Introduction to Mathematical Thinking from Coursera

Learn how to think the way mathematicians do – a powerful cognitive process developed over thousands of years.

Mathematical thinking is not the same as doing mathematics – at least not as mathematics is typically presented in our school system. School math typically focuses on learning procedures to solve highly stereotyped problems. Professional mathematicians think a certain way to solve real problems, problems that can arise from the everyday world, or from science, or from within mathematics itself. The key to success in school math is to learn to think inside-the-box. In contrast, a key feature of mathematical thinking is thinking outside-the-box – a valuable ability in today’s world. This course helps to develop that crucial way of thinking.

What's inside

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.

Traffic lights

Read about what's good

what should give you pause

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

A challenging mathematical thinking course

According to learners, this course provides a unique approach to mathematics, focusing on the *thinking process* rather than just procedures. Many found the approach challenging but ultimately rewarding, describing it as an eye-opener for understanding how mathematicians actually think. The course is often seen as a departure from traditional school math, emphasizing precise language and the fundamentals of mathematical proofs. While some students found the pace or proof-writing aspect difficult and felt the course required significant time investment, others appreciated the rigorous foundation it provided. Reviews suggest success hinges on engaging deeply with the material and assignments.

Not like traditional math courses

"Forget what you learned in school, this is a completely different kind of math."

"I was surprised how little it focused on computation and how much on logic and definition."

"This is the math I wish I learned in high school, it's about *why* things work."

"Expect a philosophical and logical approach, not just formulas."

Instructor is clear and engaging

"The lecturer is excellent, explains complex ideas simply and makes them accessible."

"Keith Devlin is a fantastic guide through this challenging and abstract material."

"Really appreciated the instructor's insights and engaging teaching style throughout the lectures."

"His passion for mathematical thinking is evident and inspiring."

Provides a rigorous analytical base

"It gave me a solid grounding for understanding higher-level mathematical concepts later on."

"This course is essential preparation for anyone serious about theoretical math or computer science."

"I feel much better equipped to tackle more advanced topics after this rigorous introduction."

"Provided the foundational logic needed for more abstract thinking."

Teaches a new way of analytical thought process

"This course isn't about equations; it's about changing how you approach problems entirely."

"I finally understand what 'mathematical thinking' means beyond just calculation or formulas."

"It helped me develop a more structured and rigorous approach to thinking through problems logically."

"The emphasis on thinking process rather than rote procedures is incredibly valuable."

Significant time commitment needed

"Be prepared to spend serious time on the problem sets, they are not easy or quick."

"This course demands significant focus and several hours per week to grasp the material."

"Don't underestimate the workload, especially when tackling the proof assignments."

"It took me much longer than the estimated times to truly understand and complete tasks."

Proof writing is difficult for many learners

"I struggled significantly with constructing proofs, they were harder than I expected."

"Writing the proofs required much more effort and understanding than solving typical math problems."

"I found the proof assignments the most frustrating part, but they forced me to really think."

"Getting the logic absolutely precise for the proofs was a major hurdle."

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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Quantitative Analyst

Quantitative Analysts use mathematical and statistical models to analyze financial data. Introduction to Mathematical Thinking can help you develop the mathematical skills needed to succeed in this role.

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Statistician

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

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Actuary

Actuaries use mathematical models to assess risk and uncertainty. Introduction to Mathematical Thinking can help you develop the mathematical 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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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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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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