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Mathematics

Date: 
Friday, September 18, 2015 to Saturday, November 21, 2015
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About the Course

Social networks pervade our social and economic lives.   They play a central role in the transmission of information about job opportunities and are critical to the trade of many goods and services. They are important in determining which products we buy, which languages we speak, how we vote, as well as whether or not we decide to become criminals, how much education we obtain, and our likelihood of succeeding professionally.   The countless ways in which network structures affect our well-being make it critical to understand how social network structures impact behavior, which network structures are likely to emerge in a society, and why we organize ourselves as we do.  This course provides an overview and synthesis of research on social and economic networks, drawing on studies by sociologists, economists, computer scientists, physicists, and mathematicians.
 
The course begins with some empirical background on social and economic networks, and an overview of concepts used to describe and measure networks.   Next, we will cover a set of models of how networks form, including random network models as well as strategic formation models, and some hybrids.   We will then discuss a series of models of how networks impact behavior, including contagion, diffusion, learning, and peer influences.
 

Course Syllabus

  • Week 1: Introduction, Empirical Background and Definitions
Examples of Social Networks and their Impact, Definitions, Measures and Properties: Degrees, Diameters, Small Worlds, Weak and Strong Ties, Degree Distributions

  • Week 2: Background, Definitions, and Measures Continued
Homophily, Dynamics,  Centrality Measures: Degree, Betweenness, Closeness, Eigenvector, and Katz-Bonacich. Erdos and Renyi Random Networks: Thresholds and Phase Transitions,

  • Week 3: Random Networks 
Poisson Random Networks, Exponential Random Graph Models, Growing Random Networks, Preferential Attachment and Power Laws, Hybrid models of Network Formation

  • Week 4:   Strategic Network Formation 
Game Theoretic Modeling of Network Formation, The Connections Model, The Conflict between Incentives and Efficiency, Dynamics, Directed Networks, Hybrid Models of Choice and Chance
 
  • Week 5:  Diffusion on Networks. 
Empirical Background, The Bass Model, Random Network Models of Contagion, The SIS model, Fitting a Simulated Model to Data

  • Week 6:  Learning on Networks. 
Bayesian Learning on Networks, The DeGroot Model of Learning on a Network, Convergence of Beliefs, The Wisdom of Crowds, How Influence depends on Network Position.

  • Week 7: Games on Networks. 
Network Games, Peer Influences:  Strategic Complements and Substitutes, the Relation between Network Structure and Behavior, A Linear Quadratic Game, Repeated Interactions and Network Structures.
 

Recommended Background

The course has some basic prerequisites in mathematics and statistics.  For example, it will be assumed that students are comfortable with basic concepts from linear algebra (e.g., matrix multiplication), probability theory (e.g., probability distributions, expected values, Bayes' rule), and statistics (e.g., hypothesis testing), and some light calculus (e.g., differentiation and integration).  Beyond those concepts, the course will be self-contained.

Suggested Readings

The course is self-contained, so that all the definitions and concepts you need to solve the problem sets and final are contained in the video lectures.  Much of the material for the course is covered in a text: Matthew O. Jackson  Social and Economic Networks, Princeton University Press (Here are Princeton University Press and Amazon  pages for the book).  The text is optional and not required for the course.  Additional background readings, including research articles and several surveys on some of the topics covered in the course can be found on my web page.

Course Format

The course will run for seven weeks, plus two for the final exam.  Each week there will be video lectures available, as well as a standalone problem set and some occasional data exercises, and there will be a final exam at the end of the course for those who wish to earn a course certificate.  

FAQ

Will I get a Statement of Accomplishment after completing this class?

Yes. Students who successfully complete the class (above 70 percent correct on the problem sets and final exam) will receive a Statement of Accomplishment signed by the instructor - and those earning above 90 percent credit on the problem sets and final will earn one with distinction.


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Date: 
Saturday, February 14, 2015 to Friday, April 24, 2015
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About the Course

NOTE: For the Spring 2015 session, the course website will go live at 10:00 AM US-PST on Saturday February 14, two days before the course begins, so you have time to familiarize yourself with the website structure, watch some short introductory videos, and look at some preliminary material.

The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand 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.
 
The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Subtitles for all video lectures available in: Portuguese (provided by The Lemann Foundation), English

Course Syllabus

Instructor’s welcome and introduction
 1.  Introductory material
 2.  Analysis of language – the logical combinators
 3.  Analysis of language – implication
 4.  Analysis of language – equivalence
 5.  Analysis of language – quantifiers
 6.  Working with quantifiers
 7.  Proofs
 8.  Proofs involving quantifiers
 9.  Elements of number theory
10.  Beginning real analysis

Recommended Background

High school mathematics. Specific requirements are familiarity with elementary symbolic algebra, the concept of a number system (in particular, the characteristics of, and distinctions between, the natural numbers, the integers, the rational numbers, and the real numbers), and some elementary set theory (including inequalities and intervals of the real line). Students whose familiarity with these topics is somewhat rusty typically find that with a little extra effort they can pick up what is required along the way. The only heavy use of these topics is in the (optional) final two weeks of the Extended Course.

A good way to assess if your basic school background is adequate (even if currently rusty) is to glance at the topics in the book Adding It Up: Helping Children Learn Mathematics (free download), published by the US National Academies Press in 2001. Though aimed at K-8 mathematics teachers and teacher educators, it provides an excellent coverage of what constitutes a good basic mathematics education for life in the Twenty-First Century (which was the National Academies' aim in producing it).

Suggested Readings

There is one reading assignment at the start, providing some motivational background.
 
There is a supplemental reading unit describing elementary set theory for students who are not familiar with the material.
 
There is a course textbook, Introduction to Mathematical Thinking, by Keith Devlin, available at low cost (US base price $10.99) from Amazon, in hard copy and Kindle versions, but it is not required in order to complete the course.

For general background on mathematics and its role in the modern world, take a look at the five week survey course on mathematics ("Mathematics: Making the Invisible Visible") Devlin gave at Stanford in fall 2012, available for free download from iTunes University (Stanford), and on YouTube (12345), particularly the first halves of lectures 1 and 4.

Course Format

The Basic Course lasts for eight weeks, comprising ten lectures, each with a problem-based work assignment (ungraded, designed for group work), a weekly Problem Set (machine graded), and weekly tutorials in which the instructor will go over some of the assignment and Problem Set questions from the previous week. 

The Extended Course consists of the Basic Course followed by a more intense two weeks exercise called Test Flight. Whereas the focus in the Basic Course is the development of mathematically-based thinking skills for everyday life, the focus in Test Flight is on applying those skills to mathematics itself.

 

FAQ: 
  • Will I get a certificate after completing this class?

    The course does not carry Stanford credit. If you complete the Basic Course with more than a minimal aggregate mark, you will get a Statement of Accomplishment. If you go on to complete the Extended Course with more than a minimal mark, you will receive a Statement of Accomplishment with Distinction.

  • What are the assignments for this class?

    At the end of each lecture, you will be given an assignment (as a downloadable PDF file, released at the same time as the lecture) that is intended to guide understanding of what you have learned. Worked solutions to problems from the assignments will be described the following week in a video tutorial session given by the instructor.

    Using the worked solutions as guidance, together with input from other students, you will self-grade your assignment work for correctness. The assignments are for understanding and development, not for grade points. You are strongly encouraged to discuss your work with others before, during, and after the self-grading process. These assignments (and the self-grading) are the real heart of the course. The only way to learn how to think mathematically is to keep trying to do so, comparing your performance to that of an expert and discussing the issues with fellow students.

  • Is there a final exam for this course?

    No. The Test Flight exercise in the final two weeks of the Extended Course is built around a Problem Set similar to those used throughout the course, and your submission will be peer evaluated by other students, but the focus is on the process of evaluation itself, with the goal of developing the ability to judge mathematical arguments presented by others. Whilst not an exam, Test Flight is an intense and challenging capstone experience, and is designed to prepare students for further study of university level mathematics.

  • How is this course graded?

    In the Basic Course, grades are awarded for the weekly Problem Sets, which are machine graded. The aggregate grade is provided in the cover note to the Statement of Accomplishment, with an explanation of its significance within the class. In the Extended Course, additional grades are awarded for a series of proof evaluation exercises and for the Test Flight Problem Set (peer evaluated). The aggregate grade is provided in the cover note to the Statement of Accomplishment with Distinction, with an explanation of its significance within the class.


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Date: 
Tuesday, January 20, 2015 to Sunday, April 5, 2015
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About This Course

This is an introductory-level course in supervised learning, with a focus on regression and classification methods. The syllabus includes: linear and polynomial regression, logistic regression and linear discriminant analysis; cross-validation and the bootstrap, model selection and regularization methods (ridge and lasso); nonlinear models, splines and generalized additive models; tree-based methods, random forests and boosting; support-vector machines. Some unsupervised learning methods are discussed: principal components and clustering (k-means and hierarchical).

This is not a math-heavy class, so we try and describe the methods without heavy reliance on formulas and complex mathematics. We focus on what we consider to be the important elements of modern data analysis. Computing is done in R. There are lectures devoted to R, giving tutorials from the ground up, and progressing with more detailed sessions that implement the techniques in each chapter.

The lectures cover all the material in An Introduction to Statistical Learning, with Applications in R by James, Witten, Hastie and Tibshirani (Springer, 2013). As of January 5, 2014, the pdf for this book will be available for free, with the consent of the publisher, on the book website.   

Prerequisites

First courses in statistics, linear algebra, and computing.

FAQ: 

Do I need to buy a textbook?

No, a free online version of An Introduction to Statistical Learning, with Applications in R by James, Witten, Hastie and Tibshirani (Springer, 2013) will be available in January 2014. Springer has agreed to this, so no need to worry about copyright. Of course you may not distribiute printed versions of this pdf file.

Is R and RStudio available for free.

Yes. You get R for free from http://cran.us.r-project.org/. Typically it installs with a click. You get RStudio from http://www.rstudio.com/, also for free, and a similarly easy install.

How many hours of effort are expected per week?

We anticipate it will take approximately 3 hours per week to go through the materials and exercises.       

Instructor(s): 
Trevor Hastie
Rob Tibshirani

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Date: 
Monday, September 29, 2014 to Saturday, December 6, 2014
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Course topic: 

The goal of the course is to help you develop a valuable mental ability – a powerful way of thinking that our ancestors have developed over three thousand 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.
 
The course is offered in two versions. The eight-week-long Basic Course is designed for people who want to develop or improve mathematics-based, analytic thinking for professional or general life purposes. The ten-week-long Extended Course is aimed primarily at first-year students at college or university who are thinking of majoring in mathematics or a mathematically-dependent subject, or high school seniors who have such a college career in mind. The final two weeks are more intensive and require more mathematical background than the Basic Course. There is no need to make a formal election between the two. Simply skip or drop out of the final two weeks if you decide you want to complete only the Basic Course.

Subtitles for all video lectures available in: Portuguese (provided by The Lemann Foundation), English

FAQ: 
  • Will I get a certificate after completing this class?

    The course does not carry Stanford credit. If you complete the Basic Course with more than a minimal aggregate mark, you will get a Statement of Accomplishment. If you go on to complete the Extended Course with more than a minimal mark, you will receive a Statement of Accomplishment with Distinction.

  • What are the assignments for this class?

    At the end of each lecture, you will be given an assignment (as a downloadable PDF file, released at the same time as the lecture) that is intended to guide understanding of what you have learned. Worked solutions to problems from the assignments will be described the following week in a video tutorial session given by the instructor.

    Using the worked solutions as guidance, together with input from other students, you will self-grade your assignment work for correctness. The assignments are for understanding and development, not for grade points. You are strongly encouraged to discuss your work with others before, during, and after the self-grading process. These assignments (and the self-grading) are the real heart of the course. The only way to learn how to think mathematically is to keep trying to do so, comparing your performance to that of an expert and discussing the issues with fellow students.

  • Is there a final exam for this course?

    No. The Test Flight exercise in the final two weeks of the Extended Course is built around a Problem Set similar to those used throughout the course, and your submission will be peer evaluated by other students, but the focus is on the process of evaluation itself, with the goal of developing the ability to judge mathematical arguments presented by others. Whilst not an exam, Test Flight is an intense and challenging capstone experience, and is designed to prepare students for further study of university level mathematics.

  • How is this course graded?

    In the Basic Course, grades are awarded for the weekly Problem Sets, which are machine graded. The aggregate grade is provided in the cover note to the Statement of Accomplishment, with an explanation of its significance within the class. In the Extended Course, additional grades are awarded for a series of proof evaluation exercises and for the Test Flight Problem Set (peer evaluated). The aggregate grade is provided in the cover note to the Statement of Accomplishment with Distinction, with an explanation of its significance within the class.


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Date: 
Tuesday, September 2, 2014 to Friday, December 19, 2014
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About This Course

The ability to reason is fundamental to human beings. Whatever the discipline or discourse it is important to be able to distinguish correct reasoning from incorrect reasoning. The consequences of incorrect reasoning can be minor, like getting lost on the way to a birthday party, or more significant, for example launching nuclear missiles at a flock of ducks, or permanently losing contact with a space craft.

The fundamental question that we will address in this course is "when does one statement necessarily follow from another" --- or in the terminology of the course, "when is one statement a logical consequence of another". This is an issue of some importance, since an answer to the question would allow us to examine an argument presented in a blog, for example, and to decide whether it really demonstrates the truth of the conclusion of the argument. Our own reasoning might also improve, since we would also be able to analyze our own arguments to see whether they really do demonstrate their conclusions.

In this course you will be introduced to the concepts and techniques used in logic. We will start right from the beginning, assuming no prior exposure to this or similar material, and progress through discussions of the proof and model theories of propositional and first-order logic.

We will proceed by giving a theory of truth, and of logical consequence, based on a formal language called FOL (the language of First-Order Logic). We adopt a formal language for making statements, since natural languages (like English, for example) are far too vague and ambiguous for us to analyze sufficiently. Armed with the formal language, we will be able to model the notions of truth, proof and consequence, among others.

While logic is technical in nature, the key concepts in the course will be developed by considering natural English statements, and we will focus the relationships between such statements and their FOL counterparts. The goal of the course is to show how natural English statements and arguments can be formalized and analyzed.

Prerequisites

This course has no prerequisites except an interest in the way in which we use language to construct arguments and justify conclusions. If that interests you, then you're all set! Go sign up.

Textbook

You will need to purchase the MOOC edition of the Language, Proof and Logic courseware package. The package contains software applications that you will use to complete exercises during the course. You will also get access to the Grade Grinder, an Internet-based assessment service for these exercises.

The MOOC edition of the courseware is offered at $10, a significantly reduced cost from the regular edition, but can only be used in conjunction with this course. You can obtain it from the Language, Proof and Logic online store. We guarantee to refund the cost of the MOOC edition of the textbook up until the end of the fourth week of the course (Oct 1 2014), so you can try the course out with no risk.

If purchasing the textbook would cause financial hardship, please write a note to the Language, Proof and Logic team, to request a free copy of the courseware.

Instructors

John Etchemendy

Professor of Philosophy and Symbolic Systems

John Etchemendy has been on the faculty of Princeton University and Stanford University where he was chairman of the Department of Philosophy and is currently Provost. He has has also served as the director of the Center for the Study of Language and Information.

Dave Barker-Plummer

Senior Research Scientist

I'm Dave Barker-Plummer. Since 1995 I have managed the Openproof project's work on educational software for teaching logic at Stanford University. I have a background in Artificial Intelligence, and have taught computer science and logic at Stanford, Swarthmore College and Duke University. In my spare time I indulge my rock-star fantasies with PAN!C, a San Franciso based reggae/pop/jazz band.


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Date: 
Tuesday, June 24, 2014 to Monday, September 1, 2014
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This course aims to provide a firm grounding in the foundations of probability and statistics. Specific topics include:

1. Describing data (types of data, data visualization, descriptive statistics)
2. Statistical inference (probability, probability distributions, sampling theory, hypothesis testing, confidence intervals, pitfalls of p-values)
3. Specific statistical tests (ttest, ANOVA, linear correlation, non-parametric tests, relative risks, Chi-square test, exact tests, linear regression, logistic regression, survival analysis; how to choose the right statistical test)

The course focuses on real examples from the medical literature and popular press. Each week starts with "teasers," such as: Should I be worried about lead in lipstick? Should I play the lottery when the jackpot reaches half-a-billion dollars? Does eating red meat increase my risk of being in a traffic accident? We will work our way back from the news coverage to the original study and then to the underlying data. In the process, students will learn how to read, interpret, and critically evaluate the statistics in medical studies.

The course also prepares students to be able to analyze their own data, guiding them on how to choose the correct statistical test and how to avoid common statistical pitfalls. Optional modules cover advanced math topics and basic data analysis in R.

COURSE SYLLABUS

Week 1 - Descriptive statistics and looking at data
Week 2 - Review of study designs; measures of disease risk and association
Week 3 - Probability, Bayes' Rule, Diagnostic Testing
Week 4 - Probability distributions
Week 5 - Statistical inference (confidence intervals and hypothesis testing)
Week 6 - P-value pitfalls; types I and type II error; statistical power; overview of statistical tests
Week 7 - Tests for comparing groups (unadjusted); introduction to survival analysis
Week 8 - Regression analysis; linear correlation and regression
Week 9 - Logistic regression and Cox regression

PREREQUISITES

There are no prerequisites for this course.

Students will need to be familiar with a few basic math tools: summation sign, factorial, natural log, exponential, and the equation of a line; a brief tutorial is available on the course website for students who need a refresher on these topics.

 

FAQ: 

Can I get CME credit for this course?

This free version of the course does not offer CME credits, but there is a fee-based CME version available as well. Go to the Stanford online CME course page for more information. You are welcome to take this free version of the course before the CME course, but note that you will still need to create an account on the CME site, pay the registration fee, and complete the CME Pre-test, Post-test, Evaluation Survey, and Activity Completion Attestation statement in order to receive your credits.


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Date: 
Tuesday, June 17, 2014
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How to Learn Math is a class for learners of all levels of mathematics. It combines really important information on the brain and learning with new evidence on the best ways to approach and learn math effectively. Many people have had negative experiences with math, and end up disliking math or failing. This class will give learners of math the information they need to become powerful math learners, it will correct any misconceptions they have about what math is, and it will teach them about their own potential to succeed and the strategies needed to approach math effectively. If you have had past negative experiences with math this will help change your relationship to one that is positive and powerful.

The course will feature Jo and a team of undergraduates, as well as videos of math in action - in dance, juggling, snowflakes, soccer and many other applications. It is designed with a pedagogy of active engagement.The course will run from May/June to the end of December, 2014.

CONCEPTS

Part 1: The Brain and Math Learning.

  1. Knocking Down the Myths About Math.

    Everyone can learn math well. There is no such thing as a “math person”. This session give stunning new evidence on brain growth, and consider what it means for math learners.

  2. Math and Mindset

    When individuals change their mindset from fixed to growth their learning potential increases drastically. In this session participants will be encouraged to develop a growth mindset for math.

  3. Mistakes and Speed

    Recent brain evidence shows the value of students working on challenging work and even making mistakes. But many students are afraid of mistakes and think it means they are not a math person. This session will encourage students to think positively about mistakes. It will also help debunk myths about math and speed.

Part 2: Strategies for Success.

  1. Number Flexibility, Mathematical Reasoning, and Connections

    In this session participants will engage in a “number talk” and see different solutions of number problems to understand and learn ways to act on numbers flexibility. Number sense is critical to all levels of math and lack of number sense is the reason that many students fail courses in algebra and beyond. Participants will also learn about the value of talking, reasoning, and making connections in math.

  2. Number Patterns and Representations

    In this session participants will see that math is a subject that is made up of connected, big ideas. They will learn about the value of sense making, intuition, and mathematical drawing. A special section on fractions will help students learn the big ideas in fractions and the value of understanding big ideas in math more generally.

  3. Math in Life, Nature and Work

    In this session participants will see math as something valuable, exciting, and present throughout life. They will see mathematical patterns in nature and in different sports, exploring in depth the mathematics in dance and juggling. This session will review the key ideas from the course and help participants take the important strategies and ideas they have learned into their future.
FAQ: 

Who is this course for?

This course is designed for any learner of math and anyone who wants to improve their relationship with math. The ideas should be understandable by students of all levels of mathematics.

Parents who have children under age 13 and who think their children would benefit from some of the course materials should register themselves (i.e., parent's name, email, username) for the course. The parent may then choose to share course materials with their child at their own discretion.

What is the course structure?

The course will consist of six short lessons, taking approximately 20 minutes each. The lessons will combine presentations from Dr. Boaler and a team of undergraduates, interviews with members of the public, cutting edge research ideas, interesting visuals and films, and explorations of math in nature, sport and design.

What is the pace of the course?

The course will be self-paced, and you can start and end the course at any time in the months it is open. It is recommended that you take no more than one session a week, to allow the ideas to be processed and understood.

How will I be assessed?

There will be no formal assessment. Participants will be asked to complete a pre-and post-survey. The course will include quizzes that combine opportunities to write, work on math and reflect. These will not be graded.

Does this course carry any kind of Stanford University credit?

No.

Instructor(s): 
Jo Boaler
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