Engage Your Students in Meaningful Mathematics Through Modeling

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Welcome!

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NCTM/SIAM/COMAP Joint Committee on Modeling Across the Curriculum

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Revealing the Brilliance of Students Through Mathematical Modeling

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Ben Galluzzo

COMAP ben@comap.org

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Joleigh Honey

J Honey Math, LLC

NCTM Board of Directors Joleighhoney@gmail.com

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Maria Hernandez

NCSSM/COMAP maria.hdz2718@gmail.com

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Megan Wickstrom

Montana State University

NCTM

megan.wickstrom@montana.edu

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Numbers in the News

100,000 stolen

Article and image from apnews.com (link)

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Numbers in the News

0. Which article?

1. What happened?

2. Where’s the math? (and if you know it, Where’s the modeling?)

3. What do you think?

• Do you believe the reported result? Why or why not?
• Why does the article matter to you (i.e., your group)? More generally, why should anyone care about this?
• Could the number(s) or mathematical method(s) in the article be used to examine other real-world problems? If so, share a few thoughts.

Let’s try it!

15 home runs & 36 runs in 3 games

Article from sports.yahoo.com (link) and image from cbsnews.com (link to associated article)

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Our Plan for the Day:

• What is Math Modeling?
• Engage in a Modeling Task
• Value of modeling: building identity, utilizing assets, implementing processes
• Share Resources
• Cross Cutting Concepts
• Assessment
• Closure

Access today’s slides, modeling resources and opportunities at:

www.comap.org/session

What does it mean to Model?

What is math modeling?

Mathematical Modeling is using mathematics to understand a real-world situation and then using math to take action or predict the future. This takes place when the math and the real world are both taken seriously. *

*This is paraphrased from a statement made by Henry Pollack

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Math Problem to Modeling Problem

Our school has 295 students. What is the best way to transport students on a field trip?

Difference between modeling and word problem

Example:

(from Siam-Guidebook: Math Modeling: Getting Started and Getting Solutions by Bliss, Fowler, Galluzzo)

Application:

The population of Yourtown is 20,000 and 35% of its citizens recycle their plastic water bottles. If each person uses 9 water bottles per week, how many bottles are recycled each week in Yourtown?

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Modeling:

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How much plastic is recycled in Yourtown?

(a)

(b)

(c)

(d)

Math Modeling Cycle Diagrams

Mathematical Modeling Mindset

This mindset is shaped by a set of interconnected habits and ways of thinking. (some of which are listed below):

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● Mathematical Versatility

● Creativity and Adaptability

● Evaluation and Revision

● Metacognition and Reflection

● Collaboration and Communication

● Tolerance for Imperfection

● Curiosity and Inquiry

● Simplification and Assumption-Making:

● Mathematical Representation

● Pattern Recognition and Systems Thinking

● Data and Computational Thinking

● Technological Fluency

More than a process

Buc-ee’s Task

• What do you notice?
• What do you wonder?
• What do you know about Buc-ee’s?

Buc-ee’s Task

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What is Buc-ee’s?

• It is a Texas-based chain of mega-gas stations/convenience stores known for clean bathrooms, extensive food offerings, and unique shopping experience. (Fun road trip stop!)

Buc-ee’s Task

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Buc-ee’s is so popular that they would like to add new locations. Here are some of the criteria that they use to determine where to place their next store:

• Traveler-focused not locals focused
• Major highways
• Avoid in or directly near major cities - people won’t

want to stop right away on their journey.

• About a “gas tank” away from store to store

Existing

Buc-ee’s

Locations

Buc-ee’s Task

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Goal: Investigate new Buc-ee’s Location.

1. Identify/specify a mathematical problem(s) that you could solve.
2. What assumptions will you make?
3. What are essential variables that you will consider?

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• Take 2-3 minutes of independent think time to brainstorm these 3 bullets.
• Break out room - 10 minutes to discuss these three bullets and report out on your slide deck.

Buc-ee’s Task

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Goal: Investigate new Buc-ee’s Location.

• Identify/specify a mathematical problem(s) that you could solve.
• What assumptions will you make?
• What are essential variables that you will consider?

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Share out!

Buc-ee’s Task

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Goal: Investigate new Buc-ee’s Location.

• Model - make a recommendation for the new location.
1. What were your assumptions for your model?
2. How did decide where to put the next location?
3. What are the strengths and weaknesses of your model?

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Break out room - 10 minutes to create your model and put in slide deck.

Buc-ee’s Task Feedback

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• Does the proposed location make sense based on Buc-ee’s criteria?
• What are the strengths of the proposal?
• What are some changes that you might suggest to make the proposal better?
• What, if anything, will you take away from your colleagues’ work to help make your model better?

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Let’s take 5 minutes. Choose one or more of the questions from above and leave your colleague a note using a call-out box.

Leave an idea!

Buc-ee’s Task

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Goal: Investigate new Buc-ee’s Location.

• Model - make a recommendation for the new location.
• What were your assumptions for your model?
• How did decide where to put the next location?
• What are the strengths and weaknesses of your model?

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Break out room - 5 minutes to read feedback and make any changes that you feel that you are able to.

Buc-ee’s Task

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Goal: Investigate new Buc-ee’s Location.

• Model - make a recommendation for the new location.
• What were your assumptions for your model?
• How did decide where to put the next location?
• What are the strengths and weaknesses of your model?

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Share out!

Buc-ee’s Reflection

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• How could you make your solution better if you had more time to work?
• How you could translate your solution to work for other companies (restaurants, hospitals, etc?)

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Break

Uncovering Assets of Modeling

Connections to Our Classrooms!

DEBRIEF

Identity

Assets

Processes

Cultivating a Positive

Mathematics Identity

Aguirre, Mayfield-Ingram, Martin

The Impact of Identity in K-12 Mathematics (2024)

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Math Identity:

“The dispositions and deeply held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the contexts of their lives.” (pg 12)

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Mathematical and Statistical Processes

Mathematical and Statistical Processes

Assets

Steele & Honey

Transform Your Math Class Using Asset-based Perspectives (2024)

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Experience

Disposition

Talents

Skills

Knowledge

Reflections

How did the modeling activity:

• Support building a positive mathematics identity?
• Promote mathematical and statistical processes?
• Highlight individual and team assets?

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Student Reflections on Math Modeling

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“I think modeling problems are my favorite in the classroom because they require us to be ‘Math investigators’ where we need to solve the problem ourselves. There is no textbook telling us if we are on the right track. Sometimes homework problems and textbook problems can feel artificial... Modeling problems serve as opportunities for us to use a skill in a way I find quite fun... Those are my favorite moments in class; when we are problem solving together as a group and everyone buys in.”

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"It [modeling] helps me remember the math because then I have some kind of example that can help me think through a problem logically and relate it to something that I know about outside of the classroom. I feel like I can apply this method to a lot of things outside of math, like sciences and literature and history.”

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Student Reflections on Math Modeling

“For me, my favorite part of the class was the open nature of the individual problems. After having gotten used to topic based math classes, having the opportunity to use my previous knowledge and intuition to solve problems instead of relying on my memory to pass tests.”

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“What I like about math modeling problems are the multiple different approaches you can take. When you work with a partner and you both have different approaches that are correct it is a very interesting situation to be in.”

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“I thoroughly enjoy math modeling because we use math skills from previous classes for real world problems. This helps me see how previous skills are applied and make me think about everyday scenarios where I can use math.”

Which currently resonates for you?

Resources

Finding Modeling Tasks!

How can we go about finding modeling tasks that are relevant and meaningful to our students?

What our our goals?

• Modeling as content*- the goal is for students to engage in the practice of modeling and related competencies.
• Modeling as vehicle* - the goal is for students to understand or interpret a mathematical idea through the task.

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*(Julie & Mudaly, 2007)

Finding Modeling Tasks!

How can we go about finding modeling tasks that are relevant and meaningful to our students?

What our our goals?

• Modeling as content*- the goal is for students to engage in the practice of modeling and related competencies.
• Modeling as vehicle* - the goal is for students to understand or interpret a mathematical idea through the modeling task.

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*(Julie & Mudaly, 2007)

Both are modeling! As teachers we have different mathematical goals and time commitments for each standard we teach.

Soap Bubbles and Distance

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What is the least distance of highway that we could construct to connect these 4 cities?

Soap Bubbles and Distance

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What is the least distance of highway that we could construct to connect these 4 cities?

Mathematical Goal - Geometric optimization and discovering Steiner points.

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Mapping out Productive Math Modeling Pathways

Where should the next Buc-ee’s go?

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Math Concepts

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Math Modeling task

�What mathematical ideas might students encounter along each branch?

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How do those ideas intersect with your curriculum?

• Skills already developed
• Emergent/grade level standards
• Inventing new ideas

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What ideas might you prioritize? Why?

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Arnold, Burroughs, Carlson, Fulton, Wickstrom

Becoming A Teacher of Mathematical Modeling K -5 & 6 - 12

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Bliss, Kavanagh, Galluzzo

Math Modeling: Getting Started & Getting Solutions

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Bliss, Galluzzo, Kavanagh, Levy Math Modeling: Computing & Communicating

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Modeling Challenges, Books, & More

Resources

Access today’s slides, modeling resources and opportunities at:

www.comap.org/session

• Sources of math modeling problems
• COMAP: https://www.comap.com/
• SIAM Modeling competition site: https://m3challenge.siam.org/
• Advancing Equity and Strengthening Teaching through Math Modeling https://www.eqstemm.org/
• Other resources:
• GAIMME Report
• Math Modeling Handbooks
• Refresh an “old” task (and win prizes!) Refresh for Relevance Challenge

Go Fishing! (10 minutes of quiet work time)

Search for a topic of interest (real-world or mathematical topic).

• Look for images, graphs, articles, or lessons related to a modeling topic for your students.
• What about the task might peak your students’ interests or relate to their lived experiences?
• What mathematical ideas will your task relate to?
• What purpose will the task serve?
• What excites you about the prospect of doing this modeling task?

Cross-Cutting Concepts

Cross-Cutting Concepts

Patterns & Generalization

Variability & Change

Functional & Structural Thinking

Comparison, Difference, & Equivalence

Making &

Interpreting Predictions

Math Modeling & Cross Cutting Concepts: GAIMME

Patterns & Generalization

Variability & Change

Functional & Structural Thinking

Comparison, Difference, & Equivalence

Making &

Interpreting Predictions

Math Modeling &

Cross Cutting Concepts: Critical Thinking

Patterns & Generalization

Variability & Change

Functional & Structural Thinking

Comparison, Difference, & Equivalence

Making &

Interpreting Predictions

Math Modeling &

Cross Cutting Concepts: EQSTEMM

Patterns & Generalization

Variability & Change

Functional & Structural Thinking

Comparison, Difference, & Equivalence

Making &

Interpreting Predictions

Assessment

Revising and Refining

Why might we engage in revising and refining our mathematical work?

Revising and Refining

Why might we engage in revising and refining our mathematical work?

• Seek new insights
• Become more detailed
• Become more illuminating
• Become more convincing

Amanda Jansen

Revising and REfining

We value revising and refining because of the message it sends about mathematics. Math is more than quick, correct procedures. It involves thoughtfulness and creativity!

How can Math Modeling Build Identity?

Empowering students to take what they learned and DO something with it.

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Teachers invite all students into the mathematical conversation.

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Students with varied math experiences are offered a chance to build their math identity and can see themselves as catalysts for change in the world.

Summary of our work today

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Together we:

• Engaged in the components of the mathematical modeling process
• Discussed strategies for teaching modeling 
• Used math modeling and asset-based perspectives as a vehicle for promoting a positive math identity and fostering curiosity in a collaborative group setting.

What about Assessment?

• Understanding our goals - For example, specific to mathematical content or implementing stages of math modeling cycle

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• Various media - Written reports, “poster” sharing, presentation or more brief class or group discussion

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• Incorporating feedback and revising models

Share Expectations - Specific to Modeling Cycle

Sample Rubric from GAIMME Appendix D

Example of Rubric for Presentation - From GAIMME

Soap Bubbles and Distance

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What is the least distance of highway that we could construct to connect these 4 cities?

Mathematical Goal - Geometric optimization and discovering Steiner points.

Minimum Road Rubric - Calculus Version

Resources

Access today’s slides, modeling resources and opportunities at:

www.comap.org/session

• Sources of math modeling problems
• COMAP: https://www.comap.com/
• SIAM Modeling competition site: https://m3challenge.siam.org/
• Advancing Equity and Strengthening Teaching through Math Modeling https://www.eqstemm.org/
• Other resources:
• GAIMME Report
• Math Modeling Handbooks
• Refresh an “old” task (and win prizes!) Refresh for Relevance Challenge

Math Modeling Sessions at NCTM Virtual Conference

• # 4 - Exploring Mathematical Modeling Through Pooled Testing

On-Demand Session:

Maria Hernandez, Lauren Siegel & Usha Kotelawala

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• #80 Experiential Math for People and the Planet- Carol Bliese

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Leaders’ Information

Joleigh Honey (Elementary, Middle and High School) Joleighhoney@gmail.com

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Ben Galluzzo (Middle, High School and College) ben@comap.org

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Maria Hernandez (Middle, High School and College) maria.hdz2718@gmail.com

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Megan Wickstrom (Elementary, Middle, High School, and College) megan.wickstrom@montana.edu

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Thank you!