All Lessons are Wrong, but Some are Useful

All lessons may be wrong, but this dancer doesn't care!

Let me pose a scenario to you, and you tell me if you’ve ever experienced it before:

You’ve contacted one of your favorite pros for a lesson. It’s going to be amazing, and you’ve got a list of questions and concepts that you’d like for them to go over. Your usual teacher is enthused that you’ll be getting some fresh perspectives on your dancing, and you’re ready to drink from the firehose of awesome that you’re sure is about to pour forth from their brain into yours.

You arrive, exchange pleasantries, and get to work. After dancing a bit so they can see your movement, you (im)patiently await their critiques and suggestions. They begin…

… and proceed to describe the exact opposite of just about everything you’ve ever been taught on some dance subject. You’ve got straight legs? They want a micro-bend. You always point your toes? They want to see more heels. You’ve focused on pushing from the standing leg? They want to see more pull from the receiving leg.


Alarm bells start clanging in your head as your self-preservation systems kick in. Could it be that they’re just fundamentally wrong about everything? Their dancing has obviously impressed you, but how could they have achieved such perfection of movement with content that seems to be the antithesis to what you’ve learned? Could it be that you and your teacher are totally off-base?

Is everything a lie?


In my opinion, the answer to that last question is an emphatic yes. Why? Because like any good, compelling discipline, dance is much more of a muddy swamp than a crystal pool when it comes to trying to see down to the bottom of it.

“All Models are Wrong; Some Models are Useful”

The title of this article is a take-off of the above quote, which happens to be one of my favorites. It was penned by George Box, an English statistician who, despite no evidence I can find to support this claim, must have danced a foxtrot or two in his day. At least, I like to imagine so.

A longer version of the same quote:

Now it would be very remarkable if any system existing in the real world could be exactly represented by any simple model. However, cunningly chosen parsimonious models often do provide remarkably useful approximations . . . For such a model there is no need to ask the question “Is the model true?”. If “truth” is to be the “whole truth” the answer must be “No”. The only question of interest is “Is the model illuminating and useful?”. 1Box, G. E. P. (1979), “Robustness in the strategy of scientific model building”, in Launer, R. L.; Wilkinson, G. N., Robustness in ...continue

In my head, here’s George after presenting the world with that gem:

George Box

Memes aside, Box is making an argument here about the way statistics attempts to understand the world by downsampling the immense complexity of reality into manageable statistical models that allow us to make predictions and test our hypotheses. Useful stuff.

But dancing is pretty simple, right? If you want to do George’s favorite foxtrot you’ve just got to put one foot in front of the other and then…

… use swing and sway to efficiently change and create shapes, rotate around your spine, use the floor to drive forward, maintain good body flight, keep good body alignment in relation to yourself and your partner…

Right. As you’re probably not particularly surprised to learn, it’s not quite as simple as all of that. In fact…

Dance is infinitely 2I’m talking functionally infinite, not literally infinite. The number of combinations is so high that for human purposes it’s infinite. complex

Warning: A bunch of somewhat esoteric anatomy/math crept into this post somehow (weird, right? Never happens when I write… ), so if you want to skip it, just buzz down to the next bold heading.


Quick quiz:

How many joints are there in the human body?

Answer: Somewhere around 200-400, depending on what you want to count as a joint and how old the individual is.

Let’s be conservative with our estimates for the sake of proving this point, and say that there are just 13 joints in the body. Here’s a list:


Human Joints

JointTypeDegrees of Freedom
Total DF:26
Ankle LHinge1
Ankle RHinge1
Knee LHinge1
Knee RHinge1
Hip LBall/Socket3
Hip RBall/Socket3
Wrist LElipsoid2
Wrist RElipsoid2
Elbow LHinge+Pivot2
Elbow RHinge+Pivot2
Shoulder LBall/Socket3
Shoulder RBall/Socket3


Now, not all joints are created equal; some allow more movement than others. Your knee works quite differently from your shoulder because human behaviors require different priorities from each: stability in the knee, mobility in the shoulder. Therefore, there are more things you can do with your shoulder than you can do with your knee, which shouldn’t be that surprising.

This is might be a little overly academic, but I’m going to represent that difference in movement capacity with Degrees of Freedom.



Whoah. Too much freedom. Wayyyyyy too much freedom.

Freedom, in this case, is the ability to change position in a given range of motion. Taking your knee as an example again, it can bend or extend in one plane of motion. Your shoulder, on the other hand, can move in several.

Each additional degree of freedom is a new plane in which the limb can move. An easy example of a joint with two degrees of freedom is the elbow, where you can either flex/extend your forearm or rotate it. Importantly, for the purposes of this demonstration, we’re going to assume that each of these planes of motion can operate independently of one another. That means that I can bend my elbow without having to rotate it and vice versa. Each can be performed independently or at the same time.

  • Aside for Stats Nerds: It may be true that some of these degrees of freedom are actually entangled with one another (try rotating your shoulder while holding your arm up to the ceiling), but for a rough estimate I’m happy assuming that they’re fully independent. We’ve already down-sampled many joints (fingers, toes, clavicle, mandible, etc) out of the equation, so I am comfortable with a little entanglement here because it’s more than made up for by the actual degrees of freedom of the human body as a whole. If you see it differently though, please comment where I’m going wrong. Back to the action.

So now that we have all of our joints lined up (hah, posture jokes!) with their respective degrees of freedom, we need to estimate the total number of different postures that these 13 joints could conceivably inhabit, a posture being any unique combination of joint configurations. Here’s where it gets a little hairy. What is a unique joint configuration? Even a change of just a few degrees can be important in dance.

For simplicity’s sake, I’m going to propose that each joint has 3 positions: Flexed, Neutral, and Extended. Again, this is a gross oversimplification of reality, but we’re just doing this to mathematically prove a point. Each degree of freedom, therefore, has 3 potential values.

We’re almost done with the boring math part, hang with me!

To find out the total number of possible whole-body configurations, we need to figure out how many different combinations there are. If there were just two joints, it’d be easy-peasy:


But if we throw in the knees, things get messy really fast:

Knees L\RFlexedNeutralExtended
Flexedankles possiblilitiesankles possiblilitiesankles possiblilities
Neutralankles possiblilitiesankles possiblilitiesankles possiblilities
Extendedankles possiblilitiesankles possiblilitiesankles possiblilities

It only gets worse from there, as we add joints with multiple degrees of freedom. Luckily, there’s a shortcut: Each additional degree of freedom added to the mix multiplies the number of total options by three.

When it was just ankles we were dealing with it was 3 x 3 = 9

With the knees thrown in we got 3 x 3 x 3 x 3 = 81

Counting all the degrees of freedom we get 3 to the 26th power equals…


2.5 Trillion possibilities.

If those were dollars, that’d be America’s GDP in roughly 1978/79. Go Carter.

2.5 trillion different positions! And that’s limiting each joint to only 3 different positions, not the multitude of different degrees to which we can actually move a limb through its range of motion. In case it’s unclear, that’s a LOT of options.

Now, I will fully admit that most of our dancing doesn’t involve a good portion of those 2.5 trillion possibilities. Somewhere in that 2.5 trillion there are several variants of the fetal position and, aside from some hilarious Jack-and-Jill antics, I think most dances don’t involve that particular position for good reason.

Cue a crappy little comic!


Just because I'm using the math to prove a point doesn't mean you should show up to your lessons with dumb stuff like this!


So I’ll admit you may have a point if you’d like to argue that a non-trivial portion of those 2.5 trillion possibilities aren’t body postures that are directly useful in dance. Yes, even my kicks.

But again, we’ve vastly downsampled the number of joints available, the range of motion that they have, and just about everything else about this. We haven’t accounted for music, movement, balance, or any other variables that would add dimensions to this hyperspace matrix.

Whoah, I’m going off the deep end again, aren’t I? Reign it in, Joel.

Suffice to say that dance is, for our intents and purposes, infinitely complex. If you still disagree with me after all of that… well, hit me up in the comments.

Coming up for air

Alright, we’ve been down in the weeds for a little while now, and I’m sure you’re wondering where I’m taking you on this somewhat mind-bending journey through anatomy and statistics that are tangentially related to dance. Well, time to zoom back up to 30,000 feet. Hold onto your pants!

Also, if you’ve made it to this point in the post let me say that you’ve got more stamina and interest than most, so I applaud you, and I’m sorry to say you’re never going to get these minutes back. But that’s life, right? For those of you who skipped to this heading, you probably made a positive life choice.

Alright, back in we go!

Dance lessons as models

I argue to you that a dance lesson is nothing more than a mental model for understanding the infinitely varied and complex tasks that a human body must perform in order to create the varied, rhythmic movements we usually refer to as ‘dance’. What a definition, huh? Gotta love semantics.

We use thousands (probably hundreds of thousands) of different metaphors to help coerce our multi-potent meat-machines (bodies) into the various forms and movements that we find most pleasing based on aesthetics, biomechanics, and social conventions. If you’ve ever heard a teacher tell you to think of your hips like a bowl full of water, you’ve experienced this. Have you heard someone describe posture as imagining you’re being suspended from the crown of your head, letting your spine hang below? It’s a great story, and incredibly effective, but ultimately it’s still just a metaphor.

Every time someone explains something in dance, they’re using an abstraction. I think that’s really important, so I’m going to say it again, but this time with fancy bolding and italics so you know I mean it: Every time someone explains something in dance, they’re using an abstraction. Every single time.

Put another way, a ‘foot’ is a foot only because everyone has made an implicit agreement to call that general region of the human body a ‘foot.’ The cells and atoms that make up said foot care very little whether they’re called a ‘foot’ or a ‘dnahgel’.

Dnahgel is ‘leghand’ spelled backward, for those of you wondering.

Not to get too woo-woo about it all, but the reality of our lives is that we’re a bunch of atoms, arranged in a particular way, hurtling through space on a rock that we all pay lip service to caring a lot about, and sometimes we shake and move those atoms based on our subjective, multifaceted perception of waves of energy traveling through the atoms of air that we interact with.

Never mind, that was way too woo-woo.

What I’m trying to say, in as many different ways as I can think of, is that every part of our experience is a construction of our imaginations, and we read meaning and expectations into everything. This includes dance.

As I’ve argued elsewhere, this kind of downsampling and simplification is actually a really good thing. It’s fantastically useful to not have to actively think about the process of creating ATP in my mitochondria every time I want to move a muscle.

Did anyone else hear a memory of a teacher whispering “the powerhouse of the cell” as you read that?


I hope you all have fond memories of college biology!

No, only me? Ah. Well then. Moving on.

I’m not trying to suggest that we think about our movements at the cellular or atomic levels. That’s just way too granular. At the same time, I firmly believe that all dance lessons are inherently wrong because words are generally inadequate to describe the infinite complexity that we see in dance. That said, they’re also often the best tools we have, and some frameworks for thinking about dance are better than others. My goal is to challenge my perception of what is true about dance to find those frameworks that take me from where I am and allow me to get to where I could be.

Of course, we have all agreed that there are some parts of a dance that are ‘correct’ and some parts that are not. I’m not arguing against that. As a matter of fact, I know of very few dance teachers who teach any kind of partner dancing by having you grab your partner’s belly-button, and I happen to think that’s a pretty great thing. I’m also not arguing for an anarchist view of dance, with no authority and infinitely branching styles; I don’t think that would be of much use to anyone. At the same time, I don’t think the dogmatic approach to learning dance is that much better. To go down one particular path without experimenting with new frameworks for your dancing is a sure way to plant yourself safely in a deep, long rut. What I do advocate for is challenging your long-held assumptions about what you’ve been taught and what you are teaching.

Your frameworks can and should change as your dancing progresses. We see this all the time as we move past the basics of a dance. You learn the rumba box as ‘simply forward, side, together, then back, side, together’ until you have that under your belt and then we upgrade your understanding to include foot articulation, weight distributions, etc. Was the first model wrong? Yes, but that’s not the point. It was useful because it got you started down the right track. Then you were taught something that was also not entirely correct, but it moved you on down the path again towards more complex, aesthetic dancing. That’s how learning works, we constantly upgrade the models we use in our understanding of dance.

I don’t believe this process of upgrading should stop, even as people pursue the highest levels of the sport. Though the techniques and ideas continually evolve, the fact that new models are needed to understand the finer points of the dance stays constant. You constantly need to engage with new material to continue advancing your dancing. I find it strange, therefore, to see people stuck in their same habits of teaching, thinking about, and executing dance. Surely the habits of engaging with new information that got them to where they are have made them receptive to getting further new information, right? Unfortunately not always the case. So it goes.

An important caveat: I’m not suggesting that you overwhelm yourself with as many new perspectives as possible either. You need to engage with many different frameworks, and having a deluge of new stuff hit you all at once is a sure way to have none of it take hold. Dig in and do the work to understand what it is and how it may be useful.

All Lessons are Wrong, but Some are Useful

So where does this leave us? Somewhat quixotically, right where we started. Take lessons, and engage with the material! Follow them to the end of their use. See how they push you. Let them dictate how you move, and how you think about movement.

Then when something new comes along, don’t be afraid to jump into the new framework and see what that does for you. If it’s not useful, it’s fine to leave it, but more often than not new ways of thinking about your movement have at least something to offer you and your dancing. And the only sure way to make no progress is to refuse to engage with new material at all. Then you’ll stay firmly entrenched in the warm rut you’ve dug for yourself. Cozy.

So go take a lesson! Think about things differently! Do the opposite of what you’ve always done, and see what that does for you! Push beyond your comfort zone!

You can always go back if it doesn’t work out, but you can’t go forward if you don’t try something new.

I’ll see you on the floor!


p.s. If you made it through the whole thing, pat yourself on the back or treat yourself to an apple or something. You just read over 2,800 words of my brain vomit, and I appreciate you immensely for that!

Footnotes   [ + ]

1. Box, G. E. P. (1979), “Robustness in the strategy of scientific model building”, in Launer, R. L.; Wilkinson, G. N., Robustness in Statistics, Academic Press, pp. 201–236.
2. I’m talking functionally infinite, not literally infinite. The number of combinations is so high that for human purposes it’s infinite.

3 Replies to “All Lessons are Wrong, but Some are Useful”

  1. So now begins my new hobby of turning joJoel 2,800 words into ten or less.

    “Everything is wrong, but thats okay, no ones right”

    1. Hahaha, I love it! Yeah, that’s the short and sweet version. 🙂

  2. Loved the science jokes. I also heard the memory of the Krebs Cy and a teacher explaining lim f(x) as x -> ∞ while reading this. I’d agree that upgrading the dance models we run on requires a more nuanced understanding of how each joint should/could move in each dance style.

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