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### Local Action Research

*In Mathematics (ages 16 – 18)*

Experiments were conducted to gauge student ability to analyze more sophisticated real-world applications of functions using a new tool and representation (icon based modeling software – Stella). Some experiments provide comparisons between student analysis using the new software and icon representation versus analysis using a traditional tool (graphing calculators) and closed-form equation representation. The following are summaries of these experiments.

#### Study of Drug Dynamics in the Human Body

I developed a mini-unit on pharmacokinetics that I have used for nearly ten years in my advanced algebra class (student ages 14 – 18) at the end of the first half of the school year. Students had studied linear and exponential functions using both the closed form equations and Stella diagrams to model function behavior, in the first half of the course. The mini-unit combines those two types of functions in a series of problem scenarios involving patients who need medical treatment. The therapeutic drug treatment is given either in the form of an IV drip, a shot, or pills. Students build simple models and manipulate the parameters to gain an understanding of the dynamics involved in each scenario. After two days of working with various model structures on the computer students are then introduced to a slightly extended model structure, one that models consumption of alcohol. Students study, also on the computer, the length of time it takes the blood alcohol level to decrease so a person could legally drive a car. There are other situations they test with this model. The next day my students and the chemistry students exchange classrooms so we can use the wet-lab stations in the chemistry classroom. The algebra students, using blue bathwater coloring pills, and some relatively simple equipment developed by Dr. Edward Gallaher,[1] perform experiments to try to keep their simulated patients alive. The day in the chemistry lab is co-monitored by both Dr. Gallaher and myself. Between 6 and 8 stations are set up for experimentation.

[1] Dr. Edward Gallaher is a research pharmacologist who has used System Dynamics Modeling and the Stella software for over 25 years in his drug research. He has taught pharmacokinetics to medical students at Oregon Health and Sciences University in Portland, Oregon. It is in conversations with Dr. Gallaher that this mini unit was developed.The following document explains the mini-unit in detail and provides data on student reaction to this set of lessons.

Fisher, D. M. (2003) How drugs work in the human body: analysis of a modeling unit used in a second year algebra class. *Proceedings from the International System Dynamics Conference*. New York, New York.

#### A Study of Different (Graphical) Rates of Change and Their (Graphical) Accumulations

Two scenarios, each represented by graphs showing their respective rates of change were given to the students. The students were to sketch what the graph of the accumulation of the rate would look like over time.

This same experiment had been given to MIT (Massachusetts Institute of Technology) students in Dr. John Sterman’s Systems Dynamics Modeling class at the beginning of fall term in 2000. He and Dr. Linda Booth-Sweeney developed the two graphical rate scenarios and they have written articles about the level of understanding (or lack of understanding) the very mathematically savvy MIT students displayed in their attempts to draw the accumulation graphs.

I was interested in determining how high school students, who had had various levels of exposure to System Dynamics (SD) modeling, might compare to the MIT students who had had no exposure to SD modeling on the same experiments. I tested my advanced algebra classes, my calculus class, and my System Dynamics Modeling class. The calculus class of another teacher at the same high school at which I teach also administered the test to his class.

What follows are two documents. The first is the handout with the two graphical rate scenarios that was given to the students. The second is a summary of the results for the high school students.

Two Bathtub Dynamics Tasks.pdf

Fisher, D. M. (2003). Student performance on the bathtub and cash flow problems. *Proceedings from the International System Dynamics Conference*. New York, New York.

#### A Study of the use of a Graphing Calculator vs the Stella software as a Student Tool in Analyzing a New Problem

Students in my advanced algebra classes use both the graphing calculator and the Stella software to set up models and analyze problems. They use the graphing calculator every day and use the STELLA software maybe once a month. I was wondering if the tool would make a difference in a student’s ability to design and analyze a new problem that was a little more difficult than the typical problems studied in class. The problem contained only linear and exponential function sub-parts. It required students to set up a population sub-model with a net exponential growth rate and a food sub-model that grew only linearly. Students then calculated the food/person over time. Students then had to alter the population growth rate and the food production rate in mid-simulation, requiring the use of piece-wise defined functions for those using the graphing calculator. (This process had been explained to all the students, and an example equation, with explanation, provided as a handout.) The assignment was called “The Malthus Problem.”

Students were asked their tool preference (graphing calculator or Stella software). Then each group of students was assigned, at random, to use either the tool of their choice or the other tool. This created 4 different student groups. The details of the experiment and the results are provided in the file below.

Fisher, D. M. (2008). Building slightly more complex models: calculators vs. STELLA. *Proceedings from the International System Dynamics Conference*. Athens, Greece.

#### A Study of Identification of Growth or Decay Patterns When Viewing Stella Diagrams

It is important for students to be able to look at a symbolic function representation and determine whether the function (output) should represent growth or decay. Additionally, it is important for students to be able to determine what pattern of growth or decay would be produced. I wondered how easily students would be able to determine these patterns when viewing a Stella model diagram. The results of that experiment are recorded in the file below.

Fisher, D. M. (2009). How well can students determine simple growth and decay patterns from a diagram? *Proceedings from the International System Dynamics Conference*. Albuquerque, New Mexico.

#### A Study of Identification of Growth and Decay Patterns When Viewing Equations versus Stella Diagrams

It has been interesting to see the response of students working with functions expressed as closed form equations versus working with functions expressed as Stella diagrams when trying to design and analyze problems in advanced algebra. Students work with equations (closed form function representations) extensively in an advanced algebra class. In my advanced algebra class students are also required to work with the Stella diagram representation of functions about once a month. In order to quantify whether students might determine the growth or decay behavior, and the particular pattern of growth or decay more easily using equations or Stella diagrams, an experiment was conducted. This was essentially the same experiment described in the previous paragraph, but this time the advanced algebra students (and another class of SD modeling students) were assigned to complete two sets of problems. At random one group of students completed a set of problems where the representation of the function was given as a Stella diagram followed by another set of problems where the representation of the function was given as a closed form equation. The other group of students used the same packets but in the reverse order. They completed the problems given as closed form equations followed by a packet using the Stella diagram functional representation. The results of this experiment have yet to be compiled, but will be added to this site when compilation is completed.

### Books, Book Chapters, and Research Papers:

Fisher, D. M. (1994). Teaching systems dynamics to teachers and students in 8-12 environment. *Proceedings from the International System Dynamics Conference*. Sterling, Scotland.

Guthrie, S, & Fisher, D. M. (1996). Systems thinking and system dynamics in the CC-STADUS high school project (How high school students become system thinkers). *Proceedings from the International System Dynamics Conference*. Cambridge, Massachusetts.

Fisher, D. M., & Zaraza, R. (1997). Seamless integration of system dynamics into high school algebra, calculus, modeling courses. *Proceedings from the International System Dynamics Conference*. Istanbul, Turkey.

Fisher, D. M. (1998). Mistakes made in the early years teaching students and teachers to create system models. *Proceedings from the International System Dynamics Conference*. Quebec City, Quebec, Canada.

Zaraza, R., & Fisher, D. M. (1999). Training system modelers: the NSF CC-STADUS and CC-SUSTAIN projects. In W. Feurzeig, & N. Roberts (Eds.) *Modeling and Simulation in Science and Mathematics Education* (38-69). New York: Springer.

Fisher, D. M., & Joy, T. (2000). What behaviors are desirable in students creating systems models? A step before assessment. *Proceedings from the International System Dynamics Conference*. Bergen, Norway.

Fisher, D. M. (2000) System dynamics models created by high school students. *Proceedings from the International System Dynamics Conference*. Bergen, Norway.

Fisher, D. M. (2000) Materials for introducing systems modeling in mathematics. *Proceedings from the International System Dynamics Conference*. Bergen, Norway.

Fisher, D. M. (2001). *Lessons in mathematics: A dynamic approach*. Lebanon, New Hampshire: isee systems, inc.

Costello, W., Fisher, D. M., Guthrie, S., Heinbokel, J., Joy, T., Lyneis, D., Potash, J., Stuntz, L., & Zaraza, R. (2001). Moving forward with system dynamics in K-12 education: a collective vision for the next 25 years. *Proceedings from the International System Dynamics Conference*. Atlanta, Georgia.

Fisher, D. M. (2002). Creating content specific lessons incorporating system dynamics. *Proceedings from the International System Dynamics Conference*. Palermo, Italy.

Fisher, D. M. (2003). Student performance on the bathtub and cash flow problems. *Proceedings from the International System Dynamics Conference*. New York, New York.

Fisher, D. M. (2003) How drugs work in the human body: analysis of a modeling unit used in a second year algebra class. *Proceedings from the International System Dynamics Conference*. New York, New York.

Fisher, D. M. (2003) Reaction to system dynamics modeling from a woman’s perspective. *Proceedings from the International System Dynamics Conference*. New York, New York.

Fisher, D. M. (2004) Women in system dynamics modeling: out of the loop? *Proceedings from the International System Dynamics Conference*. Oxford, England.

Fisher, D. M. (2007). CC-STADUS/CC-SUSTAIN projects. In D. Lyneis, & L. Stuntz (Eds.) System dynamics in K-12 education: lessons learned. *Proceedings from the International System Dynamics Conference*. Boston, Massachusetts. (Summaries from Creative Learning Exchange, Carlisle Public Schools, Catalina Foothills School District, NSF CC-STADUS and NSF CC-SUSTAIN projects, Center for Interdisciplinary Excellence in System Dynamics, Harvard Public Schools, Murdoch Middle School, Portland Waters Project)

Fisher, D. M. (2008). Building slightly more complex models: calculators vs. STELLA. *Proceedings from the International System Dynamics Conference*. Athens, Greece.

Fisher, D. M. (2009). How well can students determine simple growth and decay patterns from a diagram? *Proceedings from the International System Dynamics Conference*. Albuquerque, New Mexico.

Fisher, D. M. (2010). Modeling for high school students: Teaching critical thinking through system dynamics. In J. Richmond, L. Stuntz, K. Richmond, & J. Egner (Eds.) *Tracing Connections: Voices of Systems Thinkers* (81-91). New Hampshire: isee systems, inc.

Fisher, D. M. (2011). *Modeling dynamic systems: Lessons for a first course (3rd ed.)*. Lebanon, New Hampshire: isee systems, inc.

Fisher, D. M. (2011). “Everybody thinking differently”: K-12 is a leverage point. Presentation slides from the plenary presentation for the Lifetime Achievement Award. Proceedings from the International System Dynamics Conference. Washington, D. C.

Fisher, D. M. (2011). “Everybody thinking differently”: K-12 is a leverage point. *System Dynamics Review*. 27, (4): 394-411.

Fisher, Diana Marie. (2016). Introducing Complex Systems Analysis in High School Mathematics Using System Dynamics Modeling: A Potential Game-Changer for Mathematics Instruction. *Dissertation*. Paper 2950. Available at: https://pdxscholar.library.pdx.edu/open_access_etds/2950/

Fisher, D. M. (2017). Reorganizing Algebraic Thinking: An Introduction to Dynamic System Modeling. *The Mathematics Enthusiast*, 14(1). Article 20. Available at: http://scholarworks.umt.edu/tme/vol14/iss1/20.

Fisher, D. M., Stuntz, L., Benson, T., LaVigne, A., & Farr, W. (2017). The next 25 years in pre-college education: A move toward global understanding of complex systems. *Proceedings from the International System Dynamics Conference*. Cambridge, Massachusetts.

Fisher, D. M. (2018). Reflections on teaching system dynamics modeling to secondary school students for over 20 years. *Systems Journal Special Edition: Theory and Practice of System Dynamics Modelling*, 6(12). Available at: http://www.mdpi.com/2079-8954/6/2/12/htm.

Fisher, D. M., Gallaher, E., Schaffernicht, M., & Rooney-Varga, J. (2019). Designing assessments to judge attainment of high level learning goals: One step in propagating system dynamics in education (stage 1: Background). *Proceedings from the International System Dynamics Conference*. Albuquerque, New Mexico.

Fisher, D. M. (2020). Algebra students build stock/flow models to study non-linear, dynamic, feedback system problems. In G. A. Stillman, G. Keiser, C. Lampen (Eds.) *Mathematical Modelling and Sense Making*. Springer.

Fisher, D. M. (2021). Global understanding of complex systems problems can start in pre-college education. In F. Leung, G. A. Stillman, G. Keiser, Wong, K. L. (Eds.) *International Perspectives on the Teaching and Learning of Mathematical Modelling*. Springer.

Galbraith, P., & Fisher, D.M. (2021) Technology and mathematical modeling: addressing challenges, opening doors. In: S. Carreira, & W. Blum. (Eds*.) Mathematical Modelling in the Teaching and Learning of Mathematics: Part 1*. Quadrante. 30, (1)*: 198-218.*

* *Fisher, D. M., & Göktepe, E. (2021). Creating Assessment Questions for Systems Thinking Concepts. *Proceedings from the International System Dynamics Conference*. Chicago, Ill.