An Introduction to Vertical Learning

What Is Vertical Learning?

We learn vertically when we revise mental models by building up and drilling down. Vertical learning is a theory and set of design principles for creating learning experiences that enable and encourage us to grow as independent vertical learners in all facets of our lives.

Revising Mental Models

We learn by constructing and revising mental models. A mental model is an internal theory for how and why something works. New mental models are naive. Mental models only become sophisticated and powerful once they have been applied and revised over time. We revise mental models when they don’t work reliably, we encounter edge cases, we apply them in new contexts, or we recognize that several separate models can be unified into one general model.

Building Up, Drilling Down

A vertical curriculum enables us to learn vertically by maximizing the opportunities we have to revise our mental models. Instead of constructing new mental models to learn something new, we leverage and build on existing mental models. By constantly leveraging and building on a core set of models, we apply and revise those models repeatedly in new contexts over time, increasing the sophistication of our understanding and developing skills that we can use to revise other mental models. We build up when we leverage a mental model to learn something new, and we drill down when we revise a mental model to understand something better.

Apples and Bananas

Most of us have sophisticated and robust mental models for thinking about collections of objects. We work with collections every day and we have a wealth of experience with them. These mental models are so sophisticated that we can use them to think about apples and bananas in baskets even if we have never worked with apples and bananas in baskets before. By leveraging and building on an existing sophisticated mental model, and revising that mental model by applying it in a series of increasingly complex scenarios, students are able to construct a sophisticated understanding of the distributive property in one lesson.

Want them to start from the inside and work their way out? Want them to start from the outside and work their way in? How about starting somewhere in the middle? No problem. Their mental model is flexible and sophisticated enough to handle it.

I have taught this lesson to hundreds of students and, within 30 minutes and without exception, every one of them has been able to simplify expressions like the one below, and explain and justify their reasoning while doing it. Students are remarkably capable when applying sophisticated mental models. They are unsurprisingly less capable when applying mental models that are new and naive.

Scalability and Leverage

A vertical curriculum is designed around mental models that scale and have leverage. Mental models that scale have room to grow and can be revised and applied over many years. Mental models that have leverage can be applied in many contexts and used to learn new things.

Noted physicist and educator Richard Feynman once wrote: “If, in some cataclysm, all scientific knowledge were to be destroyed, and only one sentence passed on to the next generations of creatures, what statement would contain the most information in the fewest words? I believe it is the atomic hypothesis (or the atomic fact, or whatever you wish to call it) that all things are made of atoms—little particles that move around in perpetual motion, attracting each other when they are a little distance apart, but repelling upon being squeezed into one another. In that one sentence, you will see, there is an enormous amount of information about the world, if just a little imagination and thinking are applied.”

In Chemistry from the Ground Up, we drill down into the atomic hypothesis by investigating how and why molecules attract each other, and build up by leveraging our revised mental model to construct sophisticated understandings of states of matter, evaporation and condensation, phase transitions, chemical reactions, solubility, and cell chemistry on top of it.

Mental models go from yellow to green
to blue as they increase in sophistication

In a series of lessons I call Applying Statistical Mechanics to Model Dynamic Systems with Net Flows down Gradients, we apply the atomic hypothesis by drilling down into diffusion and modeling particles moving around in perpetual motion with pennies. Instead of leveraging an existing mental model that is already sophisticated, as in Apples and Bananas, this time we are taking a naive mental model and revising it over time so that it becomes sophisticated.

As we drill down and unpack diffusion, our understanding deepens and our mental model gains new dimensions and detail. The process of drilling down makes a model less opaque as we begin to detect and understand the underlying mechanisms. From there, we then use the mental model we have constructed to understand diffusion as a shared foundation to build up and construct a more sophisticated understanding of electrical circuits, fluid flow, and heat transfer. Along the way, we revise our mental models dozens of times and, because these mental models are constructed on top of one another, each time we revise one of them, we revise them all.

Journeying to a New Mountain

Revising a mental model can be scary and difficult. The analogy I use is journeying from one mountain to another.

Imagine that you are on a mountain. This is your existing mental model. Life is pretty good on top of this mountain, and it took a lot of hard work to get to where you are now. In the distance, you think you see a taller and better mountain, but you can’t be sure. Between you and this new mountain is a treacherous valley. Not only is the terrain in the valley incredibly difficult to traverse, the valley is also enshrouded in fog. Once you descend into the valley, you lose all sense of direction and can easily get turned around.

Providing students with opportunities to revise their mental models is not enough. We also have to encourage them to descend into the valley. Reluctant learners are reluctant to descend into the valley because the journey looks risky (it’s difficult and dangerous, and we might not make it) and the rewards uncertain (is the new mountain really taller and better than the current mountain?).

To make a journey safer, pick a mountain that is close and a path where the terrain is easy. In Applying Statistical Mechanics, we start by modeling diffusion by flipping pennies. This model is concrete and transparent. The mechanics of the model are intuitive and make sense, and students have a “360-degree” view of everything that happens. They can watch what is happening on the macroscopic scale, drill down to the molecular scale, and connect the two. The experience is also immersive. Students are surrounded by visual data with clear and immediate feedback. All of this makes analyzing and understanding diffusion easier and more accessible.

To make a journey more rewarding, pick a mountain that is tall and has a good view of a bunch of even taller mountains. We feel pleased and satisfied when we scale a tall mountain and our mental model makes more sense, and we feel powerful when our mental model has more leverage and exciting new destinations are suddenly within our reach. Leverage can’t be theoretical; it has to be experienced. Students experience that leverage by drilling down into diffusion and then leveraging their mental model to understand electrical circuits, fluid flow, and heat transfer. But they also experience that leverage by conducting their own investigations and solving their own application problems.

Think of the destinations that we can see and reach easily from our current position as a sandbox we can explore and play in. That sandbox should expand each time we scale a new mountain because our new mountain is taller (we can see farther) and we are getting more skilled at traversing the valley (we can travel farther, faster, and over more difficult terrain). Otherwise, what’s the point? If a journey is safe but unrewarding, we may go on the first journey, but we won’t go on a second.

By guiding reluctant learners on safe journeys to mountains with leverage, we supply them with experiences and new data that can shift their risk-reward calculations and help them become active learners. Active learners actively test and evaluate their mental models, seeking out edge cases and new contexts where their mental models might fail, so that they can revise their mental models and improve their understanding. Active learners don’t avoid the valley, they descend into the valley eager to reach the next mountain.

Shifting Gears

How do we know if we are learning vertically? Vertical learners pass through a number of shifts. I call them gear shifts. A gear shift is a major expansion in our ability to perceive and pursue opportunities.

The first gear shift is the shift from reluctant to active learner. My most reluctant learners typically shift into active learners in six months. “I’m not sure I understand how to do that problem. Can I come up to the board and figure it out?” “I think I understand it but I’m not sure. Can you give me a really hard problem so that I can test myself?” I hear those two requests over and over again. As active learners, my students would take the initiative to make sure they understood what we were doing, but they still relied on me to guide them. They couldn’t navigate their way through the valley to the next mountain on their own, but they were eager to follow someone who could.

After three years, some of my students shifted gears again. They shifted from active learners to sense-making learners. They knew when something made sense and when it didn’t, and they knew which way to head in the valley even though it was enshrouded in fog. But they weren’t confident in their ability to traverse the terrain. So while they would try to steer the class in the right direction, they wouldn’t strike off on their own if the class was heading the wrong way.

The next gear shift is from sense-making learner to independent learner. In this gear, we are determined to make sense of things, typically in a specific domain. We have built up and drilled down so far, and our thinking in that domain is so sophisticated, that it nags at us when something doesn’t make sense or one of our mental models isn’t integrated and grounded with the rest. None of my students shifted to independent learner while I was their teacher, but I believe that they would have in time. Consider how far they did shift in three years or less, and from inside a school system that doesn’t nurture vertical learning. Imagine what students could do in ten years in a school system and society that does nurture vertical learning.

As we learn vertically, we grow and shift gears; and as we shift gears, we learn vertically in more facets of our lives.

Vertical Learning as Theory

• We learn by constructing and revising mental models.
• Mental models only become sophisticated and powerful once they have been applied and revised over time.
• We build up when we leverage a mental model to learn something new, and we drill down when we revise a mental model to understand something better.
• Leveraging and building up from an existing sophisticated mental model enables us to construct new understandings that are also sophisticated.
• Drilling down into a mental model makes the model less opaque as we unpack it and begin to detect and understand the underlying mechanisms.
• When we construct mental models on top of one another and revise one of them, we revise them all.
• We are motivated to revise mental models when our risk-reward analysis is favorable. Our risk-reward analysis is based on data from past experiences.
• Revising mental models is easier when underlying mechanisms are concrete and transparent, and data and feedback are clear and immediate.
• We feel pleased and satisfied when our mental models make more sense, and we feel powerful when our mental models have more leverage.
• Our ability to perceive and pursue opportunities increases as the sophistication of our mental models and our ability to revise them increase.

Vertical Learning as Design Principles

• Construct mental models that scale and have leverage.
• Enable and encourage students to revise mental models.
• Apply mental models in increasingly sophisticated contexts.
• Leverage and build on a core set of mental models over time to learn new things.
• Create learning experiences that are concrete, transparent, and immersive.
• Sequence learning experiences so that, by revising their mental models, students build sophistication and leverage for the next experience.
• Encourage inquiry and enable students to conduct their own investigations and solve their own application problems.
• Provide opportunities for students to reflect on and internalize their experiences.
• Develop learning communities where students are peers and learning is explicit and the priority.
• Use the occurrence of gear shifts to evaluate your design.

Vertical Learning as Mental Model

Vertical learning builds on a number of existing learning theories. I didn’t develop vertical learning because those theories are wrong. I developed vertical learning because those theories are too naive to help me figure out how to enable and encourage students to revise their mental models and shift gears. For the past twenty years, I applied and revised existing learning theories, building up and drilling down in an attempt to construct a more sophisticated understanding of how we learn and grow. Vertical learning is the result.

Vertical learning is very much a work-in-progress. It’s still not nearly sophisticated enough. But it has reached the point where I’m ready to talk about it. But how do you talk about a theory that’s been revised for twenty years behind closed doors? Once again, I find myself following Richard Feynman’s lead. This post is a little wordier than Feynman’s atomic hypothesis, but it serves the same purpose. Think of this post as a launching point. In this one post, you will see, there is an enormous amount of information about how we learn and grow, if just a little imagination and thinking are applied.

Drill down and build up. I’ll see you on the next mountain!