Welcome back to Flipping the Focus.
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!
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 curriculum--all of it conveyed through the use of fractals as a metaphor. Specifically, the attributes of self-similarity and dynamic iteration will be considered (Fig. 1). |  Fig. 1 - Fractal as a metaphor for teaching and learning |
| Proposition:
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Spiralling Through Curriculum
A Conceptual Overview: What is teaching through spiralling?
Teaching through spiralling is a form of Connectionist Teaching--i.e., it shares a great deal of similarity with problem-based 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 problem-based teaching might follow a somewhat linear approach with topics taught in blocks--a 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 long-range planning template).
Teaching through spiralling is a form of Connectionist Teaching--i.e., it shares a great deal of similarity with problem-based 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 problem-based teaching might follow a somewhat linear approach with topics taught in blocks--a 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 long-range planning template).
Fig. 2 - Sample long-range planning template
In contrast, a modern approach to spiralling seeks to help students to see mathematics as a connected discipline and curriculum, wholly, as a coherent and cohesive set of ideas (opportunities to think, reason, and exercise procedural fluency) and concepts.
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 students--e.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 long-range 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).
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 students--e.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 long-range 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).
Fig. 3 - Sample long-range planning template for connectionist teaching
Teaching & Learning Like a Fractal! What it Means and How it Can Work
The two features that relate teaching by spiralling to fractals are self-similarity (structure) and dynamic iterations (processes). For example, a recursive mapping (structure) of a spiralled curriculum demonstrates the property of self-similarity (i.e., the cyclical nature of studying mathematical topic(s) and related, key concepts and skills over time). We can also find self-similarity (structure) in how students are supported, at various stages, to engage in deep and transfer learning (Hattie et al., 2017) through instruction that hinges on formative assessment practices and mathematical process skills.
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.
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.
Fig. 4 - Increasing complexity of learning through spiralling
| 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
|  Fig. 5 - Cycles in a Spiralled Curriculum |
Cycle 3
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.
- 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 real-life 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.
Taking a Closer Look at the Moves We Make & Why
Visualize the following: Imagine that you're working with either of the sample long-range plans featured in Figures 2 and 3, above. Let's also position ourselves to thinking in terms of cycles (Fig. 5) of teaching and learning, for example and on average, occurring every 3 weeks. In terms of secondary, semestered courses this would be equivalent to 6 cycles. These same 6 cycles, in the elementary grades, would be drawn out over the course of a school year. What's within those plans is specific to what students are to be able to do by the end of the semester/year (curriculum) and influenced in an ongoing manner by both student interactions with the mathematics, your monitoring of student thinking and student engagement with feedback.
Place yourself into this context:
- 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 working--moving in and out of shared and individual times for learning--feedback 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).
Fig. 6 - The Assessment Loop (Causarano & Coulombe, 2018)
- 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 learning--e.g., understanding, connections, skill development, needs and interests--to 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.
How Can Teachers Create & Use Complexity Science?
Teachers can create and use complexity, in part, by spiralling through curriculum (in the next post, complexity will receive further consideration in relation to Thinking Classrooms (read more about "Thinking Classrooms" here)).
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 co-relate--i.e., “create” is co-variable to “use” (see Fig. 7).
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 co-relate--i.e., “create” is co-variable to “use” (see Fig. 7).
Fig. 7 - Complexity & Implementation Principles for Spiralling through Curriculum
Consider: Let's consider what these principles might mean in the context of your own teaching, students' learning, and professional learning.
Principle - ”Neighbour” or Local Interactions
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.
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. |  Fig. 8 - Managing Flow (from Wheeler, 2016) |
Principle - Diversity & Redundancy
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.
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.
Balance 2020 Series
As you've read, much of teaching and learning mathematics requires that we create and use complexity--in this post, spiralling. The challenge that lies before us is continuously managing this complexity so that we can optimize the conditions for student learning.
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 stand-alone 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 problem-based teaching (the "5 Practices").
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 stand-alone 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 problem-based teaching (the "5 Practices").
Spring registration opening soon!
Final Remarks
In closing, I can't help but to think of the conversations that can be inspired when we take collective action to improving student learning. As this blog is a means for readers to network and gradually change the context for how they learn, teach and lead, we all benefit by drawing nearer to the perspectives shared here and shared beyond with our professional learning networks.
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
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
Sources
Causarano, J., & Coulombe, H. (2018, September 14). The Assessment Loop: Merging Assessment and Instruction. Retrieved from https://harnessassessment.com/2018/09/04/the-assessment-loop-merging-assessment-and-instruction/
Flipping the Focus. (2019, February). MAC PL - Before You Arrive: Pre-Session Learning
Activities. Retrieved from
http://www.flippedpl.ca/macpl2019_presession.html/#LearningActivityOption1
Flipping the Focus. (2019, February). MAC PL - Before You Arrive: Pre-Session Learning
Activities. Retrieved from
http://www.flippedpl.ca/macpl2019_presession.html/#LearningActivityOption2
Government of Ontario. (2004). Teaching and Learning Mathematics: The Report of the Expert
Panel on Mathematics in Grades 4 to 6 in Ontario. Retrieved from
http://www.edu.gov.on.ca/eng/document/reports/numeracy/panel/index.html.
Hattie, J., Fisher, D. B., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. L. (2017). Visible learning for mathematics: what works best to optimize student learning. Thousand Oaks: Corwin.
Liljedahl, P. (n.d.). On the Edges of Flow: Student Problem Solving Behavior. Retrieved from http://www.peterliljedahl.com/wp-content/uploads/PME-2016-Flow-and-Teaching.pdf
Overwijk, A., & So, J. (2017, May). Spiralled Curriculum. Retrieved from https://www.teachontario.ca/videos/5098
Small, M. (2009). Big ideas from Dr. Small: Creating a Comfort Zone for Teaching Mathematics:
Grades 4-8. Toronto, Ontario: Nelson Education.
Suurtamm, C. (2020). Fractals: Models for Networked Teacher Collaboration. In Teachers of Mathematics Working and Learning in Collaborative Groups. Retrieved from http://icmistudy25.ie.ulisboa.pt/
Stanley, D. (2009, February). What Complexity Science Tells Us About Teaching & Learning. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/whatWorks.html
Rainbow District School Board. (2018). RDSB Grade 8 Course Outline - Spiraled Content: Sample Long-Range Plan & Teacher Resources. Sudbury. Retrieved from https://t.co/fTvHABycVb
Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498. doi: 10.1007/s11251-007-9015-8
Wheeler, L. (2016, June 7). Sketchnote: 5 Practices for Orchestrating Mathematics Discussions. Retrieved from https://mslwheeler.wordpress.com/2016/06/07/sketchnote-5-practices-for-orchestrating-mathematics-discussions/
Flipping the Focus. (2019, February). MAC PL - Before You Arrive: Pre-Session Learning
Activities. Retrieved from
http://www.flippedpl.ca/macpl2019_presession.html/#LearningActivityOption1
Flipping the Focus. (2019, February). MAC PL - Before You Arrive: Pre-Session Learning
Activities. Retrieved from
http://www.flippedpl.ca/macpl2019_presession.html/#LearningActivityOption2
Government of Ontario. (2004). Teaching and Learning Mathematics: The Report of the Expert
Panel on Mathematics in Grades 4 to 6 in Ontario. Retrieved from
http://www.edu.gov.on.ca/eng/document/reports/numeracy/panel/index.html.
Hattie, J., Fisher, D. B., Frey, N., Gojak, L. M., Moore, S. D., & Mellman, W. L. (2017). Visible learning for mathematics: what works best to optimize student learning. Thousand Oaks: Corwin.
Liljedahl, P. (n.d.). On the Edges of Flow: Student Problem Solving Behavior. Retrieved from http://www.peterliljedahl.com/wp-content/uploads/PME-2016-Flow-and-Teaching.pdf
Overwijk, A., & So, J. (2017, May). Spiralled Curriculum. Retrieved from https://www.teachontario.ca/videos/5098
Small, M. (2009). Big ideas from Dr. Small: Creating a Comfort Zone for Teaching Mathematics:
Grades 4-8. Toronto, Ontario: Nelson Education.
Suurtamm, C. (2020). Fractals: Models for Networked Teacher Collaboration. In Teachers of Mathematics Working and Learning in Collaborative Groups. Retrieved from http://icmistudy25.ie.ulisboa.pt/
Stanley, D. (2009, February). What Complexity Science Tells Us About Teaching & Learning. Retrieved from http://www.edu.gov.on.ca/eng/literacynumeracy/inspire/research/whatWorks.html
Rainbow District School Board. (2018). RDSB Grade 8 Course Outline - Spiraled Content: Sample Long-Range Plan & Teacher Resources. Sudbury. Retrieved from https://t.co/fTvHABycVb
Rohrer, D., & Taylor, K. (2007). The shuffling of mathematics problems improves learning. Instructional Science, 35(6), 481–498. doi: 10.1007/s11251-007-9015-8
Wheeler, L. (2016, June 7). Sketchnote: 5 Practices for Orchestrating Mathematics Discussions. Retrieved from https://mslwheeler.wordpress.com/2016/06/07/sketchnote-5-practices-for-orchestrating-mathematics-discussions/
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