Below, Professor Scott Sheffield describes various aspects of how he teaches 18.600 Probability and Random Variables.
OCW: How would you describe what probability is to someone unfamiliar with the topic? And why does it matter?
Prof. Sheffield: Every aspect of life involves uncertainty. Probability is a systematic approach to quantifying that uncertainty. It enables us to think more clearly and make better informed decisions about pretty much everything.
OCW: You’ve been teaching 18.600 (formerly 18.440) since 2011. How has it changed over the years?
Prof. Sheffield: The class is typically much larger now than it was ten years ago. Thanks in part to the rise of topics like data science and machine learning, pretty much everybody recognizes the importance of probability. We now have TA recitations, an online forum, a nicer room, and a huge collection of sample problems.
OCW: The previous version of your course on OCW (18.440 from 2014) has had over 10,000 user visits to date. It usually has between 100 and 150 visits per day. What makes this subject so popular (despite not having videos!)?
Prof. Sheffield: Everybody needs to know probability. It’s a beautifully engaging subject, an exciting place to combine mathematics and storytelling with pretty much any real-world discipline.
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OCW: What kind of prior knowledge do students tend to bring to the topic of probability and random variables?
Prof. Sheffield: A lot of students have learned some probability and combinatorics for math competitions or AP statistics. So they come in knowing something about Pascal’s triangle, normal distributions, independence, permutations, etc. But it’s pretty uneven. There are topics in the beginning of the course that I would guess 80 percent of the students have seen before in some form, and topics later in the course that virtually nobody in the class has seen before.
OCW: What kinds of misconceptions do students have about probability and random variables at the beginning of the course? How do you address those misconceptions?
Prof. Sheffield: MIT students are obviously very sharp. But there are plenty of problems where the “intuitive” answer is not the correct answer, and where most people (including MIT students) would initially make the wrong guess. And these moments are actually fun. You can talk through a problem, allow people to guess the answer, and then work through it and explain why the actual answer is surprising.
OCW: In your materials, you note that “we will see many problems with applications; but each problem is the beginning of a conversation, not the end.” Tell us about the role of applications in the course, and the kinds of conversations you hope they will inspire.
Prof. Sheffield: Yes, the world is full of places where probability can be applied, but the simple probabilistic model is seldom a perfect description of real life. So there’s room to debate the answers that probability provides, to inform probabilistic models with new data, to frame entire areas in different ways. If other educators are interested in my approach, they can take a look at the 10 problem sets (or the three exams) posted on OCW.
OCW: You note that this course is in some ways even more advanced than ones that come after it, and that students may confront problems they don’t know how to do. How do you help them persevere when encountering academic challenges?
Prof. Sheffield: I encourage students to take advantage of office hours, recitations, collaboration with friends, lecture slides, and anything else that helps them master the material. It is definitely a challenging experience. People who learn algebra and calculus through the standard curriculum learn to think a certain way, and probability definitely requires new ways of thinking.
OCW: You note that “math fluency requires knowing at least a few things by heart.” What is the story sheet (PDF) and how does it help students develop math fluency?
Prof. Sheffield: Well, there is this ongoing debate in education about the extent to which exams should be “open book” and whether students should have access to very detailed formula sheets while taking the exams. And I don’t claim to have a perfect all-purpose answer. Different approaches make sense for different subjects.
But I’ve been resistant to going too far in the open book direction, because in my own experience mathematical creativity has often involved the brain making unexpected connections between things that are in the brain. I don’t think I could be creative in quite the same way if all my basic knowledge were in my phone or in a formula sheet. Some things, like the rules of basic multiplication or the Pythagorean theorem or the definition of sine, really have to be second nature.
So I try to strike a balance where there’s not a lot of rote memorization, but there are at least a few things (a page worth) that students have to know. And I try to frame it in terms of “stories” so that students feel they are learning these things in a conceptual way, not just memorizing them the way they might memorize digits of pi for a contest.
OCW: You try to make space for students to answer each other’s basic math questions on the online forum. Tell us about this decision. Why not jump in and respond to these questions yourself?
Prof. Sheffield: We use the online forum Piazza, and it is quite an amazing undertaking. Every year we have hundreds, sometimes over a thousand separate contributions: questions, answers, ideas, new ways to think about problems. Over the course of a semester, we collectively as a class (students, TAs, graders, and professor) produce several hundred pages worth of material, some of it quite well written and reasoned. Enough material to fill a whole textbook.
And then the next semester we throw it all away and start over again. Each class struggles through the material in its own way. I answer questions occasionally, but I don’t answer everything, first because there is no way I would have time for that, and second because I don’t think that is the best way for the learning to proceed.
OCW: What advice do you have for educators who may be using your materials to teach remotely?
Prof. Sheffield: Teaching a class online involves a whole set of logistical and practical challenges, and I’m still learning myself how best to do that. As far as learning online goes, I would say that Google and Wikipedia are amazingly good resources for a lot of material in mathematics. Students who become adept at finding things there (along with more scholarly works at scholar.google.com) can really learn in ways that previous generations could not. When working through a class on OCW or working through a textbook, it is really often quite helpful to supplement the education with periodic internet searches.
- 18.600 can be applied toward a Bachelor of Science in Mathematics, but is not required.
The students’ grades were based on the following activities:
- 20% Ten problem sets
- 40% Two midterm exams
- 40% Final exam
Breakdown by Year
A roughly even mix of levels, with fewer first-year students in the fall semester because many entering students take 18.02 Multivariable Calculus (the prerequisite for 18.600) in their first semester.
Breakdown by Major
Students from a wide range of majors take 18.600.
Typical Student Background
Many but not all of the students have learned some probability and combinatorics for math competitions or AP statistics.
How Student Time Was Spent
During an average week, students were expected to spend 12 hours on the course, roughly divided as follows:
Met 3 times per week for 1 hour per session; 39 sessions total; mandatory attendance.
Recitations, led by teaching assistants, met 1 time per week for 1 hour per session; 13 sessions total; mandatory attendance.
Out of Class
Course Team Roles
Delivering lectures; conducting office hours two hours per week
Teaching Assistants (3)
Leading recitations; conducting office hours two hours per week