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The Mathematical Modelling Process (Taken from the DCP)
Mathematical modelling provides authentic connections to real-life situations. 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.
The process of mathematical modelling has four key components that are interconnected and applied in an iterative way, where students may move between and across, as well as return to, each of the four components as they change conditions to observe new outcomes until the model is ready to be shared and acted upon. While moving through these components, social-emotional learning skills and mathematical processes are applied as needed.
1. Understand the problem
What questions need answering?
What information is needed?
2. Analyse the situation
What assumptions do I make about the situation?
What changes, what remains the same?
3. Create a mathematical model
What representations, tools, technologies, and strategies will help build the model?
What mathematical knowledge, concepts, and skills might be involved?
4. Analyse and assess the model
Can this model provide a solution?
What are alternative models?
The 4 Steps of Inquiry-Based Learning (adapted from Edutopia) by Heather Wolpert-Gawron
Students develop questions that they are hungry to answer. Have them develop a problem statement that requires them to pitch their question using a constructed response, further inquiry, and citation.
Research the topic using time in class. The teacher's role is to assist in guiding them and modeling methods of researching reliably.
Have students present what they’ve learned. Students should create and present a culminating artifact. Many people can understand content, but can they communicate it? Students can develop a website, slideshow, infographic, or physical model.
Ask students to reflect on what worked about the process and what didn’t. Reflection is key: reflecting on the process itself. That’s where you can work in metacognition—thinking about thinking (assessment as learning). Have students focus on how they learned in addition to what they learned.
OAME 2025 Presentation - "Real-World Connections through Mathematical Modeling in Elementary" - Oshawa, Ontario
OAME 2025 Presentation - "Real-World Connections through Mathematical Modeling in Elementary" - Oshawa, Ontario
Costello, D. (2022). Mathematizing Student Thinking Pembroke Publishers. costellomath.com
MacKenzie, T. (2019). Dive Into Inquiry: Amplify Learning and Empower Student Voice. ElevateBooksEdu. trevormackenzie.com
Mathematical Modeling General Resources
Mathematical Modeling General Resources
Taken from the RabbitMath website: "Standard school curricula emphasize technical proficiency and as a result the curriculum often feels fragmented. RabbitMath brings whole 'works of art' into the classroom. Mathematics certainly requires focused individual study but our projects bring students together to share their struggles and their conjectures. New more powerful ideas can emerge from those encounters. The RabbitMath curriculum fosters this mode of focused thought and collaborative discovery."
Jenna Laib's site Slow Reveal Graphs allows teachers to practice guided inquiry with students. Even though many of the examples can apply to all audiences, teachers can find some great Canadian content:
Real-World Problems Being Solved By Mathematicians - CEMC Waterloo
Explore some real-world problems being solved by mathematicians. For each real-world problem, try your hand at a related mathematical problem. These problems are designed to be accessible to students in grades 7-8 and later. Solutions are provided.
Mathematical Modeling Templates and Interactives
Mathematical Modeling Templates and Interactives
Mathematical Modelling Observational Chart:
Thanks to Michele Mellow for creating this observational chart with "look fors" throughout the month
Grade 3-6 - Cultural Festival
Grade 3-6 - Cultural Festival
Make a copy of the student slides here
Culture and Math
Culture and Math
It's essential for students to do a cultural exploration before engaging in this month's inquiry. By offering mirrors to their own culture, windows to observe other cultures, and sliding glass doors to understand and appreciate other cultures, students take collective ownership of the inquiry. The following is a template by the TCDSB Math Facilitator team.
MINDS ON - Notice Statements & Wonder Questions
ACTION Part 1 - Challenge Questions
Students at St Nicholas of Bari used a class Jamboard to ask their questions
ACTION Part 2a - Assumptions from the Challenge Questions
ACTION Part 2b - Where's the Math?
ACTION Part 3 - Building a Model
Mme Mancuso integrated mathematical concepts into cultural contexts so that students can plan for the cultural festival. Students had an opportunity to compare rates, learn about specific foods, and incorporate fractions into the discussion.
Mr. Scali and student teacher Ms. Povolo (TCDSB) led the Grade 3-4 class through the inquiry. As the students asked their questions, the teachers created a planning sheet to guide the students through their research. Their questions focused on four categories:
Food
Decorations
Music
Invitations
CONSOLIDATION - Presentation
Ms Kerpel (TDSB) led her Grade 3 class to design a party or festival. They consolidated their learning with these amazing designs and explanations about what they needed for their party. Students used multiplication, division, and ratios to complete their inquiry.
Mr. Scali and his student teacher led the Grade 2-3 class at St. Elizabeth Seton on a global exploration and created plans to lead a multicultural festival.
Grade 3-5 - What Does It Take To Raise A Pet?
Grade 3-5 - What Does It Take To Raise A Pet?
Make a copy of the student slides here
Recording from Mathematical Modelling Co-Planning Session: January 5, 2023
MINDS ON - Notice Statements & Wonder Questions
Notice and Wonder from the image provocations in the slideshow. Ms. O'Connor helped the Grade 3's at St Marcellus and Mr. Tinto helped the Grade 5's at D'Arcy McGee through this process.
ACTION Part 1 - Challenge Questions
Ms. Kerpel (TDSB) summarized all the open questions that the students came up with. Student-input is important in driving engagement and contextualizing the mathematics when it's needed. Mr. Tinto used open-closed question techniques to generate more specific examples from the students.
ACTION Part 2a - Assumptions from the Challenge Questions
Ms. Kerpel's summarized the Grade 3 responses into a Constants and Variables anchor chart.
Ms. O'Connor guided the Grade 3 students and Mr. Tinto guided the Grade 5 students through sorting their questions and then asked students to consider any assumptions they would have to make. This stage is important for students to understand what variables are being controlled (kept the same and would not affect the final answer) and which variables change (would need to be studied and researched).
ACTION Part 2b - Where's the Math?
Once the students have decided on the variables and constants, it's time for students to ask what math they might need to answer these questions. Students might not know right away what they need, and that's OK. Over the course of the month, a new topic might emerge where the teacher will need to teach "just in time" for the students to need that information.
Ms. Kerpel's class considered a variety of math topics and also looked at some possible tools they will need to answer their questions.
Ms. O'Connor's class went to the whiteboard to brainstorm ideas, and she let the students use the curriculum one-pagers to find connections with the inquiry. Mr. Tinto summarized the necessary math topics in a GoogleSlide.
ACTION Part 3 - Building a Model
Mr. Da Rocha created this template to help students with their research. As student questions tend to grow, the teaching team at St. Nicholas of Bari learned how to use those questions to generate new assumptions and also direct the learning in a specific way.
Ms. Kerpel helped her Grade 3 students practice geometric shapes using a virtual geoboard. Students connected these shapes to their final product: creating a pet home
CONSOLIDATION - Presentation
The Grade 3/4 Mathematicians at St. Elizabeth Seton combined data, measurement (time, volume), media literacy, and financial literacy for their culminating presentation. Students created chart-paper infographics to showcase the learning throughout the month.
Ms. Kerpel's Grade 3 class created diagrams that included measurements, shapes, and financial literacy. Lots of creative examples of what pets need!
The Grade 5 Class at St. Nicholas of Bari created websites to showcase their budget, images of items purchased, and selections of what it takes to raise a pet
Grade 5 - Inquiry Pet Project - Mr. Tinto's Class - Parrot.pdfThe Grade 5 Students at D'Arcy McGee created a presentation about raising a parrot. They chose to look at financial literacy, multiplication, and media literacy when creating their final product.
Grade 3-6 - Redesign a Room (inspired by OAME)
Grade 3-6 - Redesign a Room (inspired by OAME)
Q&A from our previous webinar:
Why 20 000$ ?
To meet the Grade 5 curriculum of Numbers to 100 000, we chose a five-digit maximum budget. For Grade 4, feel free to lower the maximum to 10 000$ or even less.
Why not choose a room in their house or a hypothetical "dream home" ?
For equity reasons, choosing a "dream" or "redesigned" personal space may be challenging for students who experience difficult living arrangements, and any discussion regarding hypothetical budget and financial choices should be placed on a level playing field (in this case, at the school)
Inquiry samples from St. Paul - 2022
Inquiry samples from St. Paul - 2022
Day 1a: Notice and Wonder
Day 1a: Notice and Wonder
Students performed a walkaround of the school, noticing how classrooms are being used and the equipment in the rooms. They used sticky notes to keep track of the questions they had about specific rooms.
Day 1b: Challenge and Assumptions
Day 1b: Challenge and Assumptions
We presented the challenge of redesigning a room with a fictitious 20 000$ grant. Students took their initial Notice & Wonder statements, and using a Question Storm technique, they created new questions to pose about the challenge.
Students were placed into groups and chose the room they wanted to remodel. They refined their questions and collected them on whiteboards. We reviewed the questions, eliminated the ones where we could assume an answer, and assumed some conditions to continue our exploration.
Day 2: Where's the Math and Data Collection (Measurement)
Day 2: Where's the Math and Data Collection (Measurement)
"Where's the math" allowed students to consider which math topics would be necessary to explore in order to solve their question. All groups determined they needed to measure the dimensions of the room, so after determining what supplies they needed, they set off to measure their rooms. With the measurements, they created both a hand-drawn scale model and a digital model using GoogleSlides.
Day 3: Shopping, Budgeting, and Tabulating (Number and Financial Literacy)
Day 3: Shopping, Budgeting, and Tabulating (Number and Financial Literacy)
With measurements complete, we had to review and make new assumptions, and students needed to explore rates and formulas to calculate the area and perimeter of the room. Then students went shopping online to find classroom furniture to compare prices, quantities, and qualities. They used GoogleSlides to organize their research.
Day 4: Presentation (Media Literacy)
Day 4: Presentation (Media Literacy)
As part of the cycle, we reviewed the questions and assumptions yet again, and students offered suggestions about the presentation. We co-created success criteria for the presentation, then students worked together to create a presentation slide and final budget table to share with the class. A final metacognitive SEL survey was completed for students to assess their own mindsets throughout the mathematical modeling cycle and to reflect about their contributions.
Day 5: Assessment of Learning
Day 5: Assessment of Learning
Grade4-5 Design a Room - Assessment.docxGrade 4-5 - Best Athlete in a Sport (inspired by OAME)
Grade 4-5 - Best Athlete in a Sport (inspired by OAME)
Grade 5-6 - Screen Time (inspired by Juicy Math)
Grade 5-6 - Screen Time (inspired by Juicy Math)
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