STEM Modeling in the Secondary Setting to Deepen Understanding

1. STEM Modeling in the Secondary Setting to Deepen Understanding AMTNYS Summer Workshop August 3, 2011 Brandon Milonovich The College of Saint Rose Syracuse University

2. Overview • What is modeling? • Why use modeling? • How does modeling relate to the Common Core? • How can I use modeling in my classroom?

3. What is modeling? • “The real situation usually has so many ‘angles’ to it that you can’t take everything into account, so you decide which aspects are most important and you keep those…Now you have a mathematical model of the idealized question” (Pollak, 2011, 7).

4. What is modeling? • Modeling differs from solving word problems because it is a process. • After modeling we have to ask, “are the results practical, the answers reasonable, the consequences acceptable” (Pollak, 2011, 7)? • Word problems focus more on solving a given problem with known variables, using a specifically aimed at algorithm and carrying it out successfully.

5. What is modeling? • “[I]mportant aspect of mathematical modeling: real-world situation comes first, the mathematics follows naturally” (Tam, 2011, 67). • “The heart of mathematical modeling, is problem finding before problem solving” (Pollak, 2011, 7).

6. Word Problem vs. Model • Word Problem: How long does it take to drive 20 miles at 40 miles per hour? • Answer: 30 minutes • Modeling Problem: When you live 20 miles from the airport, the speed limit is 40 miles per hour, and your cousin is due to arrive at 6 PM. Does this mean you leave at 5:30?

7. Word Problem vs. Model • To tackle the modeling problem, in reality, you probably won’t leave at 5:30. • For example, “This is rush hour. There are those intersections at which you don’t have the right of way. How long will it take to find a place to park? If you take the back way, the average drive may take longer, but there is much less variability in the total drive time…But don’t forget the arrival time they give you is the time the plane is expected to touch down on the runway, not when it will start discharging passengers at the gate” (Pollak, 2011, 6).

8. Why use modeling? • Modeling allows students to make deeper connections to the mathematics they are learning through relationships. • Since our brain learns best through connecting new material to old material, modeling aids in the process of learning. • Modeling promotes critical thinking skills necessary both in higher-order mathematics and in other disciplines.

9. Why use modeling? • “In sum, the main reasons for teaching modeling are that every child can benefit from its power of application, and that mathematics cannot only be learned in an isolated way but also be seen in the real-world” (Tam, 2011, 29). • Life is complicated, modeling turns a complicated situation into a simpler situation to analyze.

10. The Process • Understand and identify the issue in the real world • Formulate the structure of real-world situation • Translate to a mathematical model • Derive some mathematical facts from the model • Translate the resulting facts back to the real-world • Validate the results

11. Two Viewpoints • “Mathematical modeling provides rich examples through which students can retain the mathematics that they have learned, and can extract important mathematical content” (Tam, 2011, 30) • “…the modeling process [is] a key part of mathematical content that needs to be taught and grasped, with recognition that the process is comprised of a different set of skills from what is needed for ‘pure’ mathematics” (Tam, 2011, 30)

12. How does modeling relate to the Common Core? • “Standard 4. Model with MathematicsMathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace”(CCSS, 2011). • Although not explicitly a standard in middle school, almost all of the standards reflect steps in the modeling cycle.

13. A Deeper Look… • Make sense of problems and persevere in solving them. • Reason abstractly and quantitatively. • Construct viable arguments and critique the reasoning of others. • Model with mathematics. • Use appropriate tools strategically. • Attend to precision. • Look for and make use of structure. • Look for and express regularity in repeated reasoning.

14. How can I use modeling in my classroom?

15. Predicting the weather • Algebra—Linear Relationships • Can I approximate the unknown temperature somewhere based on three places I know? (Gould, 2011, 66)

16. The Process • Understand and identify the issue in the real world • Formulate the structure of real-world situation • Translate to a mathematical model • Derive some mathematical facts from the model • Translate the resulting facts back to the real-world • Validate the results

17. Predicting the Weather Activity • Take a few minutes to work through the modeling process, either individually or as a group, and find an approximate temperature of our unknown location.

18. My own solution… • Use GSP to create a triangle, and construct a ray from St Rose through my unknown point intersecting the line between Siena and Albany International. • We see the ratio from the Siena to the new point is REALLY close to 1/3 of the distance from Siena to ALB, therefore, accounts for REALLY close to 1/3 of the change, so must be right around 82.

19. My own solution… • Now, to do the same work on our final point. The recent point calculated, call that point C, takes the place of Siena. The ratio of C with our final point to C with St Rose is again close to 1/3. • The difference in temperature is 7, multiplied by 1/3 is about 2.31, we’ll just say 2. This suggests our final point is roughly 80 degrees.

20. An alternative… • Find the midpoints to each of the lines → 78, 80, and 83 • The average of these rounds to 80 degrees. • You notice that this average is the same if you just average the three temperatures, but finding the midpoints helps us determine the best way to round. If we are skeptical in how to round, continue finding more midpoints as smaller triangles develop.

21. Predicting the Weather • NOT a word problem! • Multiple solutions can arise, one of the keys to this lesson would be in the discussion that would follow, it’s NOT about the answer!

22. Examining the Heart • Can be used throughout high school in conjunction with science classes from grades 9-11 • The heart’s AV node decides to either beat or skip a beat based on the strength of the signal arriving from the SA node. • Students can then work to answer the question “will the heart beat?”

23. Examining the Heart The heart acts as a pump which converts electrical energy to mechanical energy. The heart is divided into four chambers, two atria and two ventricles. The atria receive blood from throughout the body. Blood is then pumped through the ventricles and out of the heart to the rest of the body. The Sinoatrial (SA) node is the pacemaker of the heart and sends signals to the Atrioventricular (AV) node. The AV node receives the signal from the SA node and sends a signal to the ventricles if the heart should beat.

24. Examining the Heart

25. Examining the Heart

26. Examining the Heart • Situation: • The threshold voltage of the AV nodes is 20 mV. • The potential at the AV node is 30 mV now. • The potential at the AV node reduces to half its original between the two signals arriving from the SA node. • The SA node sends signals of strength 10 mV.

27. Translating into the Model Substituting Symbolizing

28. Examining the Heart • We can continue this examination through modeling further activity of the heart using a basic Microsoft Excel Spreadsheet • Working with various situations leads students to finding equilibrium values allowing the heart to beat steadily.

29. Equilibrium of the Heart

30. Does Equilibrium Exist?

31. Does Equilibrium Exist? Yes, the heart does become stable (with an equilibrium value of 20 mV. The heart will steadily beat in our previous situation.

32. Final Remarks • Modeling essentially takes word problems to a new level—thereby developing deeper understanding of not only the solution, but also the problem and process itself. • When teaching lessons with modeling, our primary goal is the process of modeling, learning math follows naturally.

33. Final Remarks • Although some models may be complicated to have students to develop on their own, we can adjust models to meet all levels of students in our classroom, i.e., the heart model simplified to the high school level • Modeling provides a medium through which we can facilitate deeper thinking skills, make interesting uses of technology, and arrive at interesting mathematics.

34. References Blum, W., & Leiss, D. (2005). “Filling up”—The problem of independence-preserving teacher interventions in lessons with demanding modeling tasks. In Proceedings of the Fourth Congress of the European Society for Research in Mathematics Education (1623-1633). Presented at the European Research in Mathematics Education IV, SantFeliu de Guixols, Spain. Champanerkar, J., & Eladdadi, A. (2011). Mathematics and the Heart Workshop for High School Teachers. 2011 NHLBI-VCU World Conference on Mathematical Modeling and Computer Simulation in Cardiovascular and Cardiopulmonary Dynamics. Williamsburg, PA. Common Core State Standards Initiative in Mathematics. (2010). Common Core State Standards Initiative. Retrieved July 5, 2011, from http://www.corestandards.org/ Gould , H. (2011). Meteorology: Describing and Predicting the Weather - An Activity in Mathematical Modeling. Journal of Mathematics Education at Teachers College, 66-67. Pollak, H. O. (2011). What is Mathematical Modeling? In H. Gould, D. R. Murray, A. Sanfratello, & B. R. Vogeli (Eds.), The Mathematical Modeling Handbook. Bedford, MA: The Consortium for Mathematics and Its Applications. Tam, K.C. (2011). Modeling in the Common Core State Standards. Journal of Mathematics Education at Teachers College, 28-33. Tam, K. C. (2011). Packing Oranges. Journal of Mathematics Education at Teachers College, 67.

35. Questions and Comments? Brandon Milonovich bamilono@syr.edu Entire presentation available at http://www.bmilo.com. Feel free to visit http://matheducate.wordpress.com.