Characterizing Mental Models

Mental models are the internal theories that we use to make sense of the world and understand our experiences. We construct mental models intuitively, without realizing that we’re even doing it. When I take two steaks out of the freezer and place them in the refrigerator to defrost overnight, I expect to find two steaks waiting for me in the morning. If I woke up to find three steaks in the refrigerator, I’d be startled and I’d suspect that someone had been in my kitchen without my knowledge. I would not think that a third steak had materialized out of thin air. That’s because one of my most trusted mental models tells me that objects do not appear or disappear by themselves.

I would also expect to find the steaks soft and wet on the outside but still frozen in the center. I’d be surprised and annoyed if they were rock hard or soft all the way through. While some of my mental models about thawing food may draw from my study of thermodynamics as a chemical engineer, most are constructed from years of personal experience in the kitchen. I know that thick steaks need more time to thaw than thin steaks, and items thaw faster in the summer unless I turn down the little dial in the refrigerator.

I also have mental models that tell me where to place food in my refrigerator. From watching cooking shows, I know not to place cooked food next to raw food. I’m not sure why that’s so bad, but I have a few guesses based on my mental models for microorganisms. I typically place my thawing meat on the bottom shelf in my refrigerator. Why? I’m nervous about yucky meat juices dripping down on my fresh produce. While I know that this mental model is naive (it’s not based on any data, and potential drops would be caught in the tray I use to defrost meat), I have to place those frozen steaks somewhere, and I have no other theory to fall back on.

Our brains have evolved to detect patterns and construct mental models because we need to make thousands of decisions based on limited information every day. The prehistoric human who could predict the location of game animals, the onset of winter, or whether a stranger was friend or foe had a much better chance of survival. Even if a model is founded on little more than folk wisdom and superstition—instead of careful observations and well-reasoned theories—we generally prefer to base our decision-making on something rather than nothing. The need to detect patterns is hard-wired, as is our need to explain and make sense of those patterns.

Recently, I’ve been doing a lot of writing about mental models, and—in the process—my brain has detected a curious pattern: I tend to use the same set of adjectives over and over again to characterize them. In one context, I might describe a mental model as sophisticated and grounded; in another, as grounded and robust; and in a third, as becoming more sophisticated and robust. I’m not entirely sure how I choose which adjectives from the set to use in which contexts, but there appears to be some internal logic to it. My brain wanted to investigate further, so I’ve begun to transition from a makeshift set of adjectives that I use frequently to an organized classification system.

I characterize a mental model as well-tested when the model has been tested many times. A mental model is robust when it is well-tested and accurate. A mental model is sophisticated when it accounts for multiple variables and it has enough nuance to predict different outcomes in different situations. A mental model is intuitive when it’s been used so often that we can apply it reflexively, without having to think about it. A mental model makes sense when it explains a pattern instead of merely describing it. A mental model is grounded when that explanation is constructed on top of underlying models that are robust. A mental model is concrete when those underlying models are also intuitive.

I’m still not sure about my classification system. The adjectives in the classification system don’t quite match up with how I’ve been using those adjectives in my writing, which tells me that my classification system is imprecise or the way I’ve been writing about mental models has been imprecise. I feel like my brain needs more data before it can sort this out. But once it has, I’ll have a much better theory for understanding mental models.

Fragmented Mental Models

When we learn, we construct our own understanding using mental models. Even if that understanding is triggered by attending a lecture or reading a textbook, sense-making is always an active process. Active readers do more than read words on a page; they think about what they read, draw inferences, and make personal and meaningful connections. Reading is an experience. So, when we attempt to evaluate a learning experience, the question isn’t: “Are we constructing mental models?” The question is: “How sophisticated and integrated are the mental models we’re constructing?”

One way to avoid accommodation is to construct a number of fragmented mental models instead of revising a single integrated model. The conventional wisdom is that poor-performing students learn better when skills and concepts are separated into small, easily-digestible chunks. Most educators subscribe to this conventional wisdom even though the research literature says otherwise. We ignore the research by concluding that researchers exist in an ivory tower, and that the conditions they test under do not apply in any way to the real world where we and our students live. We have constructed two entirely separate sets of theories: one to explain the research and one to explain the classroom.

The sixth-grade linear functions unit was an effort to generate cognitive dissonance in students, and enable them to shift into an active learning mindset. But it was also an effort to generate cognitive dissonance in adults. We may be able to discount research results from an ivory tower, but how will we respond when the experiment occurs in our own classrooms and with our own students? What’ll we do when students who are seemingly incapable and uninterested in learning math are suddenly actively testing their own hypotheses, playing with problems, and out-performing eighth-grade peers taking Algebra I? I shared the impact the linear functions unit had on students in one substantially-separate math class, but the same impact was observable in every class.

In an ideal world, this data would have been an inconvenient truth, leading to cognitive dissonance and massive revisions in our understanding of both students and learning. But it didn’t surprise me at all when that didn’t happen for most people. Our ability to avoid accommodation is extraordinary. Instead of testing the conventional wisdom against this new data, most people attempted to preserve their conventional wisdom by using a separate mental model to make sense of these new experiences. They essentially said: “This is how students learn in the linear functions unit, but they won’t learn the same way anywhere else.” There was no attempt to reconcile the two theories.

Now, you may be thinking that this thin facade of two entirely disconnected theories of learning—one for the linear functions unit and one for everything else—cannot possibly stand up under scrutiny. But that’s the point. By constructing a new mental model that’s isolated from the models we use everyday, we can simply throw the new mental model into some dusty corner in our brain and forget about it. Because it only applies to one narrow scenario, we never need to access it. And if we never access it, we never have to scrutinize it, and we can hold onto our pre-existing mental models without having to revise them. This is why fragmented models tend to be naive and hard to remember.

By fragmenting our mental models, we avoid accommodation and revision, and we can ignore obvious inconsistencies. This is something that a vertical learner is unwilling to do. A vertical learner would feel compelled to reconcile the conventional wisdom with the new data from the linear functions unit, probably by hypothesizing conditions under which active learning might be more effective than the conventional wisdom, and vice versa. But since this revised mental model would be based on limited data, it would feel incredibly shaky, and the vertical learner would need to test this new hypothesis by conducting more experiments and making more precise observations. Instead of burying data that we might be wrong, vertical learners become curious—engaging in inquiry and drilling down until models are well-tested and solidly grounded.

Why We Should Learn Vertically

Mental Models

We make sense of the world by actively constructing internal theories, or mental models, to explain our experiences. The richer and more sophisticated our mental models, the more we understand and the more effectively we can navigate our lives. According to Jean Piaget, a Swiss psychologist and leading pioneer in constructivist learning theory, this learning and sense-making occurs through two cognitive processes: assimilation and accommodation.

When we assimilate, we make sense of new experiences by fitting them into existing mental models. Assimilation enables us to figure out and make sense of new experiences rapidly by drawing analogies to earlier experiences that we have already figured out. The richer our past experiences, the more likely we are to find an internal theory that matches. But one side effect of assimilation is confirmation bias. If we are too quick to conclude that a new experience matches an earlier experience, then we can overlook subtle differences and we might not notice that our internal theories aren’t working as well in newer situations. We end up filtering out evidence that our theories don’t quite fit. Using pre-existing mental models to explain a new experience is fast, but it can lead to errors.

If we start noticing those errors, then we experience cognitive dissonance. We notice that our theories don’t match up with our experiences, and this leads to accommodation. When we accommodate, we revise our mental models to fit new experiences. Instead of ignoring anomalies and edge cases where our mental models break down, we use the data to construct better theories. Those revised theories are more sophisticated. Not only do they account for our new experiences, but in hindsight, they also explain our earlier experiences more accurately.

A Failure to Accommodate

Assimilation and accommodation are essential processes. Both are needed to learn effectively. But because accommodation can take longer and requires more effort, we naturally assimilate more often than we accommodate. Unfortunately, this tendency leads some of us to avoid accommodation all together, especially as we get older. We can get attached and comfortable with a theory after using it for many years; we can get invested in a theory if a lot of other theories depend on it; and we are less likely to revise an established theory that is 75% accurate than a brand-new theory that is only 25% accurate. We convince ourselves that our current theories are good enough.

But when we accommodate less frequently, cognitive dissonance can become painful and something to be avoided. We lose confidence in our ability to revise our own thinking, and develop mindsets that are less probing and adaptable. We stop stretching and growing. Even worse, we become incapable of seeing evidence right in front of us that our theories aren’t working, and we convince ourselves that we have no desire to experience new things, limiting ourselves only to those comfortable experiences where cognitive dissonance is unlikely to show up.

This failure to accommodate has a disastrous impact on us as a society and as individuals. If people don’t accommodate, they cannot consider issues from other viewpoints, they are unable to accept any evidence that they may be wrong or that their theories may be inconsistent, and they tend to associate only with the like-minded. When evidence and consistency are irrelevant, it’s hard to have a reasoned debate; when there’s no inquisitiveness and no appreciation for other perspectives, it’s hard to have any kind of real dialogue; and when too many people lock themselves into uncompromising positions on critical issues, a society loses the ability to problem-solve and reach consensus.

People who choose to place themselves into a non-accommodating echo chamber may feel as though they are simply protecting themselves—the rest of society be damned—but failing to accommodate is even more damaging to the individual. When we stop revising our own theories, we stop believing that we can. We become rigid in our thinking and develop tunnel vision. Because we no longer recognize the need to adapt, we are blocked and stymied at every turn, unable to accomplish our goals. How can we navigate the world when the echo chamber shields out any meaningful feedback? Frustrated and resentful over our lack of progress, we fixate on obstacles and become blind to alternative pathways and new opportunities. When we fail to accommodate, we ultimately experience a loss of agency, and stop dreaming and growing.

Unfortunately, a culture of non-accommodation tends to be self-perpetuating. If children don’t see adults stretching, being flexible and inquisitive, revising their thinking, and working together, they are much less likely to accommodate themselves. Our culture teaches us that intelligence is largely fixed. Some people are simply smarter than others, some people simply have aptitudes that others lack, and there is simply no reasoning with or understanding some people. None of that is true, but many of us hold onto and operate under those beliefs. How can we break the cycle and encourage people to shift to a growth mindset if we are unable to accommodate and revise our own thinking?

Vertical Learners

After working in schools for nearly fifteen years as a classroom teacher and curriculum specialist, I have identified a specific mindset—a constellation of skills, attitudes, systems, and beliefs—belonging to learners who actively revise their own thinking. These learners welcome and seek out cognitive dissonance, deliberately probing their mental models for errors and searching for anomalies and edge cases where their models might break down. Instead of something unpleasant to be avoided, cognitive dissonance is embraced as an opportunity for self-growth and new understanding. I describe them as vertical learners.

To a vertical learner, mental models are simply unproven, working hypotheses… theories to be tested, evaluated, and refined. Theories are never proven or set in stone. They are dynamic constructs that we constantly revise to accommodate new experiences and understanding. In many ways, vertical learners approach mental models in the same way the scientific community approaches scientific theories.

In science, laws describe observations while theories attempt to explain observations. For example, a law of sunrises and sunsets might tell us that the sun rises in the east and sets in the west once a day, but it doesn’t tell us why it happens. To explain the law, we need a theory, perhaps one based on the structure of the solar system and planetary motion. But scientific theories aren’t static. They don’t just explain a fixed set of observations. They have to explain future observations that haven’t been made yet, and they have to be consistent with other related theories. This is how scientists test theories and build confidence in them.

In 1687, Isaac Newton published his three laws of motion. These laws successfully described every observation of motion for two hundred years, even as we invented instruments that allowed us to see farther and measure motion more precisely than ever before. But no one could explain how or why objects moved as they did. In 1859, astronomers detected an anomaly: Mercury’s orbit did not match Newton’s laws. The anomaly wasn’t resolved until 1915 when Albert Einstein described Mercury’s orbit using his special and general theories of relativity. Relativity is Einstein’s attempt to explain motion and gravitation, including Newton’s three laws. Since then, relativity has withstood its own vigorous testing. For example, in a series of experiments, atomic clocks were placed on airplanes and flown around the Earth. Relativity accurately predicted that the atomic clocks would run at different speeds depending on the velocities and altitudes of the planes.

This testing and revision of scientific theories leads to a natural evolution. Let’s say we start with a theory of sunrises and sunsets based on the Earth rotating as it orbits the Sun. Then, as we revise our theory using more precise observations, we notice that it also explains the timing of sunrises and sunsets throughout the year, the movement of distant objects in the night sky, and the changing of the seasons; and it merges seamlessly with a separate theory explaining the phases of the moon. At the same time, our description of the Earth’s orbit around the Sun is based first on Newton’s three laws of motion, and then relativity. Simply by testing our theory against new observations, we extend a narrow theory to explain a broader range of phenomena, and we integrate an isolated theory into a deeper framework of coherent theories.

The mental models of vertical learners go through a similar evolution using processes that I describe as drilling down and building up. When we drill down, we attempt to make sense of a mental model by analyzing and unpacking it. We are searching for underlying mental models that support and increase our confidence in the higher-level model. If those underlying models are well-tested and concrete, then the higher-level model is grounded and we have a solid foundation to build on. If not, we need to keep drilling down.

When we build up, we leverage a solid foundation of underlying models to construct more mental models on top of it. Building multiple mental models on top of a shared foundation creates a coherent framework of integrated models and strengthens the foundation at the same time. As we constantly drill down and build up, more mental models become well-tested and concrete, and our foundation grows deeper. Vertical learners will always revise their mental models so that their models are more sophisticated and flexible, more grounded and intuitive, and more integrated and coherent over time.

Interestingly, I have found that vertical learners go through a similar evolution themselves. In the first stage of this evolution, vertical learners are active learners. They embrace cognitive dissonance and actively construct and revise mental models. But they lack the perception required to evaluate models and determine if a model makes sense or not. Instead of drilling down and revising the models that are least grounded, active learners only accommodate and revise mental models as they are used and errors are uncovered.

But once active learners begin drilling down—and learn to discriminate between mental models that are solidly grounded and mental models that lack a real foundation—they enter the second stage, becoming sense-makers. Sense-making learners strive for understanding because it brings them pleasure, and they have learned that models that sit on a solid foundation are powerful and more useful. It actually pains them if something doesn’t make sense, and they feel compelled to drill down and revise their understanding until it does. By targeting mental models that need improvement for revision, sense-making learners advance and grow rapidly, increasing their self-efficacy and agency.

Initially, sense-making learners lack the agency to redesign the learning environment to support accommodation and revision, depending on others to supply the materials and resources needed to test and revise their mental models. But eventually, they recognize that it doesn’t make sense to sit back and wait for someone else to bring them the materials and resources they need to grow. Taking the initiative, they enter the third stage, becoming independent learners. As independent learners, they know what they don’t understand, they know what they need to understand it, and they know how to go out and get what they need.

But an independent learner’s confidence and capabilities are still developing, so most independent learners limit their sense-making to the domains where they feel most capable. Of course, limiting sense-making to specific domains limits growth. As humans, we have goals and aspirations in many areas, requiring diverse capabilities and understandings. Independent learners who realize they need to take a risk, and stretch to grow across all domains, enter the fourth stage, becoming coherent learners. Coherent learners have a growth mindset. Sense-making isn’t merely what they do, it’s who they are. It’s a core value. And because accommodation and revision are now fundamental to their self-concepts, coherent learners naturally develop new capabilities, constantly redefining themselves and what they can do.

When coherent learners pause to reflect and re-evaluate where they are going, they no longer see impassable obstacles blocking their paths; instead, they see demanding terrain that’s easily traversed once new capabilities are developed and internal systems upgraded. The ability to reach goals and overcome challenges simply through self-growth emboldens them to widen their perspective and look farther, revealing even more opportunities, achievable goals, and interesting pathways ahead. Given a wealth of possibilites and an increasing sense of agency, coherent learners enter the fifth stage and become strategic learners, setting goals and planning strategically. Strategic learners see a world they can reshape into a better place. For a strategic learner, reversing a culture that fails to accommodate is a worthy and intriguing challenge, not a daunting impossibility.

Vertical Learning in the Classroom

In the fall of 2008, I was the math and science curriculum specialist at the Robert Adams Middle School in Holliston, Massachusetts, and I was working with three sixth-grade math teachers to design and implement a new curriculum unit on linear functions. As educators with growth mindsets, we weren’t trying to answer the question: “Can all students become vertical learners?” We were trying to answer the question: “How do we help them do it?”

Now, a traditional middle school math class is a fairly hostile environment in which to try to nurture a community of vertical learners. Middle school math instruction tends to emphasize rote memorization. This is what students, teachers, and parents are accustomed to and expect. And the design of the curriculum is driven by high-stakes testing. Although standards like the Common Core attempt to shift the focus from procedural learning to conceptual understanding, there is still enormous pressure to cover a tremendous amount of material in a short period of time.

Located in a middle-class suburb outside of Boston, the Adams Middle School has over 200 students in the sixth-grade. I had been hired in 2007 to help guide the staff through the change process. In a typical math class, you might find five students actively constructing their own mental models; ten students trying to hide and disappear in the back of the room; and ten students open to learning, but who are sitting back passively, waiting for the teacher to tell them what to do. While half the class might perform well on a summative assessment at the end of a unit, only a few students would retain what they learned months later. We wanted to change all of that.

In January, at the end of the sixth-grade linear functions unit, we sat down as a math department to debrief and review what had happened. When the eighth-grade math teachers examined student work from the summative assessment, they were shocked at how well the sixth-graders performed. Only half our eighth-graders were enrolled in Algebra I—traditionally a high school math course—and the teachers didn’t think that their Algebra I students would perform nearly as well. We found that 60% of the sixth-graders had mastered all of the skills and concepts in the unit, and another 20% had a solid conceptual understanding but were still making minor errors.

The core of the unit focused on translating fluently among three different representations of linear functions: tables, written descriptions, and function rules (equations). Students also learned how to translate from a table to a graph. In the summative assessment at the end of the unit, students were asked to interpret a function rule, find the slope of a line, and write function rules given a set of coordinates or a graph. Writing a function rule from a set of coordinates is a skill that our Algebra I students find difficult, and few of them construct a conceptual understanding of the procedure. And writing a function rule from a graph is a skill that was never introduced in the sixth-grade unit at all.

We tested this last skill in a bonus problem to see which students could extend their mental models and reason their way through an unfamiliar situation. Over half the students solved the bonus problem correctly. Another 20% had the correct strategy but made a computational error. In a typical math class, most students will not even attempt a difficult problem that they have never encountered before; and most of the students who do attempt it will pull a strategy out of thin air. Over 70% of our students calmly applied what they knew to solve the problem. Clearly, these students were no longer sitting back and waiting to be told what to do.

What did we do differently in the unit to encourage students to learn actively? We started the unit by tapping into the pre-existing mental models that students use to extrapolate patterns. It is important to note that these are not mental models developed in previous math classes, but developed through personal real-world experiences over many years. These mental models are both sophisticated and intuitive. From there, we presented students with a series of increasingly complex scenarios designed to test and stretch their mental models. If students got stuck, they were prompted to apply what they knew. There was virtually no direct instruction in the unit.

If you have any experience with children in a school setting, you are probably picturing students getting frustrated and angry with teachers, and complaining that the unit is a huge waste of time. You are probably also picturing teachers throwing up their hands and retreating back to old habits: direct instruction and breaking problems down into step-by-step algorithms. But that didn’t happen. Given the opportunity to reason from a solid foundation, students actually had fun figuring things out for themselves. They were and felt successful. As students worked individually and in small groups, you could see them sitting on the edges of their seats, wheels turning in their heads. To help themselves learn, students asked what-if questions and made up their own problems. That’s right. Students made up their own problems because their focus was not on completing the assignment but on making sure that they made sense of the concepts. The best way to describe their behavior is to say that they were playing with the material by constructing and testing their own hypotheses.

Now, it might be tempting to conclude that 60-80% of students are capable of learning actively and 20-40% are not. But after sifting through the data, we came to a different conclusion. Ten of our sixth-graders were enrolled in substantially-separate math classes that have a separate curriculum and are taught by special educators. Placing special education students in sub-separate math classes used to be standard practice. But with the passage of the U.S. Individuals with Disabilities Act (IDEA), the law requires that students be placed in the least restrictive environment, which means students must now be educated with non-disabled peers to the greatest extent appropriate, even if that requires one-on-one support in the classroom and extra instruction outside of it. Only students with severe learning disabilities—those who are typically several grades below grade level—may be placed in a sub-separate math class and educated separately.

One of the teachers who collaborated on the design of the sixth-grade linear functions unit taught one of the sub-separate math classes. Four out of five of her students flourished in the unit, and they demonstrated solid understanding on the summative assessment. Five months later, when they took the Massachusetts Comprehensive Assessment System (MCAS) math test, they correctly answered 75% of the items in the Patterns, Relations, and Algebra strand, but only 40% of the items in the other four strands. In contrast, the five students in the other sub-separate math class all performed poorly in the unit and on the MCAS. We concluded that the difference wasn’t in the students. The difference was the nature of the instruction. While one teacher gave her students every opportunity to reason through problems on their own, drilling down and building up, the other teacher felt that she had to resort to direct instruction in order to help her students learn.

Encouraging all students to shift to an active learning mindset is difficult. It is especially difficult in a traditional school setting and in a subject that is not intrinsically interesting to them. Getting 60-80% of our students to test and revise their own thinking, and make sense of what they are learning, is a huge achievement. But the students who did not make that shift were not any less capable than their peers. We simply did not provide the opportunities they needed. When I was a classroom teacher and I could design a curriculum and a learning environment to support vertical learning for the entire year, over 90% of my students learned actively. The most math-phobic students in my classes routinely asked: “Can I do that problem in front of the class? I’m not sure if I understand how to do it. Can you give me a harder problem to try? I’m not sure if I really understand this yet.” If you have ever been a teacher, then you know how rare that is. Teachers spend years acquiring new strategies for informally assessing students because we expect them to sit back and hide.

I have also seen what happens when students are able to learn actively for more than one year. When teaching cohorts of students for 2-3 years, I would get reports from teachers of other subjects that my students were starting to push back on them. The students wanted those teachers to go deeper; they felt like they weren’t making sense of certain concepts, and it bugged them when they didn’t fully understand something. Those students had made the shift from active learner to sense-maker… and their teachers no longer knew how to support them.

But the real test of the sixth-grade linear functions unit came 16 months later. It is easy to fool ourselves into believing that our students are learning actively and constructing sophisticated mental models when they are not. The real test is seeing if our students could leverage what they had learned as sixth-graders, and revise and scale their mental models in seventh-grade and beyond. One reason why we met as a math department to review the sixth-grade unit was to give the seventh-grade math teachers a chance to re-design their seventh-grade unit based on what these students could do.

When we polled the students in the spring of 2010, the results were not promising. Virtually none of the students remembered the sixth-grade unit at all. But we went ahead and gave them a sampling of ten problems from the unit as a review. The seventh-grade teachers guided the students through the first three problems as a class, and then the students broke up into small groups to work through the remaining seven problems on their own. I was in the classroom when this happened. It was like watching students finding and putting on an old, familiar baseball mitt. Everything came flooding back to them. Not just the skills and concepts they had learned 16 months ago, but their ways of being as well. The room came alive with the buzz of industriousness and problem-solving. Students who were failing math—students who were typically disruptive—were instantly confident and capable once more.

About 70% of the students remembered everything from the sixth-grade unit as though no time had passed. Actually, some students had an even deeper understanding now that they had an opportunity to step back from the day-to-day work and reorganize what they had learned. Those students had no trouble generalizing and applying what they knew to new situations, including standard Algebra I problems involving abstract x’s and y’s. When students cannot retain what they learned after a test, it indicates they memorized the material instead of making sense of it, or they constructed a series of fragmented and isolated mental models instead of a coherent set of integrated mental models. Models that are fragmented must be recalled individually, but integrated models may be recalled collectively. If there are rich and well-worn associations among models, remembering one model will often trigger the others.

These results are impossible to achieve unless students actively try to make sense of what they are learning, play with the material, and revise their own thinking. Students who do not make the shift to an active learning mindset may be successful memorizing and building mental models in isolation at first. But as problem scenarios grow in complexity, there are simply too many scenarios to memorize and keep separate, and these students lack the understanding to identify errors or reason through variations. Their learning is brittle, and they lose self-confidence once their errors begin piling up. In contrast, students who drill down to and build up from a core set of well-tested and heavily-revised mental models are resilient. They know when something doesn’t make sense or they’ve taken a wrong turn. They have the understanding and flexibility to reason things out starting from what they know, and they have the confidence to embrace cognitive dissonance and accommodate new experiences.

Materials and Culture

In his seminal book, Mindstorms, MIT professor Seymour Papert theorizes that learning math is difficult for most of us only because we lack the building materials to construct mental models. If we grew up in Mathland—a world filled with mathematical building blocks—then we would learn math as naturally and fluently as a child learns French in France, without the need for schools, teachers, or formal instruction. In Mathland, learning math would be as easy as learning to talk or ride a bike. To test his hypothesis, Papert co-invented the Logo programming language, enabling young children to learn geometry and computer programming by directing a turtle to move on a computer screen.

Not only does the lack of necessary building materials make learning math difficult, it also creates a culture that is actively and profoundly math-phobic. We see this in adults who shy away from math and loudly assert that learning math is beyond them. But we also see it in the lowered expectations of math experts who teach math and write math curriculum. Those lowered expectations are shaped by and—despite our best intentions—perpetuate our math-phobic culture. Papert warns that, in our rush to create math building materials for young children, we often end up inadvertently reinforcing the culture we are attempting to dismantle. It is extraordinarily difficult for non-natives to create native Mathland materials because our culture influences everything we build.

I believe that the theories put forth by Papert in Mindstorms for math also apply to vertical learning. Many of us do not accommodate and revise our mental models because we do not grow up in a environment that is abundant in vertical learning materials or in a culture that nurtures sense-making, hypothesis-testing, cognitive dissonance, and revision. The materials around us encourage us to construct fragmented mental models instead of leveraging and revising the ones we already have; and our culture teaches us that it is natural for people to cling stubbornly to faulty theories, even in the face of overwhelming evidence, and that we have aptitudes that limit what we can do and who we can become. How do some people manage to become vertical learners in this environment? They might grow up in an environment that is a little richer in materials or in subculture that is a little less hostile to revision and accommodation. They also often grow up in proximity to other vertical learners who serve as role models.

It has been my mission as an educator to create materials and cultures that nurture vertical learning. That is what I did as a classroom teacher and that is what we did as a staff in Holliston. If we can help children shift to an active sense-making mindset, then they will have the capability to seek out and surround themselves with other vertical learning materials, and insulate themselves against a hostile culture. An active, sense-making, independent learner will always evolve into a coherent and strategic learner… we just have to make sure they have the materials and cultural resources to do it. But creating those materials and cultures is hard. We easily fool ourselves into believing that we are challenging students to deepen their understanding and construct more sophisticated mental models when, in reality, we are holding them back and perpetuating the status quo.

For example, some students encounter an anomaly in middle school: ^-3² evaluates to ^-9… and not to 9, as most students expect. Only some students encounter this anomaly because many teachers and textbooks deliberately choose problems so that this scenario doesn’t come up. And when it does come up, students are simply given another rule to memorize: If a number has a negative sign and an exponent, the exponent is applied first. Up until this point, we have treated negative numbers as values; suddenly, we are treating the negative sign as an operation. This reflects a significant shift in our understanding that gets brushed over in most classrooms. In fact, there is a concerted effort to make sure that students do not experience cognitive dissonance and suffer confusion. But a vertical learner would take note and ask: “Is the negative sign an operation? What does it do? How does it fit in with order of operations?”

If we investigate the anomaly instead of ignoring it as an irrelevant edge case, we discover that, when a negative sign operates on a number, it returns the number’s additive inverse. So, ^-(^-5) = 5 because the additive inverse of ^-5 is 5. Further investigation reveals that, in order of operations, negative signs are associative and have the same order as multiplication. This means we can evaluate ^-3 × 2 × ^-5 by multiplying 3 × 2 × 5 first and applying the two negative signs to the result: ^-(^-30) = 30. Just by investigating this anomaly, we generate new data that causes us to revise our understanding of both negative numbers and order of operations, integrating two separate mental models. While a vertical learner would do this naturally, students who avoid cognitive dissonance need a little help. Instead of directing students away from this anomaly or turning it into one more rule for them to memorize, we should be guiding them toward the anomaly and giving them more material to explore. Materials that focus on anomalies and trigger cognitive dissonance lead to revision and vertical learning.

There is a common misconception that progressive, student-centered materials are designed to help students construct more robust mental models. Unfortunately, progressive educators grow up in the same culture as traditional educators and share many of the same biases. In the 1990s, an activity for helping young children understand states of matter was in vogue. The children took on the role of particles, moving randomly and bumping off of one another. Then, when the teacher told them to slow down, they would gradually clump up, transitioning from a gas state to a liquid state. The activity was popular for being constructivist, experiential, and bodily-kinesthetic. Sadly, it made no sense. Particles do not clump up when they slow down unless there is some form of molecular attraction. Children only clump up because they know they are supposed to. That should be evident with a bit of thought. But, as a culture, we have decided that molecular attraction is a concept too difficult for young children to grasp, so we attempt to have them construct a theory that is flat out wrong instead. This is like using hands-on math manipulatives to convince children that 2 + 2 = 5. That would be a lousy foundation to build on. Student engagement is essential, but it does not lead to accommodation and revision if the materials and culture don’t support it.

Materials nurturing vertical learning enable students to construct mental models that are grounded and scalable, and have leverage. A mental model is grounded when it is built on top of pre-existing mental models that are intuitive and robust. The materials in the sixth-grade linear functions unit were grounded in our understandings of patterns developed through years of everyday experience. If you can extend a pattern like 10, 13, 16, and 19 forwards and backwards, then you can reason about the scenarios in the unit and solve the problems on your own. And because those pre-existing mental models are intuitive and robust, you have the confidence and ability to play and explore, developing and testing your own hypotheses. In contrast, when mental models are new and unfamiliar, we are reluctant to go off on our own because our understanding feels shaky. We are never quite sure if we are getting warmer or colder, if we are on track or hopelessly lost… and it’s hard to take risks and be open to cognitive dissonance under those conditions.

Once students had a working understanding of linear functions represented in tables, and they felt comfortable playing in and figuring out new situations, the materials encouraged them to scale their mental models by introducing additional representations: written descriptions, function rules, and graphs. Could they extend their mental models to integrate these different representations? A typical curriculum will space these representations out so that students don’t get confused and can keep each representation straight. But that encourages the construction of fragmented and isolated mental models. Our materials placed these representations in close proximity. We wanted our students to translate fluently in any direction, integrating their mental models into coherent frameworks. Each time a student scaled a mental model to fit a new situation or problem, it was another opportunity to revise that mental model and make it more sophisticated and robust.

If we ground a mental model in pre-existing models, and then scale and revise it until it is intuitive and well-tested itself, then we establish a new ground truth—a new foundation that we can leverage and use to construct other mental models moving forward. In the sixth-grade linear functions unit, students learned to leverage their understanding of rates of change and start values. They could still drill down to patterns in tables if things got confusing, but this new higher-level foundation was more powerful and sophisticated. Students could build higher and farther on top of it. Similarly, in our seventh-grade chemistry unit, we used materials that enabled students to construct an intuitive and robust understanding of particle motion—focusing on molecular attraction—and then leverage that understanding to construct integrated mental models for phase transitions, characteristic properties, physical and chemical change, and solubility. When we leverage a mental model and build up, not only are we grounding our understanding of a new concept, we are also revising the pre-existing model. The key is testing and revising our models until they are intuitive and robust enough to be leveraged.

Creating an environment rich in vertical learning materials is all that we would need to do to nurture vertical learning if our culture wasn’t so hostile to revision and accommodation. Unfortunately, most of us grow up with mindsets that cause us to avoid revision even when surrounded by materials encouraging us to scale and extend what we already know. Our culture teaches us at a young age to construct new and fragmented models instead of fixing broken ones. But new models are always less tested and less sophisticated. They are also less intuitive because we haven’t been using them as long. In the sixth-grade linear functions unit, we had to constantly prompt students to start with what they knew and revise as necessary. Their instincts, honed over many years in traditional classrooms, was to construct separate mental models for every lesson. They didn’t expect things to make sense or mental models to scale since they never had before. We basically had to create small, self-contained bubbles of vertical learning culture in each classroom specifically for this unit.

Once immersed in this new culture, most students adapted quickly. The culture encouraged revision and accommodation, and everyone was expected to contribute, experiment, make sense of things, and figure things out for themselves. In a matter of days, students of all perceived ability levels were openly testing and revising their own thinking in the middle of class. Rather than passively absorbing knowledge from the textbook or the teacher, students worked collectively as a scientific community: putting forward and evaluating new theories, sharing discoveries and observations, leveraging each other’s work, and building consensus around a core set of well-tested theories. The teacher’s primary role within the community was to guide students toward cognitive dissonance by asking probing questions, and modeling the values and behaviors of a vertical learner. Playing, exploring, and testing hypotheses quickly became the norm, and most students shifted easily into an active sense-making mindset. This indicates that we are natural vertical learners when our materials and culture support it.

Personal and Experiential Learning

The current trend in education is personal and experiential learning. As constructivists, both Piaget and Papert have argued that we learn best from doing and interacting with the world in ways that are personally relevant, and Mindstorms continues to be a touchstone and source of inspiration within the maker movement in STEM education today. But many of the STEM education reformers who cite Papert also choose to ignore one of his central theses: It is impossible to construct mental models when our environments lack the necessary building materials. While learning must be personal and experiential to construct robust and sophisticated mental models, those conditions are not sufficient. We need the materials and cultural resources to support our learning as well.

The fact that personal and experiential learning on its own will not lead to vertical learning should be evident. The vast majority of our learning occurs outside of school and is personal and experiential—from learning to walk, throw a baseball, and manage our finances to learning how to share, date, and negotiate office politics. But how many of us demonstrate the flexible thinking and thirst for cognitive dissonance characteristic of a vertical learner? Some reformers argue that schooling can impair our ability to learn actively, but there is no evidence that societies without formal schooling produce more vertical learners either.

If we are able to learn traditionally-formal subjects as naturally and easily as we learn to play video games or build snow forts, that would be a huge achievement, but it does not automatically mean that we will learn vertically. I know many domain experts who are pedantic and close-minded, including scientists who are trained to test and revise theories. To nurture an active learning mindset, materials must be explicitly designed for revision and accommodation, and learners must be guided toward cognitive dissonance. For example, the ethos of the maker movement is to enable children to make things as quickly and easily as possible. Making things takes precedence over making sense. The theory is that, once children are engaged in making the things that they want, they will be empowered to make things that require sense-making. But that rarely happens. Most children make things and never get to the sense-making phase. They would much rather move on and make something new than revise their understanding and go deeper. Children will opt to avoid cognitive dissonance as long as our culture is hostile to revision and accommodation. Our maker materials don’t encourage vertical learning because they are not designed for it.

Most of my work in vertical learning has taken place in schools because that is where the children are. But none of the work that I have done is predicated on schools, teachers, or formal instruction. In fact, it’s much easier to get students to shift to an active learning mindset when we aren’t in a classroom or required to teach to a set of mandated standards. That so many students were able to make the shift in a sixth-grade linear functions unit in a traditional school setting simply reveals how natural vertical learning can be under the proper conditions. At its core, vertical learning is experiential learning. In its current state, vertical learning is sometimes less than personal—but that is only because the materials and culture to nurture vertical learning are so hard to find. Until the culture changes and materials are abundant, our only option is to guide students toward cognitive dissonance and rich veins of vertical learning materials. This is really no different than the maker movement, which says that you can make anything that you want while steering you into computer programming, robotics, or 3D printing where maker materials are abundant.

The future is personal and experiential education. The only question is: “How do we get there?” If we cling to the notion that guiding students is coercive and ignore how limited we are by our own culture and the lack of materials in our environment, we may never get there. Not providing the materials and culture to learn vertically means depriving children of the opportunity to learn vertically. And without an active sense-making mindset, we won’t evolve into coherent and strategic learners with the vision, understanding, and confidence to reshape our culture and invent the materials we need. To reach the future, we need to start building materials and cultures to nurture sense-making, hypothesis-testing, cognitive dissonance, and revision. It won’t happen on its own.