The Quincunx

There's no one weird trick

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Consultants

A consultant came to my school a few years ago to run some professional development. He began (as many consultants do) by telling us how poor the students were at the school he turned around (they were really really really poor) and how well that school is doing now (it's doing really really really well). He then told us the secret to improving schools: turn and talks. Get kids talking to each other, and culture and classroom management will improve. He added that at the school he turned around (with the really poor students that's doing really well now) there were two teachers who refused to do turn and talks. Those were the only teachers who kept having classroom management problems after he turned the school around. The next week, all the teachers at my school started doing turn and talks, and our test scores skyrocketed overnight!

Just kidding. We mostly ignored him. Not to say teachers here don’t use turn and talks, many of us do. But that guy never talked to us again, we’ve had many more consultants come through since, and we’ve been told to do many different things. This is a common story. Some consultant comes in, doesn't do much to understand the context of the school, recommends their pet idea that they recommend to every school they work with, and then moves on. There's a whole industry of consultants like this. The point of this post is to understand why that type of advice is bad, and to build a mental model for the complexity of teaching.

The Quincunx

Here is a quincunx. It's a board with a bunch of pegs. Balls bounce down the pegs into different bins. (I learned about the quincunx idea from this blog post, here’s a footnote if you want the full story.)¹

In this mental model, the pegs are all the different things that influence learning, and the balls are students. A teacher teaches, all sorts of different factors influence learning, and students fall in this distribution of outcomes. Let's say outcomes further on the right are better, outcomes on the left are worse.

Ok but this doesn't quite make sense, in a regular quincunx all the pegs have a 50% chance of bouncing left and a 50% chance of bouncing right. But every teacher and every school is different. They do some things well, some things less well. Here's a new model. In this version, each peg gets a random probability of bouncing right — from 0% to 100%. Blue pegs mostly bounce left, red pegs mostly bounce right, and purple pegs are close to 50/50.

Here's a website where you can play around with the model if you like.

You could think of every teacher as a different quincunx, with a different set of strengths and weaknesses. At a basic level, teachers with more red pegs are good, and teachers with more blue pegs are less good.

But that's not quite right. Here's an example where a teacher has a ton of red pegs, but students still don't learn very much:

That example is a bit contrived. But the point is, the interaction between pegs is complex. In the quincunx above, most pegs don’t matter much at all.

You can find places where some peg has a disproportionate impact on the results, or where another peg has no influence at all. Here’s another model:

In this model the top peg doesn’t matter, because the two pegs below it send all the balls to the same peg no matter which side they go down. I could make that top peg red, blue, or anything else, and it wouldn’t affect the outcome. Here’s another model:

In this model, the top peg matters a ton, because the two pegs below it bounce different directions. A small tweak to that top peg would make a big difference to the final outcome.

Play around with some random setups and you'll see stuff like that happen. You'll find all sorts of weird path-dependencies. You'll find places where you could crank one pin all the way to 100% right-bouncing and it doesn't really make a difference, and other places where nudging a certain pin just a little bit would make a big change.

Back to Teaching

Now you might be saying, hey this is kind of contrived, teaching is way more complex than 36 pins in a quincunx. I agree! Absolutely. This is a toy model. Here's the point.

In this toy model, you might naively say that red pins are good and blue pins are bad. But it's way more complex than that. Some pins matter much more than others, depending on the context and everything happening around them. Some pins don’t matter much at all. A pin that matters in one context might not matter at all in another. That's true in education, where advice that works for one teacher might be totally unhelpful in another classroom. Since education is more complex, it's true to a much greater degree. Think of that veteran teacher or school leader you know who always gives the same advice to every teacher who asks for help. That advice might have made a big difference in the veteran teacher’s classroom, but that doesn’t mean it’s a good fit for everyone else. Context matters.

There's this whole industry of people who give teachers and schools advice. There are tons of school leaders who are desperate for that one weird trick that will turn their school around. And though there are some exceptions, a lot of that advice is totally divorced from the context and complexity of those teachers and schools. The quincunx is just a mental model, but I think it's a helpful mental model to get a grasp on how complex even a seemingly simple system can be.

There’s an extreme you could take this to, where everything depends on the context and there’s no such thing as good advice or bad advice. I wouldn’t make that claim. I think the quincunx model can be helpful here as well. Pegs higher up matter more, pegs lower down matter less. They are still shaped by the context around them, but a peg higher up is a better bet than a peg lower down. We can make educated guesses at some higher-leverage places for schools and teachers to focus on. But all that still depends on context, and should be shaped by a big-picture understanding of what’s happening in a school.

Another Mental Model

I was watching this fun xkcd video recently and liked the diagram they used to model meteorology. I have no idea if it’s accurate, but it’s a nice way to visualize some of the dependencies of a complex system:

I’d love to see more thinking like this in education.

Next week, I’m going to take a stab at a rough model a bit like the one above but for teaching and learning. In the meantime, if you have some smart ideas on this topic, let me know.

1. The idea for this post came from a blog by Eric Turkheimer about behavioral genetics. Behavioral genetics studies the influence of genetics on behavioral outcomes like educational attainment, mental health, personality, divorce, and more. The quincunx is a mental model for an interesting phenomenon in genetics. Take a behavioral outcome like divorce. Twin studies show that divorce is heritable, meaning that twins (who are genetically identical) are more similar than non-twin siblings. If one twin gets divorced, the other twin is much more likely to also get divorced. Twin studies show that pretty much any behavioral outcome has some genetic influence. Yet with modern methods, large and expensive studies haven’t been able to find very many genes that are associated with behavioral outcomes. There seems to be a genetic influence on divorce, yet they can’t find a divorce gene. (In these studies they typically find a number of genes that are associated with the outcome but their influence is small, much smaller than the effects in twin studies.) The quincunx metaphor basically says: if two people have the same genetic makeup (the same quincunx), they are likely to have similar outcomes (two balls are likely to end up in the same bin). But this doesn’t mean that we can pick out specific genes (pegs) and say that this gene (peg) is a divorce gene, because they interact in such complex ways and can have different influences for different people. This is just a metaphor, but it’s a nice way to conceptualize a complex system.