Instructor Insights

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.