If you've been following some of my #Balance2020 posts on social media or have seen the Balance Series campaign, you've likely noticed a progression from studying subject content matter towards exploration and discussion of pedagogies that can help students to learn mathematics more effectively.
The series continues with Spring registration for the next session: Managing Complexity Through Spiralling  Teaching & Learning Like a Fractal!
In this post, we'll consider how the complexity of teaching and learning mathematics can be better managed by spiralling through curriculumall of it conveyed through the use of fractals as a metaphor. Specifically, the attributes of selfsimilarity and dynamic iteration will be considered (Fig. 1). 
Proposition:

Teaching through spiralling is a form of Connectionist Teachingi.e., it shares a great deal of similarity with problembased teaching where there is a focus on inquiry being used to help guide students toward conceptual understanding and development/use of strategic thinking. Typically, a classical approach to using problembased teaching might follow a somewhat linear approach with topics taught in blocksa focus on connections made within lessons and/or big ideas drawn out and punctuating student learning over the course of a term or year of study (see Figure 2 for a sample longrange planning template).
Over the course of a cycle of learning, more than one strand might be under consideration with connected topics (concepts, ideas, skills) and big ideas weaving in and out to harness the power of the processes that help learning 'stick' for studentse.g., interleaving with rich tasks, spaced and deliberate practice, and environmental influences captured by complexity science (more on complexity later...).
Figure 3, below, is a sample representation of a longrange plan that might allow educators to create an internal map that will support their interaction with students as they engage in rich tasks over the course of a term or year. The development of this internal map is key to helping students create their own map of connected ideas and concepts (Small, 2009).
The graphical representation of spiralling through curriculum (see Fig. 4) can help to begin crafting insights as to how this approach to teaching and learning can be structured.
With more detail, an example of the overarching goals for each cycle of spiralling can be found in resources prepared for Grades 7 & 8 teachers in the Rainbow District School Board of Ontario (2018). Cycle 1

 Build connections within and between strands for concepts studied up to now
 Extend the concepts studied earlier; introduce the more complex aspects of each strand
 Develop and solve problems that incorporate more complex aspects; making some connections to reallife applications and some connections within strands
 Foster connections between the strands and solidify the use of these concepts in the real world through investigations.
Working in an interdependent manner to the structure of this approach is the generative process of dynamic iteration (more details to follow).
Let's consider the interplay between these features by taking a closer look at the moves an educator might be making and why as they take on this approach to mathematics teaching and learning.
 You have a particular goal (or set of goals) you’d like your students to attain.
 Both you and your students engage in rich tasks that span several days.
 Over the course of this time, you're continuously monitoring students’ thinking, managing flow within and across groups, and helping students form connections through precise and intentional use of direct instruction within and across groups.
 As your students continue workingmoving in and out of shared and individual times for learningfeedback loops continuously (dynamic process) between yourself and students, as well as between students.
 Along the way, you're also keeping the goals of student learning in mind and making visual the development of success criteria and uncovering of learning goals with your students.
 Over the course of this time, you're continuously monitoring students’ thinking, managing flow within and across groups, and helping students form connections through precise and intentional use of direct instruction within and across groups.
 From your experiences, you're seeing how the interconnected components of the Assessment Loop (Causarano & Coulombe, 2018) are influencing the structure and learning processes within and across cycles (more details concerning formative assessment and the "Assessment Loop" can be found here).
 As your students’ thinking is consolidated, conceptual understanding forms and opportunities for mastery are provided, you and your students are now better able to extract a part (or parts) of their learninge.g., understanding, connections, skill development, needs and intereststo begin the next task (dynamic iterations: feedback supporting the structure of the next iteration).
 You're also noticing over time that within and across cycles, the complexity created and used by spiralling is making room for deep and transfer learning to take place.
Each of the following principles plays a role in the use of complexity science in teaching and learning mathematics. Note that the principles as well as the variables in each pair can corelatei.e., “create” is covariable to “use” (see Fig. 7).
Example: You've chosen a rich task that provides multiple entry points for students to engage in thinking mathematically. As students engage with the task, you notice that the number of interactions between students is increasing. And as the number and quality of interactions increases, you're also documenting how students are better able to move their learning forward. Altogether, more space is being created for students to deepen their learning and generate further insights.
Principle  Decentralized Control Example: Over the course of a cycle, you've been managing flow (see Liljedahl) through hints & extensions (Fig. 8). As a result, you're noticing that student autonomy within groups is increasing. And with each day, this autonomy is spilling over into other groups as collaboration begins to occur more often between groups. Further along, new questions are being posed by several groups, which has been providing you more information for next steps in both teaching and learning. You and your students are beginning to see how this form of leadership across members of the classroom community can further influence future actions that students take to move their learning forward. 
Example: During a professional learning session, your facilitator shared an interesting article about the impact of complexity science on teaching and learning mathematics (Stanley, 2009). In that article, a particular paragraph resonated strongly with you:
“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.”
You see the value and necessity of having both diversity and redundancy and decide to use the paragraph to promote discussion with some of your colleagues as well as the students in your class. These principles also have you reflecting on how a number of strategies, including task selection and management, flexible groupings, and classroom discourse can impact the learning dynamics in your classroom.
What we know from research on improving the conditions for teaching and learning mathematics is that through collaborative, professional learning, "...new ideas and actions emerge through [the] dynamic and iterative process of reflection and action" (Suurtamm, 2020).
The Balance Series is being offered to help educators meet this challenge. It is an interactive and online, professional learning opportunity designed and offered to help educators meet the sophisticated needs of teaching, learning and leading learning in Mathematics Education, K to 12.
By participating in several parts, this series can offer educators a unique opportunity to construct their own comprehensive narrative to improving the quality of the student learning experience.
As mentioned in the introduction, this series will be continuing with a third session called Managing Complexity Through Spiralling  Teaching & Learning Like a Fractal! Future sessions are planned to incorporate Thinking Classrooms, Formative Assessment and Instructional Leadership.
If what you've read and reflected upon in this post has your interest piqued, then consider building your virtual community of practice by taking part in this session of the Balance Series. This session, like others can work as part of a true series or standalone workshop. If you've got concerns about missing previous sessions, Flipping the Focus will be circling back to run those sessions to help you create a more comprehensive experience. Previous sessions focused on deepening subject content knowledge (proportional reasoning and algebraic reasoning tied to the big ideas of number) and problembased teaching (the "5 Practices").
If at any time you have questions or comments, please feel free to share a comment to this post or reach out to me using the contact button, below.
Professionally Yours,
Chris Stewart, OCT
Educational Consultant, Flipping the Focus (c) 2020
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