It’s Only Logical to Take This Math

Image of a 4x4 square puzzle with different numbers in each square, one square empty, and spelling out the course number 6 (blank) 4 2 on the diagonal.,

6.042 serves as an introduction to discrete mathematics, probability, and mathematical thinking for computer scientists. (Image by OpenCourseWare, based on an image by Nick Matsakis.)

By Joe Pickett, OCW Publication Director

If you’re interested in computer programming, you might be familiar with OCW’s extensive resources in the discipline, ranging from our introductory classes to the graduate courses offered by the Department of Electrical Engineering and Computer Science.

Sooner or later, students must face the reality that to do serious programming, design algorithms, or become a software engineer, they need to know some math.

What kind of math?

The kind presented in 6.042J Mathematics for Computer Science, whose Spring 2015 course has just been published on OCW. Taught by Professors Albert Meyer and Adam Chlipala, the course has an extensive array of content covering a wide variety of mathematical topics. 

Tutorials, Textbook, and Much More

Chief among these resources are 130 short tutorial videos created by Professor Meyer, who drew on more than a decade of teaching experience in the subject to create them. Ever an educational innovator, and one of the first instructors to flip his classroom at MIT, Professor Meyer put his Spring 2015 course content on Residential MITx, MIT’s version of the Open edX platform. MIT students watched the videos via the online platform before coming to class, where they worked in teams to solve problems.

Screengrabs of video player, with interactive text transcript below, and interactive questions, with horizontal navigation bar above.

A sequence of two OCW pages from 6.042J. Left: Tutorial video, with interactive text transcript. Right: A pair of interactive questions that come after the video.

Now on OCW, the video tutorials are presented in the same sequential fashion, interspersed with brief interactive questions that allow you to check your understanding, over four learning units.

The in-class questions are also included on the course site, in a table showing the relevant video tutorial for each one, so you can get the same background that MIT students had when they were asked to face these challenges.

The entire course textbook is also available for free and can be downloaded.

All the non-video resources for the course are gathered together in a handy course index, so you can see in one place which resource aligns with which for each week of the semester, including the problem sets and exams.

Other Versions, Other Resources

The Spring 2015 course site is but the latest version of 6.042J to be published on OCW. Math enthusiasts can find more excitement in a Fall 2010 version taught by Professor Tom Leighton and Dr. Marten van Dijk (which features full video lectures), and a Fall 2005 version, co-taught by Professor Meyer and Professor Ronitt Rubenfield, that has solutions for its in-class problems, problem sets, and exams.

It all adds up to a very logical conclusion: OCW’s suite of 6.042J course sites is awesome!

Actively Engaged Students: The Holy Grail of Teaching

Drawing of a king meeting a giant who is roasting a pig on a spit.

King Arthur finds a giant roasting a pig. (Illustration is in the public domain from Roman de Brut by Wace, 2nd quarter of the 14th century. Courtesy of the British Library.)

By Joe Pickett, OCW Publication Director

One of the most poorly kept secrets among people who advocate flipping the classroom in the sciences and engaging students in active learning is that the humanities have been flipping the classroom and practicing active learning for decades. So it’s not an exaggeration to say that the sciences have been taking a lesson from the humanities in recent years.

It’s well known that humanities classes often consist of discussions rather than lectures. Students read the text at home, and then the instructor leads a discussion in class.

The Quest for a Lively Discussion

But just as there are good and bad lectures, so too with discussions. How does a humanities instructor ensure that a discussion isn’t just a free-for-all? Or a series of painful silences?

Enter into the lists** Professor Arthur Bahr, who has just published two literature courses on OCW.

To engage students, he has them help shape what happens in class. In the Instructor Insights of his This Course at MIT page for 21L.460 Medieval Literature: Legends of Arthur, Professor Bahr explains how he had students, working in groups of four or five, identify questions and passages that might form the basis for discussion. This priming of the pump was essential to making the class a success:

“ . . . if they were not prepared, it was not going to be a good class. I told students they were responsible for making good discussions possible, and they needed to take ownership of their learning.”

Professor Bahr typically pointed out connections between these student submissions to frame and direct the discussions. So the students were engaged to begin with, and they kept their instructor always on his toes.

In a variation of this technique, students submit questions by email before class (the submissions are required and factored into final grades). Professor Bahr is then able to focus on things that students find interesting. This strategy has the side benefit of helping the more reticent students participate in the discussion because it allows “shy students to know in advance that their questions [are] going to be the subject of conversation.”

Posters of the Round Table

If the humanities have given something to the sciences, the sciences have also lent something to the humanities. After talking with a geneticist colleague, Professor Bahr decided to offer his students the option of creating a poster for their final project, in the same manner that science reseachers create posters for conferences. So many students took up this challenge that he held a public poster session showcasing their work.

Mock Tests Make for Heroic Achievements

In 21L.705 Major Authors: Old English and Beowulf, Professor Bahr took on the ambitious goal of teaching a medieval literary classic in its original language, all in a single semester.  A key part of his instruction was another lesson learned from the sciences: low-stakes assessments in the form of mock exams.

The exams “gave [students] a sense of how difficult the real assessments would be and how they needed to pace themselves when taking the exams. Additionally, they provided an opportunity for me to model the thought processes involved in completing complex tasks, such as sight translations.” As an additional benefit, “I knew that if all of the students performed poorly on a particular item, I needed to reassess how I was teaching that concept.”

Maybe the sciences and humanities have more in common than we might think.

** “The lists” are barriers used to designate the tournament area where medieval knights jousted.

Blend your classroom with edX

Are you an educator who wants to enhance your classroom with educational technology? You may be interested in Blended Learning with edX, an online course actually created by the edX team (instead of by a university partner, like MITx).

The course intro video showcases TEAL, MIT’s Technology Enabled Active Learning classroom. We’ve published a few MIT math and physics courses that use TEAL on OCW:

In particular, the This Course at MIT section of 18.05 goes into depth about TEAL and how it’s used to teach math in an active way. You’ll see a diagram of the setup of the classroom, and a photograph of the students working. Scroll down to A Day in 18.05 to see a lecture video and a timeline that walks you through each in-class activity. We’ve written about active learning in 18.05 on the blog before, if you’d like even more information.

Three evolving thoughts about flipped learning (The Chronicle of Higher Education)

In the latest post in his excellent series on teaching in a flipped classroom, Robert Talbert describes how recent experience has led to three important shifts in his thinking about pre-class and in-class work. Here I’ll simply quote Robert’s main points; his full post is well worth reading, backing up each point with illuminating details.

…Recently I had time to reflect on how I’m implementing flipped learning in my classes, and I noticed that some of my thoughts on flipped learning have evolved over the last few years, including some breaks from things I’ve written here on the blog. Here are three of those thoughts that stood out for me.

What I used to think: Pre-class activity in a flipped learning model is about mastering content-oriented instructional objectives.

What I think now: Pre-class activity is for generating questions 


What I used to think: Students in a flipped classroom need to have some graded measure of accountability when they arrive at class (an entrance quiz, etc.) to ensure that they do the pre-class work.

What I think now: Accountability doesn’t have to look like a quiz …


What I used to think: The in-class instruction in a flipped class should focus primarily on active student work with little to no lecture.

What I think now: The in-class instruction should focus on two things: Answering questions, and engaging students in high-level tasks – and lecture can play an important role in both …

Read the full article here.

Probability and Statistics in a Very Active Setting

Image of a video player followed by a segmented horizontal bar.

A clickable video timeline showcases the different active learning components in a typical class session of 18.05 Introduction to Probability and Statistics.

OCW has just published a new multifaceted course, 18.05 Introduction to Probability and Statistics, that innovates on numerous fronts to engage students in active learning.

“We have converted the [traditional] course to a flipped classroom with active learning components and interactive online features,” say the lead instructors, Dr. Jerry Orloff and Dr. Jonathan Bloom.

In fact, the traditional course was completely rethought. In the past, students had complained that there seemed a disconnect between the units on probability and frequentist statistics. So after much discussion, Drs. Orloff and Bloom revamped the syllabus and rewrote all the course readings to create a unified curriculum:

“We wanted students to come away with a deeper understanding of the meaning of frequentist statistics, which had been the focus of the traditional course . . . [And] we wanted to introduce students to Bayesian and computational statistics, central tools in modern statistical practice.”

Beyond Blended

It’s hard to imagine a more blended learning experience. The course jettisons the traditional lecture room, along with the division between lecture and problem-solving sessions, and employs a TEAL (Technology Enabled Active Learning) classroom with stations for where teams of students work together. Each regular class session combines short lectures, online concept questions that students answered using clickers (an electronic response system), student discussions, team problem solving at the board, and demonstrations or simulations. Once a week the instructors held a “Studio” session in which students used R, a free software environment for statistical computing and graphics, to attack longer problems involving computation, simulations, and visualizations.

All course readings were hosted on the MITx platform, where students were required to answer questions about assigned readings before they came to class.

Students submitted problem sets on paper (which were examined and graded by the course staff), but they also had access to an interactive problem set checker that let them check their answers while they were working on them.

Rich Content, Dynamic Checker, Dramatic Timeline

The OCW course site has an abundance of content: all the readings, interactive reading questions, slides used in class, problem sets and solutions, exams and solutions, plus the slides, solutions, and supporting files for the Studio sessions. The interactive problem checker is also available on OCW for each problem–a first for OCW.

The OCW site has a robust This Course at MIT page which explains the context in which the class was taught and highlights the instructors’ reflections on almost every aspect of their new teaching experience. A timeline for one class session, called A Day in 18.05, shows how the different activities that took place during the session. Each colored box in the timeline is linked to a timecode in a video for that class, showing each activity took place. Class activities are also illustrated by image galleries annotated with comments by the instructors.

But a description can’t do justice to the course site. 18.05 is an amazing and innovative course taught by two dedicated and creative teachers. It’s one of the most interesting and thought-provoking courses ever published on OCW.

Joe Pickett, OCW Publication Director

A MOOC sees its greatest impact in the classroom at MIT (MIT News)

Banner image for VJx: a painting of a ship.

Japanese woodblock print of American warship, circa 1854. (Image courtesy of the Nagasaki Prefecture.)


MIT Professor Shigeru Miyagawa has been a long-time champion of online education and open sharing.  Now we can add “flipped classroom advocate” to that list.

This fall, Professor Miyagawa simultaneously taught two versions of his course Visualizing Japan to two very different audiences. He co-taught the massive online open course (MOOC) VJx on edX, and at the same time taught the residential course 21F.027 for MIT students.

The edX MOOC, which ran for 6 weeks, was a marked success. It had a completion rate of 13 percent — double the normal rate for a MOOC — and 97.5 percent of the learners said that they were satisfied to extremely satisfied with the course.

But the MIT classroom experience is perhaps the bigger story. While the MOOC was running, the MIT residential course operated largely in flipped mode. Students were assigned the MOOC video lectures and quizzes to complement their classroom work.

For both the students in class and for Miyagawa it became clear early on that something was very different. On the first day of the module “Black Ships and Samurai,” Miyagawa was set to give the lecture he had prepared with a PowerPoint presentation. Shortly into the lecture he asked the class, “What happened in 1868?” He was expecting a couple of students to raise their hands, but everyone seemed to know that this was the beginning of Meiji Restoration that put Japan on the road to modernization.

Miyagawa abandoned his lecture and pressed on with more questions. He was pleasantly surprised that most of the students were not only able to answer the questions, but also willing to engage him and the other students in discussion. “When I finished the class without showing even a single slide from my PowerPoint, I could only ask, what happened?” he remarks.

What happened was a transformative experience for both the students and the professor. Read the full story here.

Fold, Then Flip

This curved-crease sculpture, created for the opening of the National Museum of Mathematics, demonstrates the intersection of origami, design, and mathematics that is at the heart of this course. (Erik Demaine and Martin Demaine.)

This curved-crease sculpture, created for the opening of the National Museum of Mathematics, demonstrates the intersection of origami, design, and mathematics that is at the heart of the course 6.849 Geometric Folding Algorithms. (Erik Demaine and Martin Demaine.)

Quiz question: What do medical stents, car air bags, origami sculptures, robotic arms, satellite solar arrays, and morphing computer graphics have in common?

Answer: Folding, of course!

Oh, yes, and algorithms, too!

OCW has just published 6.849 Geometric Folding Algorithms: Linkages, Origami, Polyhedra, a course by Professor Erik Demaine, which explores the universe of folding in one dimension (linkages), two dimensions (origami), and three dimensions (polyhedra).

The course is itself folded in the sense that Professor Demaine flipped his classroom using videotaped “inverted lectures” that students watched outside class.  Time in the classroom was devoted to answering student questions and to pursuing deeper investigations of the multifaceted course material.  Professor Demaine also invited students to optional “open-problem sessions,” at which students worked on cutting-edge problems in a collaborative spirit espousing the belief that “there are no bad ideas.”

The course is a revealing example of the ways that video lectures can actually free up the instructor to engage with students in a more hands-on, thought-provoking way. The course publication on OCW includes all of the video lectures (from Fall 2010) and videos of the class sessions (from Fall 2012) as well.

Professor Demaine explains his thinking behind the course in his This Course at MIT page. The class sessions were shaped by forms that the students filled out online after watching the lectures, so Professor Demaine could prepare for the sessions to address student questions and incorporate the latest research, often with stunning computer graphics and animations.

Professor Demaine liked this format so much he flipped another course he teaches, 6.851 Advanced Data Structures. As he says in the course description, “You interact with data structures even more often than with algorithms (think Google, your mail server, and even your network routers). In addition, data structures are essential building blocks in obtaining efficient algorithms.”

A champion of open sharing, Professor Demaine published this course with full video lectures and a reflective This Course at MIT page on OCW. This allowed him to share a number of lessons he learned from his first flipped experience and to outline the adjustments and improvements he made for his second.

You can see many other OCW’s This Course at MIT pages here. They are part of OCW’s innovative Educator project.

— Joe Pickett, OCW Publication Director