Introduction to Mathematical Thinking from Coursera

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Syllabus

Week 1

START with the Welcome lecture. It explains what this course is about. (It comes with a short Background Reading assignment, to read before you start the course, and a Reading Supplement on Set Theory for use later in the course, both in downloadable PDF format.) This initial orientation lecture is important, since this course is probably not like any math course you have taken before – even if in places it might look like one! AFTER THAT, Lecture 1 prepares the groundwork for the course; then in Lecture 2 we dive into the first topic. This may all look like easy stuff, but tens of thousands of former students found they had trouble later by skipping through Week 1 too quickly! Be warned. If possible, form or join a study group and discuss everything with them. BY THE WAY, the time estimates for watching the video lectures are machine generated, based on the video length. Expect to spend a lot longer going through the lectures sufficiently well to understand the material. The time estimates for completing the weekly Problem Sets (Quiz format) are a bit more reliable, but even they are just a guideline. You may find yourself taking a lot longer.

Week 2

In Week 2 we continue our discussion of formalized parts of language for use in mathematics. By now you should have familiarized yourself with the basic structure of the course: 1. Watch the first lecture and answer the in-lecture quizzes; tackle each of the problems in the associated Assignment sheet; THEN watch the tutorial video for the Assignment sheet. 2. REPEAT sequence for the second lecture. 3. THEN do the Problem Set, after which you can view the Problem Set tutorial. REMEMBER, the time estimates for watching the video lectures are machine generated, based on the video length. Expect to spend a lot longer going through the lectures sufficiently well to understand the material. The time estimates for completing the weekly Problem Sets (Quiz format) are a bit more reliable, but even they are just a guideline. You may find yourself taking a lot longer.

Week 3

This week we continue our analysis of language for use in mathematics. Remember, while the parts of language we are focusing have particular importance in mathematics, our main interest is in the analytic process itself: How do we formalize concepts from everyday life? Because the topics become more challenging, starting this week we have just one basic lecture cycle (Lecture -> Assignment -> Tutorial -> Problem Set -> Tutorial) each week. If you have not yet found one or more people to work with, please try to do so. It is so easy to misunderstand this material.

Week 4

This week we complete our analysis of language, putting into place the linguistic apparatus that enabled, mathematicians in the 19th Century to develop a formal mathematical treatment of infinity, thereby finally putting Calculus onto a firm footing, three hundred years after its invention. (You do not need to know calculus for this course.) It is all about being precise and unambiguous. (But only where it counts. We are trying to extend our fruitfully-flexible human language and reasoning, not replace them with a rule-based straightjacket!)

Week 5

This week we take our first look at mathematical proofs, the bedrock of modern mathematics.

Week 6

This week we complete our brief look at mathematical proofs

Week 7

The topic this week is the branch of mathematics known as Number Theory. Number Theory, which goes back to the Ancient Greek mathematicians, is a hugely important subject within mathematics, having ramifications throughout mathematics, in physics, and in some of today's most important technologies. In this course, however, we consider only some very elementary parts of the subject, using them primarily to illustrate mathematical thinking.

Week 8

In this final week of instruction, we look at the beginnings of the important subject known as Real Analysis, where we closely examine the real number system and develop a rigorous foundation for calculus. This is where we really benefit from our earlier analysis of language. University math majors generally regard Real Analysis as extremely difficult, but most of the problems they encounter in the early days stem from not having made a prior study of language use, as we have here.

Weeks 9 & 10: Test Flight

Test Flight provides an opportunity to experience an important aspect of "being a mathematician": evaluating real mathematical arguments produced by others. There are three stages. It is important to do them in order, and to not miss any steps. STAGE 1: You complete the Test Flight Problem Set (available as a downloadable PDF with the introductory video), entering your solutions in the Peer Evaluation module. STAGE 2: You complete three Evaluation Exercises, where you evaluate solutions to the Problem Set specially designed to highlight different kinds of errors. The format is just like the weekly Problem Sets, with machine grading. You should view the Tutorial video for each Exercise after you submit your solutions, but BEFORE you start the next Exercise. STAGE 3: You evaluate three Problem Set solutions submitted by other students. (This process is anonymous.) This final stage takes place in the Peer Evaluation module. After you are done peer reviewing, you may want to evaluate your own solution. It can be very informative to see how you rate your own attempt after looking at the work of others.

Good to know

Know what's good

, what to watch for

, and possible dealbreakers

Introduces the nature of mathematical thinking and its differences from school mathematics

Develops critical thinking skills, logical reasoning, and problem-solving abilities

Taught by Dr. Keith Devlin, a renowned mathematician and author

Emphasizes the practical applications of mathematics in science and everyday life

Provides a solid foundation for further studies in mathematics or related fields

Reviews summary

Challenging yet rewarding introduction to mathematical thinking

Learners say this challenging but rewarding course can help you understand proofs and think mathematically. The course is well taught by Professor Devlin, who is described as enthusiastic, clear, and engaging. Learners say lectures are easy to follow and assignments are helpful for reinforcement. Many students report that completing assignments and participating in the course forums can boost understanding.

Exercises and assignments are helpful for learning.

"An excellent introductory course to mathematical thinking or a companion course to follow while shuffling through your first book about mathematical proofs. Professor Devlin's way of putting out lots of exercises and going through them meticulously afterwards is as challenging as it is rewarding."

"brilliant lectures! Very easy to follow and quizzes and problem sets are very helpful."

"This was a great course. The instructor Dr. Devlin was able to make an otherwise difficult topic easy to understand. His explanations were crystal clear. There is a fair amount of assignments to do. This can be challenging, both qualitatively and quantitatively. In the end, I learned to write mathematical proofs."

Course forums and discussions can be helpful for learning.

"This was an interesting course. It assumes no understanding of advanced mathematics. It teaches you how to think which is an important skill for possibly every job out there. The time you spend every week depends on your own goals."

"I am finding the course considerably less discussive than I did Peter Norvig and Sebastian Thrun’s AI course. This is partly because the video lectures are longer and more formal; and partly because there seems to be less active discussion in the course-provided discussion forums, possibly on account of the way in which students have been encouraged to make their own arrangements, which was far less the case with the AI course."

"What this course shares with the AI course is the feature that struck me so forcefully in 2011: the feeling that you are getting one-to-one personal tuition from a very skilled and interesting teacher."

The course is challenging but worth the effort.

"Very good and worth doing but hard work."

"I would say this class is more for people who want to major in mathematics than anything else."

"I learned to write mathematical proofs."

Professor Devlin is an excellent teacher.

"Tough, fun and rewarding. Prof Devlin is an excellent and enthusiastic teacher."

"brilliant lectures! Very easy to follow and quizzes and problem sets are very helpful. Fantastic professor!"

"The professor was terrific. Every lecture video was clear and had quizzes within the videos that reinforced everything we were learning."

Activities

Be better prepared before your course. Deepen your understanding during and after it. Supplement your coursework and achieve mastery of the topics covered in Introduction to Mathematical Thinking with these activities:

Review your notes from previous math courses

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This will help you refresh your memory on the基礎 concepts of mathematics and make it easier to learn new material.

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Gather your notes from previous math courses.
Review the notes and make sure that you understand the concepts.
If you have any questions, ask your instructor or a classmate for help.

Watch video tutorials on mathematical concepts

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Video tutorials can be a helpful way to learn new concepts or review old ones.

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Find a video tutorial on a mathematical concept that you are interested in.
Watch the video and take notes on the key points.
Pause the video and try to solve any problems or answer any questions that the instructor poses.

Create a notebook or binder to keep track of your notes, assignments, and quizzes

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This will help you stay organized and keep track of your progress.

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Purchase a notebook or binder.
Create a system for organizing your notes, assignments, and quizzes.
Keep your notebook or binder up-to-date.

Five other activities

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Read 'Journey Through Genius: The Theorems of Mathematics'

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This book provides a deep dive into the history and development of some of the most important theorems in mathematics, from ancient times to modern times. It will help you gain a deeper understanding of the concepts and ideas behind mathematical thinking.

View Math through the Ages: A Gentle History for... on Amazon

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Read the book in its entirety.
Take notes on the key concepts and ideas in each theorem.
Discuss the book with other students or a study group.

Solve math problems regularly

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Regular practice will help you improve your problem-solving skills and solidify your understanding of the concepts.

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Set aside some time each day or week to practice math problems.
Find a variety of problems to practice, from basic arithmetic to more complex algebra and calculus problems.
Check your answers and make corrections as needed.

Create a concept map of the main concepts in mathematical thinking

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Create a concept map that visually represents the main concepts and ideas covered in the course. This will help you organize your understanding of the material and see how the different concepts are connected.

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Identify the key concepts and ideas in mathematical thinking.
Create a visual representation of the concepts and ideas.
Label the concepts and ideas and connect them with lines or arrows to show how they are related.
Review your concept map and make any necessary revisions.

Form a study group with other students in the course

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Study groups can provide a supportive and collaborative environment for learning.

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Find a group of students who are interested in forming a study group.
Meet regularly to discuss the course material and work on problems together.
Help each other out with schwierige concepts and learn from each other's perspectives.

Write a 5-page paper on a mathematical topic of your choice

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This will give you the opportunity to explore a mathematical topic in depth and demonstrate your understanding of mathematical concepts.

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Choose a mathematical topic that you are interested in.
Research the topic thoroughly.
Write a clear and concise paper that explains the topic in detail.
Proofread your paper carefully before submitting it.

Career center

Learners who complete Introduction to Mathematical Thinking will develop knowledge and skills that may be useful to these careers:

Mathematician

Introduction to Mathematical Thinking will help you develop the mathematical thinking skills needed to succeed as a Mathematician. The course will help you build a strong foundation in mathematics and develop the critical thinking and problem-solving skills needed to solve complex mathematical problems.

See salaries and explore the career path for Mathematician

Data Analyst

Learn to analyze problems in mathematical ways with the course Introduction to Mathematical Thinking. As a Data Analyst, you will use mathematical thinking to create insight from data. This course will help you build a foundation in mathematical thinking, which will be essential for success in this role.

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Financial Analyst

Introduction to Mathematical Thinking will introduce the mathematical concepts needed for Financial Analysts to predict future financial trends. The course will help you develop the critical thinking skills necessary to succeed in this role.

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Business Analyst

Mathematical thinking is a key skill for Business Analysts, who use mathematical models to solve business problems. Introduction to Mathematical Thinking will help you develop these skills and gain a deeper understanding of the mathematical concepts used in business analysis.

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Operations Research Analyst

Introduction to Mathematical Thinking can be a great way to learn the mathematical foundations of operations research. Operations Research Analysts use mathematical models to solve complex problems in a variety of industries.

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Statistician

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Quantitative Analyst

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Actuary

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Data Scientist

Introduction to Mathematical Thinking will introduce you to the mathematical concepts and techniques used by Data Scientists. The course will help you develop the critical thinking and problem-solving skills needed to succeed in this role.

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Software Engineer

Introduction to Mathematical Thinking will provide you with the mathematical skills needed to succeed as a Software Engineer. The course will help you develop critical thinking and problem-solving skills, which are essential in this role.

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Machine Learning Engineer

Introduction to Mathematical Thinking can provide you with the mathematical foundation needed to succeed as a Machine Learning Engineer. The course will help you develop the critical thinking and problem-solving skills needed to build and deploy machine learning models.

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Risk Manager

Risk Managers use mathematical models to assess and manage risk. Introduction to Mathematical Thinking can help you develop the mathematical skills needed to succeed in this role.

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Economist

Mathematical thinking is essential for Economists, who use mathematical models to analyze economic data. Introduction to Mathematical Thinking can help you develop the skills needed to succeed in this role.

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Cryptographer

Introduction to Mathematical Thinking will introduce you to the mathematical concepts and techniques used by Cryptographers. The course will help you develop the critical thinking and problem-solving skills needed to succeed in this role.

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Teacher

Introduction to Mathematical Thinking can be helpful for Teachers who want to improve their understanding of mathematics and develop new ways to teach mathematical concepts to their students.

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