modeling allows application of math concepts to diverse real world problems
 
Alignment of the Common Core State Standards - Mathematics and System Dynamics Modeling (Student Ages 14–18)

System Dynamics (to be abbreviated SD) System dynamics is a mathematical model-building methodology for studying complex system behavior.  By analyzing feedback loop interaction one can gain a deeper understanding of the causes of systemic behavior, see how delays can create instability in a system, identify leverage points, and test policies that might mitigate undesirable behaviors or unintended consequences. 

Modeling, using equations, has been an integral part of algebra through calculus, for years. System Dynamics modeling provides a view of the structure of the system that is more revealing than many other types of modeling. Because the software is visual, because full words or phrases can be used to identify the individual icons that represent a component in the model structure, because dependencies of one part upon another can be explicitly displayed, much more information is provided to the learner. The building of models is an ACTIVE process for the students. They must understand why each component is necessary for the system to operate, how the components are connected, and the role each component has in controlling the behavior of the system. They construct the small models and/or enhance/modify mid-sized models from a smaller core model.

(Note: This modeling method can be used starting in Algebra 1, and is an effective method of developing the introductory ideas of calculus (accumulations and rates of change) conceptually.

   

MATHEMATICAL PRACTICE

Make sense of problems and persevere in solving them

  • Students explain to themselves the meaning of the problem and look for entry points to its solution

For SD, students read the modeling story scenario and decide how to start to design the model.

  • Students make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt

The SD process requires first a sketch of a potential reference behavior and a listing of possible variables to include in the model.   They also are to anticipate model behavior before simulating.

  • Students consider analogous problems and simpler forms of the original problem in order to gain insight into solutions

Transfer of model structure to new appropriate scenarios – i.e., spread of an epidemic to marketing a new product or spread of a social movement, and use of generic structures – whose behavior the student knows - applies here.

  • Students monitor and evaluate their progress and change course if necessary

Students build SD models in incremental steps, running the simulation frequently to see if the current behavior makes sense.  If it does not then the model is modified at that point.

  • Students can explain correspondences between verbal descriptions graphs or [and] draw diagrams of important features and relationships, graph data and search for regularity or trends

The SD modeling process starts with a story which students translate into a diagram containing the important features of the story, then produce a simulation graph, which they can compare to collected data. They then analyze the graphs searching for trends.

  • Students check their answers to problems using a different method, and they continually ask themselves "Does this make sense?"

Since SD modeling uses the numeric approximation of differential equations to produce simulated output, it is possible for more advanced students to work between the differential equations representation and the system dynamics stock/flow diagrams for models that are of medium size.  Students at every level of system dynamics model building always ask if the output of the model makes sense.

  • Students can understand the approaches of others to solving complex problems and identify correspondences between different approaches

The stock/flow diagramming used in system dynamics models is like a language for representing complex dynamic systems.  Students who learn to build such models are proficient at reading such diagrams and understanding the approach of others- for small to medium sized models- since the layout of the diagram indicates the logic of the dependencies, connections, and types of variables represented in the model.

Reason abstractly and quantitatively

  • Students make sense of quantities and their relationships in problem situations

In order to build an SD model students must decide which variables should be represented as stocks (accumulations) - flows (rates of change) - and converters (ancillary variables and/or parameters).

  • Students must be able to decontextualize – abstract the given situation –

In SD students use generic structures – abstract structures representing simple or more complicated behaviors – as they build their models.  These structures are decontextualized.  They also consider structures that can transfer to other applications, as they build their models. (It is a recommendation to students to try to build models that will have broader application than the specified problem at hand, so decontextualizing is part of their instruction.) Students are to abstract the given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own (translating problems into a stock/flow diagram representation requires abstraction capability in the same sense that designing an equation does, only designing a stock/flow diagram is more visual, hence easier for students).  Once in the stock/flow world students easily manipulate icons to attempt to represent the problem faithfully.

  • Students must be able to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved

Often during the building of a simulation students are required to execute the simulation and analyze the output, identifying which parts of the output are being produced by certain parts of the diagram, or what feedbacks are dominant and why.  They are to explain the output in terms of the context of the story.  They can also manipulate parameters to exercise the model to better understand how the values influence the behavior of the model.

  • Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities

All of these apply directly to SD modeling, without the need for translation.  The software even contains a unit consistency checker.

Construct viable arguments and critique the reasoning of others

  • Students understand and use stated assumptions, definition, and previous established results in constructing arguments

For SD, “definition” does not apply as it is typically used in mathematics. When creating models students are requested to explicitly state any assumptions when describing their models.   Students are expected to be able to use the assumptions made and the results of the simulation to support arguments about the scenario modeled.

  • Students make conjectures and build a logical progression of statements to explore the truth of their conjectures

As part of the SD process students are often required to anticipate the behavior of the model as it is built in stages.  They are to explain any discrepancies between their anticipated behavior and the simulation output.  The attempt is to translate what they think controls the behavior of the situation in to a true representation of their logic as they reconcile the differences between their ideas and the simulation output.

  • Students justify their conclusions, communicate them to others, and respond to the arguments of others

In SD, the computer is used as a tool to guide their theory building and modify their thinking in so far as its output disagrees with their anticipated behavior.  All models are built with the expectation that it will be used as a tool to pass along their insights from modeling to others, so the model must be designed to be easy to read and useful for explanation.  It also lays out the dependencies of the solution for all to see, aiding the student in his/her response to the arguments others might make about their solution.

  • Students reason inductively about data, making plausible arguments that take into account the context from which the data arose

SD students look for trends and patterns displayed in the model they created and use those trends to suggest possible policy arguments that might be suggested for changing the future behavior shown.

  • Students are able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and if there is a flaw in an argument – explain what it is

SD students are not alone in supporting the power of their arguments.  Their models lay out their logic and the computer displays what their logic produces.  The computer acts as an arbiter, and can be used to assist students in reconciling differences in an approach to a problem.

  • Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the argument

SD models have been used with students at different levels of understanding.  The visual nature of the software and the stock/flow diagrams help less experienced students understand connections and allows them access to problem analysis heretofore relegated to those understanding more sophisticated mathematics.

Model with mathematics

  • Students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace

A wide variety of SD modeling experiences apply to this practice.

  • Students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revisions later

The SD process requires explicitly making assumptions, drawing Behavior Over Time Graphs or Reference Graphs of the main variables before creating the model, and building and testing the model in stages. 

  • Students are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, graphs, flowcharts and formulas

The SD process requires the identification of variables relevant to the story/problem and the design of a stock/flow diagram representing the relationship of the variables, using formulas and other values to make the model operational.

  • Students routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose

All these are normal parts of the SD modeling process.

 

 

MATHEMATICAL PRACICE

Use appropriate tools strategically

  • Students consider the available tools when solving a mathematical problem

SD software provides an important tool that applies numeric solutions of differential equations to real-world scenarios but in a way that does not require a full understanding of calculus from a mathematical perspective.

  • Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations

The visual nature of the STELLA software helps a broad range of students gain access to problems normally outside their usual field of study.

  • Students analyze graphs

Graphical analysis is a big part of SD model analysis.

  • Students detect possible errors by strategically using estimation and other mathematical knowledge

SD students regularly look at graphical output of their models and try to explain what the graphs tell them. They evaluate “reasonableness” in the output, including estimating the appropriate numeric values before they are displayed.

  • Students know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data

All part of a normal lesson using SD models.

  • Students are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems

SD students may need to access data on the web, look for dynamic flowchart relationships to help formulate model variable relationships, access dynamic problems to use as a source of a problem to model.

  • Students use technological tools to explore and deepen their understanding of concepts

The core reason to use SD modeling.

Attend to precision

  • Students try to communicate precisely to others

Communication to others is a big part of SD model building, and using the model output helps students be precise.  Additionally, STELLA software comes with a story-telling feature to aid the modeler in unfolding the model a segment at a time to aid the audience in understanding the design of the structure of the model.

  • Students state the meaning of the symbols they choose

In SD, each symbol has a purpose, certain variables are stocks because they are accumulators, some are flows because they represent rates of change, some converters.  Also each icon gets a descriptive name that can be up to 64 characters long.

  • Students must be careful about specifying units of measure

The SD software has a “unit consistency” checker.

  • Students must label axes to clarify the correspondence with quantities in a problem

SD students pay close attention to labeling axes as it is essential in their analysis of graphical output.  The horizontal axis is almost always related to time.

  • Students have learned to examine claims

The purpose of building SD simulations is to be able to analyze claims made in situations that involve dynamic complex systems.  They do not have other accessible tools to do this (for complex systems) in high school.

Look for and make use of structure

  • Students look closely to discern a pattern or structure

The mantra of SD modelers is “structure determines behavior,” so students are tuned into the need to look at structure and the patterns those structures produce.

  • Students can step back for an overview and shift perspective

Another common saying for SD modelers is that they need to “see the forest and the trees.” There is also a modularity tool that assists students in thinking about the problem at a higher (modular) level, before defining the individual segments in detail.

Look for and express regularity in repeated reasoning

  • Students notice if calculations are repeated

For simple models in SD, students can be shown the recursive nature of the calculations performed in the software.

  • Students maintain oversight of the process, while attending to the details

Again, the modular tool allows oversight of the whole model structure as students define the specific details of each module.

  • Students continually evaluate the reasonableness of their intermediate results

As before, models are built and analyzed in stages, so SD students are always paying attention to intermediate results.

CCSS-M: STANDARDS

Functions Overview

  1. Understand the concept of a function, interpret functions that arise in applications in terms of the context, analyze functions using different representations
  2. Build a function that models a relationship between two quantities (although for SD problems one of the quantities is usually time), build new functions from existing functions
  3. Construct and compare linear, quadratic, and exponential models and solve problems, interpret expressions for functions
  4. Model periodic phenomena
  5. Modeling with functions: students identify appropriate types of functions to model a situation, they adjust parameters to improve the model, and they compare models by analyzing appropriateness of fit and making judgments about the domain over which a model is a good fit.

SD modeling applies to each of the “functions in algebra-based classes” standards listed in the Functions Overview shown above.

Modeling Overview

  1. Modeling links classroom mathematics and statistics to everyday life, work, and decision-making. Modeling is the process of choosing and using appropriate mathematics and statistics to analyze empirical situations, to understand them better, and to improve decisions. Quantities and their relationships in physical, economic, public policy, social, and everyday situations can be modeled using mathematical and statistical methods. When making mathematical models, technology is valuable for varying assumptions, exploring consequences, and comparing predictions with data.
  2. Real-world situations are not organized and labeled for analysis; formulating tractable models, representing such models, and analyzing them is appropriately a creative process. Some examples that were listed that are appropriate include:
    • Estimating how much water and food is needed for emergency relief in a devastated city of 3 million people; Analyzing stopping distance for a car; Modeling savings account balance, bacterial colony growth, or investment growth; Relating population statistics to individual predictions [and many more relevant problems not usually within the scope of high school students].
    • the models devised depend on a number of factors: How precise an answer do we want or need? What aspects of the situation do we most need to understand, control, or optimize? What resources of time and tools do we have?
    • Diagrams of various kinds are powerful tools for understanding and solving problems drawn from different types of real-world situations.
  3. One of the insights provided by mathematical modeling is that essentially the same mathematical structure can sometimes model seemingly different situations.
  4. In descriptive modeling, a model simply describes the phenomena or summarizes them in a compact form. Graphs of observations are a familiar descriptive model — for example, graphs of global temperature and atmospheric CO2 over time.
  5. Analytic modeling seeks to explain data on the basis of deeper theoretical ideas, albeit with parameters that are empirically based; for example, exponential growth of bacterial colonies (until cut-off mechanisms such as pollution or starvation intervene) follows from a constant reproduction rate. Functions are an important tool for analyzing such problems

SD modeling applies to all of the modeling standards listed in the Modeling Overview shown above.

Statistics & Probability Overview

  1. Summarize, represent, and interpret data on two quantitative variables.
  2. Interpret linear models
  3. Make inferences and justify conclusions from sample surveys, experiments, and observational studies.

SD aligns with #1, 2, and experiments and observational studies of #3, in the Statistics & Probability Overview shown above.

     
Some Simple Math Function Models
     

Newton's Second Law model

 

Resource Depletion model







 

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