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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

Re-inventing Teaching & Learning in Mathematics Education - Part 2: Synchronous, Online Learning and Formative Assessment Practices

12/17/2020

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Welcome back to Flipping the Focus.

This is the 2nd post in a 5-part series devoted to re-imagining how assessment practices in mathematics education can be improved, in part, with pedagogically-driven uses of technology.
Female student waving to teacher via video-teleconferencing
Part 1 of the series provided an example of pedagogical principles that can be used when making decisions about how a variety of tools and representations--including interactive, digital technologies--can be leveraged to improve student learning.

In this post, you'll have an opportunity to learn more about synchronous learning and how it can:
  • relate to classroom practice; and
  • work with hybrid and online learning models.

You'll also have the chance to begin:
  • learning about Thinking Classrooms through the vignette, below;
  • sharing your own and/or asking other educators about their experiences with different models of learning and modes of delivery; and
    • expanding your professional learning network around assessment practices, various models of learning, and modes of delivery in mathematics teaching and learning.
Introduction
Reinventing, now more than ever, how technology supports math teachers and students is so important. Educators need high-quality teaching resources, pedagogical supports, and professional learning that engender and communicate respect for equity and inclusion.

So much of the educational landscape has been changing--evidenced by the current, global paradigm; shifting priorities; and efforts to hybridize learning models and make online learning successful--for example, synchronous online learning.
What is Synchronous Learning?
​
Synchronous learning occurs in real-time--i.e., in real-time, it allows educators the opportunity to connect with their students such that the immediacy of feedback is greater. 

When this mode of delivery is done remotely and online, with the support of video-teleconferencing platforms and a host of web-based applications, we need to remind ourselves that synchronous learning is but one aspect of authentic and engaging online learning.
two speech bubbles overlapping
Just like effective teaching requires opportunities for students to engage in self- and group-directed learning activities, educators working in an online environment can help support their students' learning preferences by intentionally incorporating asynchronous learning activities into their teaching.
For example, by incorporating asynchronous learning opportunities, educators can delay some of their feedback so that students can continue thinking and re-focus their attention on reflection and metacognition (Desmos, 2016).
There are also many high-impact instructional practices in mathematics that we often associate directly with formative assessment practice--establishing learning goals, co-constructing success criteria, and interacting with descriptive feedback. On some occasions, other practices might be thought of as influencing formative assessment practices like the use of problem-solving tasks, documenting and reflecting on how students use tools and representations, and actively listening to students during math conversations; while at other times, the use of other practices inform instruction and come about as a result of ongoing, formative assessment--e.g., teaching about problem solving, intentional and well-positioned use of direct instruction, small group instruction, deliberate and purposeful practice, and flexible groupings. Of course, these distinctions are but one way to think about the complex relationship between instruction and assessment: in many situations, practices that inform instruction become those that influence assessment, and vice-versa. For example:
  • We might decide to facilitate a math conversation to better uncover students' thinking (inform); on other occasions, our assessment points to students being ready for a conversation that helps to consolidate their learning (influence).​
trail - fork in the road
  • While a small group of students is working on a task, observations and conversation  indicates that direct instruction (in the form of hints and answering keep-thinking questions) is necessary (influence); on other occasions, direct instruction with several groups and/or the whole class points to using questions that are focusing in nature, followed by encouraging students to continue working on the task in small groups (inform).
What might seem to be a dichotomy--practices that inform and/or influence--is much-needed to ensure that educators and their students are better able to interact with descriptive feedback that is based on the development and refinement of success criteria. When it comes to formative assessment, the decision to use practices in ways that make us think versus uncovering and re-purposing our thinking moves us closer to attaining learning goals.

The remainder of this post focuses on the following:
  • examining and reflecting upon a conceptual model for synchronous online teaching and learning in mathematics in Secondary grades;
  • sharing our own and connecting to the experiences of others with synchronous online teaching and learning across all divisions--primary, junior, intermediate, and senior ...
​
... and all of this to help our professional community of practice support one another by providing the best answers we have, at this time, to the following question:

 How can teachers best implement these instructional practices with fidelity
​in synchronous online learning?


Vignette: Bringing Thinking Classrooms to Life Online
 
Secondary School Example: Problem-based Learning in Mr. Stewart's Mathematics Classes
Introduction:
Mr. Stewart recognizes the value in providing space for his students to think, communicate, and make visible their mathematical ideas and struggles, and to ask questions of one another. It’s within these spaces where he’s better able to listen to conversations and observe and document his students’ thinking—all of this to provide timely, descriptive feedback to his students on how they’re working towards mathematical learning goals and monitoring their responses to feedback—feedback, generally, coming in the form of hints and questions that spur students to continue thinking.
Good Tasks:
In order that he is able to provide timely and relevant feedback to students, his students’ curiosity needs to be piqued and students need to be challenged cognitively—i.e., within the scope of something they are almost ready to do. With tasks that are low-floor/high-ceiling, Mr. Stewart is able to elicit varying degrees of student thinking—thinking that is the driver of everything that happens throughout the lessons.

Defining the Task:
From the outset of the first synchronous lesson (in a series), Mr. Stewart recognizes that students will need to develop an understanding of the context in which the problem sits before the problem is fully defined. This early part of the lesson tends to be short and is narrative in fashion—generally, a series of questions are asked to activate students’ prior knowledge, clarify the information provided, and to come to a consensus on what problem needs to be solved.

​Mr. Stewart often displays visuals associated with the problem through sharing his screen and invites students to add their ideas to a Jamboard. Students are also able to raise questions and add suggestions in the chat window. 

How Students are Grouped:
With the introduction complete, students are randomly and visibly assigned to groups of three. Using Zoom, students are aware that Mr. Stewart uses the random breakout room assignment feature.
 
Jamboard logo
How Students Share Thinking:
As students move into their breakout rooms, each group is assigned a page in the recently-shared Jamboard. Typically, Mr. Stewart asks groups to relay their thoughts to one student who can add their group’s thinking to their board. In some instances, students find it just as easy to work on the problem in a parallel manner on paper, having discussion along the way.
As groups work in their breakouts, Mr. Stewart cycles through each of the rooms to observe and respond, accordingly, to students’ questions. And prior to a fully-developed approach being completed, one of the students in the group uses their webcam to capture their written work as an image and uploads it to their page in the Jamboard. With the newly-uploaded image, students then take time to continue by adding annotations and explanations with digital sticky notes.
Managing Flow:
At key points in the lessons, Mr. Stewart intentionally interrupts student-led, collaborative group work—i.e., returning to the main meeting by closing breakout rooms—to draw the class’ attention to some key moves students have been making. The placement of direct instruction at key points in the lesson offers Mr. Stewart a chance to help students co-create a narrative from how they moved from understanding the problem, through various stages, and ending with checking the reasonableness of their answer. This narrative, really is composed of success criteria—some of these anticipated; while others occurring incidentally during the course of learning.

As he prompts groups to share their thinking, Mr. Stewart is listening and looking for conjectures, estimations, sketches, use of terminology and symbolic notation, strategies uncovered in building up towards the task, and explanations—early on, many of these being expressed in a less-than-formal way.

Each sample discussed has something of value to establishing success criteria, and as students share, he asks questions of the group about the approaches they’ve taken, as well as asking the class to discuss how they might connect the representations between different groups’ approaches.

​As students share, he begins adding these highlights to some of the examples of students’ work in the pages of the Jamboard—essentially, codifying what’s been done; validating students’ contributions; and demonstrating that students can leverage what they’ve done to move into a more formal means of analyzing and solving the problem with calculations, graphs, and algebraic representations. During this part of the lesson, Mr. Stewart uses the recording (video) feature in Zoom both a means of further documentation and a resource that students can access through their LMS.

Building and Leveraging Student Autonomy:
Knowing that formative assessment, up to this point, has been purely teacher-driven, he also recognizes the value of students providing feedback to one another. Having drawn the class together for direct instruction, students are encouraged to go back to working on the problem—this time, looking at the problem solving process through a formal lens.

Given that hints were given, keep-thinking questions answered, and the class had been brought together for discussion about their approaches, students now have a few criteria that they can use to assess their next steps in the problem solving process. Through the use of breakout rooms, students are manually re-assigned to their groups such that they can continue working together. Again, Mr. Stewart cycles through the various breakout rooms to observe and support students, accordingly.

Consolidating Student Learning:
Once Mr. Stewart recognizes a place where each student has been able to successfully engage in the task and has made some progress (which could take place over several days), he brings the class together again to have students discuss their thinking and strategies they used to solve the problem. Again, this part of the session is recorded for documentation and as a resource for students.

As he listens, he continues to annotate students’ Jamboard solutions and asks the class to review the list of success criteria the class has been developing for any refinements that can be made. As students were working, Mr. Stewart conveniently added an additional page to the Jamboard where both developing success criteria and learning goal have been posted. As the lesson wraps up, he reviews each of the criteria and how students have used them to work towards the learning goal. 

Meaningful Notes:
At the end of each lesson, students are reminded that they have the Jamboard to reflect on during periods of asynchronous study. It’s often during these times, where students are encouraged to create notes that are meaningful to themselves—i.e., if you had to explain what happened in today’s lesson to yourself or to someone else, what would you say?

Formative Assessment - Next Steps:
In order that he better understands what students know and are able to do at this time (as well as each day spent working towards the learning goal), he provides them with an exit ticket. On the ticket students will notice a problem that relates to the one they’ve been working on and/or some self-assessment questions—questions asking them to articulate what is going well, what they need to work on, and if there are any particular supports they need to keep going.
Mr. Stewart tries to provide students with options he knows will both engage and encourage continuous and shared reflection. For students who enjoy writing and diagramming their thinking, students can add to the digital notebook of the class’ LMS. For students who prefer to explain their thinking orally, Mr. Stewart uses Flipgrid.
Flipgrid logo
Through this app, students record short videos of themselves. In either case, he can provide feedback to students. In the LMS, a variety of forms of feedback can be given; in Flipgrid, both video responses and comments can be made. Either before students return to class the next day and/or at the beginning of the class, students will have had a chance to review their exit tickets.

​Much of what students have shown and have shared in their work and comments will help Mr. Stewart decide if further consolidation is required, guided groups need to be formed, student thinking needs to be challenged, and/or extensions are appropriate for students to pursue. Occasionally, these exit assignments become much like a threaded discussion with students posting and commenting on each other’s posts in Flipgrid or through the discussion board in their LMS. Altogether, Mr. Stewart finds that this ‘twist’ on the use of these tools, allows students to connect and continue conversations during times of asynchronous learning.
Check Your Understanding Questions:
Based on his ongoing observations and conversations with students, Mr. Stewart now provides his students with a set of questions (4 to 6 questions) they can use to check their understanding. These questions (posted on the LMS) are only released to students when he notes that students are in a position to practice correctly. These questions are for students’ self-assessment purposes only. Answers (only) are posted to the LMS so that students can quickly determine if they’ve made a mistake, and model solutions are only shared out once students have attempted solving each problem.

​Based on all aspects of the formative assessments that both Mr. Stewart and his students have been doing, as well as his students’ interests, further consolidation sometimes involves reviewing and discussing model solutions to the questions students have completed to check their understanding. Sometimes these solutions and their explanations are recorded using screen-casting applications and shared through the class’ LMS, where students can reflect on them during times of asynchronous work. On other occasions, these solutions are shared and discuss with students in a synchronous manner during Zoom meetings.


Professional Learning and Building Your Professional Learning Network:
​Sharing Examples: Synchronous Online, Other Models of Learning and Formative Assessment Practices
 
Through the example, you were looking through a window onto how another educator is building their practice and learning alongside their students. Perhaps, you may have even been looking into a mirror reflecting your own practice: seeing similarities in your choices and actions and/or identifying with 'moves' that you've been planning to implement. Maybe there were practices being described that you're questioning and wanting to know more about the 'why' and 'how'.
window and side-view mirror
All too often we don't have an opportunity to see ourselves reflected through the examples being shared. We might also not have an opportunity to share our perspectives. This is much like what our students experience when there are many windows onto the world and others' learning and not enough mirrors to reflect their own needs and identity.
The "Balance Series"
The "Balance Series"--re-launching in 2021 with new webinars on assessment practices--works with you and other educators from around the globe to examine the complexity of teaching and learning mathematics. This series is being offered for one, simple reason: educators, myself included, aspire to be and do more in the service of student learning. And much like our students, we, too, recognize that where there is a student learning need, we might need support in how we can best support them. You can learn more about the "Balance Series" from its Summer 2020 delivery here (information updates coming soon ...).
In the meantime, we have a great opportunity to begin sharing examples from our own context, thereby providing windows and increasing the likelihood that we'll see ourselves through the examples shared and comments made by our peers. There's so much that we can learn from one another!
Balance Series Logo
How Do I Participate?
As the "Balance Series" prepares to launch for sessions geared around formative assessment, I would like to invite you to do two things: 1 - Freely contribute to our Virtual Community Builder - Formative Assessment and as often as you like (see below) and 2 - Join the "Balance Series" of webinars related to assessment as they become available.

Webinar notifications will be sent directly to the email you provide through the community builder form, below. Please rest assured that only I will use your email only to communicate with you about professional learning opportunities I'm preparing and offering. The price of admission? The only thing I ask is that you support other math educators, starting with a contribution to the community builder.

If you have any questions about this post, the "Balance Series", and/or participating with the community builder, please feel free to contact me here. 
The Virtual Community Builder - Formative Assessment
 
Question mark
Your participation is easy and straight-forward. Click one of the survey links, below (Survey-1 OR Survey-2), to respond to a few questions--some of them personal (your name and email); the majority, about the subject of this blog post and formative assessment, in general.
​Again, name and email are purely for my email communication with you. Please read on for more information concerning survey options 1 and 2.
  • Option 1: Note that this would not be much of a learning opportunity without having access to the examples and comments made by other math educators. Here's my commitment to you: If I receive Survey-1 responses, they will be openly accessible to all respondents. Once you complete the survey, you'll notice a link to view a summary of all responses. These results will not have your name or contact email, as the form will not be collecting this information. With this option, you'll be able to read about other educators' experiences, but you won't have the option to interact with them. You'll also be able to contribute more of your experiences if and when you decide you'd like to.
  • Option 2: Respond to Survey-2 if you'd like to share your contact information with others who have responded to the same survey. I.e., You want to build your professional learning network about assessment in mathematics education. When you respond to the survey, I'll send you a link to view a "Results-2" summary of all contributions. Like Survey-1, you'll also be able to contribute more of your experiences if and when you decide you'd like to. ​As this form will be set up to collect names and emails, you'll then have a way of reaching out to any or all participants in this version of the survey to further your learning. By networking with other educators, you'll be able to raise questions, have conversations, and coordinate opportunities to communicate and collaborate.
What information is the survey collecting? Will my participation be beneficial?
Absolutely! We always want to know if what we share will be of benefit to others.

When you access and respond to the Virtual Community Builder - Formative Assessment, expect to answer questions about the following:
  1. Grades you commonly teach or have a hand in supporting (e.g., School Administrator)--e.g., Primary (K to Year 3), Junior (Years 4 to 6), Intermediate (Years 7 and 8), Secondary (Years 9 to 12).
  2. If you're engaged in synchronous face-to-face learning, synchronous online learning, a hybrid of both synchronous face-to-face and online learning, or other modes of delivery, what does your assessment practice look like? Over longer periods of time (e.g., several lessons)? What does the asynchronous aspect look like? What's working? What wonderings do you have? What supports do/might you need to reach your goals (real/anticipated)?

​In support of #2, you can describe how you provide opportunities for students to interact and construct their learning together, and how students are interacting with feedback. For example, you might choose to discuss one or more of the high-impact instructional practices listed above and/or how digital technology is being used to support your students' learning.
Learn More - Balance Series
Survey-1
Survey-2
Results-2
Recall: When you respond to Survey-2, I'll send you a link to view a "Results-2" summary of all contributions.
Contact

Final Thoughts
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, and services provided through Flipping the Focus, 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. 

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 contact me.

Professionally Yours,
Chris Stewart, OCT
Educational Consultant, Flipping the Focus (c) 2020
Learn More
contact

References
The Desmos Guide to Building Great (Digital) Math Activities. (2016, May 11). Retrieved from https://blog.desmos.com/articles/the-desmos-guide-to-building-great-digital-math/

Government of Ontario. (2010). Growing Success: Assessment, Evaluation, and Reporting in Ontario's Schools, Kindergarten to Grade 12. Retrieved from http://www.edu.gov.on.ca/eng/policyfunding/success.html

Government of Ontario. (2020). Instructional Approaches in Mathematics. Retrieved from https://www.dcp.edu.gov.on.ca/en/curriculum/elementary-mathematics/context/some-considerations-for-program-planning-in-mathematics

Gutiérrez, R. (2018). The Need to Rehumanize Mathematics [Introduction]. In I. Goffney, R. Gutiérrez & M. Boston (Eds.), ​Rehumanizing Mathematics for Black, Indigenous, and Latinx Students (pp. 1-10). Reston, VA: National Council of Teachers of Mathematics.

Liljedahl, P. (2021). Building Thinking Classrooms in Mathematics - Grades K to 12: 14 Teaching Practices for Enhancing Learning. Thousand Oaks, California: Corwin Press.
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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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    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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