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.
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.
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”, ...
- “... [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).
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.
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.
- 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.
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).
- Teach through problem solving (p11) and
- Teach students about problem solving (p13; adaptive reasoning is key to mathematical modelling)
To begin, let’s try out this task together (task available at: World Record Balloon Dog).
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).
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.
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).
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)
Questions arising from students might include:
- How many balloons has the dog popped?
- How long did it take dog to pop this many balloons?
- 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).
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).
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 watch 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?
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).
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).
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).
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.)
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-assessment.
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.
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.
(Source: Ontario Mathematics Curriculum Expectations, Grades 1 to 8, 2020, p34).
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.
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.”
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.
Chris Stewart, OCT
Educational Consultant at Flipping the Focus (c) 2020
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