Introduction to Mathematical Thinking

Syllabus - What you will learn from this course

1

Section

3 hours to complete

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....

6 videos (Total 84 min), 1 quiz

6 videos

Lecture 0 - Welcome8m

Lecture 1 - Introductory Material27m

Lecture 2 - Logical Combinators26m

Tutorial for Assignment 12m

Tutorial for Assignment 29m

Tutorial for Problem Set 19m

1 practice exercises

Problem Set 10m

2

Section

4 hours to complete

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. ...

6 videos (Total 131 min), 1 quiz

6 videos

Lecture 3 - Implication31m

Lecture 4 - Equivalence24m

SUPPLEMENT: Using the course evaluation rubric4m

Tutorial for Assignment 311m

Tutorial for Assignment 421m

Tutorial for Problem Set 237m

1 practice exercises

Problem Set 20m

3

Section

4 hours to complete

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....

4 videos (Total 128 min), 1 quiz

4 videos

Lecture 5 - Quantifiers29m

SUPPLEMENT - How to Read Mathematical Formulas27m

Tutorial for Assignment 547m

Tutorial for Problem Set 323m

1 practice exercises

Problem Set 30m

4

Section

3 hours to complete

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!)...

4 videos (Total 88 min), 1 quiz

4 videos

Lecture 6A - Working with Quantifiers 115m

Lecture 6B - Working with Quantifiers 228m

Tutorial for Assignment 616m

Tutorial for Problem Set 427m

1 practice exercises

Problem Set 40m

5

Section

4 hours to complete

Week 5

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

4 videos (Total 82 min), 1 quiz

4 videos

Lecture 7A - Proofs 122m

Lecture 7B - Proofs 224m

Tutorial for Assignment 715m

Tutorial for Problem Set 520m

1 practice exercises

Problem Set 530m

6

Section

4 hours to complete

Week 6

This week we complete our brief look at mathematical proofs...

4 videos (Total 111 min), 1 quiz

4 videos

Lecture 8A - Proofs with Quantifiers 135m

Lecture 8B - Proofs with Quantifiers 222m

Tutorial for Assignment 816m

Tutorial for Problem Set 636m

1 practice exercises

Problem Set 630m

7

Section

5 hours to complete

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....

4 videos (Total 101 min), 1 quiz

4 videos

Lecture 9A - Number Theory 121m

Lecture 9B - Number Theory 225m

Tutorial for Assignment 913m

Tutorial for Problem Set 739m

1 practice exercises

Problem Set 70m

8

Section

5 hours to complete

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....

5 videos (Total 124 min), 1 quiz

5 videos

Lecture 10A - Real Analysis 127m

Lecture 10B - Real Analysis 226m

Lecture 10C - Real Analysis 320m

Tutorial for Assignment 1012m

Tutorial for Problem Set 838m

1 practice exercises

Problem Set 80m

9

Section

6 hours to complete

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....

4 videos (Total 83 min), 4 quizzes

4 videos

Test Flight Introduction0m

Test Flight Tutorial 133m

Test Flight Tutorial 236m

Test Flight Tutorial 312m

3 practice exercises

Evaluation Exercise 10m

Evaluation Exercise 20m

Evaluation Exercise 30m

Frequently Asked Questions

When will I have access to the lectures and assignments?
Once you enroll for a Certificate, you’ll have access to all videos, quizzes, and programming assignments (if applicable). Peer review assignments can only be submitted and reviewed once your session has begun. If you choose to explore the course without purchasing, you may not be able to access certain assignments.
What will I get if I pay for this course?
If you pay for this course, you will have access to all of the features and content you need to earn a Course Certificate. If you complete the course successfully, your electronic Certificate will be added to your Accomplishments page - from there, you can print your Certificate or add it to your LinkedIn profile. Note that the Course Certificate does not represent official academic credit from the partner institution offering the course.
What is the refund policy?
Is financial aid available?
Yes! Coursera provides financial aid to learners who would like to complete a course but cannot afford the course fee. To apply for aid, select "Learn more and apply" in the Financial Aid section below the "Enroll" button. You'll be prompted to complete a simple application; no other paperwork is required.
What is the passing grade for this course?
Since the focus is to acquire a new way of thinking (as opposed to getting right answers), the passing grade for the weekly Problem Sets is 35%, and for the Test Flight Problem Sets 30%. Basically, this means that if you stick with the course and complete all the work diligently, you should get a passing grade.

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Introduction to Mathematical Thinking