Introduction To Mathematical Thinking

The Basic Course lasts for ten weeks, comprising ten lectures, each with a problem-based work assignment (ungraded, designed for group work), a weekly Problem Set (machine graded), and weekly tutorials in which the instructor will go over some of the assignment and Problem Set questions from the previous week.

The Extended Course consists of the Basic Course followed by a more intense two weeks exercise called Test Flight. Whereas the focus in the Basic Course is the development of mathematically-based thinking skills for everyday life, the focus in Test Flight is on applying those skills to mathematics itself.

The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

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Course will introduce students to the fundamentals of mathematical proof in the context of discrete structures. Topics include logic, sets and relations, functions, integers, induction and recursion, counting, permutations and combinations and algorithms.

Currently taking this course in mathematical thinking, and several students and I are confused as to why statements such as these are automatically assigned to be true. Some searching lead to this document, which on page 18, has a footnote which explains that outside of computers science, there are no practical applications to vacuous truths, and would like to know how they are applied in computer science.

I would like to very strongly urge you to take this course on an introduction to college-level mathematical thinking. It served me fantastically to bridge the gap to university when I was in a similar situation as you are now.

Oh man, I have the perfect resource for you: Introduction to Mathematical Thinking. It is all about learning to think like a mathematician and starting to do proofs. It's free, with resources created by Dr. Keith Devlin, a Stanford math prof who has somewhat specialized into mathematical education. Also, for the first time in a few years, he's planning on getting involved in answering student questions. This is the perfect time to sign up for the course! It's not a book like you asked for, but I really recommend the course!

If you already master the precalculus material and have taken an introduction to proofs as mentioned above (and perhaps read a book on the basics of calculus) then my favorite calculus text is "Calculus" by Spivak. It is very rigorous compared to introductory calculus text, in the sense that every result is proven, and he starts from the axioms of the real number line. If you read Spivak and do a lot of exercises (preferably after doing precalculus, an introduction to calculus and an introduction to proofs) you will be very well set to study mathematics in college. Do not start on Spivak until you have a firm grasp of the material in the precalculus book and Vellemans "How to Prove it". Even after those, it would perhaps be best to start with a gentle introduction to calculus.

Hey bud,

I have a few suggestions which helped master the math behind ML. I started out with the Coursera Course: Introduction to Mathematical Thinking by Dr. Keith Devlin.Link: -thinking/home/welcome

It wasn't until I realized that I needed to get an intuitive understanding of the thing I was trying to prove, and to reason/think about/diagram/play with the material to see exactly why something is the way it is. That shift in thinking, of first gaining an understanding of the material and proving to myself why something was true has made all the difference in the world for me.

Michael Starbird is Professor of Mathematics and a University Distinguished Teaching Professor at the University of Texas, Austin. He received his BA degree from Ponoma College and his PhD in mathematics from the University of Wisconsin, Madison. He has held visiting positions at the Institute for Advanced Study in Princeton, New Jersey, and at the Jet Propulsion Laboratory in Pasadena, California. He served as Associate Dean in the College of Natural Sciences at the University of Texas from 1989 to 1997. Starbird's mathematical research is in the field of topology. He has served as a member-at-large of the Council of the American Mathematical Society and on the national education committees of both the American Mathematical Society and the Mathematical Association of America.

Time: 112 hours Free Certificate The main purpose of this course is to bridge the gap between introductory mathematics courses in algebra, linear algebra, and calculus on one hand and advanced courses like mathematical analysis and abstract algebra, on the other hand, which typically require students to provide proofs of propositions and theorems. Another purpose is to pose interesting problems that require you to learn how to manipulate the fundamental objects of mathematics: sets, functions, sequences, and relations. The topics discussed in this course are the following: mathematical puzzles, propositional logic, predicate logic, elementary set theory, elementary number theory, and principles of counting. The most important aspect of this course is that you will learn what it means to prove a mathematical proposition. We accomplish this by putting you in an environment with mathematical objects whose structure is rich enough to have interesting propositions. The environments we use are propositions and predicates, finite sets and relations, integers, fractions and rational numbers, and infinite sets. Each topic in this course is standard except for the first one, puzzles. There are several reasons for including puzzles. First and foremost, a challenging puzzle can be a microcosm of mathematical development. A great puzzle is like a laboratory for proving propositions. The puzzler initially feels the tension that comes from not knowing how to start just as the mathematician feels when first investigating a topic or trying to solve a problem. The mathematician "plays" with the topic or problem, developing conjectures which he/she then tests in some special cases. Similarly, the puzzler "plays" with the puzzle. Sometimes the conjectures turn out to be provable, but often they do not, and the mathematician goes back to playing. At some stage, the puzzler (mathematician) develops sufficient sense of the structure, and only then can he begin to build the solution (prove the theorem). This multi-step process is perfectly mirrored in solving the KenKen problems this course presents. Some aspects of the solutions motivate ideas you will encounter later in the course. For example, modular congruence is a standard topic in number theory, and it is also useful in solving some KenKen problems. Another reason for including puzzles is to foster creativity.

First, read the course syllabus. Then, enroll in the course by clicking "Enroll me in this course". Click Unit 1 to read its introduction and learning outcomes. You will then see the learning materials and instructions on how to use them.

In this unit, you will begin by considering various puzzles, including Ken-Ken and Sudoku. You will learn the importance of tenacity in approaching mathematical problems including puzzles and brain teasers. You will also learn why giving names to mathematical ideas will enable you to think more effectively about concepts that are built upon several ideas. Then, you will learn that propositions are (English) sentences whose truth value can be established. You will see examples of self-referencing sentences which are not propositions. You will learn how to combine propositions to build compound ones and then how to determine the truth value of a compound proposition in terms of its component propositions. Then, you will learn about predicates, which are functions from a collection of objects to a collection of propositions, and how to quantify predicates. Finally, you will study several methods of proof including proof by contradiction, proof by complete enumeration, etc.

This unit is primarily concerned with the set of natural numbers N=\left \ 0,1,2,3,... \right \ . The axiomatic approach to N will be postponed until the unit on recursion and mathematical induction. This unit will help you understand the multiplicative and additive structure of N . This unit begins with integer representation: place value. This fundamental idea enables you to completely understand the algorithms we learned in elementary school for addition, subtraction, multiplication, and division of multi-digit integers. The beautiful idea in the Fusing Dots paper will enable you to develop a much deeper understanding of the representation of integers and other real numbers. Then, you will learn about the multiplicative building blocks, the prime numbers. The Fundamental Theorem of Arithmetic guarantees that every positive integer greater than 1 is a prime number or can be written as a product of prime numbers in essentially one way. The Division Algorithm enables you to associate with each ordered pair of non-zero integers - a unique pair of integers - the quotient and the remainder. Another important topic is modular arithmetic. This arithmetic comes from an understanding of how remainders combine with one another under the operations of addition and multiplication. Finally, the unit discusses the Euclidean Algorithm, which provides a method for solving certain equations over the integers. Such equations with integer solutions are sometimes called Diophantine Equations.

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