their learning of Mathematics but also enhances their confidence in

application of learning Mathematics. The Continuous and Comprehensive

Evaluation enables the learners to be evaluated throughout the term and at the

end of term also. The preparation of a good achievement test is a systematic

process having well defined four stages- planning the test; preparation of blue

print; preparation of the test items and try out, preparation of scoring key and

evaluation of test .There are variety of assessment tools and techniques like:

written test, observation, anecdotal records, check list, assignments, project,

rating scale, portfolio, rubrics, etc. While deciding the assessment of learners

with special needs, the specific needs and requirements, the strengths and

challenges should be taken care of. Individualized assessment of such learners

is recommended for their comprehensive assessment and evaluation.


1) Give the meaning of assessment, illustrate the role of assessment in

learning Mathematics.

2) Prepare the achievement test on any topic of your own choice.

3) Analyze the continuous and comprehensive evaluation in Mathematics.

4) Discuss different tools and techniques of assessment of learning

Mathematics;

5) Differentiate among various tools and techniques of assessment of

learning Mathematics ;

6) Analyze the assessment of learning of differently-abled children in

Mathematics


1) Assessment refers to the process of identifying, gathering and interpreting

information about learners’ mathematical learning.

2) Assessment plays an important role in following ways. (a) Guides the

student’s learning process and achievement in Mathematics. (b)

Assessment plays a major role in how learners learn. (c) Provide

information about the effectiveness of teachers’ teaching-learning

process. (d) Improves curricular activities.

3) (i)Assessment as learning, (ii) Assessment of learning

4) ( i) True (ii) False (iii) True

5) To identify difficulties areas in learning Mathematics, To improve

learners’ learning through diagnosis of their performance, To plan

appropriate remedial measures. To improve or alter instructional strategies.

To help in selecting of various tools, techniques and instructional materials.

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Assessment in

Mathematics6) Assignments, Concept Mapping, Project Work, Problem Solving,

Graphical Representation, Models including origami etc.

7) Planning of the test, Preparation of test items, Try out Preparing of and

Evaluation of the test

8) Preparation of the blue print helps the teacher to have an objective based

achievement Test giving due weightage to objectives, content area and

forms of questions.

9) At planning stage teacher must kept in the mind the following aspect:

• Analyze the course content into different content units and decide the

weightage that is to be given to each in the test.

• Decide the weightage to be given to different objectives being tested.

• Decide the weightage to be given to different forms of questions to

be used in preparing a question paper.

• Decide the weightage to be given to time and marks for different

forms of questions.

• Decide the weightage to be given to the difficulty level in the test.

10) Observation is a quantitative method of measuring classroom behaviors.

Information about a child (his/her behavior) can be collected in and

outside the class through observation. Observation can be used in a

variety of situations like debates, elocution, group work, practical and

laboratory activities, projects, and clubs activities.

11) Anecdotal notes should be used to record the day-to-day development of

students, as well as their specific behaviors, especially those that are a

cause for concern, speech patterns, language development,

social/emotional development, peer interactions, etc.

12) Portfolio can include photographs , evidences of a learner’s abilities,

thoughts and attitudes, audio-video recordings of important processes and

events, self assessment sheets, peer assessment sheets as well as parents

13) Refer section 8.6.

14) Refer section 8.6.

8.11 REFERENCES AND SUGGESTED READING

• Agrawal, S.(2004). Teaching Mathematics to blind learnersthrough

programmed learning strategies. New Delhi, India: Abhijit Publication.

• Chambers, P. (2010).Teaching Mathematics, New Delhi: Sage Publication.

• David, A.H., Maggie, M.K., & Louann, H.L. (2007). Teaching

Mathematics Meaningfully: Solutions for Reaching Struggling Learners,

Canada: Amazon Books.

• Gupta, H. N. and Shankaran, V. (Ed.), (1984). Content-Cum-Methodology

of Teaching Mathematics, New Delhi: NCERT

• James, A. (2005). Teaching of Mathematics, New Delhi: Neelkamal

Publication.


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100

• Kumar, S. (2009). Teaching of Mathematics, New Delhi: Anmol

Publications.

• National Council of Teachers of Mathematics (1980), An Agenda for

Action: Recommendations for School Mathematics of the 1980s

(Washington, DC:

NCTM,1980);availableatwww.nctm.org/standards/content.aspx?id=17278.

(retrieved on 06.03.2016 )

• National Research Council's Study Group on Mathematics Assessment

(1993), Mathematical Sciences Education Board, National Research

Council, National Academy Press Washington, DC 1993

http://www.nap.edu/catalog/2235.html

• NCTM (2000). Principles and Standards for School Mathematics National

Council of Teachers of Mathematics, Washington, DC: available at

http://www.nctm.org/standards/content.aspx?id=16909. (retrieved on

06.03.2016 )

• NCERT (2005). National Curriculum Framework. NCERT, New Delhi

• National Centre for Excellence in the Teaching of Mathematics (2008).

Mathematics Matters: Final Report [online]. Available:

https://www.ncetm.org.uk/files/309231/Mathematics+Matters+Final+Repo

rt.pdf [1 March, 2010].

• NCTE ( 2009). National Curriculum Framework for Teacher Education,

NCTE, New Delhi.

• NCERT (2013), Pedagogy of Science, Physical Science, Part II ,Text Book

for B.ED. ,New Delhi: NCERT

• NCERT (2013), Pedagogy of Mathematics,Text Book for Two Year

B.ED.Course ,New Delhi: NCERT

• Mangal, S. K. (1993). Teaching of Mathematics, New Delhi: Arya Book

Depot.

• Siddhu, K. S. (2006). Teaching of Mathematics, New Delhi: Sterling

Publishers.

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Professional

Development of

Mathematics TeacherUNIT 9 PROFESSIONAL DEVELOPMENT

OF MATHEMATICS TEACHER

Structure

9.1 Introduction

9.2 Objectives

9.3 Need of Professional Development for Mathematics Teachers

9.4 Professional Development Programmes for Mathematics Teachers

9.4.1 Seminars

9.4.2 Conferences

9.4.3 Online Sharing Communities

9.4.4 Membership of Professional Organizations

9.5 Teacher as a Community of Learners

9.6 Reflective Practices for Professional Development

9.7 Teacher as a Researcher

9.7.1 Conducting Action Research in Mathematics

9.8 Let Us Sum Up



9.11 References and Suggested Reading

9.1 INTRODUCTION

We cannot teach mathematics effectively without a thorough understanding of

content and knowledge of pedagogy, which includes acquiring knowledge and

skills for integrating technology into curriculum, instruction, and assessment.

In this unit, we will discuss the concept of professional development

programmes for mathematics teachers. The term professional development

refers to a comprehensive, sustained, and intensive approach to improving

teachers’ effectiveness in raising learners achievement. Such programmes help

teachers align their teaching techniques with the needs of their learners and

thereby ensuring better learner performance. We will discuss various ways of

promoting professional development among mathematics teachers such as

participation in seminars, conferences, online sharing; membership of

professional organizations etc. This unit will also shed light on the concept of

teachers as a community of learners, where they are actively and intentionally

involved in constructing knowledge together and disseminating it among their

collective group. Mathematics teachers act as researchers all the time in their

classrooms. Action research and innovation in teaching mathematics are part

and parcel of the job description of teachers. The importance of research and

evolving innovative practices in mathematics teaching will also be elaborated

upon in this unit.


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9.2 OBJECTIVES

After going through this unit, you will be able to:

• explain the concept of professional development programmes for

mathematics teachers;

• illustrate the purpose of professional development programmes for


• describe the importance of participation in seminars, conferences, online

sharing and membership of professional organizations in professional

development;

• justify the concept of teachers as a community of learners;

• explain the role of reflective practices in professional development of


• elucidate the concept of teacher as a researcher; and

• appreciate the importance of action research and innovation in

mathematics teaching.

9.3 NEED OF PROFESSIONAL DEVELOPMENT

FOR MATHEMATICS TEACHERS

Mathematics is often referred to as the science of numbers, quantities, and

shapes and the relations between them. Like every other branch of science,

mathematics is also constantly evolving. With technology growing by leaps

and bounds, the means available to teachers for improving the teaching-

learning process also grow. As teaching practitioners, mathematics teachers

need to keep abreast of all such developments pertinent to their learners.

Hence, the need for developing and promoting professional development

among mathematics teachers arises.

The term professional development refers to an in-service training to upgrade

the knowledge base and skills of the trainees. In the context of teachers,

professional development facilitates an in-service training wherein the teachers

get an opportunity to upgrade their content knowledge and pedagogical skills.

Usually, it is in a formal setting, though at times it can be provided informally.

It is essentially aimed at improving the overall efficacy of the teaching-

learning process by enhancing the potentials of the teachers, and ultimately, the

learners’ performance. Professional development is essentially a training

provided to teachers during the course of their service period to promote their

capacities in various aspects of content and pedagogy. To ensure learners’

bright future it is vital to support a cycle of continuous professional growth for

teachers.

Mathematics teachers are all the time hard pressed to make the subject an

interesting and appealing one for their learners At the secondary and senior

secondary level, the intricacies of mathematics begins to get more and more

complex and the mathematics teachers find it increasingly difficult to retain the

interest of the learners in the subject. It is a crucial stage for learners as it

prepares them for higher education and also for their future profession. In order

to strengthen the educational delivery models practiced by the mathematics

teachers, especially at the secondary and senior secondary level, it is vital that

the teachers undergo professional development programmes from time to time

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so that they are better equipped to deal with the growing challenges of their

classrooms.

Professional development programmes for mathematics teachers are designed

to be comprehensive and rigorous, equipping the teachers with a wide- ranging

set of pedagogical skills to become more effective in their classrooms.

Organizing and participating in such professional development programmes

help the teachers in the following ways:

• become more attune to the latest developments in the field of mathematics,

• develop greater commitment towards their learners,

• create a platform for sharing professional experiences,

• work towards continuous improvement of their teaching skills,

• keep abreast of skills required to adapt to ever- changing learners

curriculum,

• tailoring their teaching content according to the needs of their learners,

• acquiring knowledge and skills for integrating technology into everyday

teaching practices,

• to develop the knowledge and skills which they need to address learners’

learning challenges more effectively,

• integrate the latest research findings and teaching models to achieve better

learning outcomes,

• apply action research findings in diverse classrooms.

Check Your Progress


b) Check your answers with the answers given at the end of the Unit.

1) What is professional development?

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2) What is the need for professional development of mathematics teachers?

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3) What are the benefits of professional development for teachers?

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9.4 PROFESSIONAL DEVELOPMENT

PROGRAMMES FOR MATHEMATICS

TEACHERS

Professional development involves adopting a variety of strategies to ensure

that teachers continue to strengthen their pedagogical skills throughout their

career. In formal settings, professional development programmes are often

disseminated through seminars, conferences, workshops etc. Sometimes,

professional development also occurs in informal conditions, through

interactions across online platforms, peer group discussions etc.

The common modes of professional development are:

• Individual reading and/or research,

• Peer group discussions focused on a common topic or area of interest,

• Observation-teachers observing other teachers so as to improve their own skills,

• Expert advice-an expert teacher advising colleagues,

• Group meetings to learn a new strategy or teaching skill,

• Adopting reflective and exploratory practices,

• Staff meetings,

• Online courses,

• Short term courses designed by experts at local / regional/ national level,

• Seminars/ Conferences/ Workshops to learn from a variety of expertise

from around the state or country,

• Membership of professional bodies to keep oneself updated with the latest

trends , practices and issues pertaining to the field of study.

9.4.1 Seminars

Seminars are formal presentations by one or more experts in which the

participants are encouraged to discuss a particular area of interest. In the

context of professional development, it may be described as an occasion when

a teacher or expert and a group of teachers meet to interact and discuss the

intricacies of a chosen topic or subject. Mathematics as a subject is very

interesting as well as challenging. Sharing experiences with other teachers and

listening to experts’ on the concerned topics help the teachers evolve into better

practitioners of teaching skills and improve learners achievement.

Seminars may be organized at the local, regional, state, national or

international levels depending upon the needs and requirements of the teachers

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as well as the resources available. These seminars help the teachers update their

knowledge base regarding latest developments in the field of mathematics and

pedagogical skills being used. They also provide a platform for teachers to

interact with experts to discuss and evolve new, effective strategies to deal with

difficult classroom situations.

By attending seminars, not only do the teachers themselves benefit from the

discussions, but also their colleagues stand to gain from a summary of what

one has learned and obtain copies of relevant documents.

These days with technological advancements, many a times the teachers find

themselves attending webinars instead of seminars. Short for Web-based

seminar, a webinar is a seminar that is transmitted over the internet using video

conferencing software. A key feature of a webinar is its interactive elements-

the ability to give, receive and discuss information in real-time. Using webinar

software participants can share audio, documents and applications with

webinar participates. A webinar has all the advantages of a seminar without the

limitations of actually being physically present at the venue. This is a great

advantage for participates, especially in a profession like teaching which leaves

very little opportunity for the teachers to spare time for attending such

seminars.

For example, a seminar on ‘New Developments in Mathematics Teaching’

inspires Mr. X to adopt constructive approach in his classroom. As per

constructive belief, knowledge is actively created or invented by the child, not

passively received from the environment. Learners create new mathematical

knowledge by reflecting on their physical and mental actions. Ideas are

constructed or made meaningful when learners integrate them into their

existing structures of knowledge. Applying this principle in the classroom, the

teacher Mr. X asks the learners to construct a triangle, where information about

all three distances and all three angles has been given. Next the teacher asks the

learners to construct the next triangle, with fewer specifications, with less than

six. Slowly, with each step the learners learn to construct triangles with

minimum information.

9.4.2 Conferences

Conferences may be described as formal meetings in which a number of people

gather to talk about their ideas or problems related to a particular topic, usually

for a few days. It is a large gathering of individuals or members of one or more

organizations, for discussing matters of shared interest. Organizing conferences

involving members of the teaching community is an effective way to render

professional development to the teachers. Here, peer group discussions, shared

experiences regarding classroom behaviors and results of action research,

adopting reflective practices are some fruitful techniques that ultimately lead to

the participates becoming better teachers.

For example, Mr. X attends a conference on ‘Methods for Improving

Mathematical Reasoning’ among Secondary Level Learners. Here, he comes

across many other secondary level mathematics teachers facing similar

predicament- how to improve learners’ reasoning abilities? In algebra,

encouraging learners to think how changing an input (x) in an equation would

affect the output (y). Discussing among themselves, the teachers realize that in

order to solve such problems, the learners need to clearly see the abstract linear

relationships. They all concur that explaining in simpler language, sometimes

in the local language, would be beneficial. Upon coming back, Mr. X tries to


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adopt this technique in his classroom by explaining the algebraic word

problems in Hindi and then asking the learners to solve the equations. He gets

much quicker and better responses from the learners.

9.4.3 Online Sharing Communities

With the advancement of technology, sharing of resources and experiences

among like-minded people through online platforms, has become a norm.

Teachers as a community have a lot of common interests- issues related to

learners, curriculum content and its implementation, new technologies

available to the teachers, etc. Online platforms such as blogs, Facebook

communities etc provide a good opportunity to the teachers to share their own

experiences among a peer group, hold discussions on pertinent topics,

participate in online seminars ( known as webinars), even avail of online

training courses aimed at enriching their knowledge and skill base. Nowadays,

online sharing has gained popularity among the members of teaching

community owing to its ease of use, versatility, global reach and cost

effectiveness. Teachers are now transitioning to the online medium as a means

for professional development. More and more teachers are looking to these

online resources for solutions to their teaching issues and/ or training needs.

For example, The Math Forum is a community of teachers, mathematicians,

researchers, learners, and parents using the internet to learn math and improve

math education. The forum offers a wealth of problems and puzzles; online

mentoring; research; team problem solving; collaborations; and professional

development. It helps educators share ideas and acquire new skills.

(Source: http://mathforum.org/)

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9.4.4 Membership of Professional Organizations

A professional organization (also referred to as a professional body), is usually

a nonprofit organization which seeks to further a particular profession, the

interests of individuals engaged in that profession and the public interest. In the

field of education, there are numerous professional organizations that are

actively involved in facilitating better teaching- learning conditions and

learners achievement. With teachers juggling a multitude of roles, both in

professional as well as personal capacities, joining a professional organization

ought to be a high priority for them. These professional organizations offer a

plethora of benefits such as- exclusive online resources, networking

opportunities, free or discounted publications, chance to update their

knowledge of business and trade basics or acquire new skills through seminars,

workshops, conferences and online courses, develop mentoring relationships

with more experienced teachers etc. The Association of Mathematics Teachers

of India (AMTI) is one such organization in India that aims to assist teachers of

mathematics at all levels in improving their expertise and professional skills for

making mathematics interesting and enjoyable and disseminates new trends in

Mathematics Education.

For example, the Association of Mathematics Teachers of India (AMTI) has

been organizing seminars, conferences, workshops in various parts of the

country to meet and deliberate on important issues in Mathematics Education,

particularly at school level. Besides, the usual inauguration, valedictory

functions, there are endowment / memorial lectures on applied mathematics,

methodology and history of mathematics, group discussions, exhibitions,

recreations in mathematics, paper presentations, learners sessions, Quiz-written

and oral, Distinguished Mathematics teacher awards etc.

Check Your Progress



4) Name various formal and informal modes of professional development.

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5) How do seminars and conferences promote professional development?

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6) How does online sharing help teachers in their professional development?

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9.5 TEACHER AS A COMMUNITY OF

LEARNERS

Any social unit irrespective of size, that shares common values, interests is

situated in a common location is referred as a community. It consists of an

interacting population of various kinds of individuals having common

characteristics or interests. The members of the teaching community share

common academic goals and attitudes, interests as well as problems. Today

teachers have realized the need for continuous improvement through learning

new developments, to succeed in their teaching endeavours. A community of

learners may be described as a group of people who believe in the concept of

continuous learning and actively engage themselves in learning from each

other. In a learning community, teaching professionals come together in a

group, forming an informal community committed to learn. The goal of such

communities is to advance the collective knowledge and in that way to support

the growth of individual knowledge.

Teachers are today actively involved in collaborative learning, wherein the

teachers as a group of people learn or attempt to learn something together. As a

community, teachers strive to create and sustain a learning-centered, mutually

respectful and cohesive environment where they are actively and intentionally

constructing knowledge together. Such groups of teachers often meet regularly,

share expertise, and work collaboratively to improve their teaching skills and

enhance the academic achievement of their learners. Nowadays, learning

communities are being used as an alternate strategy for professional

development. Teachers are encouraged to share their expertise and experiences

with their peer group, resulting in more effective dissemination of action

research results and incorporation of effective instructional techniques being

used by other colleagues.

The ultimate beneficiary of teachers forming learning communities are the

learners, as the teachers work towards improving their skills and knowledge

through collaborative study, expertise exchange, and professional dialogue,

thereby promoting improved learner performance.

For example, when a mathematics quiz competition was announced at the

national level for secondary school learners, Mrs. Y a mathematics teacher

decided to bring together all mathematics teachers of schools of her district so

that they may deliberate on better preparing their learners and also share their

previous experiences. These weekly meetings gave the teachers hands-on

experience using practical approaches to improve their learners’ skills and also

energize them. The meetings gave teachers a chance to be learners, and to

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experience a sequence of activities designed to be just as engaging and varied

as those they might devise for their own learner Teachers had time during

meetings to consider what happened when they tried new things in the

classroom and discussed their experiences with their peer groups. As a result,

number of learners from the said district reached the state level quiz

competition.

Check Your Progress



7) Describe the concept of teachers as a community of learners.

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8) How do teachers’ learning communities promote professional

development?

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9.6 REFLECTIVE PRACTICES FOR

PROFESSIONAL DEVELOPMENT

Reflective practice is a way of studying one’s own experiences to improve the

way one works. The act of reflection is an effective method of self-

improvement. It is basically a method of assessing one’s own thoughts and

actions, to facilitate personal growth and enhance learning. Reflective practice

involves the use of self-analysis to understand, evaluate and interpret events

and experiences in which we are involved so that we may evolve and improve

our actions in future. It is an action-based skill which can be learnt and

improved, with time and sufficient practice. It is almost similar to action

research in terms of its cyclic and dynamic nature.

In a dynamic and challenging profession like teaching, the use of reflective

practices is vital to the success of the entire teaching-learning process. The

ability to reflect critically upon one’s teaching is infact crucial to become a

successful teacher. It helps the teachers become more proactive and better

skilled professionals. Reflective practices involve a process of self-observation

and self-evaluation.


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Reflective practice occurs when teachers consciously take on the role of

reflective practitioner, subject their own teaching skills to critical analysis,

reflect on their actions in the classroom, and work to improve their teaching

practices. Reflective practice can be a highly beneficial form of professional

development at the in-service levels of teaching. By adopting reflective

practices, teachers can gain a deeper understanding of their own individual

teaching styles and thus improve their effectiveness in the classroom.

Reflection in teaching may be considered to be working in two different ways -

reflection-in-action and reflection-on-action.

Reflection-in-action involves an immediate reaction. It refers to the quick

thinking and reactions that occur as one teaches in the classroom. For example-

the teacher while teaching in the classroom can observe that the pupils fail to

grasp the subject clearly. The reflection-in-action aspect allows the teacher to

realize this, contemplate why it is happening, and respond by doing it

differently. This might involve rewording one’s explanation or adopting a

different approach to teach the subject.

On the other hand, reflection-on-action involves a delayed reaction. It occurs

once the teacher exits the classroom and reconsiders the situation again. The

teacher gives deep thought as to why the pupils did not understand, what led to

the situation, what other options were available, what else could have been

done differently for a different and better result?

(Source: http://esthermyers.blogspot.in/2014_11_01_archive.html)

As a result of one’s reflection, the teacher may opt to do something in a

different way, or may just decide that the current method is the best way. Either

way, there is scope for self-analysis and/ or self improvement for the teacher,

that is what professional development, ultimately all about.

For example, Mrs. Y decided to record the proceedings of her mathematics

classroom. Later on she observed the video recording of each lesson and

discussed the observations with her peer group. She realized where her mistake

was while dealing with algebraic problems in her class. Based upon her

observations, and suggestions from peer group, she decided to opt for a

completely different approach to teach simple algebra to her learners, and

eventually, had better results. Observing a lesson enables teachers to shift their

thinking from a teaching focus to a learning focus whilst puzzling over their

learners’ mathematical thinking. As observers, they are free to focus on the

actual work the learners are doing, as well as the learners’ thought processes.

During the lesson observations the focus was on the teacher’s presentation as

well as on the learners’ content knowledge.

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9) What are reflective practices?

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10) How do reflection-in-action and reflection-on-action differ?

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11) What are the steps involved in reflective teaching?

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9.7 TEACHERS AS A RESEARCHER

Research is a voyage of discovery. It is a systematic quest for answers to

unsolved problems. Teachers are always on an unending quest for many things-

new knowledge, ways to improve learners achievement, techniques to enhance

their content delivery etc. As long as they are involved in the teaching-learning

process, all teachers act as researchers in some capacity or the other. Teaching

and research are, in fact, complementary to each other, one improving the

other’s performance.

Every day teachers actively involve themselves in research in their classrooms.

Constantly striving to attain balance between learners potentials and parental

expectations, optimizing the available resources for high learners achievement,

planning lesson delivery content and modes, evaluating learners’ performance,

and working cohesively with administrators are all part of teachers’ role as

researchers.

Teachers as researchers follow the same steps of research as followed by a

researcher in any other field of study. Be it gathering information, planning,

analyzing data, reflecting, implementing results or generalizing beyond their

classrooms, all these are actually steps involved in systematic goal oriented

inquiry i.e. research. Most members of the teaching community involve

themselves in classroom based research, aimed at improving their classroom


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practices and outcomes. In other words, almost all teachers participate in action

research with classrooms as their field of study.

Adopting reflective teaching practices in classrooms helps teachers evolve as

better practitioners of the profession as well as discover new areas of research.

With today’s teachers forming close knit learning communities, sharing of

teaching experiences and findings of action research, teachers are better

equipped to be efficacious in their professional capacity. Ultimately, research

aims to enable teachers, administrators and policy makers to make sound

decisions and effective policies regarding educational aspects which will best

serve the learners with teachers acting as researchers, these decisions and

policies become more feasible.

9.7.1 Conducting Action Research in Mathematics

Action research is a form of applied research in which the researcher works in

localized settings with a view to solve an immediate problem. It is about trying

to understand professional action from inside; as a result, it is carried out by

practitioners on their own practice, not (as in other forms of research), by

someone on somebody else’s practice. In the field of education, action research

is employed regularly by the teachers to improve their classroom practices. The

purpose of action research is to solve classroom problems through the

application of scientific method. Action research is a cyclic process. It is a

continuous process, based upon constant feedback and improvement. It is a

four step process involving:

• Planning

• Acting

• Observing

• Reflecting

Action research helps the teachers in:

• improving their teaching performance

• enhancement of learners achievement and improvement of the situation in

which the practice takes place

• better understanding of classroom problems and deriving solutions

• developing new and improved classroom practices beneficial to the learners as well as the teachers.

For example, Mr. X was facing some difficulty in teaching certain

mathematical concepts to class VI learner. He decided to opt for carrying out

an action research on his learners by adopting the cooperative learning

technique. He investigated the impact of cooperative learning on the

engagement, participation, and attitudes of his learners and also the impact of

cooperative learning upon his own teaching. He discovered that his learners,

not only preferred to learn in cooperative groups, but also that their levels of

engagement and participation, their attitudes toward math, and their quality of

work all improved greatly. Mr. X’s teaching also changed, and he found that he

began to enjoy teaching more. As a result of this research, he now plans to

continue and expand the amount of cooperative group work that happens in his

classroom.

Teaching of mathematics not only involves the knowledge of the subject, but

also the selection of appropriate content and skills of communication resulting

in improved understanding among the learners. While teaching mathematics,

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the mathematics teachers should adopt the most well suited teaching methods,

strategies and pedagogic resources that can facilitate their learners

understanding of the content being taught. The teaching of mathematics is a

complex, multi layered, dynamic activity that depends upon various factors for

its success. The nature of the content, quality of the instructional material, the

mode of presentation, the extent of subject knowledge of the teacher, resources

available to the teacher, the skills of the teacher, the classroom environment,

the level of motivation among the learners etc.

Teaching of mathematics entails developing skills like critical thinking,

analytical thinking, logical reasoning, decision making, and problem-solving

among the learners. This cannot be achieved by teaching through tried and

tested methods adopted for teaching other conventional subjects. Mathematics

demands more in terms of input from the teachers. The teachers are expected to

be well versed in conducting action research and evolving new techniques to

teach the nuances of mathematics to their learners. While teaching

mathematics, the emphasis should be more on understanding of basic concepts

and principles rather than the mechanics of it.

Different pedagogical methods are effective for different teaching contexts. In

a complex and diverse subject like mathematics, there is a plethora of methods

that can be applied. The onus is upon the teacher to choose the most

appropriate method for a given content. In certain situations, the teachers find

themselves at a loss regarding the most suitable method to employ since most

do not fit their classroom requirements. In such cases, it is imperative that the

teachers work towards innovating new methods to make their classroom

teaching more effective. Some such teaching methods that have been

developed and nowadays being employed very effectively across classrooms

for teaching mathematics are:

• Inquiry based learning

• Problem solving

• Active learning

• Cooperative learning

• Team based learning

• Participatory learning

Some activities which the mathematics teachers can adopt in their classrooms

to make certain topics more appealing to their learners:

(Source: waymadedu.org/pdf/Rachnamadam.pdf)


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12) How do teachers act as researchers?

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14) Why is action research important in education?

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15) Name some innovative practices employed in teaching mathematics.

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9.8 LET US SUM UP

The following issues/concepts are dealt in the unit:

� Professional development is essentially a training provided to teachers

during the course of their service period to promote their enrichment in

various aspects of content and pedagogy. To ensure learners’ bright future

it is vital to support a cycle of continuous professional growth for teachers.

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� Professional development programmes are often disseminated through

seminars, conferences, workshops, interactions across online platforms,

peer group discussions etc.

� In a learning community, teaching professionals come together in a group,

forming an informal community committed to learn. The goal of such

communities is to advance the collective knowledge and in that way to

support the growth of individual knowledge. Learning communities are

useful as an alternate strategy for professional development of teachers.

� Reflective practice is a way of studying one’s own experiences to improve

the way one works. Reflective practice can be a highly beneficial form of

professional development at the in-service levels of teaching. By adopting

reflective practices, teachers can gain a deeper understanding of their own

individual teaching styles, teachers thus improve their effectiveness in the

classroom. Reflection in teaching may be considered to be working in two

different ways - reflection-in-action and reflection-on-action.

� Research is a systematic quest for answers to unsolved problems. All

teachers act as researchers in some capacity or the other. Teaching and

research are in fact complementary to each other, one improving the other’s

performance. Research aims to enable teachers, administrators and policy

makers to make sound decisions and effective policies regarding

educational aspects, which will best serve the learners.

� Action research is a form of applied research in which the researcher works

in localized settings with a view to solve an immediate problem. The

purpose of action research in education is to solve classroom problems

through the application of scientific method. The mathematics teachers are

expected to be well versed in conducting action research and evolving new

techniques to teach the nuances of mathematics to their learners.


1) Explain the term professional development and its benefits for teachers.

2) Discuss the concept of teachers as a community of learners.

3) Describe reflective practices and the steps involved in reflective teaching.

4) List some innovative practices which may be employed in teaching

mathematics.


1) Professional development refers to an in-service training wherein the

teachers get an opportunity to upgrade their content knowledge and

pedagogical skills. Usually, it is in a formal setting, at times it can be

provided informally.

2) Mathematics teachers need to undergo professional development to

strengthen their educational delivery models, especially at the secondary

and senior secondary level and improve themselves in various aspects of

content and pedagogy.

3) It equips the teachers with a wide- ranging set of pedagogical skills which

helps them to become more effective in their classrooms. It keeps the


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teachers abreast of skills required to adapt to ever- changing learners

curriculum and also integrate the latest research findings and teaching

models to achieve better learning outcomes.

4) Formal modes of professional development programmes are seminars,

conferences, workshops, etc whereas informal modes of professional

development include learning through observation, interactions across

online platforms, peer group discussions, expert advice etc.

5) Seminars help the teachers update their knowledge base regarding latest

developments in the field of mathematics and pedagogical skills being

used. They also provide a platform for teachers to interact with experts to

discuss and evolve new, effective strategies to deal with difficult

classroom situations. Organizing conferences facilitates teacher peer group

discussions, shared experiences regarding classroom behaviours and

results of action research, adopting reflective practices are some fruitful

techniques that ultimately lead to the attendees becoming better teachers.

6) Online platforms such as blogs, Facebook communities etc provide a good

opportunity to the teachers to share their own experiences among a peer

group, hold discussions on pertinent topics, participate in online seminars,

even avail of online training courses aimed at enriching their knowledge

and skill base.

7) A community of learners may be described as a group of people who

believe in the concept of continuous learning and actively engage

themselves in learning from one another. Teachers as a learning

community, come together in a group, forming an informal community

committed to learn with the goal of advancing their collective knowledge

as well as support the growth of individual knowledge.

8) As a community of learners, teachers promote their own professional

development by creating and sustaining a learning-centered, mutually

respectful and cohesive environment where they are actively and

intentionally constructing knowledge together. These groups of teachers

often meet regularly, share expertise, and work collaboratively to improve

their teaching skills and enhance the academic achievement of their

learners.

9) Reflective practice is a way of studying one’s own experiences to improve

the way one works. It involves the use of self-analysis to understand,

evaluate and interpret events and experiences in which we are involved so

that we may evolve and improve our actions in future. It is an action-based

skill which can be learned and honed, with time and sufficient practice. It

is almost similar to action research in terms of its cyclic and dynamic

nature.

10) Reflection-in-action involves an immediate reaction. It refers to the quick

thinking and reactions that occur as one teaches in the classroom. The

reflection-in-action aspect allows the teacher to realize this, contemplate

why it is happening, and respond by doing it differently. On the other

hand, reflection-on-action involves a delayed reaction. It occurs once the

teacher exits the classroom and reconsiders the situation again. The

teacher gives deep thought as to why the pupils did not understand, what

led to the situation, what other options were available, what else could

have been done differently for a different and better result?

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11) Reflective teaching is a cyclic process which involves the following steps:

1) Experience

2) Observation

3) Reflection

4) Planning

12) Every day teachers actively involve themselves in research in their

classrooms. Teachers as researchers follow the same steps of research as

followed by a researcher in any other field of study- gathering

information, planning, analyzing data, reflecting, implementing results or

generalizing beyond their classrooms.

13) Research aims to enable teachers, administrators and policy makers to

make sound decisions and effective policies regarding educational aspects

which will best serve the learners. With teachers acting as researchers,

these decisions and policies become more feasible.

14) Action research helps the teachers in improving their teaching

performance, enhancement of learners achievement, gain better

understanding of classroom problems and deriving solutions and develop

new and improved classroom practices beneficial to the learners as well as

the teachers themselves.

15) Some innovative practices employed in teaching mathematics are- Inquiry

based learning, Problem solving, Active learning, Cooperative learning,

Team based learning, Participatory learning etc.

9.11 REFERENCES AND SUGGESTED READING

• Diaz-Maggioli, G. (2004). Teacher-Centered Professional

Development.Alexandria, Virginia, USA: Association for Supervision &

Curriculum Development.

• National Research Council. (2001). Adding it up: Helping learners learn

mathematics. J. Kilpatrick, J.Swafford, and B. Findell (Eds.). Mathematics

Learning Study Committee, Center for Education, Division of Behavioral

and Social Sciences and Education. Washington, DC: National Academy

Press.

• Schon, D. A. (1996). Educating the reflective practitioner: Toward a new

design for teaching and learning in the professions. San Francisco: Jossey-

Bass, Inc.

• Sharma, N.( 2015). Professional Development of Teacher. New Delhi

Publishers.

• www.htttp://waymadedu.org/pdf/Rachnamadam.pdf

• www. http://esthermyers.blogspot.in/2014_11_01_archive.html

• www. http://mathforum.org/

Documents

BES-143 Pedagogy of Mathematics