Model-Eliciting Activities Tamara Moore, Ph.D. Assistant Professor, Mathematics/Engineering Education Co-Director, STEM Education Center

Model-Eliciting Activities Goals 1. Teachers helping students develop connections among STEM disciplines through representation and translation. 2. Teachers helping students elicit their mathematical thinking through modeling.

Iterative Thinking

Lesh Translation Model Lesh & Doerr (2003)

Let’s play with Paper Airplanes Get in teams of 3-4 ◦ Each person in your team choose a different paper airplane to build. Make a target somewhere in the room with a starting point of about 10 paces. Fly your plane and the other planes in your group ◦ Can you hit the target? ◦ Can you make the plane “float” toward the target?

Let’s play with Paper Airplanes Record your observations about: ◦ What contributed to the differences in the plane flights? Pilot? Construction? Plane properties? What else? ◦ Which plane was the most accurate? ◦ Which plane was the best floater?

Paper Airplane Contest Individually: Read the newspaper article and answer the readiness questions. In your teams: Read the problem statement and answer the team questions together. Create a procedure for the judges of the paper airplane contest Be prepared to share your solutions in a 2 minute presentation.

Representations Revisited In your groups, jot down your ideas about what representations were elicited in this problem. How can you as a teacher foster multiple representations on this problem? Translations between representations? What are the “big” conceptual ideas that are elicited in this problem?

Ideas from the Field Student work from the Paper Airplane MEA ◦ You have been provided with 2 student team samples of work. ◦ What do you see in these responses? ◦ What are some of the good STEM ideas represented? ◦ What are some of the misconceptions? ◦ How well did they communicate their understandings?

Samples of student work In response to your need for an adequate equation to judge the paper plane entrees in the categories of the most accurate flyer and the best floater, we have created and tested equations for each. Each equation went through a period of trial and error and thus proved to be the most fair for the required criterion. For the first category of the most accurate flyer, we determined that the distance and the angle from the target should be factored into the equation and we realized that if those two factors went into a shape, a triangle, the hypotenuse was missing. Because of this missing component, we determined that the equation for a hypotenuse in a triangle would be most effective for finding accuracy. The equation: distance from the target 2 + angle from the target 2 = accuracy 2, was tested against sample data and it proved to be most effective. The equation proved that the Golden Flyer with the plane Hornet would be the most accurate overall. For the second category of the best floater, we determined that the average time in the air for each plane would work as the equation needed. The equation: (a+b+c)/3 was tested and proved to work because the time in the air is the only component needed for floating. The equation proved that Hornet was the best floater and Pacific Blue was the best pilot. Thank you for considering our equations to help better the judging and fairness of this prestigious competition.

Samples of student work We believe that certain measurements obtained during this competition should be brought into account in the judging, however, we believe some areas hold more value than others. Since we are looking to find the best floater and the most accurate plane, we have divided the measurements to suit the requirements of the category. For the floating competition the planes will be judged based on time in flight and the distance from the start. For the accuracy competition the planes will be judged based on the distance from the target and the angle from the target. We will take the averages of all the measurements to keep the planes from winning from one good toss. For the floating competition, we feel that the planes should be mainly judged on the time spent in the air. For this research we have decided on the equation (Average Time) 2 x (Average Distance) = Score. We incorporated distance to avoid people making planes that launch straight up, and we feel that a floater should travel a distance and not dive straight down. It should resemble the path of a glider. We decided to square the average time to put more emphasis on the time in the air. In this competition, the highest score wins. For the accuracy competition, we decided the planes should be judged on the distance from the target and the angle. For this reason we have developed the formula 2(Average Distance from Target) + |Average Angle from Target| = Score. We will take the absolute value of the angle so the score will not be affected by a negative angle and the distance will be doubled to make it more important than the angle. For this competition the lowest score wins.

a system that explains, describes, or represents another system contains elements, operations, and relations that allow for logical relationships to emerge sometimes not sufficient to completely describe the system it represents ◦ if it is a useful model, it closely approximates the system in a manner that people can use when working with the system without being unnecessarily complex What is a model?

Model-Eliciting Activities (MEAs) are client- driven, open-ended, realistic problems that involve the development or design of mathematical/scientific/engineering models These are broadening classroom experiences that tap the diversity of learning styles and strengths that students bring to the classroom Intended to make advanced STEM content and substantive problem-solving experiences accessible to a diversity of students Nature of Model-Eliciting Activities

Nature of MEAs: Realistic problems with a client Require team of problem solvers Product is the process for solving the problem ◦ End product is a model that the client can use Model-Eliciting Activities

How MEAs Have Helped ◦ Framework for constructing highly open- ended realistic problems  Require model development  Support development of teaming and communication skills ◦ Meaningful contexts for students ◦ Increase student engagement: addressing diversity and under-represented populations Motivation for Using MEAs

Who was the client? What did the client need? How did your team go about meeting that need? Paper Airplane Contest MEA

Model-Construction ◦ Description: Ensures the activity requires the construction of an explicit description, explanation, or procedure for a mathematically significant situation ◦ What is a model?  Elements  Relationships among elements  Operations that describe how elements interact What models are the students developing when they solve this MEA? MEA Design Principles

Reality ◦ Description: Requires the activity be posed in a realistic engineering context and be designed so that the students can interpret the activity meaningfully from their different levels of ability and general knowledge. ◦ Realistic contexts are constructed by:  Gathering information from actual sources  Making simplifying assumptions when information is conflicting, missing, or difficult for students to use What knowledge do students bring to this problem? MEA Design Principles

Self-Assessment ◦ Description: Ensures that the activity contains criteria students can identify and use to test and revise their current ways of thinking  Students recognize the need for model  Students use the client’s criteria to inform refinements to their model  Students must judge for themselves when they have met the client’s needs What is provided in this MEA that students can use to test their ways of thinking? MEA Design Principles

Model-Documentation ◦ Description: Ensures that the students are required to create some form of documentation that will reveal explicitly how they are thinking about the problem situation What documentation are the students being asked to produce in this MEA? MEA Design Principles

“Thought-Revealing” What can student documentation tell us? What information, relationships, and patterns does the solution (model) take into account? Were appropriate ideas and procedures chosen for dealing with this information? Were any technical errors made in using the preceding ideas and procedures?

Model Share-Ability and Re-Usability ◦ Description: Requires students produce solutions that are shareable with others and modifiable for other engineering situations ◦ Biggest challenge for students  Tendency is to create a solution only for the situation as given and only readable by the creators  We are looking for the students to construct a model that:  Someone else can pick up and use  Could be used to solve similar problems  Extent to which students can achieve this can be used in feedback and assessment strategies MEA Design Principles

Effective Prototype ◦ Description: Ensures that the solution generated must provide a useful learning prototype for interpreting other situations  Want the situations or concepts used in creating the model to be useful in future coursework & practice What are the “big” conceptual ideas? MEA Design Principles

To which standards does this MEA connect? Are there standards to which all MEAs connect? Which ones? How can this help you as a teacher trying to integrate STEM? Connections to Standards

Historic Hotels Individually: Read the newspaper article and answer the readiness questions. In your teams: Read the problem statement and answer the team questions together. Create a procedure for Mr. Graham Be prepared to share your solutions in a 2 minute presentation.

Representations Revisited In your groups, jot down your ideas about what representations were elicited in this problem. How can you as a teacher foster multiple representations on this problem? Translations between representations? What are the “big” mathematical ideas that are elicited in this problem?

Tomorrow Make sure you bring your curricular materials We will be exploring many different MEAs and making plans for implementation.