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18.05 | Spring 2014 | Undergraduate

# Introduction to Probability and Statistics

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In this section, Dr. Jeremy Orloff and Dr. Jonathan Bloom share the history behind how 18.05 was redesigned.

This course has changed dramatically in recent years in terms of both pedagogy and content. When we inherited the course, it was a traditional course with lecture and recitation, and enrollment was dwindling.

Prof. Haynes Miller has been successfully championing active learning in the MIT mathematics department for years. He had a vision for bringing active learning to 18.05, and is the Principal Investigator on the Davis Educational Grant. Over the past two years (2012–2014), with support from the Davis Educational Foundation, we have converted the course to a flipped classroom with active learning components and interactive online features.

Our particular approach to flipping the classroom involves students learning material from readings before class. During class, students are actively engaged in learning through concept questions, discussions, and board problems. Details of each of these activities can be found in A Day in 18.05, which includes a timeline, image galleries with descriptive commentary, and video of a class session. Students also have the opportunity to gain immediate feedback about their learning via problem set checkers made available through the MITx platform.

At the time we inherited the course, the math department also felt strongly that we needed to devote renewed attention to teaching probability and statistics to students who were were not mathematics majors but would need to use and understand statistics in their professional lives. To address this issue, we added a new unit on Bayesian inference to the syllabus.

In this section, we offer a brief history about our course redesign to give a sense of what an educator contemplating a similar project might expect.

### Defining our Approach to Active Learning

We kick-started the process of defining our preliminary approach to active learning by having conversations with faculty in the MIT physics department who had pioneered an active format, called Technology Enhanced Active Learning, for introductory physics. During observations of the TEAL physics class, we noticed that class sessions contained lectures that introduced content and active learning elements—such as physical experiments and problem solving at the board—that reinforced the newly learned content by getting students actively engaged in their own learning. We were inspired to adopt many of these strategies as we redesigned 18.05. We also quickly reserved the specially built TEAL classroom for our own class!

### Debating the Syllabus

Our initial plan was to lightly revise the course syllabus and to retain the original textbook, as we perceived that flipping the classroom, adding active learning components to the course, and making use of the MITx platform would present sizeable challenges. During the summer of 2012 we intended to make headway on preparing materials for the syllabus. Instead, we ended up debating the syllabus itself! The debate began when we had a brainstorming session with previous 18.05 faculty members who lamented the conceptual disconnect between the probability and statistics units in the course.

As part of the Davis grant, Sanjoy Mahajan of Olin College was retained as a consultant. Sanjoy is a strong proponent of Bayesian inference and a fierce critic of frequentist methods. He challenged the syllabus, pushing hard for a near-complete emphasis on Bayesian methods. At the same time, conversations with a number of statisticians convinced us that minimizing frequentist statistics would be a disservice to our students, who will encounter p-values and confidence intervals in their studies and professional lives. Ultimately, we decided to create a unified curriculum by incorporating Bayesian statistics. This change meant the original textbook was no longer adequate for the course. As a result, we wrote the readings ourselves. This was a huge endeavor, but it ultimately allowed us to cover precisely the content we wanted to cover, in precisely the way we wanted it covered.

### Creating a Culture of Reading Before Class

In the two remaining months, we focused on refining the class format and fleshing out readings, class slides, and problem sets for the first unit. We debated how to cultivate a culture of reading before class (a necessary condition for a successful flipped classroom), without overwhelming students with too many tasks. We decided to assign readings only for Tuesdays and Thursdays, to devote Fridays to studio sessions, and to make problem sets due on Mondays so that they would not interfere with students’ reading assignments.

We also decided to convey our expectations early and often. We reminded students of the importance of reading to success during class, and we started each class with a brief summary of the important points in the reading rather than a full lecture. Additionally, we decided to proceed to board problems after the brief introduction without first working out examples for students. Most importantly, we decided to make online reading questions count for five percent of students’ final grade!

### Planning Down to the Minute

As for materials, our productivity (and anxiety) increased with each passing week in the run-up to the start of the semester, and of course writing each lesson took much longer than we expected. With the first day upon us, we had fully prepared only for the first two weeks and we would struggle to stay just ahead of the class all semester. Nonetheless, we planned out the first class period down to the minute, writing a detailed script and rehearsing in the TEAL classroom the day before to make sure we could operate the cameras and projectors.

Our preparation paid off; the first day of class was a revelation. The energy in the room was infectious—it was unlike anything we had experienced with straight lecturing. We walked out of that first class completely exhilarated by the enthusiasm of the students.

### Making Adjustments

Despite the strong start, we had a lot left to learn about running an active learning classroom. Fortunately, this style of teaching allowed us to rapidly understand and respond to important pacing issues, such as determining the number of slides we could get through in one session (16 to 20), the optimal number of board questions to present (3), and the best sequence of clicker questions to pose (2 to 3).

The immediate and continuous feedback from students helped us make both small and large adjustments across the semester, and even within individual classes. We learned how difficult and long it took to create board questions, along with how to best support students as they solved them them; how to divide our attention among groups; how to tell when it was time to call students back from the boards; and how much summary and explanation to provide afterward.

We were surprised by how much sooner and better we got to know our students and how much students valued getting to know their teachers in this active learning setting. In the first iteration of the class it was inevitable that we would have technical glitches with the MITx system and need to make last minute additions and corrections to the readings and problem sets. MIT students usually have a very limited tolerance for such things, but we found our students to be remarkably tolerant; through the interactive format, they could see how hard we were working. They could see that we cared.

### Rethinking the Studios

For the second iteration of 18.05, Peter Kempthorne, a practicing statistician with a background in finance, joined our teaching team. Each week, the four of us (Jerry, Jon, Sanjoy and Peter) would meet to debate at length about both the statistical concepts and pedagogy underlying the course and to design projects for the coming weeks. We had a blast ganging up on one another, all in good fun. Peter does not consider himself a frequentist, but he was often driven to defend frequentist statistics against Sanjoy’s withering critiques.

With Peter’s help, we revised the readings and problem sets and refined the studios sessions. The mandatory studio sessions, held on Fridays, were designed to provide students with time outside of class to solve more complex problems. No new math content was introduced during these sessions; rather, students used and modified pre-written code to build their conceptual understanding of the week’s topics. They used software to apply statistical tools, visualize data, and explore concepts through simulations. We decided to switch studio work from Matlab to R, a free and popular programming language for statistics that is easy to install and start using. Additionally, rather than teaching syntax during the studio sessions (which was disastrously boring in year one), we created basic text-based tutorials, which interleaved example commands and scripts with explanations, and offered additional evening sessions to help students with less programming experience get up to speed.

The second year studios were a big improvement over the first year, but we were not yet satisfied with the energy level students exhibited while participating in them. Some studios worked well, but in others we still spent too much time lecturing about code and not enough in active learning. Another problem was that although the studios were designed to deepen students’ understanding of the material, they were not well aligned with the exams we gave. In future iterations of 18.05 we will continue to adjust the studios and aim to align them with our summative assessments.

### Closing Thoughts

Changing everything at once (i.e. adding the Bayesian unit, converting from lecture to active learning, adding interactive online features to the course, and writing the readings ourselves) involved an enormous amount of work, and we wouldn’t recommend it. However, having survived it, we think 18.05 is better for it. Co-teaching was both enormously fun and beneficial. We gave each other daily feedback on our teaching and challenged each other to defend our ideas about teaching, students and statistics. Our active learning format drove a rapid evolution of teaching and goals since interactions with students catalyzed a lot of self-assessment, which led us to refine our explanations. Our active learning format also allowed us to form closer relationships with more students than was ever possible with straight lecturing. We recommend educators give it a try!

## Course Info

##### Instructors

- Dr. Jeremy Orloff
- Jonathan Bloom

##### Departments

- Mathematics

##### As Taught In

Spring 2014

##### Level

Undergraduate

##### Topics

- Mathematics
  - Discrete Mathematics
  - Probability and Statistics

##### Learning Resource Types

assignment_turned_in Problem Sets with Solutions
grading Exams with Solutions
notes Lecture Notes
co_present Instructor Insights
theaters Other Video

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