MATHEMATICS LABORATORY IN SCHOOLS

It is a place where:

1)      Students do experiments with numbers and geometrical shapes and try to generalize these patterns.

2)    Students do most of their calculations with the help of scientific calculators.

3)    Students draw graphs of large number of functions with the help of scientific or graphic calculators and try to become familiar with graphs of all the functions they usually deal with.

4)    Students solve real life problems with real data because complex calculations are no longer a major consideration.

5)    Students express their answers to mathematics problems in decimal numbers and not in symbols and have a good idea about their magnitudes.

6)    Students get practice in estimating orders of magnitudes and obtaining approximate answers when exact answers are difficult to find.

7)    Students make charts and models to illustrate mathematical ideas.

8)    Students do almost all the work themselves, of course under the guidance of teachers, but the students are active all the time and are involved with what they are doing.

9)    The creativity of students is allowed free play.

10) Students solve graphically equations involving all types of functions.

11)  Students are free to discuss among themselves and with the teachers; in fact students and teachers form joint investigating teams.

12) Students find areas and volumes of both regular and irregular solids.

13) Students undertake projects both in mathematics and its applications.

14) The concepts and theorems are not given to the students; these arise naturally from their investigations.

15) Interfaces between algebra, geometry; probability; calculus etc are freely investigated and discussed.

16)  Attempts are made to interpret every symbolic solution.

17) The process of mathematics is emphasized much more than the product of mathematics.

18) Students are encouraged to find alternative solutions and alternative methods of solving problems.

19) Students enjoy learning mathematics.

Before we proceed further, let us explore as to why students do not fair well in mathematics. The reasons are not difficult to find.

It is not because:

i)                   Students are unable to solve certain problems,

ii)                Or, students are not able to memorize formulae etc.

But, it is due the fact that there are some inherent weaknesses in the teaching of present day mathematics. These are listed below:

1)      Mathematics is taught as an abstract subject.

2)    Mathematics education is far removed from applications.

3)    Mathematics is taught as an isolated subject.

4)    There is too much emphasis on symbols and their manipulations and relatively little on problem solving.

5)    Too much time is spent on routine monotonous drill type arithmetical calculations.

6)    The goal of mathematics education appears to be passing examinations in mathematics and not understanding mathematics and its applications or developing capacity to think mathematically.

7)    Instead of developing creativity, mathematics education encourages conformity to standard methods.

8)    It trains students to think that there should be only one method of solving mathematics problems.

9)    It trains students to think that there can be only one solution to a problem.

10) Mathematical proficiency is often confused with proficiency in making arithmetical calculations.

11)  The process by which mathematics is created is seldom taught or emphasized.

12) Mathematics is presented as a purely deductive science though it is also as much an experimental science as physics or biology.

13)  Geometric and Physical visualizations remain very weak.

14) Even geometric objects become just relations between symbols and are not curves or surfaces.

15)  It convinces the students that the only law which matters is the linear law.

16)  Students develop no idea of the order of magnitude of the results they get.

17)  Students are passive learners.

18)  Students do not talk mathematics, discuss mathematics or think mathematics.

19)  Mathematics is taught as a collection of topics.

20)                       The historical development of mathematics is never emphasized.

Thus the objective of a mathematics laboratory is to:

a)     Remove the weaknesses of present day mathematics education which the mathematics laboratory and the mathematics laboratory alone can do it.

b)    To develop the much needed confidence in students.

c)     To generate interest in the subject.

d)     To make the students divergent thinkers.

Having seen “WHAT” and “WHY” of a mathematics laboratory let us now discuss the “HOW” of it.

Time-table Re-scheduling:

While preparing class-wise time-table, in JNVs, the provision for mathematics practical periods may be made in the following manner:

From classes VI to X, there is a provision of one theory period of mathematics in each class in every day working time-table. Also in each class two periods for “ART” are allotted per week. It is suggested that one theory period of mathematics may be combined with one period of art and the combined periods may be re-named as “MATHEMATICS PRACTICAL PERIODS”. This way in five days each class will have an opportunity to visit the laboratory. As regards classes XI and XII are concerned, the students normally opt either mathematics or biology. The students opting mathematics can be taken to the laboratory during the practical periods for biology.

Layout of a mathematics laboratory:

The ideal mathematics laboratory will have the following sections:

1)      Section for job discussion and planning the solution.

2)    Section for making sketches, drawings for taking observations.

3)    Section for reporting the results.

4)    Section for making the working models as per job specifications.

5)    Computer section for doing experiments of mathematics on computers.

The above sections (steps to be performed by students) need be discussed by the teacher in-charge, in the laboratory. Before the children are asked for execution, the teacher should explain the planning part as well as he/she should help them in identifying the appropriate solution in respect of choice of proper tools and their use in execution. The teacher should also explain the use of computers in finding the solution and the method of checking the accuracy of the solution already found in the laboratory.

Furnishing Mathematics Laboratory:

Sufficient furniture should be provided in the laboratory to do experiments and at the same time for displaying the working models and other means of taking observations; to carry out experiments and make a clear understanding about the use of procedural tools in engineering projects.  These models are made out of discarded toys and waste articles found around us. This approach boosts the creativity, scientific development of the brain of the children, and satisfies their zeal to do something new and unique.

Raw Materials:

To enable the students to work in a mathematics laboratory, there should be a few cupboards to store raw materials, which can be issued to the students when they come to the lab for doing practical work. The list of some of the essential raw materials is as under:

I)                 Circular discs: Plates of different diameters may be cut on thermo Cole sheets or may be plastic metal discs purchased from the market to determine the value of ? or such other experiments.

II)            There should be square/rectangular plates cut from thermo Cole sheets, different solids like cone, cylinder for drawing different shapes and making observations for various computations.

III)        A thick card sheet, capable of folding, to prepare packing boxes, envelops etc or other small objects.

IV)            The students will be required to use drawing sheets, graph papers, cutting tools, thread, small balls made of different materials, ready reckoners, rubric cube, calculators etc.

Measuring Equipment:

1)      Measuring tapes 30m; 10m; 2m; 1m and smaller lengths, arrows (iron wire nails 20 to 30 cm long; 2.5 mm in diameter)- they act as pegs to mark points on the field ; pipe hole 1m long ; 10 mm in diameter, pointed at one end; painted with red and white strips- used for making solar observations and determination of N-S direction at a place and also finding out the angle of elevation of the Sun at any time.

2)    Plain mirror, plum bob suspended from a hook, drawing board , mini drafter, vernier calipers , working models to verify law of parallelogram of forces, triangle of forces etc.

Display models:

1)      Children are assigned task to imagine suitable design data to prepare models and keep them for display in the lab. This includes different types of packing boxes; tents-pyramid shaped; circular and dome shaped etc.

2)    Storing typical shaped tin cans or paper packets like a tetrahedron; prism; cylindrical shaped which are available in the market for packing milk or juice. The children are assigned task to imagine suitable design data to prepare attractive packets for liquid contents.

3)    Certain models are prepared to demonstrate the principles used in making some scientific instruments, e.g. optical square; cross staff; periscope; kaleidoscope etc. The students thus come to know the use of such scientific equipments.

Working Models:

1)      Plane co-ordinatograph: It is a model prepared in the lab and used for making observations of co-ordinates of various points in a plane. This is of great help to explain the basic concepts of co-ordinate geometry in two dimensions. Students are asked to take observations of points and write equations of incident ray; reflected ray; equations of circles; parabola; plane; straight lines; tangent lengths etc on the basis of co-ordinates observed on the working model. The students can understand the transformation of one system of co-ordinates into the other, trigonometric ratios and their applications etc.

2)    Plane Space Co-ordinatograph: It is a model prepared in the lab and used for making observations of co-ordinates of various points in the space above the surface. This is of great help to explain the basic concepts of co-ordinate geometry in three dimensions. Students are asked to take observations of points in space; write equations of straight lines in space and locate points in space. The students can understand the transformation of one system of co-ordinates into the other. With such experiments, children come to know how to determine the distances of cloud; sun; moon; space craft at the time of Arial photography etc.

3)    Dip Measurement Model: It is a model made out of transparent plastic cylinder to represent railway tanker. This demonstrates how easily the liquid contents or the volume can be determined in case of cylindrical tanker making few observations.

4)    Water analog model: It is a model to take observations for filling the pool by different taps having different rates of discharge. Such observations enable the students to formulate quadratic equations and find out their solutions. Such working models analogy can be applied in solving different types of problems related to the formation of quadratic equations on the basis of given conditions. Also the observations may be used to tackle problems based on dispersion theory and determination of the most probable value in a set of observations.

5)    Model To Make Observations of Time Periods: A pendulum is suspended in the lab and time period for the oscillations are observed. This leads to the value of g, the acceleration due to gravity.

6)    Equilibrium Forces Analog Model: This model is used to formulate the equations of equilibrium.

The Concept:

On the lines of science laboratory, the concept of mathematics laboratory may be visualized and developed. It is a place where every one should get an opportunity to establish correlation of one subject with allied subjects.

The basic linear equation answering the needs of mathematics laboratory is:

Ml = aiXi + bi Ym + ciZo ; where:

Ml denotes activities in mathematics laboratory.

Xi denotes necessary infra structure.

The coefficients are:

a1 denotes library and reference books.

a2 denotes furniture layout.

a3 denotes laboratory equipment- Computers; Calculators; Geometry Box; Cutting Tools; Letter-Stencils; Drawing Equipment; Mathematical Charts; Logarithm tables etc.

Ym denotes necessary mode of working and management tools.

The coefficients are:

b1 denotes Computations leading to desirable outputs.

b2 denotes Making drawings and sketches to explain the procedure.

b3 denotes Analysis and decision from set of observations.

b4 denotes Field layout and model making to achieve the objectives.

Zo denotes the number of objectives associated with the activity.

For example: an activity for determining the nature of ? may have the following objectives:

c1 denotes : What is ??

c2 denotes: What is the value of ??

c3 denotes : Whether ? is rational or irrational?

Now finally I suggest some activities which can be done in the mathematics laboratory:

1)      Mathematics laboratory- Definition.

2)    Activity 1: Mathematics laboratory- Introduction.

3)    Activity 2: Half Life.

4)    Activity 3: One-Less.

5)    Activity 4: Doubling.

6)    Activity 5: Span.

7)    Activity 6: Roller.

8)    Activity 7: Center-Point.

9)    Activity 8: Bigger.

10)  Activity 9: Equals.

11)  Activity 10: Side by Side.

12)  Activity 11: Paper art.

13)  Activity 12: Cut Away

14)  Activity 13: Impossible Challenge.

15)  Activity 14: Get triangle equal in area to a parallelogram.

16)  Activities 15, 16: Quick Calculations.

This is not the exhaustive list of activities to be performed in the laboratory. Many more activities may be thought of and performed in the laboratory.

The details of the above mentioned activities are available in the accompanying CD. These can be viewed using Microsoft Power Point and clicking to view slide show.