1. Mathematical Modeling: An Evolving Perspective
Cathy Martin
Denver Public Schools

2. Many Meanings of “Modeling”
In the context of mathematics teaching and learning, how have you heard/seen/used the term “modeling”?
Think individually
Share out ideas at table and introduce yourself to your table mates.

3. Objectives Extend and deepen understanding of mathematical modeling.
Explore different views and guiding principles of mathematical modeling, as described in the Guidelines for Assessment and Instruction in Mathematical Modeling (GAIMME) report
Analyze tasks to determine the aspects of mathematical modeling that are present
Explore how to modify existing tasks to create mathematical modeling problems

4. Model . . . A Word with Different Meanings

5. MP 4: Model with Mathematics
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

6. Modeling Mathematics, Mathematical Models, & Mathematical Modeling
Modeling mathematics: uses mathematical representation to communicate mathematical concepts or ideas. Key feature is that the process begins in the mathematical world rather than the real world.
Mathematical models: support students in exploring and communicating mathematical ideas and concepts. These include pictures, written symbols, oral language, real-world situations, and manipulative models.
Mathematical Modeling: links mathematics and authentic real- world questions. Essence of mathematical modeling is the translation of an authentic real-world problem into a mathematical form.
Van de Walle: a model for a mathematical concept refers to any object, picture or drawing that represents the mathematical concept. In CCSS intro to first grade, Students develop strategies for adding and subtracting whole numbers based on their prior work with small numbers. They use a variety of models, including discrete objects and length-based models (e.g., cubes connected to form length), to model take-from, put-together, take-apart, and compare situations to develop meaning for the operations of addition and subtraction.

7. GAIMME Report Joint project among COMAP, SIAM, and NCTM
Paints a clearer picture of mathematical modeling (what it is and what it isn’t)
Available (at no cost):

8. Mathematical Modeling is a process that uses mathematics to represent, analyze, make predictions, or otherwise provide insight into real-world phenomena.

9. Mathematical Modeling: A Non-Example
Computation problem: = ?
Word Problem: Jim has 6 pretzels and Suzy has 3 pretzels. How many pretzels do Suzy and Jim have in all?
Why is this word problem NOT a modeling problem?
GAIMME, p 10
Why not a modeling problem:
“First, it does not have intrinsic value or meaning for students. Other than answering this problem for homework, why would students care about how many pretzels Jim and Suzy have? The word problem is closed at the beginning and the end. While students may use a few different approaches to reach the answer, such as drawing a picture and counting or writing and evaluating an arithmetic expression, all of the necessary data is clearly provided, and there is only one correct solution.”

10. A Mathematical Modeling Problem
“Gas prices change on a nearly daily basis, and not every gas station offers the same price for a gallon of gas. The gas station selling the cheapest gas may be across town from where you are driving. Is it worth the drive across town for less expensive gas? Create a mathematical model that can be used to help understand what conditions it is worth the drive.”
GAIMME, p 16

11. What Makes a Good Mathematical Modeling Problem?
Context is relevant and based on real-life
Openness
Beginning: multiple entry points, how you define the question
Middle: multiple mathematical approaches
End: multiple solutions
Inspires questions to focus the problem through assumptions; sparks curiosity and promotes decision making
Information is needed to inform and test solutions
Someone cares about solutions; solutions are useful

12. Going Deeper: Jigsaw Reading
The Modeling Cycle and Modeling and the Math Practices (blue—secondary)
Mathematical Modeling in the HS Curriculum (green)
A Teaching Framework for Modeling in the K-5 Setting (buff)
Enacting Modeling Components and Process with Students (yellow—K-8)

13. Modeling Principles Modeling (like real life) is open-ended and messy.
When students are modeling, they must be making genuine choices.
Start big, start small, just start.
Assessment should focus on the process, not the product.
Modeling does not happen in isolation.
GAIMME, p 23

14. Criteria for Modeling Tasks
Open: at beginning, in the middle, and at the end
Sufficiently complex
Realistic or, if possible, authentic
Problematic
Accessible
Require all phases of the modeling cycle
Blum and Ferri, “Advancing the Teaching of Mathematical Modeling: Research-Based Concepts and Examples,” Annual Perspectives in Mathematics Education: Mathematical Modeling and Modeling Mathematics
(NCTM, 2016)

15. Analyzing Tasks
With a partner, read the tasks and decide whether or not the task exemplifies mathematical modeling.
Justify your thinking based on the criteria:
Open: at beginning, in the middle, and at the end
Sufficiently complex
Realistic or, if possible, authentic
Problematic
Accessible
Require all phases of the modeling cycle

16. Mathematical Modeling
Word Problems to
Mathematical Modeling
Word Problem: Jim has 6 pretzels and Suzy has 3 pretzels. How many pretzels do Suzy and Jim have in all?
Modeling Problem: You are helping to pack a picnic lunch for your family and need to determine how many pretzels to pack GAIMME, p 11
Your task: Work with a partner to modify tasks to make them mathematical modeling tasks. Use questions to evaluate your task.

17. Are My Students Modeling?
Are students using mathematical tools to solve the problem?
Did they start with a big, messy, real-world problem?
Did the students make genuine choices and assumptions?
Are students testing and revising their model?
Can students explain if/when their model makes sense?
Who cares about or is invested in the solution?
Did they ask questions to clarify or focus the problem?
Rachel Levy, NCTM Conference, 2016

18. Beliefs about Teaching and Learning Mathematical Modeling
With a partner, study the unproductive and productive beliefs about teaching and learning mathematical modeling.
Beliefs 1,2: Focus on what mathematical modeling is
Beliefs 3-6: Focus on how mathematical modeling unfolds as a process and how teachers support mathematical modeling
Beliefs 7-8: Focus on how mathematical modeling fits within school mathematics.

19. Reflection: Stand Up, Hand Up, Pair Up
Imagine that you have successfully engaged students in mathematical modeling.
What might you see differently in math classrooms?
How might this impact student achievement?
How will you take this back to your colleagues?