A Recursive Look at Vertical Learning
Constructing a grand unified theory for learning based on Dewey, Piaget, and Papert
The two core mechanisms in vertical learning are drilling down and building up. Drilling down is the process of anchoring/grounding a theory on top of more concrete, intuitive theories. Some people describe this as first principles thinking. Building up works in the other direction—it’s the process of scaling and leveraging lower level theories to construct a higher level theory. Instead of constructing a series of theories in isolation, we drill down and build up to construct theories integrated in stacks, enabling us to climb up and down the abstraction ladder and identify any holes/blindspots which may be hidden in those theories.
But what is vertical learning, you ask? Great question. Vertical learning is the development of habits of mind we use to become self-actualized, a state in which we live purposefully and in full alignment with our own coherent set of core values, reconstructing ourselves and the world around us as necessary. Believe it or not, kids can learn vertically in school as they study math, social studies, and other traditional subjects by, you guessed it, simply drilling down and building up. Of course, kids can also learn vertically outside of school or in schools with a more progressive, child-centered curriculum. The what and where are less important than the how.
Now, I’d be crazy to assume you’re prepared to accept my statements about vertical learning at face value. If you’re not skeptical, you should be. So far, I’ve presented a few abstract concepts—drilling down, building up, vertical learning, and self-actualization—without connecting any of them. How does drilling down and building up lead us to self-actualization? To answer that question, I’m going to show you how I’m using vertical learning to construct vertical learning theory. Yeah, it’s turtles all the way down!
First contact
My first introduction to John Dewey, Jean Piaget, and Seymour Papert was in the mid-1990s at the start of my teaching career. I remember reading Dewey’s The Child and the Curriculum and Democracy and Education while earning my MAT in math education at Boston University—and reflexively dismissing him as irrelevant. I read Papert’s Mindstorms: Children, Computers, and Powerful Ideas a few years later after encountering a teacher still using Logo. Although Papert’s ideas were inspiring and thought-provoking, it wasn’t at all clear how I could apply them in a more traditional school setting.
Of the three, Piaget had the clearest, most immediate impact on my thinking and practice. Dewey, Piaget, and Papert were all constructivists who believed we actively construct our own mental models/internal theories of the world through our experiences. This means we can’t simply pour knowledge into a child’s brain; we have to provide children with the experiences they’d need to construct meaning and understanding for themselves. According to Piaget, we learn and make sense of the world using two cognitive processes: assimilation and accommodation. We assimilate when we fit new experiences into existing mental models, and we accommodate when we revise existing mental models to fit new experiences. Therefore, I believed it was my role as an educator to encourage students to revise and improve their mental models, making those models more sophisticated and robust over time. But how? To help make the revision process more concrete for students, I decided we would start revising our mental models by drilling down and building up.
Second contact
In May 2016, I joined Medium and used it as a platform to publish Why We Should Learn Vertically. It was my second attempt to explain vertical learning as a theory. As you can see, while I was building on top of Piaget at the time, there was no sign of any influence from Dewey or Papert.
When Alec Resnick, a good friend of mine, asked me where vertical learning theory stood relative to Dewey and Papert, I realized I didn’t have a good answer for him. At his prompting, I went out and read Dewey’s My Pedagogic Creed and re-read the first few chapters of Papert’s Mindstorms. It quickly dawned on me that I’d badly misunderstood Dewey and Papert two decades ago. Then, in January 2017, Derek Breen, another good friend, invited me to the Thinking about Thinking about Seymour symposium in honor of Papert at the MIT Media Lab, where I was blown away by Alan Kay’s talk on powerful ideas; and in June 2017, I re-read three more of Dewey’s books after Howard Johnson pointed me to two articles on Dewey and Lev Vygotsky.
Going vertical on vertical learning theory
A few years ago, I attended a workshop run by a well-known instructional designer. Asked which learning theory he subscribed to, he declared himself agnostic. “Use whatever learning theory you think works best for the given situation,” he said. As a vertical learner, I was appalled. If I were regularly applying multiple theories in the same problem space, I’d feel driven to try and integrate those theories into some kind of coherent framework as soon as possible because I’d know that I didn’t really understand them until I did.
My drive to integrate theories operates on two levels. Subconsciously, it’s a habit I’ve developed over many years. Left on its own, my brain would work at integrating vertical learning theory with Dewey, Piaget, and Papert whenever it had any spare cycles—brushing my teeth, doing the dishes, out for a run. At this stage, it’s doubtful I could ever turn off that part of my brain. Consciously, I integrate theories for the same reasons scientists are eager to unify general relativity and quantum field theory: (1) everything we’ve learned about the universe so far tells us a unified theory must exist; (2) constructing a unified theory would be immensely satisfying and seems to be within our capabilities; and (3) a unified theory would change everything—our understanding of the universe as well as what we’re able to do within the universe. Unifying Dewey, Piaget, Papert, and vertical learning would functionalize those theories and enable us to apply and build on top of them in ways that haven’t been possible before now.
Reconstructing experiences
The first step in my process was drilling down into Dewey and Papert until I’d found a common foundation grounding both sets of theories. Without a common foundation, it’d be impossible to align and then integrate them into a coherent framework. I started with Dewey and Papert because they were both operating at similar abstraction levels, whereas Piaget appeared to be lower in the stack.
According to Dewey, we learn by reconstructing past experiences so we can apply them in new experiences. I got a jumpstart on learning how to navigate the streets in and around Boston by riding in the back seat of my dad’s car. My dad worked from 3pm to 3am, seven days a week, managing a small chain of Chinese takeout restaurants spread throughout the suburbs of Boston. Since finding quality time with him was a challenge, I treasured any opportunity to ride with him as he ran errands or dealt with work emergencies.
Once I started driving myself, I had dozens of my dad’s favorite routes stored in my brain. But I wasn’t recalling those routes as I initially experienced them. As a passenger, I never paid much attention to the various roads and turns my dad took; I was far more interested in watching people through the window, pointing out cool cars and weird buildings to my brother, and tuning into the ballgame on the radio. It was only years later that I reconstructed those earlier experiences, pulling out and focusing on the key navigational details I needed to retrace my dad’s steps.
I reconstructed my experiences, this time as a driver, yet again when I started to estimate travel time on familiar routes based on traffic conditions and time of day. Again, I did that not by recalling specific trips in the same traffic and at the same time, but by compiling multiple trips together, identifying patterns, and developing rules of thumb. This trip takes 35 minutes longer in rush hour traffic. Since it’s not rush hour yet, traffic on Route 2 will be dense but steady, though I may get stuck at the Concord rotary for a little while—let’s factor in an extra 15 minutes. By reconstructing our experiences, we’re able to draw on them in more flexible and useful ways.
Even though I’ve been driving around Boston for over 20 years now, it’s clear I’m not reconstructing my driving experiences in the same way as my dad. For example, I may know how to drive to my parents’ house in Brookline and how to drive to the Burlington Mall, but I’m not sure how to drive from Brookline to Burlington. I’ve reconstructed my driving experiences to the point where I can think of a few ways to get from Brookline to Burlington by combining legs from other routes I do know, but which of those ways is quickest? I struggle at estimating travel times on complex routes I’ve never driven before from end-to-end. My dad, on the other hand, excels at putting together and comparing new routes in his head. If he asks for directions, and someone gives them to him, he’ll immediately counter with an alternative route of his own — igniting a lively discussion, not unlike a couple of guys debating who would win if two sports teams from different eras played one another.
My dad’s uncanny ability to estimate travel times on routes he’s never driven before isn’t derived from his greater driving experience; it’s derived from his ability to make more effective use of the experiences he does have. Sure, because he spent time exploring Boston by car in his youth, while I relied far more on the carefully curated routes he showed me, he knows the back roads of Boston better than I do. But that only means he can think of more possible routes than I can, not that he knows his routes better than I know mine. If I had to, I could draw one of my possible routes down on paper, estimate the drive time for each leg, and then add those times together to get a total travel time. While my estimate might have a large margin of error, it’d still be better than nothing.
Since I clearly have the experience and knowledge to estimate travel times on possible routes, why am I not doing it? It’s because I lack automaticity; I’m not structuring and storing my experiences in such a way the knowledge I need is easily accessible to me on the fly—whereas my dad is.
Summary: We learn by reconstructing past experiences so we can apply them in new experiences. We learn more effectively when we reconstruct our experiences in ways that are more useful to us in the future.
The continuity of experience
Does this mean my dad was born a better driver than I am? For Dewey, that’d be a pointless, and perhaps even misguided, question to ask.
The central idea in Dewey’s thinking is the continuity of experience. Imagine, immediately upon being born, you set out for a walk. Every encounter along the way changes you to some degree, causing you to re-think wherever you’re heading next. Maybe you hear about an exciting new destination, discover a new interest, or figure out how to pass a previously impassable obstacle. Forty years later, could anyone say where you started from based on where you are standing now? Does it make any sense to try to analyze where you might have ended up if you hadn’t been influenced by anyone or anything along the way?
Every experience changes us, leaving us a slightly different person afterward and altering how we experience life moving forward. When my dad landed in Boston as a young man, he didn’t know his way around and he didn’t have anyone who could guide him—so, he learned to ask for directions. But simply arriving at his destination didn’t satisfy him. Was there a faster, more efficient route he could’ve taken? By asking several people, he learned different routes, which he then tested by driving and comparing travel times. Later, after learning his way around, he needed directions less often. He still asked people for directions, but only to nail down his destination’s location. Then, he’d use his internal map to generate a few possible routes to test. By the time I was old enough to ride with him on errands, he didn’t even need to drive routes to test them anymore, he could do that all in his head.
When I started driving, I already knew my way around. I’d inherited a map with dozens of the best possible routes laid out for me to use. It was pointless looking for shortcuts to shave off a minute of travel time here and there—if a shortcut existed, my dad would’ve already discovered it. While I still paid attention to travel times, it was only to plan for how long it would take me to reach my destination. My sense of time wasn’t very fine-grained and I didn’t think much about optimizing my routes. Even if I had to come up with a new route because I was going somewhere new, the first route which was merely good enough became my default. Spending time trying to nail down the best route possible wasn’t something I’d ever done—and it wasn’t something I’d ever seen my dad do either. My dad also drove the same routes over and over again: the routes he’d optimized in his youth.
Summary: Every experience changes us, leaving us a slightly different person afterward and altering how we experience life moving forward. One series of experiences may propel us in one direction; a different series of experiences may propel us in another direction. Experiences impart momentum, which can build over time.
Habits of mind and educative experiences
When I moved out to Acton, a suburb 20 miles northwest of Boston, I found myself suddenly living in unfamiliar territory. I didn’t know the way to the supermarket, the bank, the hardware store, the drugstore, or any of the local parks or restaurants. It was an opportunity to explore a new home town on my own from scratch. Twelve years later, I’ve discovered how to get to all of the places I need to go, but I’ve only explored about 25% of the back roads in the area, and I’m still more comfortable taking the routes I know best rather than the road less traveled. Why didn’t my experiences in Acton shape me as my dad’s experiences in Boston shaped him?
In some ways, living and driving in Acton represented a fresh start for me—a chance to reboot myself. For example, furnishing my own condo for the first time, after years of blindly accepting mismatched hand-me-downs from older siblings, enabled me to cultivate my own tastes in home decor. But when it came to driving, I had developed considerable momentum before ever moving to Acton. Shaped by my experiences driving in Boston, I was already inclined to prioritize comfort and familiarity over exploration and optimization. While I could undo that inclination, it would take time and focused effort; I’d have to consciously choose to act and think differently over and over again until a new inclination had been established.
Dewey described these inclinations, or ways of thinking, as habits; my coach, Sarah M. Kipp, calls them default patterns. Despite the negative connotation, Dewey felt habits could be good or bad. My preference for familiar routes is a habit, but so are my dad’s preference for optimization and his attention to travel time. We rely on habits when we need to act quickly and when we don’t have enough information to make an informed decision. Habits help guide our unconscious minds—what we pay attention to and focus on, how we filter and process sensory information. For Dewey, a good habit serves us whereas a bad habit controls us.
One habit I’ve wrestled with most of my adult life is my perfectionism. I have extremely high standards for myself. In many ways, these standards have served me well, propelling me to who and where I am now. But I experienced the dark side of perfectionism when taking the AP US History exam in high school. The essay topic was a lay up: “Describe the causes of the American Revolution.” It couldn’t get any easier than that, but I froze and never got past a partial outline. That experience was seared into my brain, traumatizing me. I’d be writing a paper for school or a report for work years later—and then my mind would go blank and I’d shut down: writer’s block.
But the writer’s block was just the most visible tip of a much larger iceberg. Even though I exulted in my drive to probe deeper, think more creatively, and solve problems no one else would even consider taking on, I worried I might never be satisfied—or happy. Was I putting myself under an unhealthy level of stress? In hindsight, after reconstructing my experiences, I can also see how perfectionism and an unwillingness to make mistakes both contributed to my habit of choosing the safer and more familiar route when driving.
Are my high standards helping or hurting me? It’s a question I’ve struggled to answer—and never could until I started working with Sarah. In the last few years, under Sarah’s guidance, I’ve developed a number of new habits. One of them is trying to turn or’s into and’s. Believe me, I know how silly that sounds. Why don’t I wish for a magic unicorn while I’m at it? But the first time I tried, it worked. Then, again and again and again. Now, whenever I’m torn between taking the path to goal A or the path to goal B, I’ll remind myself to look for a way to reach both goal A and goal B. Sometimes I’ll find one, and sometimes I won’t, but I almost always find a better option—one that resolves my internal quandary.
As I apply my new habits, the answer to my question becomes obvious: high standards can be both helpful and harmful, depending on whether they are serving me or not. The key is identifying when my drive to do better is getting in my way—and then doing something about it.
Summary: Through the continuity of experience, we gain momentum in specific directions. Over time, this momentum can develop into habits of mind — ways of thinking which influence how we reconstruct our experiences and see the world. If those habits are helpful, enabling us to make better use of past experiences, then the experiences are educative; if the habits are harmful, impeding our growth and preventing us from making better use of past experiences, then the experiences are mis-educative.
A common foundation
In the first part of this article, we’ve been examining how we learn through experiences so we can ground our understanding of learning in experiences. Here’s a summary of the mental model we’ve constructed so far:
- We learn by reconstructing our past experiences. The more effectively we reconstruct our experiences, the more we learn from them—and the more prepared we are to understand and act in future experiences.
- Experiences change us, influencing how we experience life in the future. A series of experiences nudging us in a specific direction will cause us to gain momentum in that direction over time.
- As we gain momentum, we internalize that momentum as habits of mind: ways of thinking which influence how we reconstruct our experiences and see the world.
Together, these three statements represent a common foundation for Dewey and Papert. Both men believed we learn from experiences—and by reflecting on those experiences; both men also believed we grow as learners if, and only if, the surrounding culture and environment are educative and nurturing. For Dewey and Papert, the environment can’t simply act as a neutral medium in which learners grow by following their own natural instincts. No. They saw it as an essential medium from which we draw the basic building blocks needed for growth. We actually learn from the environment—not just in it. Supply us with different building blocks and we learn and grow differently.
The power and test of a common foundation is its robustness, flexibility, and intuitiveness. Do the ideas expressed by Dewey and Papert emerge naturally from this foundation? Can we apply the foundation to predict how they might react to new ideas—or each other? How fluent are we at considering the foundation from different perspectives and building on top of it? As a vertical learner, I’m not identifying and constructing a common foundation for Dewey and Papert as a mental exercise; my goal is to create a shared understanding so we can have a three-way conversation and integrate our thinking. How do I know if I’m representing Dewey’s and Papert’s sides of the conversation at all accurately? One of my well-honed habits is an ability to sense inconsistencies. I’ll know if I’m being consistent or not once the three-way conversation begins and I start building on top of the foundation.
I should also note that, even though I’m describing these three statements as foundational, they do not form the lowest level in my mental model. Later in this article, I will drill down to the cognitive processes below this foundation, and then down to the neuroscience below that. I consider the reconstruction of experience and the development of habits of mind as foundational because this is the lowest level most of us can apply intuitively. Since I’m less familiar with the neuroscience, trying to think at that level would be more error-prone and difficult. It’s better for me to operate at the level of experience and habits of mind—and only climb down the abstraction ladder as necessary.
The purpose of education: self-actualization
Dewey believed the purpose of education is to enable us to grow as learners, so we can grow to understand ourselves and the world around us—and thus act with purpose and intelligent foresight. Because his definition of education is open to misinterpretation, depending on our beliefs about what people are capable of doing, Dewey shared a number of benchmarks so we’d know if we’re on the right track or not. For example, we are growing as learners if we develop into polymaths, experts in more than one discipline, who are able to reconstruct those disciplines to solve new problems.
Dewey isn’t saying that the purpose of education is to create polymaths; he’s merely pointing out that we become polymaths if we do grow to understand ourselves and the world around us — and thus act with purpose and intelligent foresight. Dewey sees disciplines, such as arithmetic, geography, and botany, as powerful tools which are part of our sociocultural heritage. The only reason the human race would continue honing and passing down a discipline across generations is if it found that discipline especially useful. So, if we’re trying to understand ourselves and the world around us, of course we’d pick up and use the tools which have been honed and used successfully in the past.
However, Dewey is clear that an educated mind masters its tools—and is not mastered by them. If you’re overly reliant on a tool and can only use that tool in one way, the tool becomes harmful because it limits how you see the world and the choices you’re able to make: If all you have is a hammer, everything looks like a nail. On the other hand, if you understand when to use a tool and can adapt the tool to fit new situations and solve new problems, the tool now becomes helpful. This is why Dewey believed that, if we continue growing as learners, we will eventually master multiple disciplines, and then reconstruct those disciplines to solve problems never encountered before.
In Mindstorms, instead of identifying education as a discrete process, Papert argues that a healthy culture will nurture our growth as learners naturally. After critiquing our current culture as mathophobic, where math is the Greek stem for learning and mathophobia is the fear of learning, Papert describes a healthy culture which enables us to: (1) think about our own thinking as we do and learn; (2) access disciplines via multiple pathways; and (3) construct the underlying intellectual structures we need. The first is a habit of mind used in triple-loop learning. Any mental model we construct is almost always naive at first; our mental models only become robust and sophisticated over time as we revise them to account for more experiences. The model we use to understand how we learn and what we can learn is no different — and we can only revise it by thinking about our own thinking.
The second and third items relate to the abundance and diversity of cultural materials available in the environment. It doesn’t matter how well we learn if we lack the necessary building blocks in the environment to construct robust and sophisticated mental models. It’s interesting that Dewey and Papert both cite interdisciplinary thinking as essential in our growth as learners—but for different reasons. Dewey’s focus is on utility: it’s useful learning how to use tools and to adapt those tools to fit new contexts and serve our own purposes. Meanwhile, Papert focuses on learning itself. If the only way to access physics is through algebra, what happens if we’re not interested in algebra or haven’t learned it, yet? By putting up barriers and limiting access to physics through a single pathway, we encourage people to see physics as separate and ourselves as limited learners. Unless we can also access physics through poetry, dance, toy blocks, and other experiences or disciplines, the culture limits, rather than nurtures, our growth as learners.
Papert envisioned a culture where learning is integral to, rather than separate from, life because he saw it as healthier, not more useful. In contrast, another of Dewey’s benchmarks for growing as a learner is developing the capacity to reshape society. Dewey argues we can’t be full members of society if we’re not able to reshape that society. On the surface, it appears as though Papert favors personal growth for its own sake while Dewey favors personal growth as a means to an end. However, I see Dewey and Papert describing two sides of the same coin. According to the Merriam-Webster dictionary, self-actualization is “the process of fully developing and using one’s abilities”. I like this definition because it highlights that, to be self-actualized, we must be in the process of realizing our full potential by: (1) developing our abilities fully; and (2) using those abilities. Unless we marshal our abilities to pursue goals with increasing power and scope, we’re not self-actualized.
Papert would certainly agree that a healthy culture enables us to develop our abilities fully, and Dewey would agree that the purpose of education is to enable us to develop our abilities fully so we can use those abilities to achieve our own purposes. Therefore, although Papert might object to the adjective educative, preferring healthy instead, I will describe experiences that help us develop the habits of mind needed to be self-actualized as educative—simply because more people will find that term a bit more specific.
My Mental Model for Self-Actualization
Constructing an operational definition for education
medium.com
Summary: The purpose of education is to enable us to become self-actualized, a state in which we live purposefully and in full alignment with our own coherent set of core values, reconstructing ourselves and the world around us as necessary. Experiences are educative if they help us develop the habits of mind needed to be self-actualized.
Dewey’s path to self-actualization
Using his mental model for learning, Dewey suggests a fairly straightforward path to self-actualization: seek out educative experiences, which enable us to develop the habits of mind needed to grow as learners and act with purpose and intelligent foresight, while avoiding experiences which are mis-educative. Avoiding mis-educative experiences is important because of the continuity of experience. Remember, all experiences change us, giving us a small nudge in one direction or another. To develop a habit, we need enough of those nudges to line up so we build momentum in that direction. Even if the majority of our experiences are educative, the mis-educative experiences may still prevent us from developing useful habits just by scattering our momentum.
How do we know which experiences are educative? In today’s culture, most experiences are mis-educative, especially in schools. Classroom experiences tend to be mis-educative for three reasons: (1) they aren’t designed to enable students to become self-actualized; (2) they aren’t meaningful or relevant to students—making students passive observers rather than active participants who will reconstruct those experiences to get more out of them; and (3) they are designed by teachers, not by students, which means students are actually developing the habit of not planning and not being purposeful.
Some people argue we’re born knowing when an experience is educative, but schooling causes us to override our natural instincts. While possible, Dewey argues against that theory—and I side with Dewey. Even if we’re hardwired to play, explore, and be curious, I believe we learn how to distinguish between educative and mis-educative experiences by experiencing them. Of course, by having to sample mis-educative experiences to identify them, we run the risk of developing harmful habits. When I was learning to navigate in and around Boston, it felt natural to use my dad’s routes—it made learning much easier. I didn’t notice anything awry until I started avoiding and feeling anxious about driving unfamiliar routes. I wasn’t detecting a mis-educative experience; I was detecting a harmful habit already in action. In time, I might learn to identify a mis-educative experience before a harmful habit develops, but not right away.
Even though his definition of education has nothing to do with schools, and he thought most school experiences were mis-educative, Dewey still believed schools could serve a useful role in society. First, because of human progress, we have access to an increasing number of increasingly powerful disciplines. Where it used to be possible to learn about a wide range of disciplines simply by observing the farmers, merchants, tradesmen, and artisans around town, we need some other way to learn about and explore disciplines today. Second, because distinguishing between educative and mis-educative experiences can be difficult, having someone with more experience there to guide us is always helpful.
The person I’ve hired to guide me on my own journey to self-actualization is my coach, Sarah. I believe she exemplifies many of the qualities highlighted by Dewey. When we started working together four years ago, I was very clear: I wasn’t hiring a life coach; I was hiring a coach to help me improve my communication skills so I could pitch vertical learning more effectively. In the back of my mind, I knew there were other areas of my life I wanted to work on, but I was nervous about mission creep—I wasn’t ready for an open-ended, multiyear commitment with no clear end in sight. If Sarah had said anything remotely about self-actualization at the time, I would have been out the door.
Respecting my wishes, Sarah simply provided a forum for me to discuss my goals and my plans. What are you working on this week? Where would you like to be at the end of this call? She wasn’t there to provide feedback. If I was stuck, she might ask a question or suggest a tool I might use to figure things out for myself. If it made sense, I’d try the tool out as an experiment and then report my findings back to her next week. Over time, as I got stuck less often, my goals and plans became more ambitious, and our work deepened.
One tool/habit I’ve developed since working with Sarah is carving out time and space for thinking about and writing down my intentions and reflections. I’ve always had an interest in keeping a journal, but every time I would start one, I’d abandon it after a week or two. I considered every “failure” as more evidence journaling simply wasn’t for me—or I wasn’t capable of journaling. Then, at Sarah’s suggestion, I tried journaling again, but this time focusing on the experience instead of the end result (another new habit). What worked? What didn’t work? What did I notice? What had I learned? As I learned more and more about my relationship with journaling, I got better at designing my journaling sessions to fit specific purposes. I might journal one way this week and another way next week. I’ve even gone months without journaling at all. In the past, that would alarm me, but journaling has become a trusted tool in my toolbox—I know it’s there and I pull it out when I need it.
Nothing about my work with Sarah is formulaic. She knows me incredibly well—and her suggestions are tailored to who I am and what I’m doing at the moment. I’m sure that, when she first met me, she instantly noticed habits I’d do well to undo and habits I’d gain from developing, but she followed my lead as I decided how and where I wanted to stretch. How did Sarah know which tools/habits to suggest when? Part of it is simply being able to see me from a distance I don’t have, but another part of it is hindsight. I believe that Sarah is farther down the path to self-actualization than I am, so she has a better sense of which habits are harmful and helpful — and her mental model for learning is highly robust and sophisticated. For Dewey, this is who a teacher should be: someone who knows the student and is capable of guiding the student toward educative experiences as the student pursues his or her own purposes.
While every path to self-actualization is personal and unique, Dewey does point out a few habits to develop early on. One is making plans. Remember, if we’re not planning, we’re getting in the habit of not planning. This is why Dewey advocated for project-based learning—where students would learn to organize themselves and work toward a long-range goal. Other habits involve scientific inquiry. In order to plan, we need to be able to predict the outcomes of our actions. That means understanding the world around us by identifying variables and relationships, such as cause-and-effect, formulating and testing hypotheses, and analyzing data and drawing conclusions.
Summary: We become self-actualized if we have educative experiences and avoid mis-educative experiences. A teacher, with the benefit of greater experience and hindsight, can guide us toward educative experiences while we’re still learning to identify educative and mis-educative experiences for ourselves. Habits to develop early on include planning and scientific inquiry. However, everyone’s path to self-actualization is personal and unique—and we will only become self-actualized if we’re pursuing our own purposes.
Piagetian learning versus formal instruction
One inconsistency in Dewey’s thinking is how he separates learning into two distinct categories. In one category, there are things we learn through direct participation, without being taught, simply by doing. Papert describes this as Piagetian learning. A classic example is object permanence. Initially, infants act as though objects cease to exist once they’re removed from sight. But then, at some point, infants discover that, when a parent hides a toy behind a sofa, the toy is still there. This is a theory which all infants discover for themselves naturally by interacting with the physical world.
In the second category, there are things we learn which do require teaching. Consider how doctors have been trained across history. Centuries ago, we became a doctor simply by apprenticing with a doctor. We didn’t take classes or attend medical school; we learned by actually doctoring. While the doctor supervising us might offer some instruction, we could learn most of what we needed to know just by closely observing what the doctor did—and then gradually taking on more of the doctor’s role. Today, that’s no longer possible. Doctors now spend years studying biology, chemistry, and the human body; and while an apprentice might see a doctor reading a journal to research drug interactions or applying isolated bits of scientific knowledge in diagnosing a patient, that wouldn’t be nearly enough for the apprentice to develop the full breadth of knowledge and understanding required in a doctor.
Dewey believed that, as the human race progresses, some disciplines become so advanced that we have to resort to formal instruction to learn them. Think of it this way: modern medicine sits upon the shoulders of so many giants that a time-traveling doctor from only a hundred years ago would be utterly lost in a medical practice today. Every time we reconstruct medicine to integrate new theories and ways of thinking, it grows richer and more sophisticated; but, in order to compress all that knowledge and all those insights into a form we can learn in a single lifetime, we make medicine increasingly abstract. For Dewey, this means we have to rely more and more on formal instruction, and less and less on Piagetian learning—increasing the odds of mis-education.
While categorizing learning as concrete or abstract isn’t inconsistent with any of his other theories, Dewey is inconsistent by not grounding his categories in those theories. For example, Dewey doesn’t explain why Piagetian learning is typically educative, with or without expert guidance, when formal instruction is often mis-educative without it.
Papert’s key insight in Mindstorms is that subjects which seem to be abstract may actually be concrete under different conditions. If we study how children learn French, but only observe American middle school classrooms, we might easily conclude that learning to speak French fluently requires grit, aptitude, and formal instruction. But if we had observed children in France instead, we would have discovered the precise opposite: neither grit, aptitude, nor formal instruction are required. In fact, not learning French is far more difficult than learning French in France. A child in France is able to learn French simply by engaging in Piagetian learning for the same reason children around the world are able to discover object permanence: the environment provides numerous and diverse reasons and opportunities to do so. In other words, a subject only seems abstract, requiring formal instruction, when the environment is lacking in reasons and opportunities to learn the subject naturally.
In Mindstorms, Papert describes how children learn differential geometry as they use the Logo programming language to communicate and interact with a turtle. On the surface, those children are learning to program while applying a branch of mathematics most mathematicians don’t encounter until graduate school. To the children, however, they’re simply exploring a fun microworld, a world in which learning differential geometry is easy and natural because we experience differential geometry in every interaction.
I can’t go into as much detail here as Papert does in Mindstorms, but I do want to share a small example of how a subject, normally considered abstract, can actually be concrete when experienced differently. Before learning calculus in high school, I had to take Algebra I, Algebra II, and Pre-calculus first. Without several years of formal instruction and deliberate practice in algebra, calculus would have been inaccessible to me. But that’s only true if we insist on solving problems analytically. Imagine a car traveling 25 meters per second. Once the driver steps on the brakes, causing the car to decelerate at a constant rate, the car comes to a stop after 5 seconds. How far does the car travel before it stops?
We can solve this problem analytically by integrating a function. The car will stop in 62.5 meters. However, it’s easy, and common, for students to learn to solve problems like this, using integration, without really understanding what they’re doing.
An alternative approach is to solve the problem using numerical methods. We can simplify the math, and eliminate the need to integrate, by having the car decelerate in five discrete steps. If the car travels at 25 m/s in the first second, 20 m/s in the second second, and 15 m/s in the third—the car will come to a stop in 5 seconds after traveling 75 meters. But wait, doesn’t the car stop after 62.5 meters? Yes. We only get an approximate answer when we simplify the problem and solve it using numerical methods. However, we can increase the accuracy of our answer as much as we want just by increasing the number of steps we use. If the car decelerates in fifty discrete steps, it will come to a stop in 5 seconds after traveling 63.75 meters. In a thousand discrete steps, the car stops in 62.5625 meters.
By using numerical rather than analytical methods, a fifth-grader could solve this calculus problem—and many others like it. But that may be the least important reason for using numerical methods. Besides simplifying the math, we develop a more concrete and intuitive understanding of calculus, and the problems we can solve with calculus become richer and more interesting. As a high school student, there were only a few basic physics problems that I could solve analytically. What is the stopping distance of the car if the driver presses down on the brake pedal gradually, only fully engaging the brakes after one second? What if the driver steps on the brake as the car is heading down a hill which gets progressively steeper? While I couldn’t solve any of these problems in twelfth-grade analytically, I could in fifth-grade numerically.
Why do we focus on analytical solutions in calculus when numerical methods are so much more intuitive and useful? It’s due to history and inertia. To solve a calculus problem numerically, we might have to divide a problem into over a thousand discrete steps to get an accurate answer, and each step might involve a dozen calculations. That’s a lot of arithmetic for someone using an abacus. It makes sense that early mathematicians would focus on analytical solutions first, and that algebra would develop before calculus. I imagine that, if we had computers and spreadsheets a few thousand years ago, mathematics would be very different today. Some people, most notably Bret Victor with his Kill Math project, are attempting to reconstruct mathematics, using modern technology, to make mathematics more intuitive, but change takes time.
Summary: A subject only seems abstract, requiring formal instruction, when the environment is lacking in reasons and opportunities to learn the subject naturally using Piagetian learning. We engage in Piagetian learning when we learn through direct participation, without being taught, simply by doing. In an environment that enables Piagetian learning, not learning the subject is far more difficult than learning the subject.
Mental models and mis-education
Once Papert observed that subjects aren’t inherently concrete or abstract, he developed a theory for why formal instruction is typically more mis-educative than Piagetian learning. When learning to count, children develop a number of internal theories about counting without necessarily being aware of them. Papert and Piaget refer to these theories as intellectual structures and schema, respectively. In my own training as a teacher, we called them mental models. One key theory children develop as they learn to count is that, when counting a set of objects, the order in which we count the objects doesn’t matter—we’ll always arrive at the same number.
I won’t be able to learn to count on my own, using Piagetian learning, unless I make that discovery first. If I’m counting a set of objects one way and happen to arrive at the number eleven, I can’t know for sure there are eleven objects if there’s a possibility I could arrive at a different number simply by counting the objects again in a different order. But, if someone teaches me how to count, I might be able to mimic their procedures for counting before discovering order doesn’t matter—and I’ll start counting without understanding what I’m doing.
In hindsight, I can now see that this is what happened to me when I was first learning to drive and navigate my way around the streets of Boston. Because I started by driving on familiar routes, I usually knew which lane I needed to be in well before approaching an intersection. In Boston, road signage is terrible, drivers are aggressive, and if you’re in the wrong lane, it’s easy to find yourself miles off course with no obvious way to double back. To drive successfully in Boston, you have to know how to force your way into a crowded lane of traffic at a moment’s notice. Except, because I almost never use it, my mental model for that particular maneuver is badly underdeveloped. If I’m familiar with the route, I’ll merge into a lane miles before I have to, as soon as I spot a large gap in traffic. This means I’ve never learned to spot when a gap’s just large enough for me to squeeze in without making contact—or if the other driver will back off, rather than speed up, when I do make my move. Since one of the models I need to drive in Boston is underdeveloped, I’ve developed some poor habits in my effort to compensate, resulting in mis-education.
Formal instruction is far more likely to lead to mis-education than Piagetian learning because formal instruction makes it much easier to learn something before we construct the mental models needed to really make sense of it. If I teach a student how to count, but the student hasn’t discovered order doesn’t matter yet, I could be teaching that student to follow a procedure mindlessly. Another common occurrence in formal instruction is providing students with one opportunity to make a discovery, and then moving on whether individual students make that discovery or not. In Piagetian learning, it’s not as though we only get one chance to discover object permanence, and if we miss it, well, too bad. Object permanence is discovered by everyone because we encounter it all the time.
Besides causing us to develop harmful habits, learning something before we construct the mental models needed to really make sense of it can also cause us to construct harmful mental models about learning. By the time they reach middle school, many children have learned that math doesn’t make sense and they lack the aptitude to be good at it. They experience math as a bunch of isolated facts and rules to memorize and repeat back to the teacher. As a math teacher, many adults have shared their school math experiences with me. The story I hear over and over again is how they were good at math up to a certain point—but then hit a wall where math suddenly became incomprehensible to them.
I believe we run into a wall when too many of the underlying mental models, which we need to make sense of what we’re learning, are either missing or underdeveloped—and the habits we develop to make up for them, which may seem helpful in the short-term but are harmful in the long-term, can no longer compensate. Some people may hit a wall earlier than others, but that has nothing to do with aptitude. Our ability to make sense of math is based on the opportunities we’ve had to construct key mental models in our interactions with our environment. If our environment were rich enough, we’d never hit a wall—and not learning math would be far more difficult than learning math.
Summary: Experiences can be mis-educative if we learn something before we construct the mental models needed to really make sense of it. This is more likely to occur with formal instruction than with Piagetian learning. If mental models are missing or underdeveloped, we may try to compensate by developing habits which seem helpful in the short-term, but are harmful in the long-term—and we may also construct harmful mental models about learning. Therefore, the key to avoiding mis-educative experiences is creating an environment in which we are able to construct the mental models we need.
Papert’s path to self-actualization
In describing Dewey’s path to self-actualization, I focused on the teacher’s role as expert guide, directing students toward educative experiences and away from mis-educative ones. However, Dewey believed the environment in which we learn is also important. According to Dewey, we don’t learn in order to prepare for life; we learn by living. This means that, instead of creating an artificial sandbox for students to play in, we should bring the real world into the classroom—and students should learn by actively participating in society and community life. Implicit in Dewey’s approach is the idea that the real world is more educative than any artificial world, especially a world designed specifically for children. Compared to a traditional classroom, the real world is richer in tools and cultural materials which are relevant and meaningful to all of us, encouraging and enabling Piagetian learning.
Like Dewey, Papert also opposed the creation of separate worlds for learning and living. While he did create the Logo programming language and test it in schools as an experiment, he didn’t support schooling or any sort of Piagetian curriculum: we should be able to learn by interacting with our environments and engaging in Piagetian learning, without formal instruction. Where Papert and Dewey disagree, however, is in their analysis of the health of our existing real world environments. Our environments are shaped by our cultures. If we grow up in an artistic community, we’ll be surrounded by sculptures, studios, and paintbrushes. Those cultural materials will, in turn, influence how we see and think about the world, the experiences we have, and the mental models we construct. Papert believed the real world environments we are living in are mis-educative because our culture is mathophobic—afraid of learning. In our world, even if we grew up in a community of mathematicians, we would grow up believing that the path to calculus passes through algebra and hitting walls in our growth as mathematicians is only natural. That’s what we’d learn from the people and cultural materials in our environment.
If the real world environment itself is mis-educative, merely nudging students toward educative experiences and away from mis-educative ones won’t work. Because interacting with the environment and engaging in Piagetian learning would be mis-educative, students would have to learn almost everything from formal instruction—and that would create even more problems than it would solve. Papert’s strategy is to create a healthy culture by filling the environment with healthier materials. But if we all grow up in a mathophobic culture, and construct unhealthy mental models about learning, how do we do that? Won’t all of our instincts be wrong?
The short answer to that question is: “Yes.” If we grow up in a mathophobic culture, our instincts will be to build mathophobic materials and add them to the environment—and our children will learn to be mathophobic themselves simply by interacting with those materials and engaging in Piagetian learning. That’s how cultural transmission works. Papert suggests breaking the cycle by overriding our instincts and targeting two explicit design goals when building new materials: intuition and powerful ideas. To expand on Papert’s thinking, I will draw on Alan Kay’s talk on powerful ideas at the Thinking about Thinking about Seymour symposium, held in honor of Papert after his death. Kay was a close colleague of Papert’s, and I feel that Kay’s work fleshes out key details in several theories outlined by Papert in Mindstorms.
Papert believed we have many different ways of knowing—and focusing on conscious reasoning alone is a mistake. In his talk, Kay explains how we use three brains for thinking: our body brain knows by doing, our eye brain knows by seeing, and our language brain knows by communicating and expressing things in words. If we only know something using our language brains, that’s a sign we may not have constructed the underlying mental models needed to make sense of what we know, and our experiences are mis-educative. But, if we design materials so children can think and know something using all three brains, developing understandings which are more intuitive and concrete, it’s more likely they will be able to construct the underlying mental models they need and their experiences will be educative.
Papert’s second design goal is embedding materials with powerful ideas. Kay describes powerful ideas as creating new contexts for thinking. An example of a powerful idea is the theory of evolution, which completely changed how we see and think about adaptations, classifying organisms, biodiversity, genetics, and the history of life. Another powerful idea is Papert’s observation that a subject only appears abstract when the environment is lacking in reasons and opportunities to learn the subject naturally using Piagetian learning. It causes me to rethink what I think I know about learning, aptitude, and working with struggling students. By shifting the context in which we think, we’re able to see things from a fresh perspective—gaining new insights and reconstructing past experiences in more useful ways.
How does creating cultural materials embedded with powerful ideas, which are accessible using all three brains, lead us to self-actualization? I believe Papert’s basic strategy is to add powerful ideas to a child’s environment so the child can use Piagetian learning to: (1) discover how some ideas are powerful; (2) learn to distinguish between mediocre and powerful ideas; and (3) figure out how to debug mediocre ideas so they become powerful. Once children are able to intuit powerful ideas through their experiences, they will eventually learn how to create their own—enabling them to learn and grow continuously by expanding the contexts in which they think and reconstructing themselves and what they can do.
Summary: If our culture is mathophobic, interacting with cultural materials in the environment and engaging in Piagetian learning may not be educative. We may learn mathophobia ourselves from those materials—that’s how cultures are transmitted from one generation to the next. One way to design materials so they are less mathophobic is to enable us to use all three of our brains. If we know something using our body, eye, and language brains, it’s more likely that we have constructed the underlying mental models needed to make sense of it. Another way is to embed powerful ideas into materials. If we can learn to create our own powerful ideas, then we’ll be able to learn and grow continuously by expanding the contexts in which we think and reconstructing ourselves and what we can do.
Our design challenge
It’s never been entirely clear to me whether Dewey and Papert are only trying to describe what learning looks like in a world where we’re all able to become self-actualized—or if they’re also trying to analyze how we can get there from here. So, let me be crystal clear: vertical learning is my theory for how we may transition from our present mathophobic culture to a culture of powerful ideas as described by Papert and Kay; and it’s a theory that we can implement on our own as individual learners and/or teachers. But, before diving into the theory, let’s examine the challenge in front of us:
- The purpose of education is to enable us to develop the habits of mind we need to be self-actualized. Those habits are developed by having educative experiences and avoiding mis-educative experiences.
- Distinguishing between educative and mis-educative experiences, or even helpful and harmful habits, is something we have to learn how to do. This means having enough educative and mis-educative experiences so we can identify them, but not so many mis-educative experiences that we develop harmful habits which we’re unable to undo later.
- Our present culture is mathophobic, or afraid of learning. Because of our mathophobia, most of us continue to believe that we’re born with natural abilities and talents, which we may or may not be able to overcome or cultivate through hard work and effort. It’s difficult for us to imagine that we can all become self-actualized—or that schools should be designed to enable self-actualization.
- Because we grow up in a mathophobic culture, most of the materials we build and add to the environment will be mathophobic as well. Interacting with those materials will tend to be mis-educative, even if we’re engaging in Piagetian learning. Educative experiences are rare.
- Very few people in our culture ever become self-actualized. I estimate it’s fewer than one in a million. Developing the habits of mind needed for self-actualization takes time. While it’s hard to know what the process would look like if the culture nurtured it, in our mathophobic culture, I’d guess it takes at least twenty years to become self-actualized. The pool of available expert guides or creators of powerful ideas is tiny.
Given these conditions, I’m looking for a mechanism which will enable us to achieve self-actualization, but operates on a small scale. First, it has to work for individuals acting alone. Any path which requires us to convince someone else, such as a principal or school board, before we can begin is a nonstarter. Trust me, I speak from experience. Second, if you’re a teacher who is working with students, it has to work relatively quickly. In my career, I was able to find a position where I could work with the same group of students for three years and test my theories, but that’s unusual. In general, few positions exist where we can work with students for the twenty years we need, and if we’re working alone, we can’t exactly count on the teachers who come after us. That’s not to say we have to reach self-actualization in a single school year—but we should focus on those habits of mind which will enable learners to continue growing toward self-actualization as independently as possible. Third, the path has to be perceived as an excellent choice, even if self-actualization is never reached. If the opportunity cost is high and we believe the path is only worth taking if we become self-actualized in the end, then we’ll almost certainly abandon the path for something else at some point. The path needs to make sense and feel right all along the way, regardless of the ultimate destination.
My fourth criteria is accessibility. I’d like the path to be accessible to as many teachers and learners working alone as possible. This means identifying a mechanism we can leverage using habits of mind developed early on as we’re moving toward self-actualization—well before we get there. That will enable even more people to participate. As much as I appreciate the paths suggested by Dewey and Papert, I just can’t see them working under present conditions. While I’ve experienced the effectiveness of Dewey’s approach in my work with Sarah, it’s too slow and requires too much expertise to implement in schools. Since its publication in 1980, Mindstorms has had a significant impact on how we think about learning, especially in STEM. However, nearly four decades later, the discourse on powerful ideas is nonexistent (Alan Kay is the only one I know, besides me, still talking about them) and no one has added another cultural material to the environment as intuitive or powerful as Logo. In fact, the apparent successor to Logo, Scratch, is more popular but far less powerful. Of course, that doesn’t mean there aren’t dozens of people creating hundreds of powerful and intuitive materials in obscurity, materials that will eventually disrupt our current mathophobic culture and enable a much healthier culture to emerge. But it does mean that meaningful change is decades away, and I’m not prepared to tell children that, in two or three generations, their grand kids might grow up in a culture of powerful ideas—so just wait.
My path to self-actualization
Writing this article, I finally feel like I have a basic theoretical understanding of vertical learning—twenty years after I first began using it in the classroom and forty years after I started learning vertically myself. As I child, I distinctly remember coming home from school and trying to make sense of what I’d just learned. Because my memory is poor, I’m not very good at recalling isolated facts or rules. The only way I could keep everything straight in my head was by reorganizing things around key principles or storylines, which enabled me to re-derive the facts and rules I needed on the fly. Over the years, my ability to reconstruct past experiences, reorganizing what I know so it’s more useful for me, has gotten increasingly sophisticated. But looking back, I can see how the skills and mindset I have now all originated from a habit I developed as a child forty years ago: reorganizing what I’d learned until it made sense to me.
So, how do I know if something makes sense to me or not? I have two sets of criteria: one loose and one tight. A new theory loosely makes sense to me as long as it’s consistent with my other existing mental models. That doesn’t mean I’m confident the new theory is accurate; it just means I’m not detecting any obvious contradictions, which would throw up a serious red flag for me. However, before I can have real confidence in a theory, it must meet a second, more rigorous, set of criteria: (1) Is the new theory grounded in other existing mental models which are concrete, intuitive, and well-tested? (2) Does the new theory scale so we can apply it under a wide range of conditions? (3) Can the new theory be leveraged to construct higher level theories? To make sense of a new theory, I’ll continue drilling down and building up until the theory is grounded, scales, and has leverage.
When I first started working with Sarah, we spoke on the phone once a week. But after noticing I was deferring most of my planning to our phone calls, we decided to speak by phone once a month. Between calls, I’d work on my plans independently and email Sarah regular updates. Initially, I struggled planning on my own. I had a habit of second-guessing myself, and I relied on Sarah to ask me key questions to resolve my doubts and bring clarity to whatever I was working on at the moment. Given the turn around time, exchanging emails didn’t produce the same effect. But then I noticed something weird. As I wrote to Sarah, I started hearing her voice in my head. By internalizing her, I could anticipate what she would say and we could have a conversation. This is when I knew I’d made sense of our work together. If I didn’t understand her thinking on an intuitive level, there’s no way I could hear her voice guiding me through new and highly complex situations.
Another example of my sense-making is, of course, this article. Dewey, Piaget, and Papert have loosely made sense to me for a while now; but I knew they didn’t really make sense to me because I didn’t have the intuition or flexibility to ground, scale, or leverage their thinking as I needed. This was a source of tension for me: How much longer was I prepared to apply their thinking while knowing full well I didn’t understand their thinking? In my first year teaching, when I designed my curriculum and instruction to focus on drilling down and building up, Dewey and Papert weren’t even on my radar—nor sense-making, for that matter. As a constructivist, I was only trying to encourage students to practice and appreciate, rather than avoid and fear, revising mental models. If we drill down to ground a mental model, or build up to scale or leverage it, we revise the model, making it more robust and sophisticated. It wasn’t until after I’d made sense of Dewey, Piaget, and Papert, bringing them together into the same conversation, that I could make sense of vertical learning—even though none of them ever addressed vertical learning directly.
The key to self-actualization is having educative experiences while avoiding mis-educative experiences. Dewey’s strategy is to learn in the real world, by participating in society, and to use expert guides to help avoid mis-educative experiences. Papert’s strategy is to fill the environment with cultural materials embedded with intuitive powerful ideas. If we know something intuitively, it’s more likely we’re engaging in Piagetian learning and we’ve constructed the underlying mental models we need, making the experience educative and not mis-educative. In hindsight, my strategy is to help learners identify educative and mis-educative experiences themselves. If the new theory I’m constructing makes sense to me, then my experiences are probably educative because: (1) I’ve grounded the theory in underlying mental models which are concrete and intuitive; and (2) I’ve tested and confirmed the robustness of those underlying models by scaling and leveraging the theory. On the other hand, if the theory I’m constructing doesn’t make sense to me, I have a pair of tools, drilling down and building up, which I can use to reconstruct the underlying mental models I need.
Then, over time, by observing ourselves in both educative and mis-educative experiences, two important revisions occur in our mental models for learning. First, we revise how we learn and what we know we can learn. For example, if our experiences are educative, we soon discover we have the aptitude to learn anything. Second, we stop developing harmful habits by trying to compensate for underlying mental models which are missing or underdeveloped. If we’re learning something without really making sense of it, and we recognize it, we get to choose if we want to reconstruct those underlying mental models or not. After all, none of us has time to make sense of every bit of information we encounter; we have to triage and decide what to focus on. But as long as we’re not avoiding making sense of something because we’re not sure if we can, our choices are still healthy and educative.
Summary: In an environment in which most experiences are mis-educative, it’s essential to distinguish between educative and mis-educative experiences as soon as possible, so we can avoid developing harmful habits. One way to detect if an experience is educative, or not, is by learning to sense when theories make sense. A theory will make sense when it’s grounded in underlying mental models which are concrete, intuitive, and well-tested. We can test the rigor of those underlying mental models by scaling and leveraging the theory. If we sense that a theory doesn’t make sense, there are two tools we can use to ground, scale, and leverage it: drilling down and building up. Once we select more educative experiences and avoid mis-educative ones, we also revise our mental model for learning: we update how we learn and what we know we can learn—and we stop developing harmful habits by trying to compensate for underlying mental models which are missing or underdeveloped.
Neuroscience and cognitive dissonance
To guide students toward self-actualization, I start by helping them learn to recognize when their own mental models make sense or not—and to revise a mental model by drilling down and building up when it doesn’t. Developing a habit of constructing robust and intuitive mental models, which make sense because they are grounded, scale, and have leverage, enables us to avoid mis-educative experiences as we direct our own learning. For me, sense-making is the first and most important step on the way to becoming self-actualized.
However, it’s important to point out that I’m not explicitly teaching students how to recognize when a mental model makes sense or how to drill down and build up. Even though I developed the habit of making sense of things as a child, I only figured out the criteria I’d been using later as an adult. I consider sense-making an intuitive process. While there’s value in raising an intuitive process to a conscious level, we have to develop our intuition first. That’s what I hope students are able to do in my classroom. I have even less insight when it comes to drilling down and building up. If I had to instruct students on how to drill down and build up, I wouldn’t know where to begin.
So, what does vertical learning look like in a classroom? Instead of teaching students how to drill down, build up, and recognize if something makes sense, we create an environment in which they will discover and develop those tools for themselves, simply by engaging in Piagetian learning. But before I describe this environment and how it works, let’s drill down into Piagetian learning a bit first. According to Piaget, we learn and make sense of the world using two cognitive processes: assimilation and accommodation. We assimilate when we fit new experiences into existing mental models, and we accommodate when we revise existing mental models to fit new experiences. If an animal, which is unfamiliar to me, charges at me in the woods, I’ve got to decide what to do on the spot. Do I run away, climb a tree, play dead, stand my ground, or charge at the animal while yelling and waving my arms? In order to make my decision, I actually predict the outcome for every possible action—and select the action with the most favorable outcome. But how can I predict what will happen if I have no experience with this particular animal? I force the situation to fit into the closest mental models I have (assimilation). Then, once everything is over, I compare my prediction with what actually happened, and I revise my mental models to account for the differences (accommodation).
Until recently, I was content to ground my theories in Piaget’s and leave it at that. Because I wasn’t prepared to drill down into the biology, I told myself we didn’t understand the brain well enough to shed much light on anything. But then I started conversing with Karen Kilbane and I found an intriguing article on how the brain constructs mental models by Mark Humphries, Theories of Mind, Part 5: Did I Do That? (Medium members only). Humphries explains that “our brains have a model for how the world works, and this is constantly making predictions about what should happen.” But if a prediction is wrong, a series of events is triggered. First, the brain releases a burst of dopamine. We experience this dopamine as surprise; but for the brain, it’s a signal that says: “Hey, something interesting just happened, so pay attention!” Then, the brain logs a trace of recent activities in an effort to find what caused the prediction error. While some of those activities may have caused the error, others may be coincidental. If the surprising event is recurring, links which are coincidental are discarded until only highly correlated links remain. We can speed up this process and help the brain out by experimenting with different combinations of activities, making the “right” combination easier to find.
If we assimilate and accommodate naturally, and our brain is so effective at revising mental models to account for prediction errors, why isn’t everyone self-actualized? According to Piaget, we don’t accommodate continuously, but in sporadic leaps. As long as prediction errors are small, we tend to ignore them because revising our mental models is time-consuming. It doesn’t make sense to stop what we’re doing and revise a mental model every time an error occurs unless the problem with the model is serious and consequential. But, if the errors continue piling up, Piaget believed we would experience a state of disequilibrium, which would cause us to revise the model at some point. As a teacher, I’ve always heard this state of disequilibrium referred to as cognitive dissonance, but I’m not sure if Piaget ever used that term himself or if it came later. Research into cognitive dissonance theory suggests we can minimize our sense of disequilibrium—suppressing the need to revise our mental models. If we develop habits which enable us to avoid accommodation and ignore errors in our thinking, we will never become self-actualized.
Summary: We learn and make sense of the world using two cognitive processes: we assimilate when we fit new experiences into existing mental models; and we accommodate when we revise existing mental models to fit new experiences. Our brain uses these mental models to predict the outcomes of our actions. When a prediction error is detected, the brain attempts to identify the cause of the error and revise the model by looking for patterns and making connections. This level of knowing is highly intuitive. Accommodation occurs naturally when prediction errors pile up and we experience a sense of disequilibrium known as cognitive dissonance. However, we may develop habits that minimize cognitive dissonance and suppress accommodation, impeding our growth as learners and our ability to become self-actualized.
Learning vertically in a classroom
To cultivate a community of vertical learners, we create an environment in which students are able to experiment with sense-making and discover the power and utility of drilling down and building up for themselves. It all starts with selecting or designing the right tasks. In the end, sense-making is a tool. It may be the most versatile and effective tool I own, but it’s still only one of many tools I have in my toolbox—and just as I wouldn’t pull out a hammer to chop down a tree, I wouldn’t use sense-making for every task either. So, if we want to introduce our students to sense-making and have them explore what it can do, it may be a good idea to start with tasks where sense-making really is the best tool for the job.
Unfortunately, because we grow up in a mathophobic culture, most everyday tasks we encounter are designed to minimize the need for sense-making. Instead, we go through life learning to use a series of uni-taskers. One popular uni-tasker I’ve always hated is FOIL, a mnemonic device used to memorize the steps for multiplying two binomials.
I hate FOIL for two reasons. First, we can only use it if we’re multiplying two binomials. Need to multiply a binomial and a trinomial? Sorry, this particular tool is useless. Teachers, textbook publishers, and standardized-test makers get around that limitation by creating tasks that only ever require multiplying two binomials. Since our culture promotes tasks in which the uni-tasker is the better tool, those are the tools we all use and value. Second, students already have the tools they need in their toolboxes to multiply any polynomials—and those tools are far more versatile and useful then FOIL, if we know how to use them. Why are we promoting FOIL if we can use the distributive property to multiply binomials, especially when the distributive property can do so much more and using it makes more sense? Mathophobia. Subjects in school, and perhaps even outside of school, are organized to minimize sense-making. As teachers, we can’t encourage sense-making by using common or off-the-shelf tasks and materials. We need to make sense of subjects ourselves, reconstruct them so students have an opportunity to drill down and build up, and design tasks so grounding, scaling, and leveraging what you know is a more sensible choice than adding yet another uni-tasker to the old toolbox.
Imagine you’re teaching a group of students how to cook in a world where kitchens are stocked with uni-taskers, and you’re one of the few chefs who values and knows how to use a chef’s knife. What would you do to encourage your students to pick up a chef’s knife, too? It may be tempting to ban all uni-taskers from the kitchen and to spend a week drilling everyone on the proper use of a chef’s knife; however, relying too much on formal instruction simply invites mis-education. The better strategy is to allow students to use whatever tools they choose—while focusing on cooking dishes where the chef’s knife is clearly the better tool. As you go about your business in the kitchen, use your chef’s knife as normal. If a student asks about it, explain why the chef’s knife is the better tool for you in that specific situation, but don’t make a big deal of it. At some point, either out of curiosity or because the available uni-taskers are ill-suited to the task, a student will ask you to demo the chef’s knife. Over time, more and more students will try it out, too, either learning from you or from each other. If the chef’s knife really is the better tool, one of the students will eventually note that, once you’ve mastered the chef’s knife, it’s a lot more efficient to use it, rather than three separate uni-taskers, for slicing potatoes, dicing onions, and mincing garlic—and soon, all the uni-taskers are relegated to the junk drawer.
There are several norms I establish in my classroom to facilitate tool sharing and experimentation. First, everyone participates in public demonstrations. If you get stuck and don’t know what to do, the class will guide you. Because I’m designing tasks which can’t be completed using standard uni-taskers, all students find themselves standing at the whiteboard, trying to make sense of what they know on the spot, so they can apply what they know to solve a new and challenging problem. Seeing everyone visibly engaging in sense-making, from the “smartest” to the “dumbest” kids in the class, creates a very different dynamic. By the middle of the school year, students are volunteering to go up to the whiteboard when they don’t understand something because they see it as an opportunity to make sense of things for themselves—and sense-making is powerful. Second, everyone openly shares their tools and thinking. Third, if you ask for help and the class suggests a tool, you keep an open mind and give it a try. It doesn’t take very long for these norms to take hold because the class quickly sees them as useful. It also helps that my intentions are transparent. If I had other norms which were designed for control, not for learning, it would create distrust, discouraging openness and risk-taking.
Summary: Sense-making is a tool; like all tools, it’s useful for some tasks, but not for others. To discover the versatility and utility of sense-making for themselves, students must engage in tasks where sense-making is clearly more useful than the other tools available. Unfortunately, because our culture is mathophobic, most common tasks minimize the need for sense-making. As teachers, we have to make sense of subjects ourselves, reconstruct subjects so students have an opportunity to drill down and build up, and design tasks so grounding, scaling, and leveraging what we know is more useful than memorizing something new. If sense-making is clearly the more useful tool, students will use it as long as the tool is accessible and we establish norms that facilitate tool sharing and experimentation.
Looking forward
There are two reasons why most students don’t think to use the distributive property when they see binomial multiplication for the first time. One reason arises from their mental models for learning: from firsthand experience, they have learned that, whenever a new topic is introduced, the teacher will follow up by teaching them a number of new tools. Like FOIL, most of those tools will be uni-taskers, which must be memorized and are disconnected from any of the tools they already have. So, they wait. The second reason arises because one of the underlying mental models they need to visually recognize binomial multiplication as distribution is underdeveloped. If they had learned that it’s possible to solve new types of problems using the tools we already have, and if they had a more robust understanding of order of operations, students would be far more likely to figure out how to multiple binomials themselves, without using FOIL—and binomial multiplication, the distributive property, and order of operations would all make more sense.
On its own, grounding binomial multiplication in order of operations and the distributive property doesn’t seem that important. But order of operations is a crucial underlying mental model for more than just binomial multiplication; whether we realize it or not, we use it whenever we manipulate an algebraic expression. By constructing a robust mental model for order of operations, we can then build on top of it to make sense of dozens of previously disconnected topics—changing our relationship with math and how we see and think about ourselves as learners. Suddenly, instead of passively waiting for the teacher to teach us three new tools to memorize for every new topic, which we promptly forget after the next test, we discover that we can make sense of new topics on our own simply by scaling and leveraging what we already know.
How well does sense-making, by drilling down and building up, satisfy the four criteria I established earlier for successfuly transitioning from our present mathophobic culture to a culture of powerful ideas as described by Papert and Kay? Let’s go through them one at a time.
Criteria 3: Does this path make sense and feel right, even if self-actualization is never reached? I believe the answer is a resounding yes. I can’t imagine very many people arguing we’re better off learning using rote memorization rather than making sense of what we learn. Students who drill down and build up are more inquisitive, active, and engaged—and they construct more robust and integrated mental models, enabling them to use what they know to solve complex problems. Because things just make sense, students see themselves as capable learners. Unfortunately, this growth in understanding and problem solving won’t be fully captured on state tests, which minimize sense-making; however, we’ll still see a small bump in performance just because students are able to retain and apply what they learn more effectively. Enabling students to engage in sense-making is simply the right thing to do, whether or not it leads to self-actualization.
Criteria 1: Can sense-making be implemented by someone acting alone? So far, I’ve focused on sense-making in schools because that’s where the kids are. However, we can, and should, be engaging in sense-making no matter where or what we’re learning. Learning vertically is easier as part of a community, where discoveries and tools are shared, but all it takes is one person who can drill down and build up—and reconstruct a subject. I had no trouble meeting state standards for fifteen years while cultivating communities of vertical learners, first as a classroom teacher and then as a middle school curriculum specialist. Students, parents, and administrators were impressed and happy with my instruction and how I reconstructed the curriculum. Unfortunately, the climate in schools is very different today. When I was a new teacher, I had much more autonomy. If you’re teaching in a setting where all teachers must be in lockstep, either because the administration enforces consistent practices or you’re teaching out of a textbook, you won’t be able to implement vertical learning on your own. But, if you have a bit of flexibility and just have to meet state standards at the end of the school year, this is something you can do. We don’t have disrupt the entire educational system first.
Criteria 4: Is this path accessible to as many teachers and learners as possible? To cultivate a community of vertical learners, someone has to be able to make sense of the subject, reconstruct the subject so we have an opportunity to drill down and build up, and design tasks so grounding, scaling, and leveraging the tools we already have is more useful than learning yet another new tool. I honestly have no idea how many people can do this. However, I believe sense-making is the lowest possible threshold for an expert guide. If I can’t identify if what I learn makes sense or not, then I can’t avoid mis-educative experiences, and I won’t be able to guide myself or anyone else toward self-actualization. It is the absolute minimum requirement. I’m hoping that sense-making teachers are out there—and this is the nudge they need to get moving.
Criteria 2: Does the habit of sense-making develop quickly enough, and can we use sense-making to continue growing toward self-actualization on our own, without further guidance? Because I cultivated communities of vertical learners for fifteen years, I can share some anecdotal data. In my experience, it only took a few months for most students to shift from reluctant learners to what I describe as the active learning stage. We become active learners when our brains have been utterly shocked and surprised by what we can learn and do if we drill down and build up. We still haven’t identified the activities that cause the incredible surprise—we only drilled down and built up because we were guided there, but we’re on the hunt. Since we’re focused on recreating the conditions which lead to powerful learning, we’re open to new ideas and taking risks. Instead of avoiding cognitive dissonance, we actually seek it out. Our brains have identified a connection between powerful learning, cognitive dissonance, and mental model revision. Students in this stage will frequently volunteer to demonstrate a problem when they don’t understand it, and they will ask for harder problems to test their mental models.
Within three years, many students then transition to what I describe as the sense-making stage. In this stage, students can intuit when something makes sense or not, and they can drill down and build up themselves, without any guidance, if the materials in the environment support it. Living in a healthier culture, these students may have been able to seek out educative experiences and avoid mis-educative ones on their own. However, in our mathophobic culture, where educative materials are rare, they were largely stuck with what they could find in their immediate surroundings. I frequently received reports from the English and social studies teachers that these students were pushing back—asking deeper and deeper questions, trying to get their teachers to find and use more educative materials. There was a lot of frustration all around.
I never had the opportunity to work with students for longer than three years, so I don’t know what would have happened next. But I am familiar with my own personal path—and I can see how sense-making enables other important habits of mind to develop, which do lead to self-actualization. I describe the next stage as independent learning. If we’re able to engage in sense-making in multiple domains, we eventually realize that we can make sense of anything if we have the materials we need. By paying greater attention to the materials around us, we learn to create healthier materials for ourselves, enabling us to create the conditions needed to drill down and build up in any domain. In this stage, we can continue growing toward self-actualization on our own, without further guidance, and also cultivate communities of vertical learners.
I describe the last two stages, before achieving self-actualization, as coherent and strategic learning. As we drill down and build up, we integrate mental models. Instead of constructing a series of isolated models, we begin creating an integrated stack of models—just as I’ve been constructing vertical learning theory by integrating Dewey, Piaget, and Papert. By integrating our mental models, we can detect and eliminate inconsistencies, enabling us to construct a coherent self which acts and lives purposefully. Integrating separate models into a coherent framework is also how we start to engage in systems thinking. A key aspect of systems thinking is the ability to gain insight by climbing up or down the abstraction ladder, examining the system from either a 30,000-foot-view or a 3-foot-view, as needed. This is what we learn to do by drilling down and building up. Engaging in systems thinking enables us to plan strategically.
Summary: Vertical learning is my theory for how we may transition from our present mathophobic culture to a healthy culture which enables us all to become self-actualized. I began learning vertically myself forty years ago, and cultivating communities of vertical learners twenty years ago. However, it wasn’t until I was able to drill down and build up—integrating Dewey, Piaget, and Papert into a coherent framework, that I could establish a theoretical basis for my own work in this article. Vertical learning theory is very much a work in progress. There’s still a lot of sense-making to follow. But this feels like a healthy start.
