Fractals: One Way Students Learn to Author Their Own Learning

The other day a participant from this summer's Teaching for Experience workshop (now the Teaching for Transformation workshop) wrote to ask about some comments I had made about the use of fractals as a structure for learning. I realized that while I design learning experiences with fractals in mind all the time, I had never written anything down about why...or how.  So, here goes. 

It was 1975, I was a first year teacher and Benoit Mandelbrot had just coined a word for some mathematical thinking that had been going on for centuries--"fractals."  All of my math teacher friends were raving about it. A few years later the "Mandelbrot set" was created (check out that link if you want to get a sense of the simplicity involved in complexity). When I first encountered this, it was being called "the geometry of nature...in which smaller and smaller copies of a pattern are successively nested inside each other, so that the same intricate shapes appear no matter how much you zoom in to the whole."

For example-- ferns or Romanesco broccoli.

Now I confess that I have long been a structure freak who loves seeing the way structure can be adapted to create and induce meaning. Doesn't matter if it is poetry or soccer. Or teaching and learning. Teaching that inspires transformational learning almost always is a battle between a series of tensions and two of those are chaos/order and complexity/simplicity. Fractals are a structure that allows that tension to be held and explored.

The origins of applying the idea of "fractal learning" come from an opening-day professional- development presentation that was given my first or second year of teaching at Deerfield in the mid-70’s. We had someone from Harvard Education School (I can’t remember the name) who spoke to the faculty about the importance of the first class of the year.

The Harvard professor had just posited this powerful idea that everything the student is going to deeply retain from your class will be contained in this opening class. Bold claim, right?  But it is true in a number of ways. For example, there is something about “voice” that we internalize about memorable teachers and memorable teaching (http://dkdkzone.blogspot.com/2012/09/mature-and-immature-teaching-self.html).  

Perhaps it is related to Maya Angelou’s quote, “I've learned that people will forget what you said, people will forget what you did, but people will never forget how you made them feel.” The point is that structures are a way of making people FEEL a certain way. I think that the structure of fractals produces and induces certain feelings that are hugely helpful in trying to get people to learn. (So is framing- but more on that at the end of this essay.)

This subsequently led to another more fully developed idea for a workshop that incorporated a backward design experiment starting with a teacher asking the following question, “Your student walks into their home at the end of the first day of school and their mother/father asks, ‘How was school today?'”  Thinking just about the class you had with that student, write out exactly what you want that student to say in response. Now, design your class so that you end with the desired response you just wrote out. It is a really fun workshop, and the results have been both eye-opening and terrifying on different occasions.

If this professor was right, then the first day of class was “high stakes testing,” for the teacher, and I took it as an interesting challenge to see what I could create. The other impetus for this exploration came from a University of Michigan professor named Ken Lockridge who claimed that in order for someone to learn something you had to present the information or concept in three different contexts in three different periods of time.  He was, to say the least, intuitively understanding interleaving and retrieval practice that is all the rage now in cognitive psychology.

I went looking for structures that would be sophisticated enough to make my students feel intrigued, comforted, confused, and challenged. Quite the paradox there, but it is a great formula for growth and development. And then I came upon fractals as an opening structure (after trying many other structures such as immersion, spatial disorientation, expeditions, mysteries, picaresque novels and many, many others.)

Fractals have certain principles—they are heavily detailed, they are recursive, they are infinitely self-similar, they can invoke microcosms and macrocosms so that their scale can be very small or very big, they are patterned in their self-similarity and they can expand and evolve in their symmetry. Those sounded like ideal characteristics for being introduced to something, internalizing them and then perhaps even having them make the learning more transferable (one of the gold standards of this kind of learning) to other domains. Thank you, Benoit Mandelbrot!

What follows is a recent example of how that developed in some classes at CITYterm. One of the great things about fractals is that you can expand and contract them. One example of a fractal class from Day One of CITYterm is an exploration of Learning Theory. But this class moves from practice to theory, and not from theory to practice. The focus is on the student’s exploring the concept of surface, strategic and deep learning in their own past. It is mining the student’s own experience to uncover something that caused a paradigm shift or the destruction and re-creation of a mental model previously held. Expectation failure becomes part of the vocabulary of every deep learner. Tag lines include things like:

- “Your primary obligation is to let me know when my teaching is getting in the way of your learning.”

- "The confusion of the DKDK zone( https://dkdkzone.blogspot.com/2010/07/importance-of-dkdk-zone.html) what you are trying to engage.”

- “Ambiguity is misunderstood to be vague, it is not; it is multi-layered and every text you will encounter here will be ambiguous—embrace it.”  

All of these are central concepts to CITYterm.

Structurally, this class puts students through every form of learning they will be doing during the course of the class—a mindfulness meditation of sounds, individual writing to discover and record, paired sharing, group thinking through a problem, the teachers leaving the room for an extended period of time while students are working, students writing on the board more than the teacher, and more. The chunking of time (a key ingredient in temporal experience) is fast with distinct pauses that signal transitions. The feeling is that we are introducing unfamiliar concepts but they are actually rooted in the student’s own personal experience, but the student has had no language to explore them previously. The beginning of a common language is implied in the class though not  usually explicitly stated.  Pause here to think about how different this format is from a student who goes to class and receives a cumbersome, intimidating “course expectation sheet” that includes extensive discussion of grading policies and rubrics and warnings about how plagiarism will get you kicked out of school. Now think about how that student who had THAT as an opening class would answer the question above, “How was school today?”  (No wonder, as cognitive psychologist Daniel Willingham has entitled one of his latest books, Why Student’s Don’t Like School?)

That first class is a microcosm of what will happen for the rest of the year. But remember, you can expand fractals. There are five separate classes that are interlinked and each of them together forms a still larger fractal. Paired with the Learning Theory class is one that utilizes a text by John Stilgoe from Outside Lies Magic. And that is just where students go, out of the room, out of the building, out of their comfort zone. Classrooms are everywhere being the subtext. The goal is the integration of learning in and out of the classroom and destruction of a fundamental dichotomy that defines adolescent life-- “school/NOT school.” Integrity over dichotomy—yet another key experience-based learning principle. What we have created is the intellectual kindergarten of sustained observation and questioning through “wondering and wandering.”

The second day has two other classes paired together that when put together make for a fractal of four individual classes. These two classes are based on first, Collaboration, Dialogue and Deep Listening and second, Writing in Place: exploring the relationship between our internal and external selves as they are grounded in the space we are inhabiting at any given moment. The first class has students in creative collaboration create board games based on the essay by E.B. White, Here is New York. Concepts such as: meta-cognition, being “in the balcony and on the dance floor” simultaneously and re-framing previous concepts of leadership are all the conceptual understandings that are involved in this group activity that will repeat not just three times, as Professor Lockridge suggested, but scores of times in vastly different contexts during the course of the fifteen weeks students are at CITYterm.

The second class seeks to undermine the paradigm--another example of expectation failure-- that writing is primarily for persuasion, and by introducing Barry Lopez’s genius article from Granta, “The Invitation.” Students do a piece of writing by the banks of the Hudson River (yet another form of a classroom) that asks them to be hyper-observant and mindful, to wonder and wander, to explore their observations through writing, as well as to detail the relationship between their interior life and the place they are inhabiting. There is a visceral component that relies on the concept of the “felt sense” based on the theory that what we know most deeply we know in our bodies.  In sum, all of these classes together form a fractal that asks them them to begin to understand that they are "the authors of their own learning."

And at that point, the students and faculty get on a train in Dobbs Ferry to engage in their first trip to New York City. To do what? A collaborative scavenger hunt/exploration that will ask everyone to apply all of the concepts they have been introduced to in the past two days in real time.

But before they go, students will be asked to draw their version of what they know about New York City on a piece of paper. That paper will be part of the frame for the semester that will bring both tears and laughter as they see that drawing again during the last class of the semester and realize how much their conception of place (and of themselves) has transformed. And now, perhaps, they will see "infinity" in themselves and their world because they know what it feels like and they have had a little practice in it.