Models and Modeling - Mathematics

Educator investigates the use of models and modeling as they pertain to the Common Core State Standards for Mathematics (CCSSM) and investigates resources for use in classrooms.
Made by University of Wisconsin - Milwaukee
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About this Micro-credential

Key Method

The educator studies use of models and modeling in the classroom, targeting existing resources for their specific discipline of mathematics, learns about the role of modeling in the teaching of mathematics and the distinctions between mathematical modeling and modeling with mathematics, and focuses on successful implementation of modeling in a mathematics classroom. They engage in a detailed analysis of the standards related to such, including the analysis of classroom tasks that address modeling. The educator devises a set of criteria for measuring student progress on the standards in that category appropriate to their grade level, designs and teaches a modeling lesson, then analyzes student performance using the created criteria.

Method Components

Components and Implementation of Models and Modeling

  • Examine your own personal model of teacher decision-making used for planning, teaching, and reflection of lessons.
  • Use the model of teacher decision-making from Clough, Berg, Olson to write a reflection of a recent lesson.
  • Discuss the distinctions between models, modeling, modeling with mathematics, and mathematical modeling in the context of the Common Core State Standards for Mathematics.
  • Identify critical features of modeling tasks in mathematics and the progressions related to modeling in high school mathematics.
  • Explore pedagogical aspects of modeling, including the reasons for teaching modeling and pedagogical practices that support successful student engagement in modeling.
  • Plan and teach a modeling-focused lesson that supports student learning opportunities related to other mathematics content standards.
  • Choose or create a measurement tool (rubric, standards-based grading criteria) that capture student performance related to the standards.
  • Analyze student performance related to the standards after students engage in the target lessons.
  • Write a reflection of the planning, implementation, and assessment of the modeling lesson.

Research & Resources

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 micro-credential 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 micro-credential are specifically designed for the teaching and learning of secondary mathematics. Similar distinctions exist in science; the Models and Modeling (Science) micro-credential 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 Decision-Making. International Journal of Science and Mathematics Education, 7(4), 821-847.
    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=gZJLZ-iEUys

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 decision-making 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 Decision-making 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. 57-58) 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. 7-21).
    • 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=gZJLZ-iEUys
    • 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 research-based 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 Three-Act 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 decision-making 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 small-group 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?

Submission Requirements

Submission Guidelines & Evaluation Criteria

To earn the micro-credential, you must receive a passing evaluation for Part 1 and 3 and a “Yes” for the artifacts submitted for Part 2.

Part 1. Overview Questions

Please provide reflections on a recent lesson using the Teacher Decision Making Framework from Session 1:

Part 2. Work Examples/Artifacts

To earn this micro-credential, please attach your lesson plan which includes all of the following:

  • Connections to CCSSM modeling standard and other relevant content standards
  • Connection between your task and aspects of the GAIMME framework
  • Specific plans (so that someone else could read your plan and attachments and teach the lesson)
  • How you are going to assess what students learned
  • How you are going to measure the overall effectiveness of the lesson.
  • How you assess the impact on student learning (student assessments, student artifacts)
  • Reflective narrative of efforts and results

Part 3. Educator Reflection

Reflect on the effectiveness of the lesson from your perspective as a teacher.

  • How did your lesson represent aspects of mathematical modeling?
  • What instructional practices did you use to enact the modeling task?
  • What specifically did you want students to be doing during the lesson that would be indicators of success?
  • How did the student artifacts inform your practice?
  • What was some feedback from students?
  • What would you change?

Except where otherwise noted, this work is licensed under:
Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International (CC BY-NC-ND 4.0)
http://creativecommons.org/licenses/by-nc-nd/4.0/

Requirements

Download to access the requirements and scoring guide for this micro-credential.
How to prepare for and earn this micro-credential - in a downloadable PDF document

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