Enabling Mathematics Learning Through a Technology Integrated Mathematics Laboratory

Dr. Jonaki B Ghosh,
Mathematics Laboratory Incharge
Delhi Public School, R K Puram, New Delhi

The Laboratory Approach of Teaching Mathematics.
School mathematics education is often fraught with lack of creativity and original thinking. Mathematics is taught as an abstract subject with no reference to applications where the emphasis is on routine problem solving. Mathematics text books at the school level sometimes are poorly written, unimaginative and examination centered. These drawbacks pose themselves as serious impediments to effective mathematical learning. The laboratory approach of teaching mathematics can fill this gap by providing students with the opportunity to understand and discover the beauty, importance and relevance of mathematics as a discipline. It can be expected to enhance the pupil’s understanding of the subject as taught at the school and can also provide a glimpse of what is beyond.

The primary objective of any laboratory is to perform experiments and the same is true of a Mathematics Laboratory. An ‘experiment’ in mathematics is an ‘exercise’ or ‘project’ which

· Highlights some known concept based on a well known mathematical theory.
· Sheds new light on some aspect of the topic being studied.
· Leads to some original discovery on the part of the student.
· Focuses on some interesting application of mathematics to a real life problem.

In the laboratory, students may be exposed to problems of an ‘exploratory nature’ which they do not otherwise encounter in their regular syllabus or curriculum. This does not preclude the curriculum from the laboratory. Rather it enriches the subject of study and encourages a spirit of research among the students. The emphasis must be on the process of mathematics rather than the product of mathematics.

The Mathematics Laboratory Project at Delhi Public School R K Puram, New Delhi.
The Mathematics Laboratory Project was an initiative taken by the Delhi Public School R K Puram in 1994 to improve and enhance school mathematics. The primary objective of the laboratory was to enrich mathematics at school level and to transmit the beauty of mathematics as a discipline to students by providing them with an environment, which encourages independent and original thinking through the ‘learning by discovery’ approach. The activities of the laboratory may be broadly classified as follows.

• To create and conduct projects and activities which focus on applications of mathematics to real life problems thus relating school mathematics to situations outside the classroom.
• To integrate the use of technology, specifically graphing calculators, computer algebra and other software packages in mathematical modelling activities.
• To investigate the use of technology in teaching mathematics at the middle school and senior secondary level.
• To provide training to teachers in preparing and conducting mathematics laboratory activities.

Present Status of the Mathematics Laboratory Project.
The project has now developed into a Mathematics Laboratory and Technology Centre whose primary objective is to supplement classroom teaching with innovative teaching methods. It conducts various activities, projects and supplementary courses for students from classes 6 to 12. Technology plays a key role in most of the activities conducted by the Centre. It is equipped with computers, computer algebra systems such as Mathematica, software packages such as Geometer’s Sketchpad and Autograph and graphics calculators, in particular, the Casio CFX 9850 GB plus and HP38G. All these are important tools for exploration and visualization. Presently the mathematics laboratory is a part of mainstream teaching where students meet once in a week in the allotted time and perform experiments and activities under the active guidance of the teacher.

Milestones Achieved by the Mathematics Laboratory & Technology Centre

I. Increase in Participation of Students.
Over the years the mathematics laboratory has become popular among students. Hundreds of students apply for joining the lab at the beginning of the academic session every year. Due to limited space and hardware infrastructure, however, a criteria based on merit is used to select as many students as is possible to accommodate. About 60 - 70 students from each of the classes 6 to 12 enrol for courses/projects offered by the Centre every year. Every attempt is being made to increase this number.

II. Projects by Students.
During the mathematics laboratory period, students are made to work on projects or activities, which have been planned in the annual schedule for the lab activities. Most of the projects are selected and designed in such a way that they highlight some practical use of mathematics. These projects enable students to see the relevance of mathematics to real life applications. They are allowed to choose a project of their interest. They are then divided into groups and each group is required to present their ideas about approaching the problem at hand. In most cases students have to be familiarized with the mathematical concepts on which their project is based. They are then provided with the references and materials that give some information about the problem. With the teacher acting as a facilitator they are led to define the problem in mathematical terms and obtain a solution using their knowledge of mathematics. Students are encouraged to validate their models using real data. In the end, the students also develop attractive computer aided displays and programs (C++ or Visual Basic) for their respective projects.

Over the last few years, students have worked on several projects in the Mathematics Laboratory. For example, some of the key projects done by students of year 11 and 12 in the past are: Mathematical Modeling in Genetics (based on matrices and probability), RSA: Public Key Encryption (based on Number Theory), The Brand Switching Problem (based on Matrices and Probability), Equilibrium Temperature Distributions (based on Matrix Theory), The Queueing Problem at a Vehicle Service Station (based on Calculus and Probability), Chaos and Fractals in the Classroom (based on Calculus of Complex Variables), Modelling AIDS and HIV Infection (an application of Differential Equations), Encoding and decoding of Messages Using Matrices (based on Matrix Theory), Mathematical Modeling of Forest Management (based on matrices and linear programming).

III. Participation in Mathematics Competitions and Events at the Local and National Level.
Students who opt for the Mathematics Lab are also trained to participate in various mathematics competitions and events organized by schools and various educational agencies. Many of these competitions require the participating students to work on a research project and then present their research findings to a panel of judges. The Delhi Science Fair and the Intel Science Discovery Fair are two very prestigious competitions in which students of the Centre represent the school every year. Students from all over the country participate in these competitions. The following projects by students from the Centre were accorded the best project awards in the mathematics category.

• Arbitrating Disputes Using Utility Theory (2001-2002)
• Mathematical Applications in Cryptography. (2002-2003)
• Modelling the Queuing Problem . (2003-2004)
• Markov Chains and Brand Switching (2004 – 2005)

IV. Integrating the Use of Technology in the Activities of the Centre.
World over the advent of technology has made available various educational tools which have revolutionized the teaching and learning of mathematics. This includes computers and computer software packages, handheld technology in the form of graphics calculators and Computer Algebra Systems such as Mathematica. One of the primary objectives of the Mathematics Laboratory and Technology Centre at Delhi Public School R K Puram has been to explore the use of technologies and their relevance in the Indian classroom scenario.

(a) Use of Computers and Software Packages in the Mathematics Laboratory.
A new dimension has been given to the activities of the Centre due to the use of computers and computer related software packages such as Geometer’s Sketchpad and Autograph. Serious research, experimentation and reflective criticism have been employed to explore and integrate their use in the projects or ‘experiments’ carried out by the students.

1. As a demonstration tool.
The use of these software packages has created new teaching possibilities. These have led to visualization of concepts in a manner, which would have been impossible using the traditional methods of teaching.

2. The computer encourages students to explore and discover on their own.
These packages have greatly increased the range of the students’ mathematical activities by enabling them to explore concepts and discover results for themselves. Students also explore problems by writing their own programs, which enable them to think actively about the ‘processes’ they are implementing in solving the problems. This increases their ability to think analytically.

3. Emphasis on experimental aspects of mathematics through the use of computers.
Through the use of these software packages, activities are designed so as to provide students with the opportunity to acquire skill in observing, exploring, forming insights and intuitions, making predictions, testing hypothesis etc. However care is taken that these skills must be acquired without neglecting the traditional aspects of mathematics such as proving, generalising, and abstracting.

4. Computers aid in teaching applications of mathematics.
Various projects conducted by the Centre aim at making mathematics teaching more relevant by supplementing each topic with suitable applications. In the classroom it is only possible to talk about these applications but in the lab these can be extended into projects, which the students can explore on their own.

5. Use of computers have redefined the role of the teacher.
The role of the teacher in the lab has been found to be significantly different from that of the normal classroom. The computer serves as an electronic whiteboard whose output is entirely in the teacher’s control. The teacher acts as a facilitator, ‘leading’ the students to making their ‘discoveries’ rather than providing the solutions on the blackboard.

6. Computers enhance student initiative and participation.
Research has revealed that in many ways the relationship between teachers and students is strengthened because of the use of computers in the classroom. Computers by themselves generate interest among the students. This supplemented by the teacher’s guidance leads to active participation on the part of the students. Lessons become more enjoyable and this results in a stronger link between the teacher and students.

7. Computers help in sustaining the student’s interest in mathematics.
The use of computers in the laboratory has been seen to generate greater enthusiasm among the students to learning mathematical concepts. The increased activity enables them to develop and sustain interest in the subject while at the same time increasing their confidence. It has also helped to overcome the ‘maths phobia’ many students suffer from.

(b) Use of Graphic Calculators in the Mathematics Laboratory.
The lab is equipped with Casio CFX 9850 GB plus and HP38G graphics calculators. These calculators have proved to be extremely handy for performing mathematical modelling activities in the lab. Students feel comfortable with using the calculator especially since they are hand held and portable. As a tool the graphic calculator has significant mathematical capabilities. It provides dynamic interactive working environments for exploring problems. The graphing capability and other features which do not depend on graphical representations such as working with matrices or complex numbers, symbolic integration or differentiation, data analysis etc., makes this calculator a very powerful tool ([9] – [12]). As a teaching device the graphic calculator has proved to be useful in the following ways:

1. The power of visualization is brought right onto the palm of the hand. The graphing capability of the calculator can be suitably exploited to lend a deeper physical insight into problems and concepts. The calculator also serves as a handy demonstration tool since it can be connected with an overhead projector or a display board.

2. Graphical and numerical explorations can be performed simultaneously. The calculator allows the student to view functions graphically, symbolically as well as numerically. The calculator even allows the iteration of functions. Teaching concepts in calculus and trigonometry is certainly made easy.

3. Analysis of data can be done very easily with this device. It allows the fitting of curves to data, which has proved to be extremely useful in the lab. Apart from the linear fit one can obtain a quadratic, power, exponential, logarithmic or even logistic fit to a given data. This can be used for predicting trends from the data and can even be used for comparing various sets of data.

4. Simulation of experiments. Simple experiments in probability can be easily simulated using the random number generator of the calculator [5]

(c) Use of Computer Algebra in the Mathematics Laboratory.
Computer Algebra Systems (CAS) are not a common phenomenon in India. They are predominantly used by students at the postgraduate level but are unknown in the school sector. Mathematica has proved to be an extremely useful tool in the laboratory. Its computational, graphic and symbolic manipulation capabilities have led students to explore problems, which would have been beyond their reach without this technology [4]. The use of CAS has enabled students to visualize and explore the problem through simple programs and commands, thereby allowing learning to take place through the ‘discovery’ approach ([3], [6] – [8]).
CAS may be seen as a computer aided instruction program. It offers the following advantages as a teaching aid:

1. It offers a large range of capabilities - numeric, graphic and symbolic.
2. It frees the teacher from computational details, thus allowing him to focus on insights and concepts.
3. It frees the student from tedious calculations and allows him to concentrate on understanding and forming concepts.
4. It has the capability of solving very large problems. This feature may be suitably exploited to suggest patterns, theorems or important results.

The Mathematics Laboratory as a Resource Centre for Schools
The Mathematics Laboratory and Technology Centre organises regular workshops, seminars and professional development programs for teachers and students of schools across the country. The primary objective of these workshops is to

• Popularise the concept of the Mathematics Laboratory among teachers of various schools.
• Familiarize teachers and students with the kind of activities and projects, which can be conducted in the laboratory.
• Enable teachers to interact with mathematics educators and university professors who have been actively involved in school mathematics education.
• Create awareness among teachers regarding the use of technology for mathematics teaching and to provide training to enable them to implement these technologies in their teaching.

The Centre organises short-term professional development programs for teachers on a regular basis where they are trained in the use of graphic calculators and other software packages. The Centre also provides resource materials to teachers so that they can set up mathematics labs in their own schools.

The following workshops and seminars have been conducted in the last three years for teachers and students of other schools. At least 10 to 15 schools have been represented in each of these workshops:

1. Mathematics Laboratory: A means to enrich School Mathematics (1st – 3rd May 1998)
2. Mathematics Laboratory Workshop (21st November 1998)
3. Training Workshop on Graphic Calculators (5th -6th May 1999)
4. Mathematics Laboratory, Computers and Technology: New Creative Approaches in Mathematics Teaching (29th – 30th September 1999)
5. Introductory Workshop on the use of Graphic Calculators: HP38G (8th – 9th May 2000)
6. Training Workshop on Graphic Calculators for DAV schools (20th August 2000)
7. Innovative Teaching of Mathematics through Mathematics Laboratory Activities (24th – 25th November 2000).
8. Mathematics Laboratory: A New Pradigm in Mathematics Teaching (5th-6th November 2001)
9. Mathematics Laboratory Activities (8th – 9th May 2002)
10. New Trends, Technology and Innovations in Mathematics Education (27th – 30th January 2003)
11. Enabling Mathematics Learning Through Handheld Technology (27th – 29th November 2003)
12. Mathematics Laboratory: A means to enrich School Mathematics (12th – 15th May 2004)
13. Effective Teaching and Learning of Mathematics Via Mathematics Laboratory Activities (21st – 25th February 2005)
14. Integrating Mathematics Laboratory Activities With Mathematics Teaching (25th – 27th August 2005)
15. Enhancing Teaching And Learning Of Mathematics Through Mathematics Laboratory Activities : Workshop For B.El.Ed Students (9th – 11th February 2006)
16. Redefining Mathematics Teaching and Learning Through Technology (12th – 13th May 2006)
17. Enhancing Teaching And Learning Of Mathematics Through Mathematics Laboratory Activities : Workshop For B.El.Ed Students (14th August 2006)

Research Projects Conducted by the Centre
The Mathematics Laboratory and Technology Centre conducts research projects to study the effect of the use of various technologies and teaching methods on students’ mathematical understanding and performance. These may be categorized as follows:

• Projects related to exploring the effect of hand-held technology such as graphic calculators on students’ concept formation in mathematics.
• Projects related to investigating the effect of computer technology such as computer algebra systems and other mathematical software on student’s understanding in mathematics.
• Projects related to studying the outcome of conducting hands-on activities with students at the middle school level.

How Does The Student Gain From His Association With The Mathematics Laboratory.

• Through the projects conducted in the lab, the student learns to appreciate the relevance of mathematics to real life. He no longer sees mathematics as an abstract subject.
• The student develops the potential to engage in creative and original thinking. He learns to conduct research on his own. This will eventually help him in his higher education irrespective of the career chosen by him.
• The student is made to see the importance of using technology in conducting mathematical explorations. He is made to write his own programs. This develops his analytical thinking since in writing a program for solving a problem he is actively thinking about the mathematical process behind the problem.
• The Mathematics Laboratory makes the student aware of career options in mathematics.
• It has been a general observation that the laboratory helps to sustain the students’ interest in mathematics.

How Does The Teacher Gain From His Or Her Association With The Mathematics Laboratory.

• The Mathematics Laboratory provides the teachers with new teaching possibilities through the use of technology.
• The role of the teacher in the lab is to facilitate learning and use innovative methods to help students ‘discover’ mathematics on their own. This is far more challenging than the conventional classroom teaching.
• Through the activities of the lab the teacher can explore and device new teaching methods. Serious research concerning pedagogical issues in mathematical learning is made possible through the mathematics laboratory.