Mathematics is not just a tool. It is a language, and a way of thinking and engaging with the world. Mathematical Thinking introduces students to the history, power and creative potential of mathematical and quantitative thinking and familiarizes students with some basic problem solving strategies. This course aims to give students an experience of contemporary Mathematics. One can see that Mathematics is driven by ideas, not by calculations. It is both beautiful and powerful, and it combines precision with the greatest creativity. En route, students develop a set of broadly useful problem-solving skills, gain experience in precise thinking and writing, and encounter some of history’s landmark ideas.
This course is about Mathematics and you. The course starts with the mathematical abilities you have and are unaware of. Then there is an excursion visiting numbers through the ages culminating with a discussion of the power of zero. As you progresses through the course, various concepts in Mathematics will be visited learning their use in your daily life. You will discover if elections of any sort can be fair to everyone. You will get empowered to estimate anything under (or above) the sun, such as the number of grains of sand on the beach, the number of times you blink in your lifetime and so on. You will have a lecture by the instructor in one class and will solve problems in groups in the next class each week. If you have not done serious mathematics in school or if you are scared to take a mathematics course don’t worry. You will learn it with others who will give you a helping hand.
This is an introductory course intended for students who want to learn the basics of programming. No prior programming experience is expected, though it helps to be “computer literate”. We will delve into computational thought and the principles which underlie modern programming, including rudimentary complexity theory. We will also explore the history and evolution of current computation and the internet. Students will learn how to write simple code in and express algorithms in the form of pseudocode. In terms of algorithms, we will start with some sorting and searching, as well as a number of useful computational tools and techniques. After these basics, we will learn about image and sound processing, and (if physical classes happen) working with hardware.
Note: If you already possess some coding ability, and want to improve, or to start coding more seriously in a principled manner, please take the “Introduction to Computer Programming” (CS 101) course offered in the spring. This course is not a replacement for 101. This is an FC, intended for general audiences; 101 is more intensive, and meant for people who are considering a CS major.
The course will begin with the origins of quantitative thinking vis-a-vis the number system and its evolution. This will be followed by a discussion of methods in problem solving and estimation, using real-world examples. We will then delve into the world of abstracts, i.e., set theory, geometry, graph theory, probability, and logic. Students will learn how some of these tools can be utilised to study (i) fairness in division of scarce resources, (ii) collective decisions in committees and democracies around the world (voting methods), and (iii) applications to finance- decisions regarding investments and returns. By the end of the course, students would have a basic understanding of the most widely used mathematical tools in the liberal arts. They will learn how to approach different problems from nature and society- by reducing the problems to their bare essentials and to analyse their underlying structural and logical patterns.
Some of humankind’s most powerful and beautiful ideas live in Mathematics. This ancient and most human of disciplines also happens to be the ideal arena in which to gain experience in both analytical thinking as well as creative problem solving — these twin aspects of mathematics are what will drive this course. The topics we will choose to work with will not require any technical background – you will be given all the tools you need. During this course, you will develop a set of broadly useful problem solving strategies and gain experience in making rigorous arguments while encountering landmark ideas. Among other things, we will look at the existence of non-rational lengths, the foundation of calculus, and Cantor’s set theory — ideas that we take for granted today but which represent some of the “great works” of our collective intellectual history.
There are different ways of perceiving, analysing, and expressing the world, and the foundation courses at Ashoka will hopefully introduce stu- dents to many of them. This foundation course is designed to serve as an introduction to one of the most fundamental, powerful, and beautiful modes of perception, analysis, and expression − mathematics.
In this course we will introduce a number of ideas and methods, some useful, some beautiful, some both useful and beautiful. Profound ideas in mathematics often arise through generalization of the familiar. We will give a flavour of mathematical way of thinking using topics which are accessible, without requiring overwhelming technical background, such as number theory, graph theory, calculus, probability and so on.
One of the ways in which mathematics grows is by solving tough problems. In 2000, the Clay institute listed the so-called millennium problems, and announced a prize money of one million dollars each for their solution. The objective of our course is to learn about and understand one of these problems, namely the Birch-Swinnerton-Dyer Conjecture. We will do this by reading (as a group) an expository account of this conjecture beginning with the mathematics one learns in high school. The book is Elliptic Tales by Avner Ash and Robert Gross. No background knowledge of mathematics is required. We will use computer algebra packages and graphing calculators in order to explore the intuitive side of mathematical ideas required to understand the conjecture. We will learn in groups with each other’s help.
There is a further book reading component to the course…a book review. There will be a different book assigned to each group. The objective is to gain an appreciation for mathematics and how it is used in the world around us.
This course is structured on observations of the world around us as well as data regarding it, on reasoning about these observations, and on using mathematics to advance this reasoning. What is the notion of infinity and are there different types of infinities? How do pandemics start and grow? How can we tell if data is being faked? How can we read graphs and understand them? How can we figure out if data is being presented in a way designed to fool us? How can we make intelligent guesses as to the magnitude of things, e.g. how many auto-rickshaws are there in Delhi? Which COVID-19 tests are better – the PCR tests or the Rapid Antigen tests and why? What sorts of cognitive fallacies should we be aware of? What is the idea of a function?
The course will stress estimation and approximation techniques, including order-of-magnitude arguments, the ability to understand graphs and plots, an understanding of geometric arguments, a feeling for how different functions “should” behave, probability and statistics including Bayesian methods and related questions. Some part of the course will describe models, how to construct them and how to interpret them. I will choose a range of examples and show how to reason quantitatively about them. The course itself is dynamic and its content changes from year to year in terms of the examples that will be used and the ideas that will be stressed, since I would like to use current examples as far as possible.
krishna.maddaly@ashoka.edu.in
