K.THIYAGU APH JOURNAL 2013
1
PEDAGOGICAL INNOVATIONS IN TEACHING OF MATHEMATICS:
BULLETIN BOARD AND SMART BOARD
K.Thiyagu,
Assistant Professor in Mathematics
thiyagusuri@gmail.com
9486812800
ABSTRACT
Though Mathematics being so important subject and occupying a central
position since the Ancient period still it has not been the interest of many students.
The gaps are found between aspiration and achievement. Mathematics is highly
abstract. It is concerned with ideas rather than objects; with the manipulation of
symbols rather than the manipulation of object. It is a closely-knit structure in
which ideas are interrelated. Mathematical concepts are hierarchical and
interconnected, much like a house of cards. Unless lower-level concepts are
mastered, higher-level concepts cannot be understood. Students, who discover some
of the structures of mathematics, are often impressed by its beauty. They note the
lack of contradiction, and they see how a new technique can be derived from one
that has already been learned.
The teaching and learning of mathematics is a complex activity and many
factors determine the success of this activity. The nature and quality of
instructional material, the presentation of content, the pedagogic skills of the
teacher, the learning environment, the motivation of the students are all important
and must be kept in view in any effort to ensure quality in teaching-learning of
mathematics. In this paper explores the innovations and innovative practices in
teaching mathematics, under teaching methods, strategies and pedagogic
resources. This paper also deals about the bulletin board and smart board uses in
teaching and learning mathematics.
Key words: bulletin board, smart board, math etc.,
K.THIYAGU APH JOURNAL 2013
2
BULLETIN BOARD
Math is all about logic and problem-solving skills, but that doesn't mean a math
classroom has to be boring. Use bulletin boards to teach your students about math skills,
and brighten your room.
Famous Mathematicians: You can teach your students about the history of math
by providing biographical information on mathematicians from the past. Display several at
once, or put up one biography each month for the whole school year.
Calculator Functions: In higher-level math classes, students need to learn how to
use complicated graphing calculators. Use your bulletin board to display a diagram of the
calculator your students use, and include labels for the different function keys for your
students to refer to during class.
Familiar Formulas: Sometimes math classes involve memorizing a lot of formulas
and equations. Help your students remember by displaying frequently-used formulas (such
as the quadratic equation) on the board.
Student Involvement: Have your students prepare each other for upcoming
chapters by creating a bulletin board for credit --regular or extra. The students can learn
from their research as they create the board, and other students in the class will learn as
they observe the display while it remains on the board.
Extra Time: Use your bulletin board as a tool to occupy students who finish work
early during class work time. Display difficult problems, logic puzzles or brain teasers on
your bulletin board. Then, when a student finishes work early, offer extra credit to work on
one of the items from the board. This way, you can keep all students on task all period and
decorate your classroom at the same time.
IDEAS FOR A BULLETIN BOARD FOR MATH CLASS
Brighten up your classroom with colorful and engaging bulletin boards for math
class. Bulletin board ideas should incorporate concepts the students are learning in a
visually pleasing and educational manner. Use common classroom supplies to attach
holders to the bulletin board to facilitate interactive activities during daily math meetings or
math center sessions.
Math Concepts: Create one permanent bulletin board for daily activities and others
that focus on current textbook units and math concepts. Incorporate a math meeting bulletin
board into daily learning tasks and lesson plans. Close textbooks for 15 minutes during
K.THIYAGU APH JOURNAL 2013
3
class and give the students the opportunity to be actively engaged in an interactive bulletin
board activity.
Curriculum Goals: Match the activities to ongoing curriculum goals and give each
student a turn to participate in updating the board during the math meeting. Alternate the
bulletin board border on a monthly or seasonal basis to keep the style of the board from
looking mundane. Math meeting bulletin board topics vary by age level, but typical
elementary school displays relay common critical thinking, measurement and counting
skills. Laminate any charts you place on the board so they can be wiped clean daily or
reused in future years.
Time, Money and Temperature: Staple a cardboard clock to the bulletin board
with a writing strip next to the clock. Have a student set the math meeting time on the clock
and write the time in digital format on the strip. Pin a Styrofoam cup to the board with a
different amount of pretend coins in the cup daily. Students will count the coins and write
the answer on a writing strip. Post a thermometer on an exterior classroom window and task
a student with reading the temperature and writing the degrees on the board and adjusting
the "mercury" on a cardboard thermometer attached to the board. The possible activities for
the board could also include measuring, time line, story problem and math facts activities.
Story Problems and Art: Fuse creativity and writing with math class by creating a
"problem of the day" bulletin board. Give each student a piece of construction paper and
standard notebook paper. Have each student write a story problem on the front of the paper.
On the back of the paper instruct the student to write the correct answer and draw a picture
representing the question on the front. Staple the question onto a seasonally decorated
bulletin board at the beginning of each class. The first student to answer the question
correctly gets to turn the paper over to reveal the answer and choose the question for the
next day.
ELEMENTARY MATH TECHNOLOGICAL TOOLS FOR TEACHING
Technology is becoming a larger component of the standard general education
classroom. Teachers must learn how to use the technology and determine how to integrate
the technology into their lesson plans. Elementary teachers teaching math and science
lessons can use a wide variety of technology resources to provide students with interesting
and relevant information in ways past generations never experienced.
Whiteboard: Whiteboards, also called Smart Boards, are interactive boards that
work as both as a blackboard and an interactive touch computer. A teacher can use the
K.THIYAGU APH JOURNAL 2013
4
Whiteboard for many purposes. A teacher may write a math problem on the board, touch a
link to display a video or use a program that allows him to create a graph all without having
to leave the area of the board. Whiteboards are fun to play with and websites like
Birmingham Grids for Learning offer many math and science lessons for all grade levels.
Tablet Computers: Tablet computers such as the iPad allow students to use small
hand-held computers that use touch screen. These computers can take the place of
textbooks by displaying the traditional classroom book but with the added advantage of
adding video and other links for the students to access. These tablet computers offer many
different applications, called apps, designed for a specific purpose such as identifying birds
or examining the internal organs of the body.
Digital Projectors: Digital projectors connect to a computer in the classroom to
display websites and videos that a teacher can use in lesson plans for math or science. The
projectors make presenting lessons simple using power point presentations. Powerpoint
presentations present the information in clear terms that a teacher can give to the class for
study later. She can also connect the digital projector to digital microscopes or cameras that
can display items on screen so that all students may view. This ability can help in science
class when discussing small living things such as insects.
GPS: Global positioning systems, or GPS, help people navigate without the use of
compasses and maps. Using hand-held GPS systems can provide students with
entertainment and information. Teachers can integrate games such as geocaching-type
games into lesson plans. Science lessons surrounding the GPS can involve introducing the
ideas of latitude and longitude and how the satellites determine a location. Using the GPS
can also mark the location of different types of animals or plants located on the school
grounds for review with the entire class.
SMART BOARD MATH LESSONS
The SMART Board interactive whiteboard is becoming more widely used in
schools, with teachers utilizing this technology to enhance the way they teach a variety of
subjects, including math. As use of this technology grows, so do the resources available for
teachers seeking lesson plans and other educational ideas using the capabilities of the
SMART Board.
Smart Board: Created by SMART Technologies, the SMART Board is an
interactive whiteboard that can be connected to a computer, allowing the user to manipulate
the image projected on the board by pressing a finger on the board's touch-sensitive surface.
K.THIYAGU APH JOURNAL 2013
5
Digital SMART "pens" of different colors can be used to write on the screen, with the
computer recognizing each specific color while these notes are projected on the screen. The
entire presentation, including notes made in class, can then be saved and subsequently
emailed or printed out as study notes for students.
Math Lessons for SMART Board; SMART has its own software, SMART
Notebook Math Tools that enables teachers to outline a lesson, write notes and solve
equations. Other features include a shape creator, a graph builder and an advanced equation
editor. The interactive nature of the SMART Board allows computer-generated lessons
(including high-quality graphics) to be manipulated by both teacher and student, which
makes the learning process far more hands-on than it would be if a teacher drew an
equation on a conventional blackboard while students took notes.
Where to Find Math Lessons: The growing popularity of the SMART Board
(which, according to SMART's data, is used by millions of teachers and students
throughout the world) has resulted in an explosion in online resources that allows teachers
to utilize the work of their peers in other schools--and even other countries. One excellent
resource is the website TheTeachersGuide.com, which includes a section listing numerous
links to websites containing interactive math material specifically designed for use with a
SMART Board.
Teacher-Generated Lessons: Another useful website for educators is
TeachersLoveSmartboards.com, which features online training on how to use the SMART
Board, as well as tips and tricks, Web resources and a vast array of lessons in math and
other subjects, for every age group from kindergarten to high school. In addition, many
school districts across the country share teachers' SMART Board lessons online (the
website of the Bristol, Virginia, school district, for example) as well as offering links to
other online resources. The result has been an ever-increasing virtual community of
teachers sharing their SMART Board lessons with their peers for use in the classroom.
HOW TO USE TECHNOLOGY TO TEACH ELEMENTARY MATH
Technology is a growing presence in everyday life, a modern convenience that can
be found at home, work or in the classroom. An increasingly popular teaching tool in the
classroom, technology engages students with interactive software programs that facilitate
learning. Using technology to teach elementary-level math prepares students academically,
makes them technologically savvy and aids their mathematics proficiency--all while
making learning math fun and exciting.
K.THIYAGU APH JOURNAL 2013
6
Instructions
 Consider the school budget. The use of technology in the classroom often requires
hi-tech equipment such as computers, advanced calculators or interactive software
programs. Before committing to implementing technology in the classroom as a
teaching method, approval from the school board may be required. Considerations
for the budget may include whether or not to use the school's current computer lab
or incorporate a mini-computer lab in the classroom where the students can alternate
using the computers.
 Evaluate your students' current mathematics ability. Depending on the age of
your students and where they stand academically, they may or may not be ready to
accept the new teaching method. If they are advanced, perhaps they would like a
new challenge. But if they have a low attention span, you may want to develop a
new strategy for incorporating technology into the classroom. This may include
bringing in one donated computer at a time to test the system.
 Prepare the students. It is important to communicate changes to the students and
to prepare them for alterations in classroom activities, expectations and
responsibilities. Give them some time to adjust and eventually embrace the new
technology.
 Discover the technology that matches your personal teaching style and your
classroom culture. There are many types of technology that aid in teaching math to
elementary students, such as ASSISTments, Smart Tech and Cognitive Tutor. If you
want a program for the classroom, Smart Tech is a practical solution. It uses the
classroom whiteboard, connected to the computer, to display Smart Tech computer
screen information directly to the whiteboard, allowing students to drag, enlarge or
create objects. ASSISTments and Cognitive Tutor programs challenge students by
personalizing questions and creating math scenarios based on students' previous
answers or current math abilities.
 Personalize the technology. Many of the technology programs available allow the
teachers to personalize the programs to match the students' academic level.
Depending on the program you choose, you may be able to create your own math
questions. You can be as creative with this process as you like.
K.THIYAGU APH JOURNAL 2013
7
 Start teaching math using your chosen technology. Your first time will be a trial
and error situation as you learn what works and what does not. It is okay to make
mistakes. Just make the corrections as you go along and have fun with the process.
CONCLUSION
In this paper discussed the innovations and innovative practices in teaching
mathematics, under teaching methods, strategies and pedagogic resources. This paper
highlighted the two importance concepts: bulletin board and smart board. The process of
innovation is generally described as consisting of three essential steps, starting with the
conception of an idea, which is then proposed and is finally adopted. Though many ideas
have been conceived to bring about change in the teaching of mathematics, it is yet to be
proposed and adopted. So, the innovations discussed may not be new in terms of the idea
but is new in terms of practice.
REFERENCES:
 Bhatia, K. (1992). Identification and Remedy of Difficulties in Learning Fractions
with Programmed Instructional Material. Indian Educational Review, 27(3). 102-
106.
 Canny, M. E. (1984). The Relationship of Manipulative Materials to Achievement
in Three Areas of Fourth-Grade Mathematics: Computation, Concept Development
and Problem Solving. Dissertation-Abstracts International, 45 A. 775-776.
 Ducharme, R. E. & Ducharme, M. K. (1999). Using Teacher Reflective Practice to
Evaluate Professional Development in Maths and Science. Journal of Teacher
Education. 50 (1). 42.
 Dutta, A. (1990) Learning Disability in the Reasoning Power of the Students in
Geometry. An Unpublished Ph.D. Thesis, University of Kalyani. In J. P. Sharma
(Ed.) Fifth Survey of Educational Research. New Delhi: NCERT.
 Heddens, J.W. Improving Mathematics Teaching by Using Manipulatives. Kent
State University.http://www.fed.cuhk.edu.hk/~fllee/mathfor/13hedden.html
 .NCERT (1971) Education and National Development, Report of The Education
Commission 1964-66. New Delhi: NCERT.
 Sidhu, K. S. (1995) The Teaching of Mathematics. New Delhi: Sterling Publishers
Pvt.Ltd.

PEDAGOGICAL INNOVATIONS IN TEACHING OF MATHEMATICS: BULLETIN BOARD AND SMART BOARD

  • 1.
    K.THIYAGU APH JOURNAL2013 1 PEDAGOGICAL INNOVATIONS IN TEACHING OF MATHEMATICS: BULLETIN BOARD AND SMART BOARD K.Thiyagu, Assistant Professor in Mathematics thiyagusuri@gmail.com 9486812800 ABSTRACT Though Mathematics being so important subject and occupying a central position since the Ancient period still it has not been the interest of many students. The gaps are found between aspiration and achievement. Mathematics is highly abstract. It is concerned with ideas rather than objects; with the manipulation of symbols rather than the manipulation of object. It is a closely-knit structure in which ideas are interrelated. Mathematical concepts are hierarchical and interconnected, much like a house of cards. Unless lower-level concepts are mastered, higher-level concepts cannot be understood. Students, who discover some of the structures of mathematics, are often impressed by its beauty. They note the lack of contradiction, and they see how a new technique can be derived from one that has already been learned. The teaching and learning of mathematics is a complex activity and many factors determine the success of this activity. The nature and quality of instructional material, the presentation of content, the pedagogic skills of the teacher, the learning environment, the motivation of the students are all important and must be kept in view in any effort to ensure quality in teaching-learning of mathematics. In this paper explores the innovations and innovative practices in teaching mathematics, under teaching methods, strategies and pedagogic resources. This paper also deals about the bulletin board and smart board uses in teaching and learning mathematics. Key words: bulletin board, smart board, math etc.,
  • 2.
    K.THIYAGU APH JOURNAL2013 2 BULLETIN BOARD Math is all about logic and problem-solving skills, but that doesn't mean a math classroom has to be boring. Use bulletin boards to teach your students about math skills, and brighten your room. Famous Mathematicians: You can teach your students about the history of math by providing biographical information on mathematicians from the past. Display several at once, or put up one biography each month for the whole school year. Calculator Functions: In higher-level math classes, students need to learn how to use complicated graphing calculators. Use your bulletin board to display a diagram of the calculator your students use, and include labels for the different function keys for your students to refer to during class. Familiar Formulas: Sometimes math classes involve memorizing a lot of formulas and equations. Help your students remember by displaying frequently-used formulas (such as the quadratic equation) on the board. Student Involvement: Have your students prepare each other for upcoming chapters by creating a bulletin board for credit --regular or extra. The students can learn from their research as they create the board, and other students in the class will learn as they observe the display while it remains on the board. Extra Time: Use your bulletin board as a tool to occupy students who finish work early during class work time. Display difficult problems, logic puzzles or brain teasers on your bulletin board. Then, when a student finishes work early, offer extra credit to work on one of the items from the board. This way, you can keep all students on task all period and decorate your classroom at the same time. IDEAS FOR A BULLETIN BOARD FOR MATH CLASS Brighten up your classroom with colorful and engaging bulletin boards for math class. Bulletin board ideas should incorporate concepts the students are learning in a visually pleasing and educational manner. Use common classroom supplies to attach holders to the bulletin board to facilitate interactive activities during daily math meetings or math center sessions. Math Concepts: Create one permanent bulletin board for daily activities and others that focus on current textbook units and math concepts. Incorporate a math meeting bulletin board into daily learning tasks and lesson plans. Close textbooks for 15 minutes during
  • 3.
    K.THIYAGU APH JOURNAL2013 3 class and give the students the opportunity to be actively engaged in an interactive bulletin board activity. Curriculum Goals: Match the activities to ongoing curriculum goals and give each student a turn to participate in updating the board during the math meeting. Alternate the bulletin board border on a monthly or seasonal basis to keep the style of the board from looking mundane. Math meeting bulletin board topics vary by age level, but typical elementary school displays relay common critical thinking, measurement and counting skills. Laminate any charts you place on the board so they can be wiped clean daily or reused in future years. Time, Money and Temperature: Staple a cardboard clock to the bulletin board with a writing strip next to the clock. Have a student set the math meeting time on the clock and write the time in digital format on the strip. Pin a Styrofoam cup to the board with a different amount of pretend coins in the cup daily. Students will count the coins and write the answer on a writing strip. Post a thermometer on an exterior classroom window and task a student with reading the temperature and writing the degrees on the board and adjusting the "mercury" on a cardboard thermometer attached to the board. The possible activities for the board could also include measuring, time line, story problem and math facts activities. Story Problems and Art: Fuse creativity and writing with math class by creating a "problem of the day" bulletin board. Give each student a piece of construction paper and standard notebook paper. Have each student write a story problem on the front of the paper. On the back of the paper instruct the student to write the correct answer and draw a picture representing the question on the front. Staple the question onto a seasonally decorated bulletin board at the beginning of each class. The first student to answer the question correctly gets to turn the paper over to reveal the answer and choose the question for the next day. ELEMENTARY MATH TECHNOLOGICAL TOOLS FOR TEACHING Technology is becoming a larger component of the standard general education classroom. Teachers must learn how to use the technology and determine how to integrate the technology into their lesson plans. Elementary teachers teaching math and science lessons can use a wide variety of technology resources to provide students with interesting and relevant information in ways past generations never experienced. Whiteboard: Whiteboards, also called Smart Boards, are interactive boards that work as both as a blackboard and an interactive touch computer. A teacher can use the
  • 4.
    K.THIYAGU APH JOURNAL2013 4 Whiteboard for many purposes. A teacher may write a math problem on the board, touch a link to display a video or use a program that allows him to create a graph all without having to leave the area of the board. Whiteboards are fun to play with and websites like Birmingham Grids for Learning offer many math and science lessons for all grade levels. Tablet Computers: Tablet computers such as the iPad allow students to use small hand-held computers that use touch screen. These computers can take the place of textbooks by displaying the traditional classroom book but with the added advantage of adding video and other links for the students to access. These tablet computers offer many different applications, called apps, designed for a specific purpose such as identifying birds or examining the internal organs of the body. Digital Projectors: Digital projectors connect to a computer in the classroom to display websites and videos that a teacher can use in lesson plans for math or science. The projectors make presenting lessons simple using power point presentations. Powerpoint presentations present the information in clear terms that a teacher can give to the class for study later. She can also connect the digital projector to digital microscopes or cameras that can display items on screen so that all students may view. This ability can help in science class when discussing small living things such as insects. GPS: Global positioning systems, or GPS, help people navigate without the use of compasses and maps. Using hand-held GPS systems can provide students with entertainment and information. Teachers can integrate games such as geocaching-type games into lesson plans. Science lessons surrounding the GPS can involve introducing the ideas of latitude and longitude and how the satellites determine a location. Using the GPS can also mark the location of different types of animals or plants located on the school grounds for review with the entire class. SMART BOARD MATH LESSONS The SMART Board interactive whiteboard is becoming more widely used in schools, with teachers utilizing this technology to enhance the way they teach a variety of subjects, including math. As use of this technology grows, so do the resources available for teachers seeking lesson plans and other educational ideas using the capabilities of the SMART Board. Smart Board: Created by SMART Technologies, the SMART Board is an interactive whiteboard that can be connected to a computer, allowing the user to manipulate the image projected on the board by pressing a finger on the board's touch-sensitive surface.
  • 5.
    K.THIYAGU APH JOURNAL2013 5 Digital SMART "pens" of different colors can be used to write on the screen, with the computer recognizing each specific color while these notes are projected on the screen. The entire presentation, including notes made in class, can then be saved and subsequently emailed or printed out as study notes for students. Math Lessons for SMART Board; SMART has its own software, SMART Notebook Math Tools that enables teachers to outline a lesson, write notes and solve equations. Other features include a shape creator, a graph builder and an advanced equation editor. The interactive nature of the SMART Board allows computer-generated lessons (including high-quality graphics) to be manipulated by both teacher and student, which makes the learning process far more hands-on than it would be if a teacher drew an equation on a conventional blackboard while students took notes. Where to Find Math Lessons: The growing popularity of the SMART Board (which, according to SMART's data, is used by millions of teachers and students throughout the world) has resulted in an explosion in online resources that allows teachers to utilize the work of their peers in other schools--and even other countries. One excellent resource is the website TheTeachersGuide.com, which includes a section listing numerous links to websites containing interactive math material specifically designed for use with a SMART Board. Teacher-Generated Lessons: Another useful website for educators is TeachersLoveSmartboards.com, which features online training on how to use the SMART Board, as well as tips and tricks, Web resources and a vast array of lessons in math and other subjects, for every age group from kindergarten to high school. In addition, many school districts across the country share teachers' SMART Board lessons online (the website of the Bristol, Virginia, school district, for example) as well as offering links to other online resources. The result has been an ever-increasing virtual community of teachers sharing their SMART Board lessons with their peers for use in the classroom. HOW TO USE TECHNOLOGY TO TEACH ELEMENTARY MATH Technology is a growing presence in everyday life, a modern convenience that can be found at home, work or in the classroom. An increasingly popular teaching tool in the classroom, technology engages students with interactive software programs that facilitate learning. Using technology to teach elementary-level math prepares students academically, makes them technologically savvy and aids their mathematics proficiency--all while making learning math fun and exciting.
  • 6.
    K.THIYAGU APH JOURNAL2013 6 Instructions  Consider the school budget. The use of technology in the classroom often requires hi-tech equipment such as computers, advanced calculators or interactive software programs. Before committing to implementing technology in the classroom as a teaching method, approval from the school board may be required. Considerations for the budget may include whether or not to use the school's current computer lab or incorporate a mini-computer lab in the classroom where the students can alternate using the computers.  Evaluate your students' current mathematics ability. Depending on the age of your students and where they stand academically, they may or may not be ready to accept the new teaching method. If they are advanced, perhaps they would like a new challenge. But if they have a low attention span, you may want to develop a new strategy for incorporating technology into the classroom. This may include bringing in one donated computer at a time to test the system.  Prepare the students. It is important to communicate changes to the students and to prepare them for alterations in classroom activities, expectations and responsibilities. Give them some time to adjust and eventually embrace the new technology.  Discover the technology that matches your personal teaching style and your classroom culture. There are many types of technology that aid in teaching math to elementary students, such as ASSISTments, Smart Tech and Cognitive Tutor. If you want a program for the classroom, Smart Tech is a practical solution. It uses the classroom whiteboard, connected to the computer, to display Smart Tech computer screen information directly to the whiteboard, allowing students to drag, enlarge or create objects. ASSISTments and Cognitive Tutor programs challenge students by personalizing questions and creating math scenarios based on students' previous answers or current math abilities.  Personalize the technology. Many of the technology programs available allow the teachers to personalize the programs to match the students' academic level. Depending on the program you choose, you may be able to create your own math questions. You can be as creative with this process as you like.
  • 7.
    K.THIYAGU APH JOURNAL2013 7  Start teaching math using your chosen technology. Your first time will be a trial and error situation as you learn what works and what does not. It is okay to make mistakes. Just make the corrections as you go along and have fun with the process. CONCLUSION In this paper discussed the innovations and innovative practices in teaching mathematics, under teaching methods, strategies and pedagogic resources. This paper highlighted the two importance concepts: bulletin board and smart board. The process of innovation is generally described as consisting of three essential steps, starting with the conception of an idea, which is then proposed and is finally adopted. Though many ideas have been conceived to bring about change in the teaching of mathematics, it is yet to be proposed and adopted. So, the innovations discussed may not be new in terms of the idea but is new in terms of practice. REFERENCES:  Bhatia, K. (1992). Identification and Remedy of Difficulties in Learning Fractions with Programmed Instructional Material. Indian Educational Review, 27(3). 102- 106.  Canny, M. E. (1984). The Relationship of Manipulative Materials to Achievement in Three Areas of Fourth-Grade Mathematics: Computation, Concept Development and Problem Solving. Dissertation-Abstracts International, 45 A. 775-776.  Ducharme, R. E. & Ducharme, M. K. (1999). Using Teacher Reflective Practice to Evaluate Professional Development in Maths and Science. Journal of Teacher Education. 50 (1). 42.  Dutta, A. (1990) Learning Disability in the Reasoning Power of the Students in Geometry. An Unpublished Ph.D. Thesis, University of Kalyani. In J. P. Sharma (Ed.) Fifth Survey of Educational Research. New Delhi: NCERT.  Heddens, J.W. Improving Mathematics Teaching by Using Manipulatives. Kent State University.http://www.fed.cuhk.edu.hk/~fllee/mathfor/13hedden.html  .NCERT (1971) Education and National Development, Report of The Education Commission 1964-66. New Delhi: NCERT.  Sidhu, K. S. (1995) The Teaching of Mathematics. New Delhi: Sterling Publishers Pvt.Ltd.