Certainly, here’s a sample syllabus for a course on Mathematical Thinking (Introduction to Proofs):

Course Title: Mathematical Thinking (Introduction to Proofs)Course Description: This course serves as an introduction to mathematical thinking and the art of constructing and writing mathematical proofs. It focuses on developing the foundational skills needed for advanced mathematics and fostering critical thinking.Course Objectives: By the end of this course, students should be able to:

## Free Textbook

There are several free textbooks and open educational resources (OER) that you can use to study for a Mathematical Thinking (Introduction to Proofs) course. Here are a few options:

**“Book of Proof” by Richard Hammack**:- This textbook is available for free online and is designed for students learning about mathematical proof writing and reasoning.
- Website: http://www.people.vcu.edu/~rhammack/BookOfProof/

**“A Transition to Higher Mathematics” by D. Smith, M. Eggen, and R. St. Andre**:- This book provides a bridge from computational mathematics to more theoretical mathematics and includes sections on proof writing.
- Website: https://www.pearson.com/us/higher-education/program/Smith-Transition-to-Higher-Mathematics-A/PGM1013034.html

**“How to Think Like a Mathematician: A Companion to Undergraduate Mathematics” by Kevin Houston**:- While not a traditional textbook, this resource provides guidance on developing mathematical thinking and proof-writing skills.
- Website: https://www.cambridge.org/core/books/how-to-think-like-a-mathematician/68F97CE474D1A64F3C3A50764D71E09C

**OpenStax College Pre-Algebra**:- OpenStax offers free math textbooks, including a Pre-Algebra book that covers foundational concepts often encountered in a Mathematical Thinking course.
- Website: https://openstax.org/details/books/prealgebra

**“Proofs and Concepts: The Fundamentals of Abstract Mathematics” by Dave Witte Morris and Joy Morris**:- This book focuses on proof techniques and mathematical thinking, making it suitable for an introductory proof-writing course.
- Website: https://www.proofsandconcepts.com/

**“An Interactive Introduction to Mathematical Analysis” by Jonathan Lewin**:- While it’s primarily an analysis textbook, it includes chapters on mathematical logic and proof techniques, which are relevant for proof-writing courses.
- Website: https://www.mathcs.org/analysis/reals/book.pdf

These resources cover various aspects of mathematical thinking, including proof writing and reasoning. Be sure to explore them and choose the one that aligns with your course’s curriculum and your learning style.

## Free MOOCs

As of my last knowledge update in January 2022, you can find free Massive Open Online Courses (MOOCs) that cover topics related to Mathematical Thinking and Introduction to Proofs. While these MOOCs may not cover the entire curriculum of a formal course, they can be valuable for self-study and improving your mathematical thinking skills. Here are some MOOC platforms where you can explore relevant courses:

**Coursera**:- Coursera offers courses related to mathematical thinking and proof writing. You can audit many of these courses for free, but there may be fees for certificates or additional features.
- Website: https://www.coursera.org/

**edX**:- edX provides courses in mathematics and related topics, some of which may cover aspects of mathematical thinking and proofs. You can audit courses for free, with the option to pay for certificates.
- Website: https://www.edx.org/

**Khan Academy**:- Khan Academy offers a comprehensive set of free math courses and tutorials that cover foundational topics related to mathematical thinking, logic, and problem-solving.
- Website: https://www.khanacademy.org/

**MIT OpenCourseWare (OCW)**:- MIT OCW offers free access to course materials from Massachusetts Institute of Technology (MIT) courses, including those related to mathematical thinking and proofs.
- Website: https://ocw.mit.edu/index.htm

**Stanford Online**:- Stanford University offers some free online courses in mathematics and mathematical thinking. Check their website for current offerings.
- Website: https://online.stanford.edu/courses

**UC Irvine OpenCourseWare (OCW)**:- UC Irvine’s OCW includes courses in mathematics, including those that focus on proof-writing and mathematical reasoning.
- Website: http://ocw.uci.edu/

Please note that the availability of specific courses may change over time, so I recommend visiting these platforms and searching for relevant courses. Additionally, when enrolling in a course, check whether there are any fees associated with certification or additional features.