Fractals, Self-Organizing Principles, and Self-Segregation

Fractals are all around us

Came across this study on planting patterns, “Fractal planting patterns yield optimal harvests, without central control” that bears some closer review.

What’s fascinating about this is that it seems to present a real-world solution to a problem commonly referred to as the “tragedy of the commons,” in which the self-interest of individuals leads to the destruction and overuse of shared natural resources.

Spatial patterning often occurs in ecosystems as a self-organizing process caused by feedback between organisms and the physical environment. “The centuries-old Balinese rice terraces are also created by feedback between farmer’s decisions and the ecology, which triggers a transition from local to global scale control,” explains Lansing. “Our model shows for the first time that adaptation in a coupled human-natural system can trigger self-organized criticality.”

This is absolutely fascinating, and it’s interesting to consider this in relation to other ideas on a “self-organizing criticality” that we’ve explored here before, such as in brains and sand piles.

For proponents of ed reform who argue against centralized control, this seems like it could be worth digging into further.

The aspect of fractals here is also tantalizing. After chatting with a colleague about the article, he referred me to Ron Eglash, a mathematician who has studied fractals in African history and culture. Do yourself a favor and watch his TED talk, it will blow your mind.

Eglash raising an interesting point about self-organizing principles: they can be wonderful, as in Google search, or our brains. But there is also a dark side, such as in the spread of HIV or the damaging effects of capitalism. He suggests that the fractal algorithms employed in Africa could present us with “robust” “ways of doing self-organization — of doing entrepreneurship — that are gentle, that are egalitarian.”

I’d love to explore more about how the fractal design of school structures and systems could be utilized for a productive purpose. Please share if you’ve got more on this.

Another interesting angle on the Balinese rice farmers is suggested in an earlier study reported also on, “Phase transitions of rice farmers may offer insight into managing natural resources.

Their study and modeling seems to suggest that smaller self-segregated communities within a society are desirable in the long-run.

They found that the downside of the segregation is that it increases the social disharmony throughout the society as a whole. The upside, however, is that the social disharmony within each community becomes very low. In some communities, individuals are more likely to keep cooperating with each other—using the shared resource fairly—compared to the situation without segregation. These results were very similar to what the researchers observed in the segregated society of the Balinese subak.

This seems to be a dark side unmentioned in the more recent study on how the Balinese farmers exemplify a self-organizing society in harmony with nature. Or perhaps this isn’t a dark side — it’s a suggestion that some self-segregation can be positive.

But I don’t know anything about the Balinese subak, either, so not sure how much to read into this. Certainly worth going deeper into this. If any readers have knowledge of African fractal algorithms or Balinese subaks, please share!

A Brain is More than the Sum of its Parts

“Why does the brain transcend bell-curve averages?  One possible explanation is that the brain lacks a privileged scale because its functioning cannot be reduced to component parts (i.e., neurons).  Rather, it is the complex interactions between parts which give rise to phenomena at all spatial and temporal scales. . . . Like averages, reductionism is deeply ingrained in our scientific thinking.  Water is explained in terms of molecules, molecules in terms of atoms, etc.  If the brain is reducible to simpler parts, it should also exhibit a privileged scale of organization.

And yet, it does not.  A unifying mechanism for power law behavior in the brain and other systems is that of self-organized criticality (SOC).  According to this model, systems such as the brain operate on the brink of instability, exhibiting slow processes that build energy and fast processes that dissipate energy.  In such systems, small causes have effects of many sizes. Imagine you are at the beach building a sand pile.  As you add sand, the pile gets taller until its slope reaches a critical angle where it can barely support more sand.  Steadily adding more sand will result in avalanches ranging in size from a few grains to significant portions of the pile.  The avalanches are a scale invariant emergent property. Studying individual grains of sand tells you little about avalanches.”

—Joel Frohlich, “Scale Invariance: A Cautionary Tale Against Reductionism” on Knowing Neurons (HT Alexis Madrigal’s newsletter)

Economic Opportunity + Education


I recently wrote about the need to provide options for children and families living in communities impacted by poverty.

Sometimes it can be frustrating as an educator to talk about poverty, because there’s a tone in the sector that this is something we shouldn’t talk about. “Keep your focus in the classroom,” some people seem to say, “don’t worry about what you can’t control.”

Thankfully, there’s educators and activists out there refusing to keep their heads down and ignore the problems of the community they work within.

Here’s an example of a teacher, Stephen Ritz, who’s not only helping to educate his students in the South Bronx, but creating economic opportunities for them within their community. Take a look:

What’s intriguing is that Ritz presents us with a way out from the polarization of poverty vs. education: we can tackle both, and turn the equation into the win-win of economic opportunities + education.

This is the kind of innovation that I’d like to see more of. Because the children are learning skills and knowledge that will empower them with more options. They are gaining capital: social, natural, psychological, and economic. And that’s the kind of wealth that we need to cultivate in all our communities.

Check out the Green Bronx Machine for more on this great work going down in the South Bronx.

Slouching Towards Progressivism


You would not likely mistake me for a Dewey eyed constructivist. I get kind of dogmatic against the teaching of skills over literary content, for example, though I’m not quite so hard line against progressive instructional precepts as some, such as Harry Webb (my favorite ed curmudgeon. Read his blog regularly). If you’re really bored, you can read some of my circuitous expositions against constructivism here or here for further confirmation of my anti-constructivism.

But I’m moving into my 6th year of teaching special education, and my thoughts on progressive instructional approaches have shifted. Let me explain why, then let’s examine three articles that might provide some corroboration of my reasoning.

Coro New York ELC: Experiential Learning at its Finest

I took part in the first cohort of Coro New York Education Leadership Collaborative last year. The Coro methodology is experiential based, and in this sense progressive. I’ll admit I was skeptical at first when I sniffed constructivist signifiers at the outset of the program in words such as “self-discovery” and “skills.”

But after experiencing how effective well structured experiential group learning could be, I shed some of my cynicism. My co-teacher and I later brought the methodology to our own students and planned what we called an “Inquiry Day” on the issue of segregation (you can see the full unit we developed here), providing our students an opportunity to learn directly from group-guided interviews with inspiring thought leaders and activists. This experience for my students was just as powerful as it had been for me as an adult on Coro ELC strategy days.

I work with students who are often so accustomed to failure that they have long given up putting forth effort. Finding methods to include and engage them are just as critical as providing them with essential domain-specific knowledge. This is why I have shifted in my thinking.

Supporting Ideas for Embracing Progressive Elements of Instruction

Idea #1: “You do, y’all do, we do”

Elizabeth Green, one of the sharpest minds in the realm of education journalism, recently published a book, “Building a Better Teacher.” In a controversial sample from the book on NY Times, “Why Do Americans Stink at Math?,” Green presents a challenge to the traditional lesson format many teachers use. The “I do, we do, you do” format (something I utilize in my own lesson plan template), stems from principles of direct and explicit teaching: the teacher models and demonstrates, the students practice with guidance and feedback, and then practice is conducted independently.

How could you teach math in school that mirrors the way children learn it in the world? That was the challenge Magdalene Lampert set for herself in the 1980s, when she began teaching elementary-school math in Cambridge, Mass. She grew up in Trenton, accompanying her father on his milk deliveries around town, solving the milk-related math problems he encountered. . . .

She knew there must be a way to tap into what students already understood and then build on it. In her classroom, she replaced “I, We, You” with a structure you might call “You, Y’all, We.” Rather than starting each lesson by introducing the main idea to be learned that day, she assigned a single “problem of the day,” designed to let students struggle toward it — first on their own (You), then in peer groups (Y’all) and finally as a whole class (We). The result was a process that replaced answer-getting with what Lampert called sense-making. By pushing students to talk about math, she invited them to share the misunderstandings most American students keep quiet until the test. In the process, she gave them an opportunity to realize, on their own, why their answers were wrong. . . .

Over the years, observers who have studied Lampert’s classroom have found that students learn an unusual amount of math. Rather than forgetting algorithms, they retain and even understand them [Bold added].

Definitely constructivist elements in play here, which set alarms ringing in anti-constructivists minds. Tom Loveless wrote a scathing rebuttal to combat these constructivist implications.

But this flip from “I do, we do, you do” to “You do, y’all do, we do” is an instructional approach I find very compelling.

The idea of allowing students to process and struggle with something based on their current level of understanding, then slowly unpack those misconceptions and insights via group discussion, is something that I’ve witnessed take place very powerfully in classrooms.

Idea #2: Foster Productive Confusion

In an article by Steve Kolowich, “Confuse Students to Help Them Learn,” there’s support for Magdalene Lampert’s approach to instruction:

“It seems that, if you just present the correct information, five things happen,” he said. “One, students think they know it. Two, they don’t pay their utmost attention. Three, they don’t recognize that what was presented differs from what they were already thinking. Four, they don’t learn a thing. And five, perhaps most troublingly, they get more confident in the ideas they were thinking before.”

“Common wisdom holds that confusion should be avoided during learning and rapidly resolved if and when it arises,” wrote a team of researchers in a paper published earlier this year. While this might be true when it comes to superficial tasks such as memorizing facts and figures, “Confusion is likely to promote learning at deeper levels of comprehension under appropriate conditions.”

In other words: If teachers want students to learn the really important stuff, like comprehending difficult texts and modeling complex systems, they should put their students in confusing situations.

This is where harnessing the power of confusion becomes difficult—and risky. Some students are likely to snap sooner than others. “We still don’t know how soon to rescue them from the confusion,” Mr. Graesser told The Chronicle in an interview, and the right answer undoubtedly varies from student to student. Researchers say that confusing students works only if instructors can track and moderate the confusion.

There’s the rub: Detecting confusion is hard, especially in the moment. So is controlling it. For an instructor standing before a sea of faces, it’s virtually impossible [Bold added].

This caveat at the end is important: it suggests that to balance confusion with success is extremely difficult to achieve. This can explain in part some reluctance to adopting such an approach wholesale for classroom pedagogy. Great teaching is hard. I’ve had the honor of working with some great teachers, and I’ve seen this approach work. I’ve also seen it fail miserably—most especially when I try to implement it with my own students in a self-contained setting.

With students who struggle the most in the classroom, achieving this balance is tricky. We want to “confuse” and challenge our students to unveil their misconceptions, but we also need to engage and motivate them. This is where our next idea comes in.

Idea #3: Make Your Audience Care

In a seemingly unrelated article about making films, some advice from Pixar’s Andrew Stanton stood out to me in relation to this:

Since we’re all natural problem solvers, it brings us great satisfaction to solve problems put in front of us. Contrary to what it might seem, we actually like to work for results rather than be given them, and this goes for watching films, too. Audiences don’t tend to enjoy films with a lot of exposition and over-explanation or over-simplification of plot and character motive, because it takes the fun out of putting the pieces together themselves; it denies them the chance to engage in the story, to participate in it, which, in the end, doesn’t inspire them to care [bold added].

In other words, engaging our students—making them care—could be viewed as part and parcel of allowing them struggle through something before we provide them with any answers. This beautiful struggle occurs when we figure things out on our own. But in a classroom, such struggle must be carefully designed: “The elements you provide and the order you place them in is crucial to whether you succeed or fail at engaging the audience.” It’s not something that just happens. “Discovery” learning must be rightly critiqued as BS. But a well-crafted problem, designed intentionally to surface deep-seated misconceptions and create productive confusion, can be simultaneously engaging and enlightening.

What do you think?

A Self-Organizing Criticality, Somewhere Between Boredom and Chaos

By Emadrazo (Own work) [GFDL ( or CC-BY-SA-3.0-2.5-2.0-1.0 (, via Wikimedia Commons
In our last post, we examined an interesting analogy between quantum physics and human behavior made by George Soros. While we’re on the physics tip, here’s an intriguing article on a theory of the brain that incorporates understandings from physics in describing how systems balance  order and disorder, “Toward a Theory of Self-Organized Criticality in the Brain” by Jennifer Ouellette in Quanta Magazine.

Termed a “self-organized criticality” by Danish physicist Per Bak, the gist of the idea is that dynamic systems—such as the brain, a school, or traffic—maintain a semblance of stability by spontaneously transitioning between states of order and disorder. These spontananeous transitions are akin to small avalanches, and the concept is explained most concretely by an analogy to a pile of sand:

Think of sand running from the top of an hourglass to the bottom. Grain by grain, the sand accumulates. Eventually, the growing pile reaches a point where it is so unstable that the next grain to fall may cause it to collapse in an avalanche. When a collapse occurs, the base widens, and the sand starts to pile up again — until the mound once again hits the critical point and founders. It is through this series of avalanches of various sizes that the sand pile — a complex system of millions of tiny elements — maintains overall stability.

For close readers of this blog, this description will bring to mind our exploration of the concept of emergence and thresholds, and indeed, these ideas are interrelated.

How a self-organized criticality factors into the equation lies in the term “self-organized.” As the article explains it:

The precise moment of transition — when the system is halfway between one phase and the other — is called the critical point, or, more colloquially, the “tipping point.”

Classical phase transitions require what is known as precise tuning: in the case of water evaporating into vapor, the critical point can only be reached if the temperature and pressure are just right. But Bak proposed a means by which simple, local interactions between the elements of a system could spontaneously reach that critical point — hence the term self-organized criticality (bold added).

In other words, what appears to the observer as something incredibly complex and based on a fragile and precise set of conditions can be adequately described as an accumulation of small, local interactions. The elegance of this idea lies in the implication that there is an underlying simplicity behind seemingly unintelligible processes, and that such simple mechanisms can be determined in a mathematical sense.

“Self-organized criticality” still remains more of a tantalizing idea than an applicable theory to organizations such as brains and schools, but even still, there is an underlying understanding that we can draw from it now. Let’s return to the article:

A complex system that hovers between “boring randomness and boring regularity” is surprisingly stable overall, said Olaf Sporns, a cognitive neuroscientist at Indiana University. “Boring is bad,” he said, at least for a critical system. In fact, “if you try to avoid ever sparking an avalanche, eventually when one does occur, it is likely to be really large,” said Raissa D’Souza, a complex systems scientist at the University of California, Davis, who simulated just such a generic system last year. “If you spark avalanches all the time, you’ve used up all the fuel, so to speak, and so there is no opportunity for large avalanches.”

This is a wonderful way to describe a high functioning school. In any school or classroom, regularity, rituals, and structure are key to providing a positive learning environment. But educators also know that children also require events and changes that mix it up every now and then. But if there’s too many random and chaotic events, effective teaching and learning is difficult—and I can attest to this, as many other educators can, having worked in a school where schedules were upset so frequently that I walked in each day assuming chaos, and announcements were made over the loudspeaker all throughout the day, interrupting teaching and learning.

In other words, there should be a healthy balance between boring but safe regularity, and taxing randomness.


One concept which I (surprisingly) haven’t discussed much here on Schools as Ecosystems is emergence. Something unforeseen created as the synthesis of constituent parts. A “higher-order complexity arising out of chaos in which novel, coherent structures coalesce through interactions among the diverse entities of a system” (Peggy Holman).

In perceiving a school as a complex adaptive system, we can view emergence in the manner that a school culture is formed, as one example. While school leaders play a critical role in the establishment of that culture, there is also a quality of fortuity to it, dependent on the teachers, students, parents and their many dynamic interactions.
Put a bunch of very complicated and diverse people together in a school, and something unforeseen will certainly emerge.
The question is how effectively we can manage and structure this seeming chaos to promote growth, positive adaptations, and the emergence of strong relationships and healthy interactions.

There’s many facets to this, of course, but there are three main things that stand out in my mind: the importance of leadership in establishing shared values and vision, the academic and non-academic content taught both in the classroom and in the hallways, and the physical environment created by infrastructure, furnishings, and decor.

Without those pillars in place, what may arise from a school can be insidious, a toxic culture that settles on each new group of students and teachers that arrive at its doorstep and pervades its hallways, nearly impossible to dispel.

Some may argue that the very concept of emergence entails that what inevitably emerges from any given environment is uncontrollable, patterns that are self-sustaining and without any directly attributable cause. I would agree that once something new emerges, it may be well nigh impossible to stuff it back into its Pandora’s box. But just as what kind of ecosystem ultimately emerges from a human created garden may be unforeseen, the intentional design and actions taken in laying the foundation for that garden and cultivating it have a tremendous influence on the quality and content of what emerges.

So while we may never know what exactly will emerge from any given community, we must take responsibility for creating the conditions that will most promote learning, health and well-being.