Hindsight is Stats 2020, Part I: Fractal Course Design

This summer (2020) I taught Statistics for the Behavioral Sciences.

The course was unusual for a number of reasons. I’ve wanted to teach stats for a long time, so I came into this class with a collection of unorthodox ideas that I’ve been sitting on for a few years.

Things went really well. I had high expectations, but more than half (!) of my students got an A or higher. I didn’t shift my expectations, or make the class easier halfway through. These grades mean that most of the students either mastered the material to my satisfaction or came very close to doing so. This approach worked and I would definitely recommend it.

1. Course Format

1.1 Being Online

The big curveball for this class was the pandemic, which made it necessary to teach the class online. I’ve never taken a course online, and I had never expected to teach one that way. Going into this, I had almost no experience with online classes. When we transitioned to online instruction in March, I was TA’ing for a class, so I got to see how that went. But that was about it.

I’m confident in my skills, but there were a few things in particular that I was worried about.

One of the really rewarding parts of teaching is getting to know your students. But Zoom isn’t that great, so I was worried that there might be no personal connection. Partly I was worried that the class would be less enjoyable. People like making friends and knowing that the instructor cares about them. But part of it was also practical. Without that sense of the classroom and knowledge of the students, I was concerned that I wouldn’t be able to tell when students didn’t understand the material. Maybe I wouldn’t be able to explain things as well when they had questions.

The other major concern I had was cheating. I knew that in the transition towards online classes brought on by the pandemic, many schools forced students to install unsettling exam-monitoring software on their personal devices. This sort of thing is pretty evil. While I would never consider spying on my students, it did make me worry about cheating on exams. With online exams, it seems like it could be a real problem. But I also know from being a TA that students cheat a lot less than professors think they do. In the end I took no special steps to prevent cheating. I don’t really care about or believe in grades, and I decided to trust the students.

1.2 Personal Connection

It turns out that both of these concerns were unfounded.

Admittedly, there was very little personal connection. I didn’t get to know most of my students. I would recognize their names, but I never even saw most of their faces.

But no one seemed to suffer for it. In the end we still developed the rapport that you need for good teaching. In their evaluations, students said things like:

“Ethan was a great teacher! He clearly loved the subject, and wanted to try and teach it in a more accessible way”

“Ethan specifically explained things very well and was so real. It was nice hearing examples in ‘layman’s terms’ that were more approachable”

“I really felt as if this teacher wanted us to do well, and helped us learn as much as possible in the clearest way possible. … Great great teacher!”

and

“Ethan is cool”

This experience has changed my mind about classroom engagement, and makes me doubt some of the common wisdom about teaching.

Is getting to know your students a reasonable expectation? Certainly we can get to know our students. But is it appropriate? Students aren’t in your class to be your friends, and you’re not there to be their pal. People are in the classroom to, hopefully, learn something.

“Personal connection” often seems to be used as a proxy for respecting your students and treating them like human beings. But — surprise! — you can respect your students and treat them like human beings without necessarily having a friendship with them, or even knowing their names. Students are sensitive to this difference. They care about being treated with respect, but don’t seem to care about the other stuff.

A cynical take would be that professors use the excuse of “getting to know their students” to push students into having an unnecessarily friendly relationship. But pretending to be equals when you are in a position of power over someone is at best dishonest, and at worst is a way of denying that you have a responsibility to them.

I do think there are things you can do to drive engagement. But I don’t know if it really matters. My students got really good grades and displayed surprisingly deep understanding of the material, so it didn’t hurt their education. And many of them told me that this was one of the most enjoyable courses they have ever taken, so it didn’t seem to make learning any less fun.

1.3 Cheating

I was even more wrong about cheating. I didn’t see any evidence of cheating on exams or assignments, and there was plenty of evidence that they weren’t cheating. Students made lots of simple mistakes, which they could have avoided if they were cheating. Exam scores improved incrementally over time, just as you would expect from honest learning. Their assignments and answers on the tests were idiosyncratic, not the carbon copies you might expect if they were sharing answers. If students were cheating, they didn’t leave any trace of it, and so I’m inclined to believe that they didn’t.

The lack of cheating is a little weird. When I was a TA, I would catch students cheating all the time. They usually do a bad job of it — they forget that I was a student not too long ago, and so they don’t realize that I know most of the tricks. So the fact that we didn’t see any of the classic signs is strong evidence that there wasn’t any cheating.

So why didn’t they cheat on my class, when they do cheat during the semester? I think it has to do with trust. In the exit survey for the class, one student wrote down, “no feeling of being ‘cheated’ by the prof”. Another student wrote, “My biggest fears for this course revolved around completing it and not only doing poorly, but also learning nothing.”

Students tend to stoop to cheating when they think, often correctly, that there is no other way to do well in the course. When professors are unclear about expectations, or make examinations needlessly difficult, the students feel cheated by the professor, and will cheat themselves. When you see an exam filled with trick questions, it’s hard not to feel like the game is rigged. But to their credit, even in this situation, most students still won’t cheat.

Teachers have a lot to learn about cheating. If you don’t cheat your students, most of them won’t cheat on your assignments. It’s about trust. Not your trusting that they won’t cheat on assignments — their trusting that you won’t cheat them in their education.

This all makes it especially disappointing that, during this pandemic, so many schools are engaging in unethical surveillance of their students in the name of academic honesty. Students just don’t cheat all that much, even when they definitely could get away with it.

2. Course Content

So much for the course format. What was I actually teaching?

2.1 What’s Wrong with Stats?

Statistics education is pretty terrible, and everyone knows it. All the professors who teach stats agree: students come into class, usually manage to pass, and retain almost nothing.

Everyone is looking for the magic bullet. But even so, no one thinks it’s a great mystery. Professors and TAs will all tell you the same thing: the problem is motivational. The majority of students, they say, simply aren’t interested in learning this esoteric form of math. As a result, most of the proposed solutions are motivational as well: find a way to make it fun and interesting, or at least find the right set of rewards and punishments.

But when I was a TA for intro stats, I noticed that this didn’t match what I saw at all. The students in my recitations were engaged, and really wanted to understand stats. They asked insightful and sophisticated questions, and were always pestering me for more detail. Yet somehow they seemed to come back every week having forgotten everything we discussed the week before. This isn’t the behavior of students who are checked out — this is the behavior of students who are trying, and repeatedly failing, to build a model of what is going on around them.

Even if I had been wrong about most students, there were a few of them who were clearly both able and motivated. These students got perfect scores on multiple tests and assignments, regularly came to my office hours, and discussed many of the concepts in great detail. They showed me the extensive, meticulous notes they had taken in lecture. But when it came to answering simple questions about the material in a new context, they always came up blank.

These students weren’t lacking in motivation or intelligence. So it must be external; something about the class was failing them. Even if everyone in the class were as motivated as these high-achievers, we would still be having trouble with comprehension and retention.

2.2 Driver’s Ed

I think the motivation story is all wrong. The problem is that the subject is taught at the wrong level.

Imagine you are taking a driver’s ed course, and have just shown up to the first day of class. The professor gets up and says, “Hi everyone, in this class you’re going to learn all about cars. Cars are really amazing. Some people use cars to get to work. Some people use them to get to school. Some people use them to go on vacation! There are a lot of kinds of cars. The big ones are called trucks. Those ones carry things like fruit and gravel. In this course you’ll learn all the different kinds and their uses, and we’ll talk a bit about the history of cars.”

You raise your hand, “Excuse me, professor. I’m here because I want to learn how to drive. I didn’t come here to learn about the types or history of automobiles. I’m sure that knowledge will come in handy in some ways, but it’s really not my focus. How do you actually drive?”

“Worry not,” he says, “To drive, move the wheel back and forth.”

So you leave that course and you sign up for a different one. You show up to the new class, and the professor gets up and says, “Hi everyone, in this class we’re going to learn all about cars. We’re going to be starting with the drivetrain. It’s important that you be able to describe and identify all the parts. Look at this diagram. Here’s the gearbox (which you can see is constant-mesh), clutch mechanism, the flywheel, the differential…” You get up and walk out of the room.

Neither of these classes will teach you how to drive. And sadly, this is a pretty good metaphor for how statistics is usually taught. Some statistics courses give students an overview of probability theory and a brief sense of the history, without teaching them how to actually conduct an analysis. Others throw the equations right on the board and start discussing the terms without any context. All too often, a single class will try to include both of these approaches. This is probably worse than either of them alone.

Students don’t want to learn a list of tests, the life history of Ronald Fisher, or the exact meanings of the terms in the formula for the pooled standard deviation. All these are things one naturally picks up over time, but none of it is useful without the core knowledge. Students want to learn what statistics is and how we actually use it. But somehow they seem to come away from our courses without having been taught either of these things.

Driver’s ed focuses on the point of contact: how to use the car. Similarly, the main goal of this class was statistical skills and how to use them.

I wanted students to become statistically literate. Most students won’t end up being researchers or statisticians in the same way that most people who take driver’s ed won’t end up being auto mechanics or engineers for GM. We still benefit from knowing what a car is and how to operate it. Similarly, students benefit from knowing what statistics is and how to use it. For those students who do want to go on to use statistics professionally, this will still give them a strong foundation. Auto mechanics don’t suffer from having taken driver’s ed in high school.

The focus was limited and practical. Students were taught how to recognize different kinds of variables and data, interpret standard plots and graphs, read and understand statistical reports, and conduct basic analyses using statistical software. I alluded to other subjects of interest in lectures, but in the lessons and the evaluations, I focused on these basic skills.

We can also talk a little bit about what I didn’t want to cover. The history of stats is interesting, but most of the time it doesn’t help you be a better statistician. The most important thing to know about the history is that these tests and concepts were just invented by a few guys not all that different from you and me. Anyone can make up a concept or design a new test. You assign it a Greek letter and suddenly it sounds official, but for all we know, Fisher came up with it while sitting in the bathtub. Besides that, most of the details don’t matter. Aside from a couple of helpful examples, I didn’t teach them anything about the history of statistics.

You do need to know a few symbols to be able to interpret tests, but I didn’t want to cover much in the way of formatting. I don’t care if students report a number as 0.02 or .02 or 0.0212; I don’t care if they write “p-value” or “p-value”. Time is limited, and I don’t want to waste their time or my time going over this nonsense. If by the end of the class, they know the concepts but not the formatting, then I have succeeded. If they know the formatting but not the concepts, I have definitely failed. So I decided to focus on the concepts and, as much as possible, ignore the formatting.

2.3 Fractal

So that’s what I wanted to teach. How do you actually teach something like this?

Most courses take a cumulative approach. You start with the basics, and the material slowly becomes more and more complex. Each lesson builds on all the previous lessons. At the end you finally tackle the most advanced material. Then you take the final.

In my experience, this falls apart by the second week of class. Students who miss even a single lecture are cut adrift, left to founder or drown. Even if you make it to every class, your safety isn’t guaranteed. If you don’t understand the explanation they give in lecture, you’re out of luck, because the class is never going to come back to that topic again.

Rather than being cumulative, my course approach was fractal. A fractal is a figure or function where every part has the same character as the whole. Every part contains copies of the whole thing. That’s how I structured the course: every part of the course was nested within other parts of the course.

A photo of my stats course from space. jk it’s fractal broccoli

You could be the best teacher who ever lived, with the most beautiful slides imaginable. It doesn’t matter — students just can’t learn something in one go. This is especially true in statistics. The classic learning pattern for the subject is brief flashes of insight, a feeling of sudden understanding, and then losing your hold on it and slipping back into confusion. This is normal.

For some reason, people don’t understand this. Everyone thinks there is going to be a shortcut explanation for these ideas, but we don’t think that way about other skills. We don’t think that painters will master three-point perspective in a single session, and we don’t expect programming students to master for loops in a single day. Maybe you can get the gist after the first introduction, but really understanding these topics takes time. Somehow we see stats differently. In particular, there is a whole genre of articles and blog posts all about how to explain p-values. These assume that the concept can be distilled into a single statement, or a single lesson. But that’s crazy. You can’t understand p-values in one hour, no matter how good the explanation is.

I think of statistics as really being three closely-related topics: a language for talking about data in general, descriptive statistics for talking about individual variables, and inferential statistics for making educated guesses about the world on the basis of limited samples.

The structure was built around these topics. The first day of class was an overview of the entire course, introducing all three topics in very general terms. Day 2 and Day 3 were another microcosm: again we covered the whole course, this time in slightly more detail.

Week 2 covered data in more detail. Week 3 covered descriptive statistics. Weeks 4 and 5 covered inferential statistics. Finally, in week 6, we went even deeper into inferential statistics, exposing exactly how the math behind the tests works.

This means that students see every single topic many times before the end of the course. For example, the two-sample t-test appears a total of six times in the lectures. It appears first in day one, during the complete overview, again in the lectures for day three, and then again in weeks three, four, and six.

It doesn’t matter if you don’t understand the two-sample t-test the first time, or the second time, or even the third time you see it. It doesn’t matter if you miss a few classes. It doesn’t matter if one of the examples I use doesn’t make sense to you. We will come back to this concept again, in a new context, with new examples. By the end of the class, you will get to see it from every angle.

These things take time. Mastery of a subject comes only when you return to an idea over and over, seeing it in new situations and becoming more familiar with it, building your own understanding. The structure of the class needs to support this, or students won’t be able to learn a damn thing.

2.4 Context

My influences in this were the Snowflake Method, and Progressive Rendering from It’s Time For An Intuition-First Calculus Course. Both of these perspectives emphasize understanding the gist of an idea before getting stuck in the details. To quote the reasoning from It’s Time For An Intuition-First Calculus Course:

The “start-to-finish” approach seems official. Orderly. Rigorous. And it doesn’t work.

What, exactly, do you know when you’ve seen the first 20% of a portrait in full resolution? A forehead? Do you even know the gender? The age? The teacher has forgotten that you’ve never seen the full picture and likely can’t appreciate that you’re even seeing a forehead!

Progressive rendering (blurry-to-sharp) gives a full overview, a rough approximation of what the expert sees, and gets you curious about more. After the overview, we start filling in the details. And because you have an idea of where you’re going, you’re excited to learn. What’s better: “Let’s download the next 10% of the forehead”, or “Let’s sharpen the picture”?

Let’s admit it: we forget the details of most classes. If we’ll have a hazy memory anyway, shouldn’t it be of the entire picture? That has the best shot of enticing us to sharpen the details later on.

Sometimes I think of this course as Intuition-First Statistics. “Intuition-first” doesn’t mean our goal is to teach good statistical intuitions, though hopefully students do get some of that. It means that we should start by working with intuitions, and that everything else will follow from that. Because, although it may sound surprising, students actually have pretty strong statistical intuitions.

The problem is context. The cumulative or start-to-finish approach makes perfect sense to the instructor, but only because they already know what is coming. They can see the context; how everything is connected.

The students don’t have any of that. They just get hit in the face with new material that they never saw coming. Every day it’s some new bullshit. They have no idea what is up next, what it means, or how it all is related. They’re always being knocked off-balance by new topics you didn’t prepare them for, and they never have time to figure out how it’s all connected.

Your Students
Your Students

This is a huge problem, because context really matters for comprehension and memory. A great example comes from research by Bransford & Johnson (1972). In their studies, participants heard a paragraph like the one below. Take a look at this passage and see if you can figure out what it is all about:

The procedure is actually quite simple. First you arrange things into different groups. Of course, one pile may be sufficient depending on how much there is to do. If you have to go somewhere else due to lack of facilities that is the next step, otherwise you are pretty well set. It is important not to overdo things. That is, it is better to do too few things at once than too many. In the short run this may not seem important but complications can easily arise. A mistake can be expensive as well. At first the whole procedure will seem complicated. Soon, however, it will become just another facet of life. It is difficult to foresee any end to the necessity for this task in the immediate future, but then one never can tell. After the procedure is completed one arranges the materials into different groups again. Then they can be put into their appropriate places. Eventually they will be used once more and the whole cycle will then have to be repeated. However, that is part of life.

One third of the participants heard the paragraph without any context. It didn’t make much sense to them, and they had trouble recalling what they had heard.

The next third of the participants, before hearing the paragraph, were told that it was about doing laundry. To these participants, the paragraph made perfect sense, and they had very little trouble recalling the details.

The final third learned the topic only after they’d heard the entire paragraph. These participants also found the paragraph confusing, and even having been given the context, weren’t able to recall much about it. Context alone isn’t enough; you need to see the context up front.

Something similar happens in class. Without context, even the most motivated students have trouble remembering the material. They have a hard time memorizing tests or equations because they don’t understand what a test is used for, let alone how it works. I don’t have trouble with the equations, but only because I understand what the tests were created to do. It’s easy to put things into their proper categories them if you have a good grasp of the system; it’s impossible if you don’t even know what categories there are.

The fractal approach solves this problem. The first two or three times I went over the material, I didn’t expect them to remember any of it. We cover all the material early on, because being introduced to everything at a shallow level prepares students to understand the material in depth once it comes back around again.

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