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Flipping the Focus

This blog and discussion forum is being moderated as an expression of servant leadership in teaching & learning. As a collaborative tool for brainstorming enriching experiences for students, teacher learning groups, and district learning teams, we can inspire and build experiences to help empower each of us to personal leadership in learning. 

From the Archives

Mathematical Modelling: A Process for Making Sense of the World Around Us -  Part 1

9/2/2020

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Welcome back to Flipping the Focus. In the last post, readers had an opportunity to consider how spiraling seeks to help students see mathematics as a connected discipline and curriculum as a coherent and cohesive set of ideas and concepts.
​The current post not only relates but takes the element of a "connected discipline" a bit further--i.e., exploring the intersection of math and real worlds in K-12 Education. Educators have realized, for far too long, that there's been an imbalance--much more effort and time devoted to teaching and learning in the first.
Image of process model for problem solving
A process model for problem solving
The graphic to the right--a starting point for reconciling the divide between these two worlds--depicts how educators might teach the process of problem solving (Creativity in Mathematics). Readers might also be familiar with how these 4 stages have been used previously to describe the mathematical process of problem solving in the 2005, revised version of Ontario's elementary curriculum, Grades 1 to 8. Similarly, we can uncover aspects of these stages in both intermediate and senior math curricula--Grades 9 & 10 (2005, revised) and Grades 11 & 12 (2007, revised)--where connections to real-world contexts is shared as a way to help students to achieve expectations. These steps have been drawn from George Pólya's 4-step problem solving process: 1 - understand the problem; 2 - devise a plan; 3 - carry out the plan; and 4 - look back. Notably, the significant distinction between these two frameworks relates to how the current version (see graphic) seeks to position problem solving as a means to representing mathematics in reality.

​And now, with the introduction of Ontario's Mathematics Curriculum, Grades 1 to 8, 2020, there is an incredible opportunity to "settle the score"--through Mathematical Modelling. For the purposes of this post, we can consider the linear-sequential process captured above as a starting point. But as you'll see throughout the remainder of this post, the process of mathematical modelling is far from a linear one. In fact, modelling is a connected process--connecting elements 1, 2, 3 and 4 in a variety of ways and across diverse contexts.
For far too long, there's been an imbalance ... "settle the score" through Mathematical Modelling.
Introduction
In mathematics teaching and learning, we frequently ask questions of our students that require specific answers. Essentially, learning mathematics from the student perspective has been more about answering questions rather than ...
  • “... [making] connections between math and everyday contexts to help...make informed judgments and decisions”, ...
as well as
  • “... [working] through challenging math problems [and] understanding that their resourcefulness in using various strategies to respond to stress is helping build [their] personal resilience” (see social-emotional learning, SEL). 
What other views of mathematics do we want our students to develop? What if a primary goal of Mathematics Education was to have students engage with tasks that require them to ask questions and identify underlying assumptions first?

Think about the number of applications we are exposed to and/or use frequently that rely on our ability to ask good questions and identify assumptions: 
  • Consumers' buying habits (E.g., When is the last time you used a mobile app while grocery shopping?)
  • Mapping geographical regions (E.g., Through your notifications, think about how the technologies we use for mapping travel continuously work to optimize our plans while driving.)
  • Population growth and epidemiology (E.g., How is it that a prediction is made on one day, and then updated  at a later point in time?)
  • ...aspects of game design, financial planning, weather forecasting, and so on.

Each of the developments, above, were borne out of the process of mathematical modelling. The 'best' questions at the time needed development; assumptions about the information collected had to be identified, leaving room for imperfection...the need to assess the model...a need to go back and improve upon the question and/or the tools, representations and strategies used to solve the problem.  
What is Mathematical Modelling?
​When we look to Ontario’s new, Elementary Mathematics Curriculum--Grades 1 to 8, we read: “The process starts with ill-defined, often messy real-life problems that may have several different solutions that are all correct. Mathematical modelling requires the modeller to be critical and creative and make choices, assumptions, and decisions. Through this process, they create a mathematical model that describes a situation using mathematical concepts and language, and that can be used to solve a problem or make decisions and can be used to deepen understanding of mathematical concepts” (p34). Taking this approach, and integrating mathematical modelling across all strands through intriguing tasks, can allow students to be in the "driver’s seat"--encouraging them to explore and learn more about mathematics and problem solving.
Developing Our Understanding: What is Mathematical Modelling?
  • This blog post is about the process of mathematical modelling--what it could look like …
    • for students to create a mathematical model in response to a real-world scenario; and
    • in terms of its scope--i.e., how modeling might play out over long-range integration of curriculum expectations (coming soon ... Part 2)
  • This post is also about what mathematical modelling is not ... more on that later.
Unpacking Mathematical Modelling through a 3-Act Math Task
Let’s unpack these components through the example of a 3-Act Math Task. These tasks, made famous by Dan Meyer, can be intriguing and provide opportunities for students to develop their mathematical identity--seeing themselves as capable and confident mathematics learners and developing an appreciation for the beauty of mathematics beyond the classroom and seeing connections to the type of mathematical and wider-context work they might like to do later in life (Curriculum Context, Meyer).
NOTE: Not all 3-act math tasks are mathematical modelling tasks. The tasks that educators curate, as I have done here, and create need to reflect the 4 elements of mathematical modelling and their relationships (see graphic, below). More to come on this framework and its intersection with this task a bit later...
Picture of the process of mathematical modelling
The Process of Mathematical Modelling (Source: The Ontario Curriculum, Grades 1–8: Mathematics – 2020)
Let’s consider the “World Record Balloon Dog” 3-act math task to help us understand both the structure of such tasks, as well as how it can be used to incorporate High-Impact Instructional Practices in Mathematics (HIIPM) into teaching and learning. Take for example:
  • Teach through problem solving (p11) and
  • Teach students about problem solving (p13; adaptive reasoning is key to mathematical modelling)
(Note: References to HIIPM can be found throughout this post in an effort to highlight what these components of effective mathematics instruction can look and sound like for teachers and their students.) 

​
To begin, let’s try out this task together (task available at: World Record Balloon Dog).
ACT 1
Let's start by watching the video to the prologue. While viewing, what are you noticing? And what are you wondering? Feel free to add your ideas in the shared doc (button, below).
Link to video for fastest time to pop 100 balloons
Fastest Time to Pop 100 Balloons
Notice & wonder
Imagine your students sharing their observations and wonderings--some of these, mathematical in nature. As their teacher, you hone in on questions related to the learning goal (HIIPM, p7) you’ve selected.
Learning Goals
In this case, the goal (e.g., Grade 7 - B2.10 and Grade 8 - B2.8) might be for students to “...apply proportional reasoning to solve problems” as well as a goal for mathematical modeling, which requires that students “...apply [this] process ... to represent, analyse, make predictions, and provide insight into real-life situations.” And more importantly, for students to achieve this goal, they’ll need to continue developing and refining their social-emotional learning skills (i.e., through the mathematical processes).
SEL Considerations
As you continue to work through the task and the remainder of this post, consider how any or all of the following SEL outcomes can be highlighted with your students:
  • Understanding the feelings of others
  • Working through challenging problems to build resilience
  • Testing approaches and learning through mistakes
  • Working collaboratively with others
  • Identity: Seeing themselves as capable math learners, strengthening their sense of ownership
  • Making connections to everyday contexts (helping to make informed judgments and decisions)
What Might Your Students Say?
Questions arising from students might include:
  • How many balloons has the dog popped?
  • How long did it take dog to pop this many balloons?
Through the discussion with and among your students, the conversation might tend towards one of recognizing and needing to identify a rate. Collectively, you’re defining the problem (one that you’ve thought carefully about crafting ahead of time) and also leading towards posing a question--one that will spur students on to creating and using a mathematical model. 

How long will it take the dog to pop all of the balloons?

With this question set as their task, and well-before students engage in solving their problem, it’s important that we ask students to offer up an estimate. Not only will estimation help students to develop mental math skills, it can:
  • In a variety of related scenarios: Give a teacher insight into what knowledge and skills students already have and may apply (formative assessment, opening possibilities for your interaction with students, but more importantly, not making assumptions about what students are capable of doing … focusing on students’ thinking, connections to developmental continua and most importantly … possibilities that emerge for later discussion (Direct Instruction; HIIPM, p9).
  • Help students to identify information that is required, and possibly, any assumptions that might need to be made … assumptions that may impact the mathematical model they create. In this specific scenario, students will identify that they are not only needing to identify a rate, but that the rate can be used to determine an unknown value in this situation (proportional).
What Do You Say?
Next, take your own experience with Act 1 a bit further by offering up a too-low, as well as a too-high estimate, for how long it will take the dog to pop all of the balloons. Both of these can be shared in the doc provided (see button, right).
ESTIMATE
Although there is a lot happening and much information is being shared and considered in a short amount of time, developing this narrative with your students takes up only a small fraction of your shared time together AND is critical to setting up for the next Act. Act 2 is where students, in flexibly-organized groups (e.g., visibly random groups; read more about the affordances of visible random groups), will spend most of their time--collaborating on the development and use of their mathematical model.
ACT 2
You and your students are now ready to work with the question: How long will it take the dog to pop all of the balloons? And have possibly identified a variety of underlying assumptions, as well as additional information required to solve the problem (i.e., time it takes the dog to pop 25 balloons). In this Act, you also have the previous world record time. From this, students can gauge if the dog will beat this record, and this might also encourage students to work further with the assumptions they’ve identified. Check out both the 25-balloon time and previous world record in Act 2, here. During this time
How About Your Own Mathematical Model
Before continuing with this post, take time to have some fun with this problem! Create your own mathematical model and incorporate any assumptions (adjustments to your model) and share that in the doc provided (e.g., try inserting an image of your model and some of your work leading up to it, see right).
MY MODEL
Coming back to the time and experience you’re sharing with your students, this is where you ‘walk away’. Perhaps, at times, you're appearing to be intentionally less helpful--using hints and extensions with students--to maintain an optimal learning experience. And in some situations, you're also 'walking into' by generating and asking questions of students--sometimes in the moment--to help keep students thinking. Altogether, you're working on managing flow (read more about flow here). During this time where students are engaged in active learning, you’re documenting and enacting some of the 5 Practices (monitoring, selecting and sequencing; summary) that will help to prepare you and your students for a consolidation of their thinking, leading towards students making sense of the mathematics and developing procedural fluency in meaningful ways (Anthony & Walshaw, 2009).
ACT 3
Alright, the final Act! In Act 3, it’s time to let students know that there is a big reveal! This is always a great moment to celebrate with your students.

For fun! (if you’ve been playing along with the balloon pop task), take a moment to w​atch the short clip and the time taken to pop the remaining balloons. Also, compare your model and answer to what’s shown. You’ll see that a graph of the number of balloons popped vs. time is being generated dynamically--i.e., as the balloons are popped (a representation that can be effective in subsequent discussion of students’ results, models and assumptions that have played into the model they’ve created).
So ... How About Your Own Mathematical Model?
For fun! and to inspire reflection, consider adding both your result (time to pop all of the balloons) and comment on why your model either fell short, was too much or just right. And if you’re off from the actual time, what changes might need to be reflected in your model?
Reflect
Reflecting on Your Experience
Now that you’ve watched the reveal, how did it make you feel? Do you feel resolved? Ecstatic over meeting the challenge? Inspired to keep trying … a different approach? Consider now how you’ll prepare to work with students’ reactions to the result and experiences with this task--i.e., through a consolidation of students’ thinking (using their contributions in both a respectful and intentional manner) and using a number of high-impact practices (HIIPM, pp.9, 15, 17).
Math Conversations: What Are We Discussing?
You now have a wonderful opportunity to engage students in discussing both subject-matter and social-emotional learning goals (as mentioned, above; repeat below). This can be through a teacher-facilitated discussion of sequenced students' approaches beginning with an example that all groups were able to reach.

What are you discussing? The math conversation (HIIPM, p17) then follows a progression that reflects the hints and extensions you provided while students were working. You and your students are focusing on identifying and recognizing criteria--e.g., types of mathematical thinking and problem solving strategies; examples (see sample continuum, below)--that can become part of the success criteria co-developed for the content goal. Moving forward, these criteria will afford you the opportunity to provide descriptive feedback to students--precise information on how they can navigate towards the learning goal (HIIPM, p7).
Picture of continuum of mathematical thinking and problem solving strategies
Sample continuum of mathematical thinking and problem solving strategies
Social-Emotional Learning Skills...What Are We Discussing?
And based on the SEL goal(s) chosen, alongside the mathematical modelling experience students have been occasioning (i.e., real-world scenario) you might choose to focus on the mathematical process skills (HIIPM, p21) that your students were using as a means for helping to develop their SEL skills (see the table, below).
Picture of a table of social emotional learning goals and outcomes. Their development being through the mathematical processes.
Source: Ontario Mathematics Curriculum Expectations, Grades 1 to 8, 2020 - Strand A
Developing and Refining Success Criteria
As students progress--informed by your documentation (observations of and conversations with students)--you'll want to ensure that there are opportunities for them to go deeper with their learning. When it comes to success criteria, we want to consider some next steps in both teaching and learning. 
​
For example, you might choose to focus on any one or all of the following in your next steps (HIIPM, p8):
  • define what successful attainment of learning goals looks like
  • make connections to other strands or content areas
  • promote self-reflection
    • Ask your students: “What are some of the key characteristics of mathematical modelling?” What are some of the ‘moves’ you’ve made when learning about and using the process of mathematical modelling? (Notice: Consider how this line of questioning wraps around to SEL and mathematical process skills.)
Next Steps
As with many 3-Act Math Tasks, we might choose to have students focus on a ‘sequel’. 
As students build their model and convince themselves, a peer and their teacher, a sequel could be an extension. With the balloon popping scenario, several suggestions were provided by educators (here) on a possible next step. 

A next step could be for students to complete a set of “Check Your Understanding” Questions--related questions for self-assessm
ent.

And if the extension involves students convincing themselves, their peers and their teacher, this is a perfect opportunity for students to redress their mathematical models--to think about why their model didn’t match the reveal, and to re-analyze, adjust and re-assess the appropriateness of their improved model.

A good point to remember here, and it’s not about getting an answer--the answer of being either right or wrong. It is about taking a balanced approach to instruction by using a variety of high-impact instructional practices to help our students become better thinkers. It’s really about students developing a positive identity as a math learner: All mathematical models are imperfect; some are more useful than others (George Box) and this creates the need and allows us to demonstrate the importance of valuing mistakes and imperfection of mathematical models.


Lastly, as you introduce the next task and students begin working, moments will arise where you and your students can connect to other strands or content areas and gather additional information from students’ insights, further refining what successful attainment of learning goals looks like. These moments will not only come about because of your intentional planning, but more often than not, they can and will be incidental in nature--either as a carry-over from the previous or an earlier activity, an extension, and/or check your understanding questions.
Unpacking Your Experiences: A Framework for Mathematical Modelling
From our position as educators, let’s review and unpack just what has occurred with this 3-Act Math Task and how it relates to a definition and framework for mathematical modeling.
The process starts with ill-defined, often messy real-life problems that may have several different solutions that are all correct. Mathematical modelling requires the modeler to be critical and creative and make choices, assumptions, and decisions. Through this process, they create a mathematical model that describes a situation using mathematical concepts and language, and that can be used to solve a problem or make decisions and can be used to deepen understanding of mathematical concepts
(
Source: Ontario Mathematics Curriculum Expectations, Grades 1 to 8, 2020, p34).​
Consider the two graphics provided below. The 3-act task that you've engaged through this post--i.e., its solution process--could be represented quite well by the problem solving process on the left (below). And if we contrast this process against the description of mathematical modelling (above), we might find that students' decision-making skills and/or understanding of mathematical concepts may not have deepened as a result of working with this particular task--i.e., they have not been engaged in learning about and using the process of mathematical modelling.
Image of a process model for problem solving
A process model for problem solving
Image of the process of mathematical modelling
The Process of Mathematical Modelling
All of this said, I would ask that you take a few moments to identify and reflect on the pedagogical 'moves' shared throughout the course of this post and how they might relate to the process graphic for mathematical modelling (above, right). What intentional decisions were made to adapt this task to reflect the overlapping nature of the 4 mathematical modelling process elements? How did you come to wrestle both with creating the question, as well as following a process that led you to sharing an answer? 
Table: 3 - Act Math Tasks & Mathematical Modelling
To further support your reflection, the table that follows--"3-Act Math Tasks & Mathematical Modelling"--attempts to unpack each Act in the task and relate it to one or more elements of The Process of Mathematical Modelling.

Note that the document is open to suggesting edits--i.e., in the final column of the table, if you have other ideas about how this task or tasks, in general, can be used to both teach students about and use the process of mathematical modelling, please feel free to share. 
This version of the document is linked here. ​If you'd like to use the document in a read-only fashion, I've embedded it below.
suggest edits
Download table
Conclusion
What is mathematical modelling? At the start of this post, you read about what it could look like for students to create a mathematical model in response to a real-world scenario. You also took time to engage and reflect upon a task and types of experiences that are representative of the process of mathematical modelling.

Further to this, it's important to consider what mathematical modelling is not.
  • Mathematical modelling (a process) can incorporate the use of mathematical models (e.g., discrete/set, linear, area, etc.) and a variety of tools and representations to organize and communicate thinking and understanding, but ...
    • The use of models in the mathematics classroom is not, in and of itself, mathematical modelling. Mathematical modelling is a process that uses these models as one part of an interconnected and iteratively-applied set of components (see graphic, p35).

As mathematical modelling is an integrated process that can support students' conceptual understanding, development of big ideas, and ability to make decisions, it's incredibly important that we incorporate mathematical modelling across the course of a year of programming--seeking to help students make connections within and across strands.
“When students construct a big idea, it is big because they make connections that allow them to use mathematics more effectively and powerfully. They will be able to use this knowledge to solve a range of problems flexibly…” (Government of Ontario, 2004). 

Dr. Marian Small (2009) reminds us that “Our job is to build their [students’] confidence and reassure them that they really can understand the new ideas with which they are dealing. This is accomplished by helping them create an internal map of how these [essential] ideas connect to what they already know. This means we, as teachers, need that internal map, too, in order for our students to recognize our comfort and confidence.”
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. 

In the next post, we'll continue to develop our understanding of what mathematical modelling could look like 
in terms of its scope--i.e., how modeling might play out over long-range integration of curriculum expectations (coming soon ... Part 2).

I am more than happy to collaborate with you and make our learning visible, here. If at any time, you have questions or comments, please feel free to comment to this blog and/or reach out to me here. 

Professionally Yours,

Chris Stewart, OCT
Educational Consultant at Flipping the Focus (c) 2020
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References
Anthony, G., & Walshaw, M. (2009). Characteristics of Effective Teaching of Mathematics: A View from the West . Journal of Mathematics Education, 2(2), 147-164.

Creativity in Mathematics. (2018, July). Guest Blogger: Scott Kim, III [Web log post]. Retrieved from https://cre8math.com/2018/07/

Ehlert, D. (2015, October 26). Promoting Growth Mindset with 3 Act Math. Retrieved October 06, 2020, from https://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/Blog/Promoting-Growth-Mindset-with-3-Act-Math/

Fetter, A. (2015, June 8). Ever Wonder What They’d Notice? Retrieved from https://youtu.be/a-Fth6sOaRA 
​
Government of Ontario. (2020). High-impact Instructional Strategies in Mathematics. Retrieved from https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics/resources

Government of Ontario. (n.d.). Teaching and Learning Mathematics – The Report of the Expert Panel on Mathematics in Grades 4 to 6 in Ontario, 2004. Retrieved from http://www.edu.gov.on.ca/eng/document/reports/numeracy/panel/index.html

Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, J. Kilpatrick, & E. Pekhonen (eds.) Posing and Solving Mathematical Problems: Advances and New Perspectives. New York, NY: Springer. [ResearchGate, Academia]

Liljedahl, P. Building Thinking Classrooms - Keynote. OAME Leadership Conference, Ottawa (Nov 10, 2017). Retrieved February 12, 2018, from http://www.peterliljedahl.com/presentations

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

​Liljedahl, P. (2014). The affordances of using visually random groups in a mathematics classroom. In Y. Li, E. Silver, & S. Li (eds.) Transforming Mathematics Instruction: Multiple Approaches and Practices. New York, NY: Springer. [ResearchGate, Academia]

Meyer, D. (2019, March 8). All Learning Is Modeling: My Five-Minute Talk at #CIME2019 That Made Things Weird [Web log post]. Retrieved from https://blog.mrmeyer.com/?s=mathematical+model

Meyer, D. (2017, March 9). The Difference Between Math and Modeling with Math in Five Seconds [Web log post]. Retrieved from https://blog.mrmeyer.com/2017/the-difference-between-math-and-modeling-with-math-in-five-seconds/

Pearce, Kyle. (2018, October 21). The 3 Act Math Beginner's Guide: Spark Curiosity to Fuel Sense Making. Retrieved from https://tapintoteenminds.com/3act-guide/


Pólya, G. (1957). How to solve it (2nd ed.). Princeton N.J.: University Press.
​

Small, M. (2009). Big ideas from Dr. Small: Creating a Comfort Zone for Teaching Mathematics: Grades 4-8. Toronto, Ontario: Nelson Education.

​Smith, M. S., & Stein, M. K. (2011). 5 Practices for Orchestrating Productive Mathematical Discussions. Reston, VA: The National Council of Teachers of Mathematics.

The Ontario Curriculum, Grades 1–8: Mathematics – 2020. (n.d.). Retrieved October 06, 2020, from https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics/
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Teaching & Learning Mathematics: Like a Fractal!

2/18/2020

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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!
Balance 2020

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).
Fractal as Metaphor
Fig. 1 - Fractal as a metaphor for teaching and learning
Proposition: 
  • What if we took fractals, as a metaphor, and applied them to the teaching and learning of mathematics through spiralling--i.e., with a structure of self-similarity and a generative, learning process based on dynamic iterations?
    • Would it help us to better understand the complexity that this type of teaching and learning creates and uses to support student learning?
Fractal - Spiral
Sierpinski Triangle
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). 
Sample Long-Range Planning Template - Topics Blocked
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).
Sample long-range planning template for spiralling
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.
Cycles in Spiralling
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​
  • Initial building blocks; focus on prior knowledge, building representations, and making some connections
Cycle 2
  • Extend 1st cycle with applications; explicit discussion of how representations can be used to develop conceptual understanding of operations and earlier applications​
Cycles of a Sprialled Curriculum
Fig. 5 - Cycles in a Spiralled Curriculum
​Cycle 3
  • Build connections within and between strands for concepts studied up to now
Cycle 4
  • Extend the concepts studied earlier; introduce the more complex aspects of each strand
Cycle 5
  • Develop and solve problems that incorporate more complex aspects; making some connections to real-life applications and some connections within strands
Cycle 6
  • 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.
  • 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).
The Assessment Loop
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).
Complexity & Implementation Principles - Spiralling through Curriculum
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.
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.
Managing Flow - Thinking Classrooms
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.
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").

June 2020 registration now OPEN!
Register

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
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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/  
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    Chris Stewart, OCT Educational Consultant, Flipping the Focus

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    I am passionate about leadership for learning and teaching and learning through inquiry. Through collaborative exploration of high-yield, pedagogical strategies, I have been able to further engage students to deepen their learning and fellow educators in continuously growing their practice--Flipped Learning, Thinking Classrooms, culturing Student Voice, and balancing approaches to instruction in Mathematics--as examples.  I hope that this site serves you well in your educational journey through teaching and learning by moving professional learning into your time ... your space. If you have questions or feedback, please feel free to contact me. Sincerely, Chris Stewart (OCT).

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I am passionate about leadership for learning and teaching and learning through inquiry. Through collaborative exploration of high-yield, pedagogical strategies, I have been able to further engage students to deepen their learning and fellow educators in continuously growing their practice--Flipped Learning, Thinking Classrooms, culturing Student Voice, and balancing approaches to instruction in Mathematics--as examples. I hope that the sites and resources I have created serve you well in your educational journey through leadership for learning, teaching and learning by moving professional learning into your time ... your space. If you have questions or feedback, please feel free to contact me. Sincerely Yours, Chris Stewart |OCT | Founder & Educational Consultant, Flipping the Focus.

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