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18.05 | Spring 2014 | Undergraduate

# Introduction to Probability and Statistics

In this section, Dr. Jeremy Orloff and Dr. Jonathan Bloom describe how they have changed the course, in terms of both pedagogy and content.

This course has changed dramatically in recent years in terms of both pedagogy and content.

We first taught the course in Spring 2012. When we inherited the course, it was a traditional course with lecture and recitation, and the enrollment was dwindling.

The math department felt strongly that we needed to devote renewed attention to teaching probability and statistics to students who are not mathematics majors but will need to use and understand statistics in their professional lives. Many such students had been taking statistics courses run by other departments or 18.440 Probability and Random Variables, which is a more theoretical course that doesn’t cover statistics.

Over the past two years (2012–2014), with support from a grant from the Davis Educational Foundation, we have converted the course to a flipped classroom with active learning. The syllabus has evolved significantly, most notably with a new unit on Bayesian inference.

### Converting to a Flipped Classroom with Active Learning

Prof. Haynes Miller has been championing active learning in the MIT mathematics department for years with some success. He had the vision for bringing active learning to 18.05, and he is the Principal Investigator on the Davis Educational Grant.

Our particular approach to flipping the classroom involved students learning material from readings before class. During class, students were 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 a video of a class session.

The Davis Foundation and the MIT Department of Mathematics, along with the rest of MIT and much of the world, are interested in online learning and how the internet can be leveraged to improve university learning. To this end, we incorporated interactive reading questions and problem set checkers on the edX platform into our course.

### Unifying the Curriculum by Adding a Bayesian Statistics Unit

The course used to consist of a unit on probability and a unit on frequentist statistics, which included standard concepts such as confidence intervals and p-values. We heard from previous instructors that students felt a strong disconnect between the units; in particular they felt that in the statistics unit they were just learning computational procedures.

We revised the curriculum with several goals in mind. First, we wanted to unify the course by building a much stronger connection between the probability and statistics portions. Second, we wanted students to come away with a deeper understanding of the meaning of (classical) frequentist statistics, the focus of the traditional course. Third, we wanted to introduce students to Bayesian and computational statistics, central tools in modern statistical practice.

As far as we can tell, our syllabus is unique. In brief, we start with the basics of probability with an emphasis on Bayes’ Theorem. We then transition to statistics by analyzing data when all the pieces of Bayes’ Theorem are known; this is still pure probability. Finally we teach both Bayesian and frequentist statistical inference as methods for dealing with the case when the prior is unknown: if you’re a Bayesian, you use a subjective prior; if you’re a frequentist, you do the best you can using only the likelihood.

Taught in this way, Bayesian and frequentist statistics are mutually reinforcing. First, by emphasizing that both are responses to the lack of a known prior, students see clearly that statistical inference is an art involving practical compromises rather than pure deductive mathematics. Second, students intuitively assign a Bayesian interpretation to frequentist constructs such as p-values and confidence intervals. By first showing them Bayesian inference we give them a solid base to avoid such misinterpretations.

We also believe that Bayesian statistics is important because of its exploding role in applications; much of machine learning, big data, and cutting edge work on genetics and neuroscience is done with Bayesian methods.

With the new Bayesian statistics unit, we have one-third more material than the course used to have. In practice, we’ve had no problem covering all this material, and students are doing at least as well on exams as they did in the past.

### Changes for the Instructors

All these changes took an unbelievably large amount of work, partially because we implemented everything at once: adding the Bayesian unit, converting from lecture to active learning, adding interactive online features, and writing the readings ourselves. In retrospect, we wouldn’t recommend changing so much at once, though we’re glad it’s done.

We did find it to be less work to teach the class once we were in the classroom. We spent a lot of time talking with students, which was energizing.

### Flipping the Dice: An Active-Learning, Technology-Enhanced, Up-to-Date 18.05

On April 7, 2014, Dr. Orloff and Dr. Bloom gave the following talk on the transformation of 18.05 at the HHMI Education Group Seminar. A summary is posted on the OEIT website.

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> "The peer instruction happens because the students have these discussions at their tables because they’re working together in groups… You see a lot of students explaining [the problems] to each other and students being able to look over at the next group if they’re stuck."

— DR. JONATHAN BLOOM

This video contains the seminar talk that was given on April 7, 2014 about the transformation of 18.05.

Slide deck for HHMI seminar: Flipping the Dice (PDF)

## 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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