In a 2003 paper*, Brent Davis and Elaine Simmt write about the application of principles of complexity to the teaching of mathematics. Complexity science is essentially the study of living systems that are adaptive and emergent – and so it is a wide-ranging branch of scientific study. I enjoyed their article as their consideration of a school, of a classroom, of a department – of learning in general – as a living system really highlighted some important aspects of teaching and learning for me. Again, I know that this article is over 10 years old, and this may be considered “old” information, but I think the article falls into the “oldie but a goodie” category and I wanted to reflect on it here because it is such a good read.
Davis and Simmt outline several conditions that must be present for a complex system to emerge. While reading these, think of the different layers of a school – a concept to be learned, a group of four students working on an activity, a classroom as a whole, a mathematics department, a grade level team, a middle-high school faculty, an entire school staff (K-12), a school community as a whole, the administrative team, etc.:
1) Internal Diversity: members of the system must be diverse enough to be able to contribute different things to their purpose.
2) Redundancy: members must have enough in common in order to interact more favourably.
3) Decentralized Control: members’ results are collective, not the result of one member or a central “leader”.
4) Organized Randomness: Proscriptive rather than prescriptive. Members come to a result by “living in the boundary defined by the constraints, but also using the space to create something greater than the sum of its parts” (Johnson, 2001, 181 – quoted in Davis and Simmt).
5) Neighbour Interactions: Members must be allowed to interact with one another for new results to emerge.
Concerning change, there are so many things that come out of this article for me. One of the main thrusts of the article is that complex systems need to be self-similar – such a sweet idea that is incredibly mathematical. In the context of a school and educational change, for me this means that all layers of a school, from administration to faculty and staff to students to parents, need to be operating in self-similar ways. If teachers are meant to be distributing leadership to students to develop their confidence and independence, then so should department heads and grade level leaders distributed leadership for this purpose, and so should administrators do the same for grade level leaders and department heads – and so should school structures be set up to develop the confidence of community members such as parents to contribute their expertise to the learning process. If we wish for students to respect the value of working together with one another in groups, so should teachers and administrators and staff and the school community also demonstrate their respect for this through meaningful action. In short, my feeling is that meaningful change does not happen unless the school as a complex system is self-similar.
My feeling also is that change is most ideally achieved if a school becomes a complex system as Davis and Simmt outline. A school working towards positive change is able to maintain a balance of redundancy and diversity so that people will be motivated to come together yet there will also be enough diversity among people to allow them to develop new ideas. Members of the complex system will be allowed to freely interact so that new ideas can be developed and they should be free from outside control to do this. We can see this latter characteristics of a complex system in the common complaint from teachers about the lack of planning time with other teachers to properly put together a unit of inquiry, or to work in grade level teams on interdisciplinary work.
The most challenging of these aspects – and where the article hits home regarding educational change – is the idea of control. Too often, change makes people nervous and can rattle their confidence. Using a teaching example, often when changes take place or a teacher is not feeling confident or comfortable, they will seek to gain control in other ways, and this is usually when one might see in their classroom more teacher-centered lessons and other such decisions that take independence away from students. Interestingly, this in turn takes confidence away from students. If a teacher doesn’t let students answer a question, or doesn’t let them freely explore a problem, maybe this means the teacher feels they can’t do it. And so we see here that the desire for control in education can powerfully usurp positive change.
Sorry this isn’t a completely thought out post – your thoughts on how to grow this idea are appreciated!
*Davis and Simmt. Understanding Learning Systems: Mathematics Education and Complexity Science. Journal of Research in Mathematics Education. 2003. Vol. 34, No. 2, 137-167.