Dr. V.K.Maheshwari, M.A. (Socio, Phil) B.Sc. M. Ed, Ph.D.
Former Principal, K.L.D.A.V.(P.G) College, Roorkee, India
Manjul Lata Agrawal. M.A. (History) B.T.
Former Principal S.K.V, Delhi Cantt. Delhi.
The mathematics laboratory is a place where anybody can experiment and explore patterns and ideas. It is a place where one can find a collection of games, puzzles, and other teaching and learning material. The materials are meant to be used both by the students on their own and with their teacher to explore the world of mathematics, to discover, to learn and to develop an interest in mathematics. The activities create interest among students or in anybody who wants to explore, and test some of their ideas, beliefs about mathematics.
The maths lab provides an opportunity for the students to discover mathematics through doing. Many of the activities present a problem or a challenge, with the possibility of generating further challenges and problems. The activities help students to visualize, manipulate and reason. They provide opportunity to make conjectures and test them, and to generalize observed patterns. They create a context for students to attempt to prove their conjectures.
Mathematics laboratory is a place to enjoy mathematics through informal exploration. It is a place where anyone can generate problems and struggle to get a answer. It is a space to explore and design new mathematical activities. So, the maths lab should not be used to assess students’ knowledge of mathematics. Often mathematics lab takes students knowledge beyond the curriculum.
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.
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 incharge 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 incharge 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
i. Paper folding
ii. Collage (Paper cutting & pasting)
iii. Unit Cubes (wooden or any material)
iv. Geo–board, rubber band
v. Transparency sheets, cello tape
vi. Graph paper
vii. Pins & threads
viii. Broom sticks
ix. Chart papers, glazed papers, sketch pens.
x. Stationery
Example:
List of activities
1A. To carry out the following paper folding activities:
Finding –
1. the mid point of a line segment,
2. the perpendicular bisector of a line segment,
3. the bisector of an angle,
4. the perpendicular to a line from a point given outside it,
5. the perpendicular to a line at a point given on the line,
6. the median of a triangle.
1B. To carry out the following activities using a geoboard:
1. Find the area of any triangle.
2. Find the area of any polygon by completing the rectangles.
3. Obtain a square on a given line segment.
4. Given an area, obtain different polygons of the same area.
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
a. inside for an acute-angled triangle.
b. on the hypotenuse of a right-angled triangle.
c. 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
a. inside for an acute angled triangle.
b. at the right angle vertex for a right angled triangle.
c. 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
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”.
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 equipments and other useful teaching aids related to mathematics are available.
For example, equipments related to geometry, mensuration, mathematical model, chart, balance, various figures and shapes made up of wood or hardboards, graph paper etc.
Objectives of Lab method:
• Develop intuition and deepen understanding of concepts.
• Apply concepts learned in class to new situations.
• Experience basic phenomena.
• Develop critical, quantitative thinking.
• Develop experimental and data analysis skills.
• Learn to use scientific apparatus.
• Learn to estimate statistical errors and recognize systematic errors.
• Develop reporting skills (written and oral).
• Practice collaborative problem solving.
• Exercise curiosity and creativity by designing a procedure to test a hypothesis.
• Better appreciate the role of experimentation in science.
• Test important laws and rules.
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
Circumference of a circle, area of circle
Area of square, rectangle,, parallelogram, and trapezium
Area of triangle, right angled triangle, isosceles right angles triangle
Total surface area of cone, cylinder
Volume of a sphere
Volume of a cone
Expansion of identities such as (a+b) 2, (a-b) 2 , (a+b+c) 2
Verification of
Properties of certain geometrical figures like parallelogram, rhombus etc
Angle sum property in a triangle
Congruency postulates
Theorems relating to triangles, circles and transversal properties.
Merits:
The method is based on the principle of learning by doing.
This method is psychological as we proceed from known to unknown.
It is based on the student’s self pacing.
It helps in making clear certain fundamental concepts, ideas etc.
It develops the self-confidence and teaches the students the dignity of labour.
The children learn the use of different equipments, which are used in laboratory.
It develops in the child a habit of scientific, enquiry and investigation.
This method presents mathematics as a practical subject.
It stimulates the interest of the students to work with concrete material.
It provides opportunities for social interaction and co-operation among the students.
It is child-centred and therefore it is a psychological method.
It helps the students to actively participate in the learning process and therefore the learning becomes more meaningful and interesting.
Demerits:
This method can be used for a small class only.
It requires a lot of planning and organization.
This method is suitable only for certain topics.
This method it is not possible to make progress quickly.
This method requires laboratory equipped with different apparatus.
All mathematics teachers cannot use this method effectively.
It is an expensive method. All schools are not able to adopt this method.
This method has very little of theoretical part in it.
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.
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.