Introduction to Mathematical Thinking

Go to Course: https://www.coursera.org/learn/mathematical-thinking

Introduction

### Course Review: Introduction to Mathematical Thinking on Coursera #### Overview Mathematics often evokes memories of complex equations and rigid problem-solving methods encountered during our school years. However, the **Introduction to Mathematical Thinking** course on Coursera doesn't merely focus on traditional mathematics; instead, it provides a unique perspective into the "how" and "why" of mathematical reasoning. This course, developed over many years of teaching and research, invites learners to engage deeply with the cognitive processes that mathematicians utilize to dissect problems – both conventional and real-world. Designed for anyone fascinated by mathematics, this course is especially valuable for those who have struggled with mathematics in the past and want to change their perspective. The content emphasizes mathematical thinking as a more flexible, dynamic approach compared to rote problem-solving. #### Syllabus Breakdown The course is structured over ten weeks, encompassing various facets of mathematical thought. Here's an overview of what you can expect: - **Week 1: Introduction and Foundations** The course begins with a welcome lecture that sets clear expectations. This week aims to orient participants to the course's unique approach, emphasizing the importance of taking time to digest the material. It includes background reading assignments and a focus on set theory. - **Weeks 2 to 4: Language and Formalization** The subsequent weeks delve into the analysis of language as it pertains to mathematics. You will explore how language serves as a tool for formalizing ideas, ultimately building toward an understanding of infinite concepts crucial to calculus. - **Weeks 5 and 6: Entering the Realm of Proofs** The course then shifts focus to mathematical proofs, which establish the foundation of modern mathematics. These weeks encourage students to comprehend the meticulous structure underlying mathematical arguments. - **Week 7: Number Theory** A journey through number theory will highlight its historical significance and contemporary ramifications in technology. Here, fundamental concepts will be examined to illustrate the thinking processes embraced by mathematicians. - **Week 8: Real Analysis** The final instructional week engages with the essential subject of Real Analysis, laying down a rigorous foundation for calculus while applying the previously discussed language analysis. - **Weeks 9 and 10: Test Flight** These weeks facilitate a peer evaluation process designed to mimic authentic mathematical critique. Participants will engage actively with mathematical arguments from their peers, enhancing critical evaluation skills. #### Course Evaluation **Strengths:** - **Intellectual Empowerment:** The course's framework empowers students to reshape their understanding of mathematics, transitioning from apprehension to appreciation. - **Community Support:** The recommendation to form study groups highlights the importance of collaborative learning and aids retention of complex concepts. - **Practical Application:** This course emphasizes real-world applications and encourages learners to think critically about everyday phenomena through a mathematical lens. **Challenges:** - **Time Commitment:** The course is rigorous in its material and requires significant time investment beyond the machine-generated estimates. This can be daunting for students balancing multiple commitments. - **Abstract Concepts:** For beginners, some topics, particularly in formal proofs and Real Analysis, may come across as challenging without prior exposure. #### Recommendation I wholeheartedly recommend the **Introduction to Mathematical Thinking** course. Whether you're a student looking to gain a deeper understanding of mathematical principles, a professional seeking to enhance problem-solving skills, or simply a curious learner, this course will enrich your cognitive abilities and appreciation for mathematics. The foundations you build in this course could transform not only how you approach mathematics but also how you tackle everyday challenges. With its thorough exploration of the language of mathematics, this course prepares you to face complex ideas without fear and encourages a robust intellectual curiosity. Embark on this mathematical journey — your future self will thank you!

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.

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.

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.

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!)

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

This week we complete our brief look at mathematical proofs

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.

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.

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.