Dewey, Society, and Cognitive Development

[This post is part of a series: Piaget, Dewey, and Vertical Learning.]

Dewey makes another critical point that I also brushed aside twenty years ago. He stresses that, for education to be effective, growth must occur along two parallel tracks: the sociological and the psychological. The sociological track is where we learn by participating in society, eventually contributing to society and rebuilding it as an individual. The psychological track is where we come into our own power, growing our capabilities as an individual. When we begin our education, we don’t and can’t know who we want to be or what we want society to be, yet. We need to grow to figure those things out, and we can’t grow psychologically if we don’t grow sociologically, and vice versa.

In Dewey and Society, I wrote that a strategic, coherent, independent, active, sense-making learner will want to rebuild all of the communities and relationships in his life, that he will want to change society. I’m sure that some of you balked at that: “Hey! Not everyone wants to change society. People have different interests, goals, and passions in life.” While that is true, it is also true that you cannot know all of your future interests, goals, and passions based on where you are right now. Are you happy with society as it presently exists? Is there anything you would change about society if you could?

Most of us feel as though we can’t, on our own, change society; we lack the capability. So, we don’t try. And to avoid any cognitive dissonance that might come from that decision, we tell ourselves that we aren’t that interested in changing society. But how can you know whether or not you want to change society if you haven’t even tried to envision what those changes might look like? Our sociological growth is limited by our individual capacity. We won’t consider trying to change society if we don’t think we have the capacity. And, if we can’t change society, we aren’t full members of society, so we hold back from participating and investing in society. But our psychological growth is also limited if we aren’t allowed to change society. If we aren’t allowed to change society, then we limit our aspirations to what society does allow. We dream smaller, and that limits our growth as individuals.

In My Pedagogic Creed, Dewey writes: “I believe that the psychological and social sides are organically related and that education cannot be regarded as a compromise between the two, or a superimposition of one upon the other.” He warns against letting one track take precedence over the other. I ignored that warning twenty years ago, and I ignored it again when I wrote An Introduction to Vertical Learning two months ago.

I define vertical learning as a process that takes us through a series of five developmental stages: from reluctant learner to strategic, coherent, independent, active, sense-making learner. I haven’t defined those stages yet (that will have to happen in a separate post), but those stages represent cognitive development, or what Dewey would describe as the psychological track, where we grow into our own power. In defining vertical learning, I didn’t discuss a parallel sociological track. That’s not because one doesn’t exist—my students wouldn’t have gone vertical unless I also nurtured their sociological growth—but because I didn’t recognize that psychological growth cannot occur without sociological growth. (By definition, a strategic thinker is operating in a social context and a full member of society.) I plan on rectifying this in the near future.

Dewey and Society

[This post is part of a series: Piaget, Dewey, and Vertical Learning.]

When I read Dewey twenty years ago and he said that education is a social process, that made sense to me. While we can and do learn on our own, we learn more effectively when we interact with other people. It opens us up to new experiences and viewpoints that can generate cognitive dissonance, which causes us to revise our mental models and learn.

Consider the game Minecraft. Millions of people play Minecraft and learn from each other every day. Now, imagine that the social component of Minecraft did not exist: you could still play Minecraft, but you couldn’t share what you were doing or know what others were doing. People would still play Minecraft, but probably not as many. People would still learn to do new things in Minecraft, but again, probably not as many. Our education in Minecraft is dramatically enhanced when we are an active member of a larger community, even if that community is just one other person.

But Dewey took his thinking much farther than that. He didn’t just say that individuals learn by participating as members of a society; he said that individuals must also participate in creating that society. In My Pedagogic Creed, Dewey writes: “In sum, I believe that the individual who is to be educated is a social individual and that society is an organic union of individuals. If we eliminate the social factor from the child we are left only with an abstraction; if we eliminate the individual factor from society, we are left only with an inert and lifeless mass.” Educating ourselves to participate in a society that we cannot change is a pointless exercise. We aren’t full members of a society we cannot change.

Twenty years ago, I didn’t know what to do with that line of reasoning. I couldn’t translate it into classroom practices, so I assimilated it and brushed it aside. Today, it makes sense to me. For learning to go vertical, you need to be in a state of constant cognitive dissonance. To tolerate a state of constant cognitive dissonance, you need to experience continuous growth and increasing agency. It takes a very special and a very rare community to support and nurture that. If you spend a lot of time with people who constantly make you think and help you grow and do, you notice. And if you are inquisitive, you wonder about it. What makes this community and these relationships different? Why aren’t all of my communities and relationships like this? How can I make all of my communities and relationships better?

Can we learn vertically if we aren’t members of a community that we can change? Of course. You can learn vertically on your own. You can learn vertically using a community as a resource, drawing from it what you need. But if you are participating as a member of a learning community that nurtures vertical learning, eventually you are going to want to change it… and society in general. When you are a reluctant learner, you may ignore the differences between the vertical learning community and the other communities in your life. But those differences will generate cognitive dissonance once you are an independent, active, sense-making learner; and you will want to rebuild your communities once you are a strategic, coherent, independent, active, sense-making learner. If you sense that changing the community is not an option, it places a ceiling on your potential growth and agency, and you won’t tolerate the state of constant cognitive dissonance required to go vertical. In fact, the community will cease being a vertical learning community at all.

Cognitive Dissonance, Agency, and Growth

[This post is part of a series: Piaget, Dewey, and Vertical Learning.]

Consider cognitive dissonance for a moment. You have a theory that you think is accurate. But now you have to admit to being wrong: your theory is inaccurate. You may have held that theory for years. Other theories may be based on it. It may be central to your world view. It may have taken years for you to develop and you may be immensely proud of it. And now you have to let it go.

When Alec prompted me to reexamine Piaget and Dewey, and I realized that I needed to revise my understanding of their theories, I was happy. I’ve learned to recognize cognitive dissonance as an opportunity for growth and new learning. But that is a learned response. At a much deeper level, cognitive dissonance is still a powerful blow to the ego. Alec and I get together every few months, and I enjoy our conversations because he challenges me to think in new ways. But if he was challenging me daily or weekly, I might grow to resent it. I have resented it. I resent it when I feel like I’m not challenging him in the same way, or when I feel like revising my thinking isn’t enabling us to interact at a higher level. In other words, I resent it when I feel like a little kid learning at someone’s knee and I’m not making much progress.

Alec and I are peers, so that affects how I experience that relationship. Sarah has been my coach for two years. Even though there is no expectation that she will learn from me as I learn from her, it is important to me that she has. But more important is my sense that I’ve been able to internalize the things that I’ve learned, which enables us to take on work that has a much wider scope and impact.

When I first hired Sarah, I had concrete goals in mind. For example, I had the opportunity to pitch one of my ideas to a hundred people in the edtech community. I wasn’t that worried about the pitch itself, but I was scared of the networking that would follow. Sarah suggested that I take some time to think about who I wanted to be at the event and to visualize what that would look like. I then made a plan for myself. While the evening didn’t go exactly according to plan, it went well. At past networking events, I would try to decide what to do in the moment. As the wheels in my head spun, I probably looked like a deer in headlights. By developing a plan ahead of time, I took a lot of thinking (and second guessing) out of the equation and let my reflexes take over. I felt active instead of passive, and this carried over into the energy I brought to each interaction.

If all we had done is work through my concrete goals in a year, I would have considered my time with Sarah very well spent. But as we worked together, I was emboldened to stretch farther and farther, to goals that I had put on the back burner years ago and to goals that I had never even admitted to myself. After working through less than half of my concrete goals, I knew that I had the tools to finish those goals myself and I started setting my sights on bigger targets.

For decades, I have known that I’m at my best when I’m in the moment. I can be very analytical and tend to over think things, hampering my ability to act. When I can turn off that part of my brain for a little while and just be, I’m smarter and happier. There’s always been some tension between the spontaneous person I envision myself to be and the work that I’ve done with Sarah to be more intentional and to use pre-planning to break away from default patterns. I haven’t experienced cognitive dissonance because I’ve viewed this work as transitional: in the future, once I’ve broken old habits and established new ways of being, I’ll be present with myself enough to make all of my decisions in the moment. But recent experiences have made me question that.

It is hard to be in the moment when thinking about everything at the same time. It is hard to work on this blog post if I’m thinking about whether or not blogging is my best, most strategic use of time. And it’s probably not the best time to re-open that discussion when I’m in the midst of struggling with a difficult thought or passage. So how do I know what to think about when?

A week ago, Sarah gave me a little nudge that made me see things differently. What if it’s not about my conscious mind knowing when to plan and when to be in the moment? Maybe it’s about integrating the different parts of myself. At times, I feel like the different parts of myself are battling for dominance. This cacophony makes it hard to function and I rarely feel at peace with my decisions. I’m often happy with an experience after I’ve had it, but almost never happy going into it. This battling takes a huge toll on me. But maybe the parts of myself are battling because they don’t trust that they will get what they need when they need it? If I feed the parts of myself whenever they stand up, maybe they will be happy and learn to manage themselves. This is a new line of thinking for me, but it makes sense. I’m running some experiments to see how well my experiences fit this new mental model.

I wouldn’t have been receptive to Sarah’s nudge if we hadn’t built trust over the years. But I wouldn’t have trusted Sarah if I didn’t sense that she was capable of nudging me in the right places. Being in a state of constant cognitive dissonance is difficult, even when you know that it is good for you. I would say that it is impossible unless you can sense yourself growing and taking on more things. If there was any hint that I was dependent on Sarah and not growing, I would resent and reject her nudges, and our coaching relationship would break down. To be in a state of constant cognitive dissonance, we need to experience agency and growth.

Piaget and Cognitive Dissonance

[This post is part of a series: Piaget, Dewey, and Vertical Learning.]

One of the core principles of vertical learning is that we learn by constructing and revising mental models. This comes straight out of Piaget’s constructivist learning theory. When I entered the teaching profession in 1995, constructivism was all the rage. Well over half the math teachers I met self-identified as constructivists. But then TERC Investigations and CMP happened, igniting the math wars. The public backlash was enormous, and constructivism was an early casualty.

The failures of TERC Investigations and CMP had nothing to do with constructivism itself. Constructivist learning theory is a theory for how we learn, not a theory for how we should teach. We construct our own understanding whether we are playing with blocks or sitting in a lecture. One learning experience is not inherently superior to the other.

So, what does constructivist learning theory tell us?

• We construct our own understanding by developing internal theories (mental models or schema) to explain our experiences.
• Two people having the same experience may not use or develop the same mental models to explain it. We explain our experiences differently.
• We force new experiences to fit into existing mental models using a process called assimilation.
• We experience cognitive dissonance when we notice discrepancies between our experiences and our mental models.
• If we experience cognitive dissonance, we attempt to revise our mental models to better fit our experiences using a process called accommodation.

Imagine that you have a friend who believes that Democrats always vote to increase the size and power of government. Now, imagine that this mental model is naive: your friend believes that there are no exceptions to his theory even though there are clear cases where his theory does not line up with reality. When your friend is confronted by a discrepancy, he can either deny the discrepancy (assimilation) or note the discrepancy and revise his theory (accommodation). If he is able to revise his mental model so that it fits better and explains more, then his mental model has become more sophisticated.

I believe that the primary role of a teacher is to encourage us to actively test our mental models as widely as possible, and to take note of discrepancies where they exist. If we experience cognitive dissonance, then we will feel compelled to revise and improve our own mental models. But do we need teachers in this equation at all? Can’t we test our models and find discrepancies on our own?

As we grow as learners, we get better at testing our models and detecting discrepancies. Where reluctant learners avoid cognitive dissonance, independent active learners seek out and embrace cognitive dissonance. But I don’t think we ever reach a point where we can see everything ourselves. We may not need a formal teacher, but we do need to interact with people who can point out our blind spots. For example, I thought I knew Piaget and Dewey well. It wasn’t until my friend Alec gave me a nudge that I questioned that belief and uncovered that I didn’t know Piaget and Dewey nearly as well as I thought. I don’t think I was being defensive and avoiding discrepancies; I just thought I was too busy to go back and double check every little thing. Now I’m glad that I did.

Piaget, Dewey, and Vertical Learning

I had dinner with my friend Alec a few weeks ago. After reading An Introduction to Vertical Learning, he asked me how I saw vertical learning sitting relative to Piaget and Dewey. Jean Piaget was a Swiss psychologist and philosopher who believed that we generate meaning through the interaction of our experiences and ideas, and that cognitive development occurs through a series of stages. John Dewey was an American psychologist and philosopher who had a profound belief in democracy. He believed that education and learning are social and interactive processes, and that all students should take an active role in their own learning.

Vertical learning is firmly grounded in Piaget’s constructivist learning theory. It is designed to encourage students to become independent active learners by seeking out and embracing cognitive dissonance. Dewey’s influence on vertical learning is less direct. Like Dewey, I believe that the purpose of education is not to acquire a pre-determined set of skills, but to realize one’s full potential. On the surface it may have appeared as though my students were studying math and science, but they knew that they were actually building their capacity to learn in general. Without that sense of relevance, they would have never taken ownership of their own learning and gone vertical. You could say that I was using Piaget’s engine to drive us to Dewey’s destination.

Alec’s question prompted me to brush up on Piaget and Dewey. Ideas that I thought I knew well took on new meanings once I examined them again through twenty years of experience and a more sophisticated set of mental models. This cognitive dissonance is causing me to revise my own thinking once more. This is still a work in progress, but here are some of my recent thoughts on Piaget and Dewey.

Piaget and Cognitive Dissonance
Cognitive Dissonance, Agency, and Growth
Dewey and Society
Dewey, Society, and Cognitive Development
Vertical Learning and Constraints

On My Birthday

I celebrated my 46th birthday two weeks ago. I haven’t celebrated my birthday on my birthday in over ten years. I usually have dinner with my family on the weekend. We’ll have a cake, and my niece and nephew will make me cards, but my birthday is folded in with Mother’s Day and I am resolutely not the center of attention. I usually spend a day with my friend Kerry some time in May. Our birthdays are both in May, and we use the occasion to slow down, hang out, and catch up. But nothing on the day itself.

In the weeks leading up to my birthday, well-wishers inevitably ask me what I’m doing that day. I’ll deflect and say that I’m too old to celebrate my birthday, that getting one year older isn’t something that I want to celebrate. They’ll tell me that I’m being silly and that I should do something fun. I’ll smile and shake my head ruefully. But this year was a little different. Relatives flew into town for a graduation, and I don’t know if it’s because they grew up in Mississippi, but they were shockingly warm and caring. The number of people insisting that I do something special on my birthday doubled, and it must have had an impact on me because, when I went to bed the night before my birthday, I suddenly wanted to mark the occasion. How? I had no idea.

In the morning, I woke up to find a lengthy email from my sister Carol, and a half dozen birthday wishes posted on my Facebook wall. Over the course of the day, a few more emails and another half dozen Facebook posts came in. I also got a phone call from my parents. I had the second of two birthday cupcakes from my sister Lisa for breakfast. Then at around noon, I sat down to work at my computer. I had recently taken on two freelance projects, and both projects were coming down to the wire. Deadlines were tight and team members were depending on me. I worked until 7:50 PM, taking a few lengthy breaks for lunch, to go on a walk, and to clean my kitchen. After work, I had dinner, responded to everyone who had sent me birthday wishes, took a midnight stroll, and then wrote in my gratitude journal.

As I sat down to write in my gratitude journal, I felt a warm, radiating sensation in my body. I was glowing with happiness. When I tried to identify the source of my happiness, the first thought that popped into my head was that I felt “well-taken care of.” What did that even mean? As I probed deeper, a second thought popped into my head. This second thought was so far outside my normal way of being that I was reluctant to commit it to paper. But I did. I wrote that I felt “loved and well-connected.”

Let me tackle the second thought first. A few months ago, I reconnected with an old friend through Facebook. Zmira is Israeli. We worked together at the Jewish Community Day School and, even though she was the visitor in a foreign land, she sort of adopted me. The whole community at JCDS sort of adopted me. I was constantly being invited over to Shabbat dinner by people whom I barely even knew as work friends. But when Zmira returned to Israel and I left JCDS, we fell out of touch and I disconnected from the JCDS community. Reconnecting with Zmira through Facebook restored this old network. And this old network didn’t know or care that I had stopped celebrating my birthday. Old friends just posted birthday wishes on my wall. But here’s the weird thing. Current friends who had stopped trying to send me birthday wishes over the years were suddenly posting on my wall, too. It was this sudden and completely unexpected outpouring that made me feel incredibly loved and well-connected.

Now back to the first thought. It is hard to reconcile feeling “well-taken care of” while working on my birthday, going through my regular routine, and not speaking to a soul all day other than my parents. When I woke up that morning, I was concerned that my little voice was going to throw a tantrum because I decided to mark my birthday the night before, but I had already committed to a full day of work in my weekly plan. My little voice was going to be upset no matter what I chose to do. So, I woke up that morning determined to be extra kind and attentive to my little voice. I let him know that his needs took priority. I planned to work, but we would spend time together doing a few special things, and I was prepared to ditch work at a moment’s notice.

I am normally a bit dismissive and disdainful of my little voice. He shows up to nag me at the worst possible times. So, my concern and attentiveness must have been disarming because he agreed that working made sense as long as I took frequent breaks and promised to stop working by 8:00 PM. Together, we outlined some of the special things that we would do together. The top priority was responding to each birthday wish with warmth and gratitude. We really looked forward to it. We also planned an extra long midnight stroll and a special birthday dinner. My parents had given me spareribs and two beautiful fresh green peppers. I love green peppers, but hadn’t prepared them with black bean and garlic sauce in ages. It was the perfect comfort food for my birthday. With the day mapped out, my little voice was content, but I was still concerned. So I checked in on him constantly, maintaining a running dialogue with him as I worked. This made my little voice very happy. And I was happy, too. We both felt “well-taken care of.”

There are a few things that I took away from my birthday celebration. The first is that I’m rarely present with myself. I’m happiest and at my best when I’m present with other people, especially kids. But when I’m by myself, I associate being in the moment as getting lost in the moment and entering a flow state. When I started my midnight strolls, going for a walk at 3:00 AM on a warm summer night in a t-shirt, shorts, and sandals, I wrote about the serenity I feel as my senses open wide to take everything in. But I am there, too. Noting my own pleasure makes me smile and laugh inside. I need to be present with myself more often.

The second thing that I took away was the need to acknowledge the positive things that I have in my life. The love and connections represented by those birthday wishes didn’t just materialize out of thin air. They existed the day before my birthday, too. But I hadn’t acknowledge them until they manifested themselves in an unmistakable way on my birthday. What if I acknowledged and experienced that love and those connections on a daily basis?

My third takeaway is something that I’ve been working at for a long time, but I still have a long way to go. One of my daily practices is writing in my gratitude journal. For 15 minutes a day, I am present with myself as I acknowledge the things that make me grateful. I am grateful for many things, but I tend not to acknowledge the role that I play in any of it. I noticed that I was doing this again as I was writing the fifth paragraph of this blog post. I had a hand in creating those connections, too. Zmira and I adopted each other, and the JCDS community and I embraced each other. It wasn’t by accident that they decided to make the secular math teacher the 7th- and 8th-grade homeroom teacher for the school. I can be a bit odd, but I’m a good friend. Sometimes I need to acknowledge how awesome I am.

Happy birthday, kid! Be good to yourself and stop deflecting! :)

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!