Dr. V.K.Maheshwari, M.A(Sociology, Philosophy) B.Sc. M. Ed, Ph. D

Former Principal, K.L.D.A.V.(P.G) College, Roorkee, India

Mathematics as a subject is indispensable in the development of any nation with respect to science and technology since mathematics itself is the language of science. In this 21st century where virtually all attentions are shifted towards technological advancements and the mathematics education into the 21st century project is waxing stronger in objectively achieving all its goals in which mathematics is a veritable tool.

The past fifteen years has witnessed wide-spread activity relative to new mathematics curricula for elementary, junior high, and high school pupils. New school mathematics differ very little from old ones as far as subject matter is concerned. Only a few old topics have been de-emphasized and only a few new topics have been added. The chief difference between the old and the new is the point of view toward mathematics. Now there is an equal emphasis on an understanding of the basic concepts of mathematics and their interrelationships, i.e., the structure of mathematics.

The concept of a mathematics laboratory has become very popular in recent years. The phase has blossomed in to popularity so fast that the variety of meanings which have been attributed to it all are struggling to co-exist.

Mathematics laboratory is a room wherein we find collection of different kinds of materials and teaching/learning aids, needed to help the students understand the concepts through relevant, meaningful and concrete activities. These activities may be carried out by the teacher or the students to explore the world of mathematics, to learn, to discover and to develop an interest in the subject.

As defined by Adenegan (2003), the mathematics laboratory is a unique room or place, with relevant and up-to-date equipment known as instructional materials, designated for the teaching and learning of mathematics and other scientific or research work, whereby a trained and professionally qualified person (mathematics teacher) readily interact with learners (students) on specified set of instructions.

**The functions of mathematics laboratory**

Thus, as highlighted by Adenegan (2003), the functions of mathematics laboratory include the followings: – Permitting students to learn abstract concepts through concrete experiences and thus increase their understanding of those ideas. – Enabling students to personally experience the joy of discovering principles and relationships.

- Arousing interest and motivating learning.
- Cultivating favorable attitudes towards mathematics.
- Enriching and varying instructions.
- Encouraging and developing creative problems solving ability.
- Allowing for individual differences in manner and speed at which students learn.
- Making students to see the origin of mathematical ideas and participating in “mathematics in the making”
- Allowing students to actually engage in the doing rather than being a passive observer or recipient of knowledge in the learning process.

** Design and general layout.**

A suggested design and general layout of laboratory which can accommodate about 30 students at a time. The schools may change the design and general layout to suit their own requirements.

**Physical Infrastructure and Materials**

It is envisaged that every school will have a Mathematics Laboratory with a general design and layout with suitable change, if desired, to meet its own requirements. The minimum materials required to be kept in the laboratory may include all essential equipment, raw materials and other essential things to carry out the activities included in the document effectively. The quantity of different materials may vary from one school to another depending upon the size of the group.

** Human Resources**

It is desirable that a person with minimum qualification of graduation (with mathematics as one of the subjects) and professional qualification of Bachelor in Education be made in charge of the Mathematics Laboratory. He/she is expected to have special skills and interest to carry out practical work in the subject. It will be an additional advantage if the in charge possesses related experience in the profession. The concerned mathematics teacher will accompany the class to the laboratory and the two will jointly conduct the desired activities. A laboratory attendant or laboratory assistant with suitable qualification and desired knowledge in the subject can be an added advantage.

**Time Allocation for activities.**

It is desirable that about 15% – 20% of the total available time for mathematics be devoted to activities. Proper allocation of periods for laboratory activities may be made in the time table.

**List of materials used in the mathematics laboratory**

- Broom sticks
- Chart papers, glazed papers, sketch pens.
- Collage (Paper cutting & pasting)
- Geo–board, rubber band
- Graph paper
- Paper folding
- Pins & threads
- Stationery
- Transparency sheets, cello tape
- Unit Cubes (wooden or any material)

**General Pedagogy for Mathematic Lab usage**

** **

The pedagogy must be developed in such a manner that it helps in the all round development of a learner at the same time it is easily understood by all. That is why, in developing the mathematics pedagogy following must be specified -

- Content to be made more relevant to the children’s life and experiences .
- Effective approach of teaching learning process of the subject must be specified as activities based, learner centered, load free, stress free, enjoyable and effective.
- It should be aimed at the learner’s 100% competency development and making her/ him a competent person covering her/his total mental growth, which can be, reflected in the day to day actions and whole life activities.
- Mathematics Pedagogy must be more realistic, practical, useful, suitable and justified upto the learner’s mental ability according to its level rather than stereo type, theoretical and traditional.
- Scope should be there for removing “fear psychosis” among the learners on the subject with proper instruction for creating interest and love for the subject.
- Scope to be provided for the use of the essence of mathematics against the success of any type of programme at home as well as outside.
- To make mathematics more understandable, enjoyable & permanently retained in the mind of the learner more use of fun and activities .

Mathematics lab can be maintained with the help of the community for spreading the message that there is no other subject like mathematics, which is so interesting, enjoyable and useful and that nothing can be done in this world without mathematics.

**Activities in the Math Lab**

Activities in the Math Lab have a great role in mathematics learning and developing various skills in solving problem as well as developing creative and logical thinking. Activities in the Math Lab is also a pure mathematics . The pedagogic value of activities in the Math Lab is now widely recognized which in turn help the low achievers of mathematics and converting them into a lover of mathematics by removing fear-psychosis from their minds towards the subject. For a lover of mathematics, there is all beauty. One finds a huge treasure of pleasure after getting success in the solution of a Mathematics problem. It was the reason why Pythagoras sacrificed hundred oxen to the Goddess for celebrating his discovery of the theorem that goes by his name. In the same way, Archimedes had also forgotten his nakedness after discovering his principle.** **

The activities in the math’s lab should be appealing to a wide range of people, of different ages and varying mathematical proficiency. While the initial appeal is broad-based, the level of engagement of different individuals may vary. The maths lab activities listed here have been done with students and teachers of different grade levels. The activities are intended to give children an experience of doing mathematics and not merely for the purpose of demonstration.

**List of activities **

Thiyagumath in his blog ( thiyagumath.blogspot.com ) has suggested some interesting activities. Here is a list of activities for young mathematics students:

1A. To carry out the following paper folding activities:

**Finding –**

- Bisector of an angle,
- Median of a triangle.
- Mid point of a line segment,
- Perpendicular bisector of a line segment,
- Perpendicular to a line at a point given on the line,
- Perpendicular to a line from a point given outside it,

1B. To carry out the following activities using a geoboard:

- Find the area of any polygon by completing the rectangles.
- Find the area of any triangle.
- Given an area, obtain different polygons of the same area.
- Obtain a square on a given line segment.

2. To obtain a parallelogram by paper–folding.

3. To show that the area of a parallelogram is product of its base and height, using paper cutting and pasting. (Ordinary parallelogram and slanted parallelogram)

4. To show that the area of a triangle is half the product of its base and height using paper cutting and pasting. (Acute, right and obtuse angled triangles)

5. To show that the area of a rhombus is half the product of its diagonals using paper cutting and pasting.

6. To show that the area of a trapezium is equal to half the product of its altitude and the sum of its parallel sides and its height, using paper cutting and pasting.

7. To verify the mid point theorem for a triangle, using paper cutting and pasting.

8. To divide a given strip of paper into a specified number of equal parts using a ruled graph paper.

9. To illustrate that the perpendicular bisectors of the sides of a triangle concur at a point (called the circumcentre) and that it falls

- Inside for an acute-angled triangle.
- On the hypotenuse of a right-angled triangle.
- Outside for an obtuse-angled triangle.

10. To illustrate that the internal bisectors of angles of a triangle concur at a point (called the incentre), which always lies inside the triangle.

11. To illustrate that the altitudes of a triangle concur at a point (called the orthocentre) and that it falls

- At the right angle vertex for a right angled triangle.
- Inside for an acute angled triangle.
- Outside for an obtuse angled triangle.

12. To illustrate that the medians of a triangle concur at a point (called the centroid), which always lies inside the triangle.

13A. To give a suggestive demonstration of the formula that the area of a circle is half the product of its circumference and radius. (Using formula for the area of triangle)

13B. To give a suggestive demonstration of the formula that the area of a circle is half the product of its circumference and radius. (Using formula for the area of rectangle)

14. 1) To verify that sum of any two sides of a triangle is always greater than the third side.

2) To verify that the difference of any two sides of a triangle is always less than the third side.

15. To explore criteria of congruency of triangles using a set of triangle cut outs.

16. To explore the similarities and differences in the properties with respect to diagonals of the following quadrilaterals – a parallelogram, a square, a rectangle and a rhombus.

17. To explore the similarities and differences in the properties with respect to diagonals of the following quadrilaterals – a parallelogram, a square, a rectangle and a rhombus.

18. To show that the figure obtained by joining the mid points of the consecutive sides of any quadrilateral is a parallelogram.

19. To make nets for a right triangular prism and a right triangular pyramid (regular tetrahedron) and obtain the formula for the total surface area.

20. To verify Euler’s formula for different polyhedra: prism, pyramids and octahedron.

21. Obtain length segments corresponding to square roots of natural numbers using graduated wooden sticks.

22. To verify the identity a3 – b3 = (a – b) (a2 + ab + b2), for simple cases using a set of unit cubes.

23. To verify the identity a3 + b3 = (a + b) (a2 – ab + b2), for simple cases using a set of unit cubes.

24. To verify the identity (a + b)3 = a3 + b3 + 3ab (a + b), for simple cases using a set of unit cubes.

25. To verify the identity (a – b)3 = a3 – b3 – 3ab (a – b), for simple cases using a set of unit cubes.

26. To interpret geometrically the factors of a quadratic expression of the type x2 + bx + c, using square grids, strips and paper slips.

27. To obtain mirror images of figures with respect to a given line on a graph paper.

** The Mathematics Laboratory method**

Though mathematics may deal with some abstractions. But, it is the mathematics teacher who should try to inculcate sufficient interest in the class room transaction so that the subject mathematics may not be treated as dull and tiresome. And the main tool to answer all these for converting mathematics into a loved and enjoyable subject to all is the curriculum and text book which directs the teacher to apply recreational activities for proper and effective mathematics learning.

** **

The term “**laboratory method**” is commonly used today to refer to an **approach** to** teaching** and learning of **mathematics** which provides opportunity to the learners to abstract **mathematical** ideas through their own experiences, that is to relate symbol to realities.** **

Laboratory method of teaching mathematics is that method in which we try to make the students learn mathematics by doing experiments and laboratory work in the mathematics room or laboratory on the same lines as they learn sciences by performing experiments in the science rooms or laboratories. It is based on psychological principles of learning such as ‘learning by doing’, ‘learning by observation’ and so on. Laboratory method is quite competent to relate the theoretical knowledge with the practical base. This approach makes the learning process more interesting, lively and meaningful.

The success of the laboratory method depends on an able skilled mathematics teacher as well as the availability of a well-equipped mathematics laboratory. According to J.W.A young “ a room specially filled with drawing instruments, suitable tables and desks, good black boards and the apparatus necessary to perform the experiment of the course is really essential for the best success of the laboratory method”.

Laboratory method is a teaching method which allowed the pupil to work with manipulative materials. As far as possible, the pupil has an active role to play. The usual pattern is 10 to have pupils manipulate physical objects, then describe a pattern or rule based on an inductive sequence

The Mathematics Laboratory method is a method of teaching whereby children in small groups work through an assignment/ taskcard, learn and discover mathematics for themselves. The children work in an informal manner, move around, discuss and choose their materials and method of attacking a problem, assignment or task.

- This method is based on the maxim “learning by doing.”
- This is an activity method and it leads the students to discover mathematics facts.
- In it we proceed from concrete to abstract.
- Laboratory method is a procedure for stimulating the activities of the students and to encourage them to make discoveries.
- This method needs a laboratory in which equipment and other useful teaching aids related to mathematics are available. For example, equipment related to geometry, mensuration, mathematical model, chart, balance, various figures and shapes made up of wood or hardboard, graph paper etc.

**Objectives of Lab method:**

- Apply concepts learned in class to new situations.
- Better appreciate the role of experimentation in science.
- Develop critical, quantitative thinking.
- Develop experimental and data analysis skills.
- Develop intuition and deepen understanding of concepts.
- Develop reporting skills (written and oral).
- Exercise curiosity and creativity by designing a procedure to test a hypothesis.
- Experience basic phenomena.
- Learn to estimate statistical errors and recognize systematic errors.
- Learn to use scientific apparatus.
- Practice collaborative problem solving.
- Test important laws and rules.

**The basic principles of the new pedagogy for teaching mathematics**

More recently, mathematics educators in colleges and universities, along with classroom teachers, have become concerned with the methods used to teach the new content which evolved from the “revolution.” Scott (56, p. 15) has summarized the basic principles of the new pedagogy for teaching mathematics in the following statements :

1. The structure of mathematics should be stressed at all levels.

2. Children are capable of learning more abstract and more complex concepts when the relationships between concepts are stressed.

3. Existing arithmetic programs may be severely condensed because children are capable of learning concepts at much earlier ages than formerly thought.

4. Any concept may be taught a child of any age in some intellectually honest manner, if one is able to find the proper language for expressing the concept.

5. The inductive approach or the discovery method is logically productive and should enhance learning end retention.

6. The major objective of a program is the development of independent and creative thinking processes.

7. Human learning seems to pass through the stages of preoperations, concrete operations, and formal operations.

8. Growth of understanding is dependent upon concept exploration through challenging apparatus and concrete materials and cannot be restricted to mere symbolic manipulations.

9. Teaching mathematical skills is regarded as a tidying-up of concepts developed through discovery rather than by a step by step process for memorization.

10. Practical application of isolated concepts or systems of concepts, particularly those drawn from the natural sciences, are valuable to reinforcement and retention.

Since 1965, discovery teaching, guided discovery, programmed instruction, computer assisted instruction, and inquiry training have been the subject of much study and 3 research. Still more recently, individually guided instruction and activity programs in mathematics have been the subject of action research in many classrooms.

**Procedure:**

Aim of The Practical Work: The teacher clearly states the aim of the practical work or experiment to be carried out by the students.

Provided materials and instruments: The students are provided with the necessary materials and instruments.

Provide clear instructions: Provide clear instructions as to the procedure of the experiment.

Carry out the experiment: The students carry out the experiment.

draw the conclusions : The students are required to draw the conclusions as per the aim of the experiment.

**Example 1:**

Derivation of the formula for the volume of a cone.

**Aims:** to derive the formula for the volume of a cone.

Materials and instruments: cone and cylinders of the same diameter and height, at lease 3 sets of varying dimensions, sawdust, water and sand.

**Procedure:** ask the students to do the following activity.

Take each pair of cylinder and cone having the same diameter and height

Note down the diameter and height

Fill the cone with saw dust / water or sand and empty into the cylinder till the cylinder is full.

Count the number of times the cone is emptied into the cylinder and note it down in a tabular column.

Repeat the same experiment with the other two sets of cone and cylinder and note down the reading as before.

S.NO. DIAMETER OF CONE / CYLINDER HEIGHT OF CONE/ CYLINDER NO. OF MEASURES OF CONE TO FILL THE CYLINDER

1 3 CM 5 CM 3

2 5 CM 7 CM 3

3 6 CM 10 CM 3

**Drawing conclusions:**

Each time, irrespective of the variations in diameter and height it takes 3 measures of cone to fill the cylinder.

Volume of cone = 1/3 volume of cylinder

But volume of cylinder = r2 h

Volume of cone =1/3 r2 h

Example 2:

Sum of three angles of a triangle is 180 degree. “How we can prove this in the laboratory.

**Aims:**

To prove that sum of the three angles of a triangle is equal to two right angles or 180 degree.

**Materials and instruments:**

Card board sheet, pencil, scale, triangle and other necessary equipments.

**Procedure:**

In the laboratory pupils will be given on cardboard sheet each and then they are told how to draw triangles of different sizes on it. After drawing the triangles they cut this separately with the help of scissors.

Observation:

Student will measure the angles of the triangles drawn and write these in a tabular form

Figure no. Measure of different angles Total

Angle A +B+C

Angle A – Angle B- Angle C

1 90 60 30 180

2 120 30 30 180

3 60 60 60 180

Calculation: after measuring the angles of different triangles in the form of cardboard sheet. We calculate and conclude their sum.

In this way by calculating the three angles of a triangle the students will be able to conclude with inductive reasoning that the sum of three angles of a triangle is 180 degree or two right angles.

**Some More Topics for Laboratory Method**

Derivation of the formula for the

Area of square, rectangle,, parallelogram, and trapezium

Area of triangle, right angled triangle, isosceles right angles triangle

Circumference of a circle, area of circle

Total surface area of cone, cylinder

Volume of a cone

Volume of a sphere

Expansion of identities such as (a+b) 2, (a-b) 2 , (a+b+c) 2

Verification of

Angle sum property in a triangle

Congruency postulates

Properties of certain geometrical figures like parallelogram, rhombus etc

Theorems relating to triangles, circles and transversal properties.

**Merits:**

Based on the principle of learning by doing.

Based on the student’s self pacing.

Child-centred and therefore it is a psychological method.

Develops in the child a habit of scientific, enquiry and investigation.

Develops the self-confidence and teaches the students the dignity of labour.

Helps in making clear certain fundamental concepts, ideas etc.

Is psychological as we proceed from known to unknown.

Presents mathematics as a practical subject.

Provides opportunities for social interaction and co-operation among the students.

Stimulates the interest of the students to work with concrete material.

Children learn the use of different equipments, which are used in laboratory.

Helps the students to actively participate in the learning process and therefore the learning becomes more meaningful and interesting.

**Demerits:**

Is not possible to make progress quickly.

All mathematics teachers cannot use this method effectively.

An expensive method. All schools are not able to adopt this method.

Can be used for a small class only.

Has very little of theoretical part in it.

Requires laboratory equipped with different apparatus.

Requires a lot of planning and organization.

Suitable only for certain topics.

It is important to note that while in science experiments provide evidence for hypotheses or theories, this is not so in mathematics. Observed patterns can only suggest mathematical hypotheses and conjectures, not provide evidence to support them. (Sometimes, they may help to disprove a conjecture through a counter-example.) Mathematical truths are accepted only on the basis of proofs, and not through experiment.

In conclusion we can say that this method is suitable for teaching mathematics to lower classes as at this stage teaching is done with the help of concrete things and examples. Math lab makes teaching and learning activity based and experimentation oriented at school stage. It exhibits relatedness of mathematics concepts with everyday life.

REFERENCES

Adenegan, K. E. (2001). Issues and Problems in the National Mathematics Curriculum of theSenior Secondary Schools level. Pp.4-5. Unpublished paper.

Unpublished B.Sc.(Ed.) Project, Adeyemi College of Education, Ondo.

Adenegan, K. E. and Balogun, F.O. (2010): Some proffered Solutions to the Challenges ofTeaching mathematics in Our Schools. Unpublished Seminar Paper at Ebonyi for Pricipals

Alao, I. F. (1997). Psychological Perspective of education, psychology and education series(pp. 48, 91). Ibadan: Revelation Books, Dugbe.

Balogun et al (2002). Mathematics Methodology in Approaches to Science Techniques, Yinka Ogunlade and R. O. Oloyede (Ed).

Ekhaguere, G.O.S. (2010). Proofs and Paradigms: a Mathematical Journey into the Real World. Inaugural lecture, Ibadan University Press, Ibadan. Pp 1-30.

Olademo J. O. (1990). Mathematics and Universe. Journal of NAMSN ACE, Ondo Ifeoluwa (NOD), Ent. Ltd. Pp 30.

Oyekan, S. O.(2000). Foundation of teachers education (pp.17, 240). Ondo: Ebunoluwa Printers (Nig.) Ltd