Where does math impostor syndrome come from?

If you ever sat through an equations-dense class and felt stupid, I’d like to share a secret with you: the other students are probably faking it. Turns out the human brain can’t handle a typical math lesson.

I taught math and statistics for over a decade and today I’m often asked to speak about technical topics at conferences. Although I come from a field that loves equations, you’ll find almost none in my talks. I also avoided them as a statistics lecturer. Here’s why.

The myth of superhuman working memory

Before heading to grad school in mathematical statistics, I was a PhD student in neuroscience and psychology. I was fortunate enough to get hands-on research experience on the topic of human attention and memory, which brought me to a hilarious realization.

Anyone who claims to be following an equations-based mathematical lecture is probably faking it.

To boil the cognitive science down to a point, anyone who claims to be following an equations-based mathematical lecture is probably faking it. There’s one exception: those who have already learned most of the material. Mathematicians are just as human as anyone and their working memory capacity works similarly too. It turns out that standard lectures overload students’ working memory by defining too many new symbols and equations for even the brightest students to keep track of what’s what.

Mathematicians aren’t superhuman and they don’t have superhuman working memory.

Think you’re special? Read this once, close your eyes, and say it back to me:

AHGJBSKEIFDDRHWSL

Psychologists would suggest that AHGJBSK is about as much as you should expect humans to handle. Every time a speaker adds a new symbol to their talk, the audience has to devote working memory capacity to tracking what it symbolizes and how it fits with the other new stuff. That’s the same working memory needed for remembering the previous slide and tracking the logical argument. AHGJBSK highlights just how little capacity there is to go around.

Before presenting equations and jargon, do this

Professors and technical speakers, don’t take my word for it, try it yourself:

1. Pick an alphabet you wouldn’t be able to name characters from. Chinese characters are my favorite choice when I practice. 熟能生巧
2. Ask yourself which symbols in your equations or jargon terms in your slides might not be immediately familiar to every human in your audience.
3. Replace each one of those with a random letter from the alphabet you’ve chosen.
4. For best results, have a friend reformat your slides so that things have also moved around a little bit visually.
5. Try giving your talk/lesson.

If you stumble, you’d also have lost your audience at this point. If the cognitive load of remembering what all those new symbols mean is too much for you (the expert!) then it’s definitely too much for your poor audience.

Once you’ve lost your audience, all they will absorb is summaries and explanations that you give in plain language.

Where math impostor syndrome comes from

I’ve had some professors who didn’t even summarize; they just read the textbook and wrote it out on the board. In a class like that, you get three main species of student:

1. Daydreamers, who sit through class with eyes glazed over, confident in their ability to go learn it from the textbook later. Class is a waste of time for them, but they attend just in case there’s an announcement about the exam.
2. Fake geniuses, who study and practice before class to become fluent in all the new terms and symbols before attending. They’re not learning it all for the first time. They are the ultimate trolls, because they make it look easy and give the professor fake feedback about lesson quality.
3. Self-diagnosed impostors, who worry that they can’t understand anything because they’re stupid. They hear the “geniuses” asking intelligent questions and take that as proof that they don’t belong in this group. Unlike the daydreamers, they assume that they don’t understand because of their own ability to learn rather than the teacher’s ability to teach. They are terrified of speaking up in class in case someone finds out they don’t belong there.

What a farce! Turns out we don’t grow out of this nodding and smiling. I’ve amused myself frequently by asking the smiler next to me at a technical workshop to share what they learned. Typical responses? The first slide’s message / the last slide’s message / something about the point conveyed by the flashiest illustration / “oops, sorry, I was thinking about other things.”

Growing up, I was lucky to be socialized into extreme levels of self-confidence, so I usually inhabited the first two groups. (I can remember two intense classes that made me seriously contemplate the possibility that I might be an idiot, so I know what that feels like too.) Friends with lower self-confidence found themselves in the third group, eventually spiraling down and out of math. I’ve seen those who stayed anyway spend brilliant careers drowning in impostor syndrome.

Boring or difficult?

My parents (physicists, both with PhDs) brought me up to believe math is easy. They treated sentences like “math is hard” or “I’m bad at math” as ridiculous: math is just like cooking, there’s no more and no less honor in it. Equations are like recipes, you only need them when you’re actually cooking. Until you’re cooking, you need a conceptual understanding and knowledge of where to find the right cookbook. If you’re not even sure if you want to make the dish, fiddly details about oven temperature are boring and you should be bored. My parents would say things like, “Numbers-worship is silly. I understand math so well I no longer respect it.” Using math was like knowing how to read — talking about it as magical or something only for intelligent/talented people would have been laughed right out of my living room.

I understand math so well I no longer respect it.

I grew up with the expectation that I belonged in my math classes and if I was bored, well, that was my teacher’s fault. My parents encouraged me to sneak out of class and go study from my own textbook when that happened. (Of course I’d come back scary-fluent in next semester’s symbols and theorems— I’d played with them already. While everyone else spent their time trapped in a bad lesson, I spent my time learning. That meant I would be one of the kids who ruined everyone else’s day by making it look so darned easy.)

Huxley thinks this textbook is boring. I thought so too, so I bought a different one.

Folks who were brought up on the myth that math is hard (in some sense other than occasionally boring) set themselves up for thinking that their inability to follow the lesson is their own fault. That gears them up for impostor syndrome.

What this means for diversity

Difference — diversity! — should be welcomed and celebrated. Unfortunately, this whole setup isn’t very welcoming.

I’ll hazard a guess that if you feel different from what you think of as the standard math nerd, you’re more likely to assume that not understanding what’s going on is your fault, rather than the teacher’s. If you start out feeling like an impostor, you’re less likely to ask your classmates whether they’re suffering too. Every time one of those “geniuses” knocks one out of the park, you’re shrinking deeper into your shell, making it harder for you to come to the correct conclusion: “These humans are just like me and they probably feel what I feel. If I’m not following along, everyone else is lost too.”

In that moment when you’re doubting yourself because it feels like you’re the only one who doesn’t follow, remember: the others either learned most of it already or they’re faking it. You’re not alone, so be kind to yourself.

In the sense that matters for math, you’re the same as everyone else in the room; you’ve covered the concepts listed as prerequisites for the class, so you belong in the class. Simple as that.

Let me remind everyone that the skills that make a great mathematician are not the skills that make someone an entertaining presenter or effective teacher. It takes an incredible (not just decent, incredible) teacher to turn their back on generations of “writing the cookbook out on the blackboard” as a means of educating the next crop of learners. Most will just do it the way it has always been done.

What does that mean for students?

On behalf of math teachers and professors everywhere: “It’s not you, it’s me.”

If you’re struggling to understand, it’s probably because the teaching quality is poor and/or the speaker doesn’t know their audience. They’re throwing too many new symbols at you all at once and you can’t hope to hold them all in working memory. Be kind to yourself. You’re not going to follow all of it and that’s okay.

If the entire session sounds like birdsong to you, that’s the speaker’s fault, not yours. Asking questions in class doesn’t help that much: it’s too little too late. The problem lies not with a factoid or two but with the speaker’s entire teaching philosophy. If you’re brave, talk to them after class about their presentation style. If not, find other resources (YouTube? Books? Tutors? Puzzles? Blogs? Friends?) that teach the same concepts in a different way, then take it at your own pace. Personally, I learn better by going straight to solving problems without even reading the chapter. Instead, I use it as a resource that I can dig into when I (inevitably) get stuck on a problem.

To me, math and programming feel a lot like LEGO. A lesson or textbook chapter gives you some new LEGO pieces, which you add to your bucket and construct a sturdy solution out of your collection. I’ve always found it amazing that someone would want to pay me to play LEGO all day. (Yet they do!)

What does that mean for speakers and teachers?

“This is how it has always been done” is a terrible reason for anything, including the way you choose your communication style.

Instead, know your audience and keep human working memory constraints in mind. If every single human in the room can’t look at x̄ without thinking “sample average” (a decent assumption if your audience is statistics professors), then go ahead and use x̄. It saves space and is a powerful symbol… for that specific audience. If not, drop x̄.

Give yourself a budget: no more than 7 new things (symbols, theorems, concepts, equations, etc.) in working memory. If your audience is seeing something for the first time, finish up with it now (and tell your audience when you’re done with it) or pay for it out of the budget. Don’t lean on it later unless you’ve kept your audience’s working memory free of clutter.

If you’re honest with yourself, you’ll see that very few of the details that look so beautiful to you actually help your audience. Don’t waste their time with equations they can’t absorb right now. Instead, tell them how to use that equation when they’re hunched over it with pen and paper. Tell them why they should be excited about it and how it fits into the greater picture. Tell them why it was hard to derive / discover and what the key insight that drove that discovery was. Point your audience to any equations they will need later by indicating the place to look and what they will want to use that equation for. Tell them why they should care.

It’s just like cooking, after all. You won’t convince your audience to cook mansaf by reading them the recipe, especially if they’ve never heard of it. They don’t care how many onions go into it. Your job is to tell them what they won’t learn by reading the recipe themselves. Which one is more useful as an intro to mansaf, the nitty gritty equations or the conceptual overview? You should only cook it in front of your student after you’ve convinced them that they want to do it themselves. Get them excited or they’ll think cooking is boring or, worse, that they’re bad at it.

Where does math impostor syndrome come from? was originally published in Hacker Noon on Medium, where people are continuing the conversation by highlighting and responding to this story.