ncert project based learning through mathematical modeling project based learning through... ·...

MATHEMATICS ACROSS DISCIPLINE: MATHEMATICAL MODELING

PROJECT BASED LEARNING THROUGH MATHEMATICAL MODELING


Himani Asija Sr. Math Teacher

Delhi Public School Vasant Kunj Phone: +9717160042

Email: [email protected] "Project-based learning is... focused on teaching by engaging students in research and investigation. Within this framework, students pursue solutions to nontrivial problems by asking and refining questions, debating ideas, making predictions, designing plans and/or experiments, collecting and analyzing data, drawing conclusions, communicating their ideas and findings to others, asking new questions, and creating artifacts”

Abstract

Children today are digital natives and we teachers have to keep pace with them: we were born before the web arrived and the students of today are web masters. Project based learning allows students to acquire 21st century skills in the context of real world scenarios.We have to bridge the digital divide- it is a necessity today for the schools to be equipped and not scared of technology.

As a mathematics teacher teaching senior secondary level to the age groups of 16-18 years, I emphasize on the creativity, project based learning, innovation, and love for the subject. Project Based Learning in Mathematics gives the students an exposure to correlate the subject Mathematics that they consider as dry and dull; divorced from real life, with the real life applications. They try to explore and find out a hypothesis and finally justify it to be true. Since all these projects are based on some new ideas, it promotes critical and analytical thinking. To mention one of the research projects, “Mathematical Modeling in the diagnosis of Cancer”, the project was undertaken with students after giving the basic knowledge of fractal structures like Koch snowflake. The idea behind the project is to make the students appreciate the applications of mathematics in anything we look around; our bodies being one of the finest examples. It is also to appreciate how technology increases the applications of the mathematical content in its simplest form in complex scientific situations. Another project “Mathematical Modeling of the Coastlines in Tsunami Wave heights” talks about how the fractal dimension of the coastline affects the disaster caused due to the tsunami waves.

Keywords

Project Based Learning, 21st century skills, Fractals, Koch Snowflake, fractal dimension, box counting method



Introduction

Project Based Learning is a model for teaching that involves an in depth study and investigation of a real world problem. It is a structure that transforms teaching from ‘teacher telling’ to ‘student doing’. This type of learning does not focus on ‘learningsomething’, but on ‘doingsomething’. The learner centered characteristics of PBL contribute to the learner motivation and involvement, thus developing a positive attitude for the subject. Project based learning engages the students in learning while they get a chance to explore complex challenging problems closely related to their real life. It encourages higher level thinking and put the students in an active role of a problem solver, decision maker, investigator and a researcher.It creates opportunities for the students to investigate meaningful questions that require them to gather information and think critically.

Project based learning nourishes best in technology rich environment. The use of technology makes it easier for the student and the teacher to conduct investigation, research and collaborate with others beyond the boundaries. The use of technology in PBL is important in two aspects: the investigation and research of the topic of interest, and secondly communication and collaboration.In contradiction to the role of teacher as a knowledge giver in traditional methods, project based learning demands from the teachers

• Enhanced professionalism to be able to recognize situations that make good projects. • Creativity in framing the problem • Ability to motivate the students • To be Resourceful • Collaboration among colleagues • Integrating technologies where ever applicable • To adopt new methods of assessment: to evaluate not on the basis of proven

hypothesis, but the process followed and presentation of the work done. • To teach the students to be patient because there is no pre determined answer to the

problem undertaken. There should be space for tolerance of error and change.

Execution of the projects

Projects are the primary vehicle for instruction in the Project Based Learning; they vary greatly in depth of the questions explored, clarity of the learning goals, content and structure of the activity and guidance from the teacher. Projects undertaken may be multidisciplinary or single subject; they may involve the whole class, a small group or some individuals. As a mathematics teacher teaching senior secondary level, I take up investigatory and research projects; which have been prize winning at various levels, both at inter school and national level. To name some of them are: “Mathematical Modeling in the diagnosis of Cancer”. Simpli-fly- an Optimization model in aviation industry”, “Aviation Safety”, “Mathematical Modeling of earth quake resistant buildings ,“Fractal Horizons : Future use of Fractals as new frontiers of Science and Mathematics”. In all the above mentioned projects and others, students get an exposure to



correlate Mathematics with real life. They try to explore and find out a hypothesis to be tested and finally justify it. Since all these projects are based on some new ideas, it promotes critical and analytical thinking. This, I believe is one step in preparing the students for the 21st century skills.

With my deep interest in the field of fractals and its applications in real life which arouse while framing holiday homework on finding perimeters and areas of fractals Koch Snowflake and Sierpinski’s triangle and carpet at different stages and generalizing it to the nth stage, I asked my students to find out different applications of fractals in real life. The discussions and collaborative works resulted in beautiful presentations made by students. While exploring for fractals, we came across how fractals are closely related to human bodies and nature. The clouds, the coastlines are all fractal natured. This is where we actually felt the applicability and beauty of mathematics in nature. To mention one of the research projects, “Mathematical Modeling in the diagnosis of Cancer”, the project was undertaken with students after giving the basic knowledge of fractal structures like Koch snowflake. The idea behind the project is to make the students appreciate the applications of mathematics in anything we look around; our bodies being one of the finest examples. It is also to appreciate how technology increases the applications of the mathematical content in its simplest form in complex scientific situations. The project is based on the observation that for a Koch snowflake, the perimeter increases in a geometric progression with common ratio more than one while the area increases in a geometric progression with common ratio less than one. So, at an infinite stage, ratio of perimeter squared and area approaches infinity. The project is based on two hypothesis:

Hypothesis 1: Ratio of the perimeter squared and the area is least for a normal cell and is maximum for a malignant cell.

Hypothesis 2: Higher the stage of malignancy, more is the fractal dimension.

Not only does the project reflect on the application of math in cytology, but because of its simple approach, it can be used with the school students thus proving to be a very practical and simple application of mathematics in their lives. This brings forth to the students the results of very complex scientific and mathematical principles in very simple mathematical terms.

Beginning with an equilateral triangle of side 12 cm, when a Koch snowflake was constructed; the first three iterations gave the following results:

Stage Perimeter Area Ratio (P^2/A) 0 36 62.35 20.8 1 48 83.13 27.7 2 64 92.37 44.3 3 85 96 75.3

OBSERVATIONS



While both the perimeter and area are increasing, the perimeter increases in a geometric progression with the common ratio more than 1 thus tending to infinity at an infinitely large stage. The area on the other hand also increases; but in a geometric progression with common ratio less than 1; thus tending to a finite sum even at an infinite stage.

The invasiveness of a tumor is related to the shape of its two dimensional cross-section near its base. A less invasive tumor in the initial stages of development has a smoother, oblong shape, whereas the more invasive tumor has more of a spiky sort of shape with more fingers like projections. This has been a repeated observation in many cases of cancer. So it was postulated that the invasiveness of a tumor is directly proportional to the ratio between the square of the perimeter and the area.

R= (Perimeter2)/Area

This hypothesis was verified using dynamic software ‘Geometer’s Sketchpad’ by drawing a polygonal boundary and finding out its area, perimeter and the required ratio.

Gsp snapshot of an advanced stage cancer cell: polygonal boundary has been drawn; area and perimeter

of this cell have been calculated The second hypothesis was based on Fractal dimension of the cells. The boundary of the cells drawn by using Geometer’s Sketchpad was pasted on an MS Excel worksheet and the method of finding fractal dimension using Box Counting was used.

P1

Perimeter P1( )2

Area P1( ) = 1558.10

Area P1 = 100.95 cm2

Perimeter P1 = 396.60 cm

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Following table describes that both the hypothesis are true.

STAGE R = P^2/A FRACTAL DIMENSION

NORMAL CELL 14.82 1,025

PRELIMINARY STAGE 23.94 1.27

INTERMIDIATE STAGE 119.36 1.352

ADVANCED STAGE - 1 1558 1.48

ADVANCED STAGE - 2 1809.84 1.56

ADVANCED STAGE- 3 1858.05 1.62

INCREASING INCREASING

FURTHER APPLICATIONS

This project was undertaken in the year 2010; the year when H1N1 virus was a pandemic. Another group of students confirmed the same hypotheses for the H1N1 virus cells. The ratio of perimeter squared and area is higher for the H1N1 virus cells and low for a normal viral cell.

Gsp snapshot of an H1N1 virus cell

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Ofunato(2011) 11-13m 1.16 Miyako had lesser wave heights as compared to Ofunato as the fractal dimension and thus fractal nature of Miyako is more. Miyako(2011) 8-10m 1.24

Onahama(2011)

just below 20 1.15 Soma, as discussed above is in the worst hit region so it is expected to have a higher wave height as compared to Onahama but it was almost the same as that of Onahama as the fractal dimension of Soma is more than Onahama.

Soma(2011) approximately 20 1.17

Nagapattinam --- 1.07 Fractal nature of Nagapattinam is more than that of Chennai but amplification off its coast is more than Chennai due to width of the Ocean floor below the coastline i.e. a wide ocean floor is required for more amplification.

Thus the assumed hypothesis was verified.

Role of the teacher

In the above mentioned projects and others, the role of teacher changes from a knowledge giver to a mentor and a facilitator. The students’ experience with projects reduced themath anxiety and resulted in more positive attitude towards math. Since Project Based Learning works best when done in collaboration, the teacher needs to provide a platform for this.The teacher needs to be updated with the information and resources required for the projects so that the students donot deviate from the right path.

Limitations of Project Based Learning

Project Based Learning has never been a part of the mainstream curriculum, teachers have little training or experience in the approach. Though PBL is a great way of learning for a student, it is a difficult process for the teachers; they need to take extra time and effort beyond the hours of the school day to create PBL’s. In addition, it requires high degree of professionalism and collaboration amongst the teachers. The amount of time involved in the projects, the availability of resources, pressure to cover the curriculum topics and the constraints of the examination system discourage many teachers from venturing into PBL.



Suggestions from the author

Project Based Learning plays a very vital role in the development of 21st century is skills. In fact, it should be a mandatory change in education. I believe it is our responsibility as educators to change with the time and incorporate more PBL choices into our teaching. My urge to the curriculum developers is that our students shall learn to ‘live Math’ if there is a space for Project Based Learning for them as it helps in reduced student math anxiety and results in more positive attitudes towards mathematics. The teachers need to fully understand the concepts involved in their projects and be able to model the strategies effectively. This calls for a formal in service teacher training programme which should be an ongoing process. Project Based Learning is not the ‘be all’ of the curriculum but can be made a component, a compulsory part of the curriculum and student assessment in the examination.

References

1. http://pbl-online.org/ 2. A review of researchon project-based learning, John W. Thomas, Ph. D., March, 2000. 3. Project-Based Learning for the21st Century: Skills for the Future, Stephanie Bell. 4. 21st Century Learning in Schools – A Case Study of New Technology High School in

Napa, CA 5. Designing Effective Projects: Characteristics of Projects Benefits of Project-Based

Learning 6. Work that matters the teacher’s guide to project-based learning, Published by the Paul

Hamlyn foundation February 2012 7. The Power of Project Learning with Think Quest, Prepared by: SRI International Menlo

Park, CA

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