Introduction to Mathematical Thinking
Stanford University
Об этом курсе
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.
Приобретаемые навыки
Strategic ThinkingProblem SolvingAlgebraWolfram Mathematica
Syllabus - What you will learn from this course
1
Section
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 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
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 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
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
SUPPLEMENT - How to Read Mathematical Formulas27m
Tutorial for Assignment 547m
Tutorial for Problem Set 323m
1 practice exercises
Problem Set 30m
4
Section
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 6B - Working with Quantifiers 228m
Tutorial for Assignment 616m
Tutorial for Problem Set 427m
1 practice exercises
Problem Set 40m
5
Section
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 7B - Proofs 224m
Tutorial for Assignment 715m
Tutorial for Problem Set 520m
1 practice exercises
Problem Set 530m
6
Section
Week 6
This week we complete our brief look at mathematical proofs
4 videos (Total 111 min), 1 quiz
4 videos
Lecture 8B - Proofs with Quantifiers 222m
Tutorial for Assignment 816m
Tutorial for Problem Set 636m
1 practice exercises
Problem Set 630m
7
Section
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 9B - Number Theory 225m
Tutorial for Assignment 913m
Tutorial for Problem Set 739m
1 practice exercises
Problem Set 70m
8
Section
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 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
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 Tutorial 133m
Test Flight Tutorial 236m
Test Flight Tutorial 312m
3 practice exercises
Evaluation Exercise 10m
Evaluation Exercise 20m
Evaluation Exercise 30m
4.8
38%
83%
12%
Top Reviews
By NR•Mar 5th 2018
An awesome course. Very easy to follow at the start, becomes more challenging at the end. I have a PhD in economics yet I struggled with the real analysis at the end. And that's just intro level! :-D
By MF•Aug 26th 2017
Great course, it's really shows the way how to think in mathematical rigor. It gave me an intuition about what Mathematics is itself.\n\nThank you Profesor Keith Devlin, you are a great teacher !
Instructor
Dr. Keith Devlin
Co-founder and Executive DirectorAbout Stanford University
The Leland Stanford Junior University, commonly referred to as Stanford University or Stanford, is an American private research university located in Stanford, California on an 8,180-acre (3,310 ha) campus near Palo Alto, California, United States.
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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