The end goal of all knowledge is the upliftment of society by providing solutions to problems that plague us and are threatening our very existence. Today problems such as climate change, pollution, the spread of diseases, immigration control, and finding renewable energy among others are cutting across boundaries and globally relevant. It is incumbent upon people from diverse backgrounds to come together to address these issues. This need places a more significant responsibility on us educators to prepare and groom the present and future generations to take on these problems. As educators, it becomes necessary for us to expose our students to these problems of societal relevance in the classrooms and apply our discipline to solving them.
One cannot overstate the role of mathematics in understanding and helping provide solutions to complex problems of the world. Right from modeling the motion of fluids (which gave us the technology to fly aircraft and land a rover on Mars) to modeling the volatile ups and downs of financial markets, the spread of diseases, social networking, the complex network of neurons in the brain and the predicting of weather, mathematics has touched our lives in the most profound ways. It is essential to bring these aspects of mathematics into the classroom so that students not only appreciate them, but also get first -hand experience in practicing them.
Formulating a real-world problem in mathematical terms, solving it and then interpreting the results in ordinary language is loosely called "Mathematical Modeling". The first step in this process- formulating a real-world problem in mathematical terms- is probably themost challenging. It is also one of the most creative aspects of mathematical modelling where the students might have to draw on their knowledge of other disciplines and then use all the mathematical knowledge they have acquired to write down an equation or many equations describing the problem. One needs to make reasonable assumptions, identify relevant parameters, and try to include them in the model. It is also vital to check the mathematical correctness or what in mathematical terms is called the "well-posedness" of the equations. After that, identifying mathematical methods, numerical techniques, and efficient algorithms that can help solve the equations is the next challenging step. Here one has to dig into the arsenal of mathematical tools and skills that one has acquired over years of mathematical study. Often one may have to learn new mathematics at this stage for the sake of solving the problem. So, students may try a particular line of approach and reach a dead end from where they have to start all over again. Mostly the model needs to be revisited again and again and modified to take into account aspects that had been ignored. After obtaining the results, they need to be interpreted to find physical meaning. Lastly, the results have to be presented to lay-people in ordinary language. These steps, in short, constitute the process of Mathematical Modeling.
My own students have worked on a range of such questions, from the spread of disease to assessing environmental impact.
For instance, the 2014 Western African outbreak of Ebola (which was the worst epidemic of this disease and claimed more lives than all of its previous outbreaks combined) prompted some students to investigate, via mathematical modelling, how the spread of Ebola virus can be contained using vaccination and hospitalisation. Some other students tried to analyse how electric vehicles compare with conventional automobiles in terms of their environmental impact from an economic point of view. In the famous case of the Kariba dam, built on the Zambezi river, which is on the verge of collapse, students worked out whether it is better to break it down and erect a new one or make a number of smaller dams instead and in the case of smaller dams, how many should be built. In another project, students traced the path of garbage debris in the Pacific and North Atlantic Oceans using data from NASA to understand the formation of garbage patches. For this, they used the knowledge of various mathematical software as well as GIS tools. In solving such problems, students’ knowledge from various other disciplines such as economics and environmental studies came in handy, making their learning interdisciplinary, more meaningful and holistic. In their own words, working on these projects was the most challenging as well as satisfying part of their learning experience.
It is amply clear that today we need to impart skills to our students, which will help them positively impact the world in which we live. In this regard, mathematical modelling paves the way to providing solutions for a better future.
- Prof. Renu Dhadwal, Associate Professor – Mathematics