Mathematical Modeling Across the Grade Levels and Content Domains

Mathematical Modeling Across Grade Levels and

Content Domains

17th Annual CMP Users ConferenceMichigan State University – Feb. 22, 2014

Speaker IntroductionAll current Ph.D. students in mathematics education at MSU and working on CMP

Nic Gilbertson Former middle school and high school math teacher (OR, WI, IL)[email protected] focus on technology and classroom discussions

Kevin Lawrence Former high school math teacher (Indiana), [email protected] focus on technology, transition from K-12 to college

Jamie Wernet Former high school math teacher (Michigan), [email protected] focus on algebra teaching and learning, student engagement and motivation to learn, and the use of contextual tasks in math classrooms

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

What we hope you take-away from this session…

1) Seeing more opportunities to help students develop modeling skills / abilities

2) Some different (possibly new) ways of thinking about modeling

3) Being able to identify “good” modeling tasks

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Agenda• Intro & Thinking about Modeling• Background of ways of thinking about modeling• Three approaches to thinking about modeling• A look at tasks – where are opportunities for modeling• as written? • as implemented?

• Summary • Questions & Comments17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson,

Lawrence & Wernet

Opening QuestionAt the grade level you teach,

Where do you see opportunities for students to learn about mathematical

modeling?

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Opening QuestionAt the grade level you teach,

Where do you see opportunities for students to learn about mathematical

modeling?

17th Annual CMP Users’ Conference – Student Work Session – Gilbertson, Lawrence & Wernet

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

What Researchers Say About Modeling

• Generally speaking, modeling involves taking a real-world scenario and translating it into the mathematical world (Niss, Blum, & Galbraith, 2004) by finding a model to represent the scenario.

• Modeling problems are different than application problems, where students start with a mathematical concept and move toward a real-world situation (Niss et al., 2004).

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

What Researchers Say About Modeling

• Connection between modeling and real world– Modeling gives students the opportunity to engage in real-world contexts that are new to them situated outside of the mathematical world (Blum, 2002).

– Modeling serves purposes in the classroom as well as everyday experiences and future work in different careers (Zbiek & Conner, 2006).

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

One Model of Modeling• Lesh & Zawojewski (2007)   

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

1. The cycle typically begins in the real world position on the diagram.

2. Create mathematical model through description.

3. Mathematical manipulations are made within the model.

4. Predictions are made based on the model and brought back to the real-world scenario to solve the original problem.  

5. Predictions are verified by seeing how accurately the solution based on model relates back to the real world.  

6. If insufficient, revise.

Three Approaches to Modeling

• Using the full modeling cycle• Using parts of the modeling cycle• Opening up a problem

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Three Approaches to Modeling

• Using the full modeling cycle• Using parts of the modeling cycle• Opening up a problem

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Using the Full Modeling Cycle

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Using the Full Modeling Cycle

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Using the Full Modeling Cycle

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Three Approaches to Modeling

• Using the full modeling cycle• Using parts of the modeling cycle• Opening up a problem

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Using Part of the Modeling Cycle

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Using Part of the Modeling Cycle

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Using Part of the Modeling Cycle

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Three Approaches to Modeling

• Using the full modeling cycle• Using parts of the modeling cycle• Opening up a problem

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Opening up a Problem

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Opening up a Problem• Ask students how they would collect data first.

• Give data not organized in a table, focus on prediction, representation

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

A look at the Locker Problem

A look at the Locker Problem• 6th grade, Prime Time Problem 3.4

• As you explore this problem, think about:– Is this a modeling task (as written)?– If so, does it go through the entire modeling process? If not, why not?

– If it has potential as a modeling task, how could you implement it in a way that highlights aspects of mathematical modeling?

• Disclaimer: Various CMP team members disagreed about whether this is a modeling task—so whatever you think, you have friends!

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

A look at the Locker Problem

As you explore this problem, think about:

– Is this a modeling task (as written)?– If so, does it go through the entire modeling process? If not, why not?

– If it has potential as a modeling task, how could you implement it in a way that highlights aspects of mathematical modeling?

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Your turn!• On your own or with a team, select any CMP problem--we have sample units available for each grade, or you can select a task from the DASH.

• Select something relevant for you!

• As you explore the problem, think about:– Is this a modeling task (as written)?– If so, does it go through the entire modeling process? If not, why not?

– If it has potential as a modeling task, how could you implement it in a way that highlights aspects of mathematical modeling or opens the problem more to support the full modeling process?

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Summary• We have discussed three approaches to incorporating modeling tasks without needing to “tack on” extra tasks in the curriculum.

• From the Guide to CMP3: “Mathematical modeling is, in some sense, what every unit in CMP is about.”

• Regardless of the approach, additional aspects of a “good” modeling task highlighted in literature include:– Opportunities to build understanding of content and mathematical practices

– Realistic context– Requires student sense-making

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Summary• We have discussed three approaches to incorporating modeling tasks without needing to “tack on” extra tasks in the curriculum.

• From the Guide to CMP3: “Mathematical modeling is, in some sense, what every unit in CMP is about.”

• Regardless of the approach, additional aspects of a “good” modeling task highlighted in literature include:– Opportunities to build understanding of content and mathematical practices

– Realistic context– Requires student sense-making

17th Annual CMP Users’ Conference – Mathematical Modeling Session – Gilbertson, Lawrence & Wernet

Thank you!• Nic Gilbertson: [email protected]

• Kevin Lawrence: [email protected]

• Jamie Wernet: [email protected]

17th Annual CMP Users’ Conference – Student Work Session – Gilbertson, Lawrence & Wernet