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# Mathematics

Go to Course## Course Overview

### You will learn:

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## COURSE SYLLABUS

### Introduction to quantum mechanics

### Schroedinger’s wave equation

### Getting "quantum" behavior

### Quantum mechanics of systems that change in time

### Measurement in quantum mechanics

### Writing down quantum mechanics simply

### The hydrogen atom

### How to solve real problems

## PREREQUISITES

## COURSE STAFF

### David Miller

## FREQUENTLY ASKED QUESTIONS

### Do I need to buy a textbook?

### How much of a time commitment will this course be?

### Does this course carry any kind of Stanford University credit?

### Will I get a Statement of Accomplishment?

Go to Course## ABOUT THIS COURSE

## REQUIREMENTS

## COURSE INSTRUCTORS

### Jeff Zwiers

###

### Phil Daro

###

### Shelbi Cole

## FREQUENTLY ASKED QUESTIONS

### What web browser should I use?

### Any additional textbooks or software required?

### Will I receive any completion document upon finishing the course?

### How many professional development hours is this course equivalent to?

### Can my work in this course be accepted by a school district??

Go to Course## About the Course

## Instructor

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## COURSE OVERVIEW

### You Will Learn

## WHO SHOULD ENROLL?

## COURSE STRUCTURE

## TUITION

## PROFESSIONAL DEVELOPMENT HOURS

## QUESTIONS?

## Important Note: Course Postponed

## About the Course

This course is an introduction to General Game Playing (GGP). Students will get an introduction to the theory of General Game Playing and will learn how to create GGP programs capable of competing against other programs and humans.## Recommended Background

Go to Course## About the Course

## FAQ

Go to Course## About the Course

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

## Recommended Background

## Suggested Readings

## Course Format

## FAQ

## Pages

Date:

Tuesday, October 10, 2017 to Friday, December 15, 2017

Course topic:

This course is designed to introduce students to advanced MATLAB features*, syntaxes, and toolboxes not traditionally found in introductory courses, using in-class examples, demos, and homework assignment involving topics from scientific computing. The MATLAB topics covered are advanced graphics (2D/3D plotting, graphics handles, publication quality graphics, animation), MATLAB tools (debugger, profiler), code optimization (vectorization, memory management), object-oriented programming, compiled MATLAB (MEX files and MATLAB coder), interfacing with external programs, toolboxes (optimization, parallel computing, symbolic math, PDEs). Scientific computing topics will include: numerical linear algebra, numerical optimization, ODEs, and PDEs.

- Advanced graphics and MATLAB tools
- Code optimization and object-oriented programming
- MATLAB interfacing with external programs

*Participants will acquire a MATLAB license for the duration of the course, courtesy of MathWorks.

Prerequisite(s): None

Registration Fee: Free

Date:

Tuesday, October 3, 2017 to Friday, December 15, 2017

Course topic:

This 9 week course aims to teach quantum mechanics to anyone with a reasonable college-level understanding of physical science or engineering. Quantum mechanics was once mostly of interest to physicists, chemists and other basic scientists. Now the concepts and techniques of quantum mechanics are essential in many areas of engineering and science such as materials science, nanotechnology, electronic devices, and photonics. This course is a substantial introduction to quantum mechanics and how to use it. It is specifically designed to be accessible not only to physicists but also to students and technical professionals over a wide range of science and engineering backgrounds.

How quantum mechanics is important in the everyday world, the bizarre aspects and continuing evolution of quantum mechanics, and how we need it for engineering much of modern technology.

Getting to Schroedinger’s wave equation. Key ideas in using quantum mechanical waves — probability densities, linearity. The "two slit" experiment and its paradoxes.

The "particle in a box", eigenvalues and eigenfunctions. Mathematics of quantum mechanical waves.

Time variation by superposition of wave functions. The harmonic oscillator. Movement in quantum mechanics — wave packets, group velocity and particle current.

Operators in quantum mechanics — the quantum-mechanical Hamiltonian. Measurement and its paradoxes — the Stern-Gerlach experiment.

A simple general way of looking at the mathematics of quantum mechanics — functions, operators, matrices and Dirac notation. Operators and measurable quantities. The uncertainty principle.

Angular momentum in quantum mechanics — atomic orbitals. Quantum mechanics with more than one particle. Solving for the the hydrogen atom. Nature of the states of atoms.

Approximation methods in quantum mechanics.

The course is approximately at the level of a first quantum mechanics class in physics at a third-year college level or above, but it is specifically designed to be suitable and useful also for those from other science and engineering disciplines.

The course emphasizes conceptual understanding rather than a heavily mathematical approach, but some amount of mathematics is essential for understanding and using quantum mechanics. The course presumes a mathematics background that includes basic algebra and trigonometry, functions, vectors, matrices, complex numbers, ordinary differential and integral calculus, and ordinary and partial differential equations.

In physics, students should understand elementary classical mechanics (Newton’s Laws) and basic ideas in electricity and magnetism at a level typical of first-year college physics. (The course explicitly does not require knowledge of more advanced concepts in classical mechanics, such as Hamiltonian or Lagrangian approaches, or in electromagnetism, such as Maxwell’s equations.) Some introductory exposure to modern physics, such as the ideas of electrons, photons, and atoms, is helpful but not required.

The course includes an optional and ungraded “refresher” background mathematics section that reviews and gives students a chance to practice all the necessary math background background. Introductory background material on key physics concepts is also presented at the beginning of the course.

David Miller is the W. M. Keck Foundation Professor of Electrical Engineering and, by Courtesy, Professor of Applied Physics, both at Stanford University. He received his B. Sc. and Ph. D. degrees in Physics in Scotland, UK from St. Andrews University and Heriot-Watt University, respectively. Before moving to Stanford in 1996, he worked at AT&T Bell Laboratories for 15 years. His research interests have included physics and applications of quantum nanostructures, including invention of optical modulator devices now widely used in optical fiber communications, and fundamentals and applications of optics and nanophotonics. He has received several awards and honorary degrees for his work, is a Fellow of many major professional societies in science and engineering, including the Royal Society of London, and is a member of both the National Academy of Sciences and the National Academy of Engineering in the US. He has taught quantum mechanics at Stanford for more than 15 years to a broad range of students ranging from physics and engineering undergraduates to graduate engineers and scientists in many disciplines.

You do not need to buy a textbook; the course is self-contained. My book “Quantum Mechanics for Scientists and Engineers” (Cambridge, 2008) is an optional additional resource for the course. It follows essentially the same syllabus, has additional problems and exercises, allows you to go into greater depth on some ideas, and also contains many additional topics for further study.

You should expect this course to require 7 – 10 hours of work per week.

No.

Yes, students who score at least 70% will pass the course and receive a Statement of Accomplishment. Students who score at least 90% will receive a Statement of Accomplishment with distinction.

We recommend taking this course on a standard computer using Google Chrome as your internet browser. We are not yet optimized for mobile devices.

Date:

Tuesday, October 3, 2017 to Tuesday, March 6, 2018

Course topic:

New learning standards for math emphasize the importance of developing students’ abilities to reason and articulate reasoning across a variety of topics in math. Starting October 3, Jeff Zwiers, Phil Daro, and Shelbi Cole will offer a free online professional development course, Integrating Language Development and Content Learning in Math: Focus on Reasoning, to help teachers improve their design and development of learning activities that foster mathematical reasoning and its language.

The sessions will zoom in on different language modes and how they can foster students’ abilities to reason and describe their reasoning. Sessions focus on listening, speaking, whole class conversations, small group and pair conversations, reading, and writing. We provide a design tool and plenty of practice using it in order to strengthen the language development potential of a wide range of math teaching activities. The course also includes instructional activities and routines to be used across lessons and units to meet the linguistic and cultural needs of English learners and other students who struggle with the language demands of learning math.

This course consists of seven online sessions, with three weeks or so between each session. Each session includes expert video screencasts, reflection prompts, classroom video clips, readings, resources, and assignments that will prompt participants to use what they learn in their classrooms and reflect on student language use. Participants are free to complete the session tasks at their own pace as long as they finish them within the allotted time. Teachers (K-12) will learn how:

- Authentic communication in the math classroom accelerates both language and content development
- To model and strengthen activities for reasoning and its language
- To integrate and leverage different language modes in one activity
- To formatively assess language and reasoning to improve instruction

There is no pre-requisites for the course. Classroom teachers and instructional coaches from grades K to 12 who teach math are welcome and encouraged to take this course together with their colleagues.

Dr. Jeff Zwiers is the Director of Professional Development at Understanding Language at Stanford Graduate School of Edcuation. He has worked for more than fifteen years as a professional developer and instructional mentor in urban school settings, emphasizing the development of literacy, thinking, and academic language for linguistically and culturally diverse students. He has published books and articles on reading, thinking, and academic language. His most recent book is *Academic Conversations: Classroom Talk That Fosters Critical Thinking and Content Understandings*. His current work focuses on developing teachers’ core practices for teaching academic language, comprehension of complex texts, and oral communication skills across subject areas. He holds a BA in Psychology from Stanford, an MAT in Language and Reading from Stanford, and a PhD in Education from USF.

Dr. Phil Daro is a mathematics educator who most recently co-directed the development of the Common Core State Standards for mathematics. He has also directed large-scale teacher professional development programs for the University of California including the California Mathematics Project and the American Mathematics Project. He is Site Director of the Strategic Education Research Partnership (SERP) at the San Francisco Unified School District. Steering Committee, Math Work Group (chair), and District Engagement Committee.

Dr. Shelbi Cole is a Senior Content Specialist on the Mathematics team at Student Achievement Partners. Prior to joining the team, Shelbi was the Director of Mathematics for the Smarter Balanced Assessment Consortium. She was a high school mathematics teacher and has worked on a range of projects in curriculum development, teacher professional learning, and pre-service teacher education. Shelbi holds bachelor’s and master’s degrees in Mathematics Education and a doctoral degree in Educational Psychology from the University of Connecticut.

To optimize your learning experience, please use the current versions of Chrome and Firefox.

No.

Participants who complete the course requirement will be eligible to receive a Statement of Accomplishment after the course ends.

We estimate that the coursework is equivalent to approximately 40 professional development hours to individuals who complete the course and gain a Statement of Accomplishment.

Stanford University makes no representations that participation in the course, including participation leading to a record of completion, will be accepted by any school district or other entity as evidence of professional development. Participants are solely responsible for determining whether participation in the course, including obtaining a record of completion, will be accepted by a school district, or any other entity, as evidence of professional development coursework.

Go to Course## Overview

## Topics Include

## Instructors

## Units

## Prerequisites

Course topic:

*Application and fee apply.*

Integrating computation, visualization and programming with MATLAB is a powerful approach to model and control systems. This course builds on the fundamentals of calculus to explore vector analysis techniques that are essential for engineers. Using examples drawn from various engineering fields, it introduces differential and integral vector calculus and linear algebra to analyze the effects of changing conditions on a system.

- Analytic geometry in space
- Green's, divergence and Stokes' theorems
- Integrals in Cartesian, cylindrical and spherical coordinates
- Lagrange multipliers
- Matrix operations
- Partial derivatives
- Unconstrained maxima and minima

- Hung Le
*Lecturer*,*Institute for Computational & Mathematical Eng.*

5.0

10 units of AP credit (Calc BC with 4 or 5, or Calc AB with 5), or Math41 and 42.

Go to Course## About this course

## Who is this class for

## Instructor

### Matthew O. Jackson

Course topic:

Learn how to model social and economic networks and their impact on human behavior. How do networks form, why do they exhibit certain patterns, and how does their structure impact diffusion, learning, and other behaviors? We will bring together models and techniques from economics, sociology, math, physics, statistics and computer science to answer these questions. 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. You can find a more detailed syllabus here: http://web.stanford.edu/~jacksonm/Networks-Online-Syllabus.pdf You can find a short introductory videao here: http://web.stanford.edu/~jacksonm/Intro_Networks.mp4

This course starts every four weeks. The next session begins October 10.

The course is aimed at people interested in researching social and economic networks, but should be accessible to advanced undergraduates and other people who have some 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). Beyond those concepts, the course is self-contained.

Professor, Economics

Date:

Monday, September 18, 2017

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.

Date:

Tuesday, June 7, 2016

Course topic:

Explore the new research ideas on mathematics learning and student mindsets that can transform students' experiences with math. Whether you are a teacher preparing to implement the new Common Core State Standards, a parent wanting to give your children the best math start in life, an administrator wanting to know ways to encourage math teachers or another helper of math learners, this course will help you. The sessions are all interactive and include various thinking tasks to promote active engagement - such as reflecting on videos, designing lessons, and discussing ideas with peers.

- New pedagogical strategies
- An understanding of high quality math tasks
- Questions to promote understanding
- Messages to give students
- Inspirational messages from educational thought-leaders

Teachers of math (K-12) or other helpers of students, such as parents. An accompanying course for students is also available here.

The course comprises 8 sessions, each with videos and activities that require approximately 1.5 to 3 hours to complete.

What is the course pace?

The course will be self-paced, you can start and end the course at any time in the months it is open.

$99 per person

Group enrollment is available at $99/person by purchase order, company check, or wire transfer by emailing stanford-educ@stanford.edu.

A discounted rate is available for groups of 150 or more, at $75 per person. Please contact stanford-educ@stanford.edu for more information on groups of 150 people or more

In the first run of the course many school districts in the US gave 16 professional development hours to the teachers who took the course – which means finishing the course and completing most of the assignments. Stanford University makes no representations that participation in the course, including participation leading to a record of completion, will be accepted by any school district or other entity as evidence of professional development.

Participants are solely responsible for determining whether participation in the course, including obtaining a record of completion, will be accepted by a school district, or any other entity, as evidence of professional development coursework.

Please contact stanford-educ@stanford.edu or call 650-263-4144

Instructor(s):

Jo Boaler

Course topic:

The latest offering of the course on General Game Playing is postponed. The next scheduled session will take place in the Spring of 2017.

Although the MOOC will not be running this Spring, materials will be made available on the website for the currently running Stanford version of the course. Just click on the link shown below. The materials can be found via the links at the top of the page. You should be able to access everything except the Piazza newsgroup.

General game players are computer systems able to play strategy games based solely on formal game descriptions supplied at "runtime". (In other words, they don't know the rules until the game starts.) Unlike specialized game players, such as Deep Blue, general game players cannot rely on algorithms designed in advance for specific games; they must discover such algorithms themselves. General game playing expertise depends on intelligence on the part of the game player and not just intelligence of the programmer of the game player.

GGP is an interesting application in its own right. It is intellectually engaging and more than a little fun. But it is much more than that. It provides a theoretical framework for modeling discrete dynamic systems and for defining rationality in a way that takes into account problem representation and complexities like incompleteness of information and resource bounds. It has practical applications in areas where these features are important, e.g. in business and law. More fundamentally, it raises questions about the nature of intelligence and serves as a laboratory in which to evaluate competing approaches to artificial intelligence.

This course is an introduction to General Game Playing (GGP). Students will get an introduction to the theory of General Game Playing and will learn how to create GGP programs capable of competing against other programs and humans.

Students should be familiar with Symbolic Logic and should be able to read and understand program fragments written in a modern programming language. This background is sufficient for understanding the presentation and for configuring players to compete in competitions (using software components provided by the instructors). Students who wish to modify the standard components or who wish to build their own players also need the ability to develop programs on their own. This latter ability is desirable but not required.

Date:

Monday, September 28, 2015 to Saturday, November 21, 2015

Course topic:

Logic is one of the oldest intellectual disciplines in human history. It dates back to the times of Aristotle; it has been studied through the centuries; and it is still a subject of active investigation today.

This course is a basic introduction to Logic. It shows how to formalize information in form of logical sentences. It shows how to reason systematically with this information to produce all logical conclusions and only logical conclusions. And it examines logic technology and its applications - in mathematics, science, engineering, business, law, and so forth.

The course differs from other introductory courses in Logic in two important ways. First of all, it teaches a novel theory of logic that improves accessibility while preserving rigor. Second, the material is laced with interactive demonstrations and exercises that suggest the many practical applications of the field.

**Will I get a statement of accomplishment after completing this class?**Yes. Participants who successfully complete the course will receive a statement of accomplishment signed by the instructor.

**What is the format of the class?**The class consists of videos, notes, and a few background readings. The videos include interactive demonstrations and exercises. There are also standalone quizzes that are not part of video lectures. Workload: one to two hours of video content per week.

**What should I know to take this class?**The course has no prerequisites beyond high school mathematics. You should be comfortable with symbolic manipulation techniques, as used, for example, in solving simple algebra problems. And you need to understand sets, functions, and relations. However, that's all. If you have this background, you should be fine.

**Do I need to buy any textbooks?**None is required, as the course is self-contained.

Date:

Monday, September 21, 2015 to Friday, November 27, 2015

Course topic:

NOTE: For the Fall 2015 session, the course website will go live at 10:00 AM US-PST on Saturday September 19, 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.

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

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

10. Beginning real analysis

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).

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 (1, 2, 3, 4, 5), particularly the first halves of lectures 1 and 4.

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 (1, 2, 3, 4, 5), particularly the first halves of lectures 1 and 4.

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.

- 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.