MATH 300 – Mathematical Thinking

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:

  • Understand and apply fundamental concepts of logic and set theory.Construct and write mathematical proofs using various proof techniques.Recognize and work with common mathematical structures and properties.Develop problem-solving skills and mathematical creativity.Communicate mathematical ideas clearly and effectively.
  • Textbook:
  • [Insert Mathematical Thinking Textbook Title and Author(s)]
  • Materials:
  • Notebook or binder for class notes and assignments.Pencils, erasers, and a ruler.Access to a computer and mathematical software (if required).
  • Grading:
  • Homework and Class Participation: XX%Quizzes: XX%Midterm Examinations: XX%Final Examination: XX%Proof Writing Assignments: XX%
  • Course Outline:Unit 1: Introduction to Mathematical Thinking
  • The nature of mathematics and mathematical reasoning.Basic principles of logic, including propositions and truth values.Set theory and set operations.
  • Unit 2: Mathematical Proof Techniques
  • Direct proofs, contrapositive, contradiction, and mathematical induction.Proving by cases and counterexamples.Understanding and using quantifiers (universal and existential).
  • Unit 3: Sets and Functions
  • Set notation and operations.Cardinality and countable sets.Functions, injective, surjective, and bijective functions.
  • Unit 4: Mathematical Structures
  • Relations and equivalence relations.Algebraic structures: groups, rings, and fields.Lattices and Boolean algebras.
  • Unit 5: Proof Writing and Advanced Topics
  • Developing clear and structured proof writing skills.Introduction to additional topics such as graph theory, combinatorics, and number theory.
  • Unit 6: Review and Final Exam PreparationNote: This is a general example of a syllabus for an introductory Mathematical Thinking course with an emphasis on proof writing. Please adapt it to meet the specific needs and standards of your educational institution, and consider any prerequisites or learning outcomes required by your curriculum.

    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:

    1. “Book of Proof” by Richard Hammack:
    2. “A Transition to Higher Mathematics” by D. Smith, M. Eggen, and R. St. Andre:
    3. “How to Think Like a Mathematician: A Companion to Undergraduate Mathematics” by Kevin Houston:
    4. OpenStax College Pre-Algebra:
    5. “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/
    6. “An Interactive Introduction to Mathematical Analysis” by Jonathan Lewin:

    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:

    1. 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/
    2. 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/
    3. 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/
    4. 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
    5. 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
    6. 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.

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