Supporting Research
Prior to the advent of recent standards documents (e.g., CCSSM) and detailed reports (e.g., GAIMME), modeling in the context of teaching mathematics has had a wide array of meanings. In particular, the constructs of modeling with mathematics and mathematical modeling have been used fairly interchangeably by some, leading to confusion among teachers about what components are necessary to meet modeling standards and how to design a modeling standard. This microcredential is designed to untangle the modeling confusion and discuss the important components of a modeling lesson as reflected in current research and policy. You will engage in readings about mathematics and modeling using the resources noted above to deepen your knowledge of modeling, then plan, implement, and study a lesson related to that content with your students.
The ideas in this microcredential are specifically designed for the teaching and learning of secondary mathematics. Similar distinctions exist in science; the Models and Modeling (Science) microcredential contains those resources.
Mathematics Resources

Hirsch, C. R. & Roth McDuffie, A. (2016). Annual Perspectives in Mathematics Education 2016: Mathematical Modeling. Reston, VA: National Council of Teachers of Mathematics.

Common Core Standards Writing Team. (2013, July 4). Progressions for the Common Core State Standards in Mathematics (draft). High School, Modeling. Tucson, AZ: Institute for Mathematics and Education, University of Arizona.
http://bit.ly/2nC4TQp

Society for Industrial and Applied Mathematics (SIAM). (2016). Guidelines for Assessment and Instruction in Mathematical Modeling Education. Philadelphia, PA: SIAM. Accessed on 1 March 2017 from
http://www.siam.org/reports/gaimme.php

Clough, M. P., Berg, C. A., & Olson, J. K. (2009); Promoting Effective Science Teacher Education and Science Teaching: A Framework for Teacher DecisionMaking. International Journal of Science and Mathematics Education, 7(4), 821847.
http://bit.ly/2xCkaK0

Zbiek, Rose M. (2016, October.) IGNITE: Teaching for Learning Mathematical Modeling. Presentation at the NCTM Regional Conference and Exposition, Philadelphia, PA. Accessed 1 March 2017 from
https://www.youtube.com/watch?v=gZJLZiEUys
Resources
Learning Opportunities
It is suggested that the work be completed in a small learning community to accomplish and discuss the learning activities.

Session 1

Examine a model to guide thinking about planning for instruction, while teaching, and reflecting on your teaching.

Create a diagram that represents a model of the decisions you make as a teacher when planning for lessons, teaching lessons, and then reflecting on your success regarding the impact you have on the learner.

When you think about planning a lesson and teaching a lesson, there are considerations that guide or affect your thinking. The same considerations affect youwhen teaching a lesson, or reflecting on a lesson. Those considerations have been shaped into something called your model for your teacher decisionmaking framework.

What is your current model or framework? If a colleague, principal, or parent asked you to explain your thought processes as you build a new lesson, how would you create a graphic that would help them understand your thinking and the considerations that go into your planning, or teaching of a lesson. Draw your model. Share your model and a short explanation.

Read the Teacher Decisionmaking article (Clough, Berg, & Olson, 2009) with a focus on the diagram on page 8

How does the model on page 8 compare with your initial model you drew?

What are the strengths and limitations of the model?

How would having a more sophisticated and complex model affect the utilitarian usage of the model when it comes to understanding your teaching, or explaining the act of instruction to someone else such as a peer, administrator, or parent?

Take a recent lesson and reflect on that lesson using the diagram on page 8 (two pages of reflection).

Modeling in the context of mathematics

Considering the general concept of modeling

Examine the high school conceptual category fromthe Common Core State Standards for Mathematics (pp. 5758) and identify what you notice about the concept of modeling and what you are wondering

Examples of successful modeling: The Cruise Ship Task

You have won a trip to Europe on a cruise ship. The day arrives for you to fly to New York City to board the cruise liner. Unfortunately, your plane has mechanical difficulties, which delays your departure. By the time you arrive at the harbor in New York City, the cruise ship has already left for England. In fact, it has left 5 hours ago. You are really disappointed because the next cruise doesn’t leave for another two weeks! What should you do?

Solve and discuss solutions for the Cruise Ship task

What aspects of modeling did you notice in this task?

Review the Modeling conceptual category in CCSSM.

What aspects of our work on the Cruise Ship task connect to the Modeling standard and aspects of the modeling cycle?

Read the GAIMME Report (Chapter 1, pp. 721).

How does the Cruise Ship task relate to the GAIMME framework for identifying modeling tasks (Figure 1.1)?

What aspects of our work on the Cruise Ship Task connect to mathematical modeling process (Figure 1.2)?

Modeling in Mathematics

View and discuss the Zbiek IGNITE talk

https://www.youtube.com/watch?v=gZJLZiEUys

What is the distinction between“modeling with mathematics” and “mathematical modeling”?

“When students are modeling they need to be making genuine choices.” What do you think the GAIMME report authors mean by this? What constitutes a genuine choice in teaching high school mathematics?

What are some connections between characteristics of modeling tasks, content goals, and aspects of the modeling cycle

Summary discussion: what criteria do we want to prioritize as we consider tasks with respect to modeling?

Session 2

Read:

GAIMME Report, Chapter 1

GAIMME Report, Chapter 2, 3, OR 4 (depending on your grade level)

CCSSM Progression document on Modeling

Hirsch & Roth McDuffie (2016) Chapters 1 and 16

Meyer (2015)

Anhalt & Cortez (2015)

Wendt & Murphy (2016)

How do the readings help to clarify the nature of a good mathematical modeling task and how might we plan for lessons that use those tasks?

What did you take away from each of the three types of readings (GAIMME report, APME researchbased reports, practitioner articles)?

What are you still wondering about related to mathematical modeling?

Session 3

Debrief the readings:

Why use modeling? What are the affordances for student learning?

Unpacking the modeling cycle: What are characteristics of modeling tasks based on GAIMME and a closer read of the CCSSM and modeling progression?

What is the distinction between mathematical modeling & modeling with mathematics (Standard for Math Practice 4)?

Work through the ThreeAct Task Coin Carpet (at http://threeacts.mrmeyer.com/coincarpet/)

Act 1:

What questions could you ask about the picture?

If you were going to carpet the room with a single kind of coin, which would be the cheapest option?

Write down a guess.

Write down a coin that you're surewouldn'tbe the cheapest option and how you know.

Act 2:

If you were going to carpet the room with a single kind of coin, which would be the cheapest option?

What information would be useful to know in order to answer the question?

Once you have the information you need, try to find an answer to the question.

Act 3:

If you could use any coin in the entire world, which would be the cheapest option?

If you could use any kind ofpaper currencyin the entire world, which would be the cheapest option?

Share your best thoughts from Act 2 and Act 3.

What criteria did you use to make decisions?

How would you visually represent that decisionmaking process?

What aspects of modeling did you notice in this task?

Discuss mathematics teaching practices that support modeling (Principles to Actions, NCTM, 2014)

Consider the eight effective mathematics teaching practices (NCTM, 2014).

Which practices connect most strongly to modeling?

Planning for a modeling lesson

What resources will you be using (or will need) to identify a strong mathematical modeling task?

What instructional practices will you prioritize as you plan and enact the lesson?

What aspects of student performance will you measure, and what tools will you use to measure them?

Session 4

Produce a written summary of modeling in mathematics key points from Session 3 smallgroup discussions

Read GAIMME Appendix B on assessment

Plan and teach a lesson on modeling, including a rigorous data collection component

Be sure to design data collection instruments related to modeling that incorporate student outcomes and teacher actions

Session 5

Analyze data from taught lesson(s)

Final Reflection and Critique

What type of model did you use? What modeling strategy did you use?

How did you integrate it into the lesson?

What specifically did you want students to be doing during this lesson that told you it was successful?

What was some feedback from students and artifacts?

What would you change?