High School & Mathematical Modeling

Marta MagieraRET Fellow 2007

Chicago Mathematics & Science Teachers Research Program

University of Illinois at ChicagoJuly 28, 2008

RET Summer Conference

Mathematical Modeling

“A model can function as a tool to makesense of something that we try to

understand. Where do we begin? I think webegin with the world, in particular the world of the

student’s experience. Mathematics has apower to put a lot of order into that world

and help us understand a lot of itsbehavior” (Taylor, 1992, p. 7)

• Express, sort out, modify, test, integrate, and refine clusters of mathematical concepts from topics within and beyond mathematics while progressing through problem-solving cycles

• Interpret complex problems, make assumptions about complex systems, select and develop own mathematical tools for solving problems, test the accuracy and efficacy of developed mathematical models.

• Improve written and oral presentation skills

Mathematical Modeling

Illinois Learning Goals Addressed• IL J 8A.3 Represent relationships arising from various

contexts using symbolic expressions

• IL J 7B.4 Apply informal concepts of successive approximation, upper and lower bounds, and limit in measurement situations

• IL I 8.D Use algebraic concepts and procedures to represent and solve problems.

• IL J 7B.1 Analyze precision, accuracy, and approximate error in measurement situations.

• IL I 9.C Students who meet the standard can construct convincing arguments and proofs to solve problems

Expected Learning Outcomes1. Provide mathematical descriptions to real life situations

2. Identify assumptions and constrains for a given situation

3. Collaborate in small group an design of mathematical procedure, description, model for a given situation

4. Provide constructive feedback to the members of other groups

5. Draw on received feedback to improve once own solution

Mathematical Modeling Projects

Materials Used

Mathematical-problem solving as a scientific process

• Students will investigate given phenomena, acquire new, correct or integrate existing mathematical knowledge.

• Students will collect data through observations and measurements.

• They will identify problem, make assumptions about, and select their own tools (mathematical concepts, procedures, etc.) for solving the problem.

• Students will formulate and test hypothesis about problems.

• Students will experience problem solving as a process and develop holistic view of mathematics

1. Given a coin, a tuna-fish can and a soup can: Devise a definition of disc-ness that allows you to say which object is the most disc-like and which is the least .

2. Given a mailing tube, a straw, and a piece of uncooked spaghetti: Use your definition of disc-ness to determine which object is the most disc-like and which is the least.

3. Write a formula (or algorithm or algebraic sentence) which expresses your measure of disc-ness. You may introduce any labels and definitions you like and use all the mathematical language you care to.

4. Make any measurements you need, and calculate a numerical value of disc-ness for each of the six items.

5. Discuss whether these numbers seem reasonable in light of your notion of disc-ness.

6. How would you change your answers to these questions if you were asked to write a formula for cylinder-ness rather than disc ness? Balanced Assessment Problem No. HL003

from Center for Twenty First Century Conceptual Tools (TCCT), Purdue University.

Using the three microscopic pictures of the samples of aluminum below, determine the typical size of crystal in each sample.

• ..\..\old laptop files\PowerPoint Presentations\Case studies\mrs. magiera3rd per.ppt

Grading: solution

Shows no understanding of the question’s mathematical ideas and processes; or Fails to respond to question

0

Shows minimal understanding of the question’s mathematical ideas and processes. Omits significant parts of the question and response has major errors. Communication of ideas is unclear

1

Shows some understanding of the question’s mathematical ideas and processes but with considerable flaws or gaps in understanding Presents arguments but incomplete. Communication of ideas not expressed clearly

2

Shows a good understanding of the question’s mathematical ideas and processes, with some minor flaws or gaps in understanding Contains a good, solid response with some of the characteristicsabove

3

Shows clear and complete understanding of mathematical ideas andprocesses and provides a complete response with clear, coherent,unambiguous, and elegant explanation.

4

Individual grade

Takes InitiativeOn task Share ideas Exhibits cooperative effort

Introduction & Conclusion No spelling errors Correct grammar Typed or neatly handwritten Thorough

Professional form Visual organization Logical flow Posture & aye contactCorrect grammar

CollaborationWrite-upPresentation

Resources

• Balanced Assessment http://balancedassessment.concord.org

• Mathematical Case Studies: Center for Twenty First Century Conceptual Tools (TCCT), Purdue University and the Small Group Mathematical Modeling (SGMM) Project Purdue University

• Taylor, P.,D. (1992). Calculus: The analysis of functions. Torronto: Wall & Emerson

Acknowledgments

• The National Science Foundation, NSF EEC-0502272 Grant

• The University of Illinois at Chicago

• Dr. Andreas Linninger, RET Program Director