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 historyâs landmark ideas.
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.
For many mathematicians and scientists, life is about solving puzzles.
Much of high school mathematics can be viewed as a series of puzzles to be solved. Sometimes solving a puzzle requires one to experiment and find patterns. At other times just thinking logically or geometrically is enough.
You will work with your group to solve puzzles, and then write up the solutions on your own. No prior knowledge of mathematics is required to be able to solve the puzzles. But you will be required to use some computer algebra and some basic programming (which you have to pick up with the help of your group) to experiment and find patterns. The objective is to develop a taste for problem-solving and learn (with the help of some friends) the pleasures of getting creative ideas.
There is a further book reading component to the courseâŠa 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. In addition, the group has to present their book. Past presentations have involved making movies, writing and enacting plays, writing poetry, and various unique ways of storytelling.
At the end of the course, I expect students to gain an appreciation for mathematics, learn something about its culture, and experience the joy that accompanies a creative idea.
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 âshouldâ 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.
We all encounter prime numbers for the first time in secondary school: a natural number is called a prime if it has exactly two divisors, viz. 1 and the number itself eg. 31. We also learn that every natural number can be written as a product of its prime factors â for example 2022 = 2 Ă 3 Ă 337.
This is merely the beginning of a long story in which mathematicians working in the area of âNumber theoryâ have been unravelling the properties of prime numbers for thousands of years and yet the story is far from over âmany questions about prime numbers remain unsolved!
Determining whether a number is prime or not (primality testing) has interested mathematicians for long. In recent times, attention has focused on tests that run efficiently on a computer, because such tests are an integral part of several widely used systems for encrypting data on electronic devices e.g. for e-commerce transactions or internet banking. Primality testing plays a crucial role in the widely used RSA algorithm, which we shall learn about in this course, whose security relies on the difficulty of finding a numberâs prime factors. In the summer of 2002, computer scientist Manindra Agarwal and his then students Neeraj Kayal and Nitin Saxena, all from IIT Kanpur, discovered an efficient and deterministic test for the primality of a natural number (âPRIMES is in Pâ) which subsequently appeared in the Annals of Mathematics in 2004.
The objective of this foundation course is to understand the AgarwalâKayalâSaxena algorithm (AKS algorithm) without requiring any prior knowledge beyond general quantitative skills and the ability to think mathematically. As part of this course, we will develop the prerequisites from mathematics and theoretical computer science required to understand the AKS algorithm and to appreciate its elegance and importance.
References
Rempe-Gillen, Lasse and Waldecker, Rebecca, Primality Testing for Beginners, Student Mathematical Library Vol. 70, American Mathematical Society, Providence, RI (2014).
Agrawal, Manindra and Kayal, Neeraj and Saxena, Nitin, PRIMES is in P, Ann. of Math. Vol. 60(2), 781-793 (2004).
Bornemann, Folkmar, PRIMES is in P: a breakthrough for âEverymanâ, Notices of the AMS, Vol. 50(5), 545â552 (2003).
Robinson, Sara, New Method Said to Solve Key Problem In Math, New York Times, Section A, Page 20, August 8, 2002.
Faculty Name: Ravindra B Bapat
Department: Mathematics | Semester: Monsoon 2022
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 historyâs landmark ideas.
The course starts with an excursion into visiting numbers through the ages culminating with a discussion of the power of zero. As you progress 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 learn about the notion of randomness and how it can be used in studying uncertain phenomena. You will know about the power of modelling a situation using graph theory and a variety of uses it can be put to.
If you have not done serious mathematics in school or if you are scared to take a mathematics course donât 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 computing and computational thinking. 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 computer science and programming. There will also be some discussion of the history and evolution of current computation and the internet. Students will learn how to write simple code and express algorithms in the form of pseudocode. In terms of algorithms, we will cover some sorting and searching, as well as a number of basic computational tools and techniques. We will also discuss the inherent complexity of problems, the limits of computation, and the scope and limits of modern AI and ML.
Note: Even if you have prior exposure to programming, this course may have something new to offer – especially on computational thinking.
In this course, we will try to understand each of the words in the course title carefully. We will try to figure out how to quantify objects, how to reason, and what it means to reason. We will try to see if we can pinpoint what Mathematics is and maybe learn some Mathematics in the process. We will also try to figure out if “Mathematical” thinking is really a thing as opposed to just thinking!
Along the way we will pass through the following turns:
mihir.bhattacharya@ashoka.edu.in
