The Literacy and Numeracy Secretariat is committed to providing teachers with current research

on instruction and learning. The opinions and conclusions contained in these monographs are,

however, those of the authors and do not necessarily reflect the policies, views, or directions of

the Ontario Ministry of Education or The Literacy and Numeracy Secretariat.

February 2009

Research Monograph # 17

What Complexity Science Tells Us

about Teaching and Learning

Many teacher candidates grapple with concerns pertaining to the management

of a classroom and express an overwhelming sensibility about controlling its

many aspects – students, curriculum, assessment, etc. As far as learning goes,

the natural inclination is to simplify as much as possible what students are to

learn. But how can teachers create and use complexity rather than manage it,

not just for their students’ benefit but for their own? Although complexity is

often perceived as a liability, this monograph considers how it can be viewed as

an asset and how the ideas behind complexity science might inform pedagogical

Many people would agree with the notion that the world is complex. In terms of

something quite specific, especially in terms of learning and learning systems.

Complexity science principles suggest that educational matters might be or

Can we be less prescriptive in our

classrooms – and more successful

with our students?

Complexity Science Research

Tells Us

• The diversity of knowledge and experience

is an important source of intelligence in

learning systems.

• Rich learning engagements (and the

making of seemingly abstract connections)

occur when knowledge is seen as shared

and distributed.

• Local interactions within small groups of

students (and the power of self-organization)

create greater shared classroom coherence

and understanding.

• Complexity science principles allow teachers

to think about and imagine ways in which

their classrooms can become healthier and

more democratic learning organizations.

WHAT WORKS?

Research into Practice

A research-into-practice series produced by a partnership between The Literacy and

Numeracy Secretariat and the Ontario Association of Deans of Education

professor at the University of Windsor.

His research interests are in the areas

of healthy learning organizations;

the dynamics of, and possibilities for,

mathematics classrooms where

diversity can be encouraged and

embraced rather than controlled.

If we return to my classroom on campus, we notice a few things. When we

consider, for instance, the kinds of things that we do when we look at fractions,

we see a diversity of ways in which we understand certain ideas and make connections

among different concepts; some ideas and discoveries are shared while

others are not and many insights are arrived at through some kind of hands-on

engagement. For sure, our interactions are not so easily prescribed by me or

anyone else. We discover that our learning as a whole class is very non-linear

and full of redundancy and diversity in how we interact and what we learn.

We are, I tell my students, a reflection of certain complexity science principles

that underlie a healthy learning organization.

A complexity science perspective focuses on the relational qualities of organizations

of all types and scales – complex organizational bodies like neuronal

assemblies, biological bodies, social collectives, bodies of knowledge, governance

with attempting to isolate the parts of an organizational system as with understanding

there is a need to focus on a different scale, that of the classroom, rather than

the individual students. Moreover, it is the activity of the classroom that needs

our attention.

Such an assertion may seem paradoxical. How can one tend to the needs of individual

students by focusing on the classroom as a whole? This runs contrary to

centuries of thought, which has tried to reduce life’s complexities to something

knowable and controllable by tending to the smallest parts. In the classroom,

these “parts” are individual students. Teachers are often concerned about managing

Complexity science, however, suggests that teachers can be less prescriptive;

rather than thinking of learning as linear and sequential, teachers could be

encouraged to imagine it as a web of playful possibility, where their role is to

outline the “playing area,” allowing for connections and insights to arise through

shared class activities. When the classroom is thought about and organized in

this way, it is the interactions among students and ideas that propels learning

forward, what complexity science refers to as the principle of “neighbour” or

local interactions.

Healthy complex organizations arise from particular conditions and organizational

the sharing of ideas, too much interaction can overwhelm the classroom. A highly

controlled classroom can become too rigid and fixed, whereas a classroom where

power and authority are distributed ensures that, collectively, students can

learn what they need to learn. Complexity science refers to this as the principle

of decentralized control.

A diverse collection of students and teachers, activities and ideas affords the

possibility of novel or creative prospects and the capacity to adapt. This is referred

to as the principle of diversity. Too many diverse ideas, however, can bring a class

to a halt. Thus, a certain level of redundancy or repetition and recursiveness of

activities and ideas helps bring groups of students, or even the entire classroom,

to a place where the larger collective has the capacity to move forward (to learn!),

bringing into play the principle of redundancy. Our classrooms should be diverse

places, but we all have shared experiences that allow us to relate to one another.

We now have a few complexity principles with which to work: neighbour or local

are either absent or are present in unbalanced ways (e.g., too much diversity,

too little redundancy), the learning organization and the people therein are

unable to share, adapt and evolve.

A new mindset is needed ...

To be sure, classrooms are neither

mechanical nor anarchical – that is,

learning is not a matter of either

building knowledge or of simply sitting

back and letting anything happen.

Classrooms of children – like all healthy

complex learning systems – are, or

should be, more like schools of fish,

flocks of birds or colonies of bees.

An entirely different mindset, language

and set of metaphorical images are

needed to understand the classroom

and its constantly unfolding dynamics

when one views a classroom as a

complex system. Indeed, the images

of factories and production lines –

something carried over from the

Industrial Revolution – are simply

inappropriate for contexts where learning,

as a web of interconnections, unfolds.

February 2009 3

Creating conditions for

learning ...

What is oftentimes needed is not some

tidy arrangement of isolated and isolatable

concepts, but a rich mathematical engagement

where students can interact with one

another, share different ideas and make

connections with seemingly disconnected

ideas and concepts. To do otherwise,

would entail prescribing the activities and

outcomes of teaching, rather than creating

playful possibilities. It is the difference

between causing learning and creating

conditions for learning. The actions of

teachers do not so much determine what

students learn; rather, what students learn

is dependent upon our collective presence

and participation in the classroom.9

A complexity science approach to mathematics opens up a greater possibility

for playfulness, creativity and diverse perspectives. And, naturally, what we find,

as a result, are classrooms that are “messy,” tentative and all abuzz – classrooms

which break down persistent negative experiences and apprehension about math.

In the mathematics classroom, the concept of “local interactions” refers not so

we can see a blending of new and old ideas, expressed individually or collectively,

that opens up opportunities for further elaborations of mathematical ideas and

structures. When the students in my class look at fractions, for instance, they

engage in paper-folding exercises as a way to understand a variety of concepts and

share with one another what they discover – like the commutativity law, equivalent

fractions and lowest terms. More conventional, teacher-directed/textbook-based

approaches break down these concepts into isolated ideas without real meaning

and present them to students in a controlled and deliberate manner. Rather than

relying upon a single node in a web of connections to disseminate what is required,

the whole web of ideas is present, contributing to growth in understanding.

“Decentralized control” stands in opposition to “centralized control” – like

“top-down” management where a single person is (apparently) in control of

everything. Decentralized control reflects the idea that the actions of a group

and the directions that it takes are shared and distributed. Students and teachers

students work together, the direction their study takes is determined by ideas

that arise locally rather than “from above.” A seemingly small insight from one

child can suddenly change the whole focus for the class. Thus, the leadership

of and direction in a class is distributed and often not held by the biggest,

the loudest, the smartest or the most popular person in the class.

Diversity refers to the range of possibilities that are present when generating,

identifying and evaluating new ideas and possible actions. It is what allows a

local working group to learn something new. When all students are required to

produce the same solution with the same method at the same time, new and

useful insights are hard to come by. On the other hand, too much diversity makes

it quite difficult for a group to “stick together.” So, for instance, when students

explore the notion of “half-ness” by folding their pieces of paper (“hotdog”

folds, “hamburger” folds, folds along the diagonal), a variety of representations

arise and open up new questions and insights into children’s ideas about halves

of things. Why and how would folding a piece of paper in one way possibly be the

same kind of “half” as folding it in another way? Using the concept of equivalent

fractions allows us to easily make these kinds of diverse connections.

Of course, we don’t find 30-something unique examples of paper folded into

two halves. In fact, we find that the classroom, through its shared knowledge,

my students notice, for instance, that many will have folded “hamburger” folds

while others will have done “hotdog” folds. In other words, and as paradoxical

as this might seem, classrooms need to have enough shared mathematical

if a student or teacher does not contribute to the class’s mathematical understanding,

others in the class can still contribute productively.

Implications for Practice

What Works? is updated monthly and posted at: www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/whatWorks.html

ISSN 1913-1097 What Works? Research Into Practice (Print)

ISSN 1913-1100 What Works? Research Into Practice (Online)

Unpublished doctoral dissertation,

University of Alberta.

New York: Scribner.

9. Towers, J., & Davis, B. (2002). Structuring

10. Kubota-Zarivnij, K. (2005). President’s

44(2), 1.

11. Davis. B., & Simmt, E. (2005). Mathematicsfor-

teaching: An ongoing investigation of the

mathematics that teachers (need to) know.

293–319.

References

Learn More about LNS

Resources ...

Visit The Literacy and Numeracy Secretariat

Guide to Print and Multi-media Resources at

http://www.edu.gov.on.ca/eng/literacynumeracy/

PrintMultiMediaResources.pdf

Call:

416-325-2929

1-800-387-5514

Email:

LNS@ontario.ca

1. Axelrod, R. M., & Cohen, M.D. (1999).

New York: Free Press.

2. Davis, B., Sumara, D. J., & Luce-Kapler, R.

Lawrence Erlbaum.

4. Davis, B., & Simmt, E. (2003). Understanding

learning systems: Mathematics education

137–167.

5. Weaver, W. (1948). Science and complexity.

University Press.

Some students will do different kinds of folds to show an understanding of a

“half.” But, even more, many are able to make other important connections

with seemingly abstract mathematical ideas. For instance, some are able to

. And, moreover, they realize that the order of the operations

(in the form of paper folding alone) does not matter. Thus, while the processes

used to generate these results and representations will vary, we find some

measure of redundancy in the class’s work, remembering that the place of

control in the classroom can be decentralized and local interactions help

to facilitate the processes of discovery.

This model for learning mathematics may be quite different from what teachers

experienced themselves in the past where classrooms were less interactive, filled

with little activity and conversation. Teachers were generally in control, directing

all aspects of what was to be learned; different points of view and approaches

seldom brought to the surface new ideas and insights and a high degree of

redundancy meant that everybody learned the exact same thing at the exact

same time.

We would do well to let go of being completely in control in the classroom

and take advantage of the diversity and redundancy principles of complexity

science. This is not to suggest that we need to abdicate our responsibility

for planning and facilitating learning. On the contrary, it is all the more

important for the teacher to know how to enhance the learning potential

of the classroom as a healthy learning collective that is much more than the

mere intelligence of any single student. In other words, it is up to the teacher

to understand how to create the conditions, drawn from complexity science

principles, for everyone in the class (as a whole, living breathing thing) to

learn mathematics most effectively.

For those interested in learning more

about education and complexity ...

http://www.complexityandeducation.

ualberta.ca/