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Unit 5: Analyzing Complexity

Deborah Gross (Carleton College), Lisa Gilbert (Williams College), and Karl Kreutz (University of Maine)


This unit has students build on a system diagram, to include new knowledge about quantitative values and relationships. They will also write about and discuss what they know about their systems, the questions that still remain, and how to find answers to their questions.

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Learning Goals

  • Students will describe quantitative relationships among components of a complex system.
  • Students will sketch graphs of the behavior of measurable quantities in their system as a function of time.
  • Students will discuss with their peers their current level of knowledge about the components of a systems diagram.
  • Students will write about the components of a systems diagram in an optional homework assignment.

Context for Use

This lesson should occur at a point in a course when students know enough information to look at a system diagram with some quantitative understanding of the processes within it. This will allow them to use skills developed with STELLA models in Units 3 and 4 in an application directly relevant to their course. If Units 3 and/or 4 are not used in the course, this exercise can follow directly from Unit 2 or be used alone. This unit is designed to include class discussion, in-class activities, and an optional homework assignment that involves writing about complex systems.

Description and Teaching Materials

To carry out these activities students will need:

  • This handout of The Carbon Cycle (available as a PDF (Acrobat (PDF) 175kB Jun21 16) or PowerPoint slide (PowerPoint 2007 (.pptx) 233kB Jun21 16)), their original systems diagram created in Unit 2, or any other systems diagram of the instructor's choice
  • Blank half-sheets of paper

The first goal of this unit is to encourage students to think about the items which they designate as "quantifiable" in a system diagram and to predict the form of the measured quantity, as a value (flux/rate or quantity in a reservoir) or as a graph of value versus time. Doing this will require bringing in outside knowledge about the components of a system, and the level of detail desired for this will depend on the instructor's course goals.

Part 1. Discussion of a Complex System and Quantifiable Components (10 min)

The instructor will provide the students with a diagram of The Carbon Cycle and will lead a discussion about specific content within the diagram related to the course. Examples of data for many quantifiable fluxes and reservoir amounts are provided (Carbon Cycle Quantities PDF (Acrobat (PDF) 666kB Jul24 20) or PowerPoint slide (PowerPoint 2007 (.pptx) 855kB Jul24 20)), and those which do not directly measure carbon can be converted into carbon fluxes or reservoirs when combined with measurements of, for example, carbon concentrations in reservoirs. If the students have done Unit 2, labeling values for which quantitative information can be obtained will be review. If the class has included discussions or activities related to quantitative information in a systems diagram, this portion may be skipped.

Part 2. Addition of Quantifiable Data to Original Systems Diagrams (20 min)

Students should be organized into groups of 3–5 people. If they did Unit 2, these can be the same groups. The groups should be given at least five half-sheets of blank paper.

Student groups are asked to identify a small number of quantifiable items in the diagram of The Carbon Cycle and describe them quantitatively (fluxes or rates) and/or graphically (for values that change with time), with appropriate units. Students should quantify at least one flux/rate, at least one reservoir, and at least one variable that changes with time. Groups should sketch graphs on pieces of paper and attach them to their system diagrams. The level of quantitative detail requested of the students will vary depending on the instructor's goals for the course. If the students do not have the content knowledge to generate absolute numbers, they can be asked to propose relative values or sketch trends graphically but without numerical values.

For a more advanced activity, this task can be expanded to include a pre-activity homework or in-class assignment for students to find relevant data sets to use in developing these values and graphs.

Students will then do a very brief "gallery walk" where they look at all or a subset of the graphs sketched by other groups. Students will reconvene with their groups at their own diagram and add to it anything that they saw in the other diagrams that they think is critical to include, but which they had missed. Students should photograph their graphs and, if possible, upload their photographs to an online space (course management system, etc.) so that there is a permanent record of them.

Part 3. Brainstorming About Sources of Information and Knowledge (20 min)

This activity allows students to explore the ideas that they have been working with in their groups, and simultaneously prepare for the optional homework assignment described in Part 4. In addition, it provides an opportunity for students to connect the concepts they have explored in the system diagram with experiences from outside of the course. If the homework will be assigned, the text of the homework assignment should be available to students at this time, either as a handout or electronically, so that the students see the questions that they will be asked to address. A template for a handout describing the homework is provided in Part 4, below.

Students should continue to work with their groups for a small-group discussion that will allow them to brainstorm about ways to fill in the gaps in their system diagram and gain further knowledge. Students should categorize and explore how to answer questions that they have about the source(s) of the knowledge that goes into making The Carbon Cycle diagram quantitative. The assignment does not include answering the questions they pose, but that could be added by the instructor if desired, in an expanded homework assignment.

Guided by the discussion handout (Discussion Handout PDF (Acrobat (PDF) 85kB Sep14 16) or Word doc (Microsoft Word 2007 (.docx) 20kB Sep14 16)), the members of the groups should identify two questions that they have about the content of The Carbon Cycle diagram. They should focus on the topics they feel least secure about, and the type of knowledge that would be necessary to gain to become more secure (questions 4 and 5 in the optional homework described in Part 4). They should sort their questions/answers into the following categories:

  • Those that can be answered by another member of the group.
  • Those which can be answered within the context of this course, either from past or anticipated future classes.
  • Those which can be answered by taking a specific course about which the students are aware.
  • Those which can only be answered by researching the primary literature.
  • Other, including specific topics designated by the instructor or those which do not easily fit into a defined category.

The students should designate one group member to be a note-taker and one group member to report out into a subsequent larger group discussion. The instructor should provide a printed copy of the discussion sheet to each member of the group, with an additional sheet for each group that the designated note-taker can submit at the end. After the groups spend about 10 minutes working together on their questions, the instructor should facilitate a discussion (~10 minutes) that either includes the whole class or subsets of the class. In this discussion, the instructor should ask students to share their results from categorizing their thoughts in the above exercise, by posing any or all of the following discussion questions:

  • What understanding of The Carbon Cycle do you gain from using the systems diagram that you would not get by looking only at individual relationships within the system?
  • Which discipline or disciplines contribute the most to your current understanding of The Carbon Cycle, and which one(s) will be most instrumental in helping to increase your understanding of it?
  • How has your understanding of The Carbon Cycle grown or changed by labeling components and/or by including quantitative measures of reservoirs, fluxes, and/or values that change with time?
  • Would the use of a system diagram such as The Carbon Cycle Diagram help you address questions about the impact of a change in policy, and thus human activity, on The Carbon Cycle? For example, what would happen if coal-fired power plants were suddenly banned, or if regulations on fisheries were suddenly lifted?

As desired by the instructor, the students can be asked to address any of the topics identified within the system diagram, or the questions above, with further research.

Part 4. Writing About the Quantitative Aspects of the Complex System (Optional Homework)

As a summative assessment after Unit 5, students will individually write a two- to three-page narrative description of The Carbon Cycle diagram, describing the same five topics discussed in the homework assigned in Unit 2, but with emphasis on the quantitative aspects of the diagram which they have recently engaged with:

  1. The components of the diagram.
  2. The connections between components of the diagram. The students should use correct terminology throughout the discussion.
  3. The areas of their diagram which they feel most secure about (i.e. which do a good job of representing what they are trying to represent).
  4. The areas of their diagram which they feel least secure about (i.e. which do a poor job of representing what they are trying to represent or which they do not fully understand). There should be at least five specific areas identified in this section.
  5. The type of knowledge they believe would be necessary to gain to be able to improve the areas of the diagram they are least secure about, and some thoughts about who generates this type of knowledge (i.e. geologist, chemist, political scientist, etc.).

A template for a student handout for the homework assignment is provided here (Writing About Complexity Assignment PDF (Acrobat (PDF) 88kB Oct17 16) or Word doc (Microsoft Word 2007 (.docx) 18kB Oct17 16)). This assignment sheet includes a rubric for evaluating the students' work; the rubric is given to them to guide their writing.

Teaching Notes and Tips

Choice of System Diagram For This Unit: This unit is described above with The Carbon Cycle as the central system diagram, however there are various options for carrying out this unit:

  • Option 1: Use The Carbon Cycle, as described above.
  • Option 2: Use The Whale Pump or The Cryosphere diagrams, examples of which are provided in the presentation created for Unit 2 (How to Label a System Diagram PDF (Acrobat (PDF) 789kB Oct17 16) or as a PowerPoint slide (PowerPoint 2007 (.pptx) 823kB Oct17 16)). Note that these examples include information about variables which can be quantified, but rates/fluxes and examples of measurable quantities as a function of time are not provided.
  • Option 3: If the class has done Unit 2, have the students use the system diagram that they drew in Unit 2 as the diagram for this unit. This will require retrieving their submitted system diagram from Unit 2 as a starting point
  • Option 4: Use any other system diagram that is relevant to the course.

If the class has done Unit 2, Option 3 is recommended, but any of these options will be appropriate. If a system diagram other than The Carbon Cycle is used, just replace "The Carbon Cycle" in the activities and handouts with the appropriate name.

Group Work: This unit suggests that students reform the groups that they were in when they drew and worked with systems diagrams. If the instructor did not do Unit 2, or if course enrollment has changed significantly, groups should be created such that all students are in groups of 3–5 students. For the larger group discussion, larger classes can be broken up into subsets, or, in smaller classes, the whole class can discuss together. The goal is to have students articulate their answers to others and to hear from other groups.

Gallery Walk: A gallery walk provides an opportunity for students to circulate through the room and look at the products of the other groups' efforts; in this unit, Part 2, the products are posters containing system diagrams that now have graphs attached to them showing how one or more quantities change with time. More information about how to implement Gallery Walk activities is available, although in this version of the activity, no questions are posted around the room — students are instead asked to look at and evaluate the components included by their peers as they evaluate the completeness of their own diagrams, providing them input for revision. If time is a constraint or if there are many groups, each student in a group can be asked to look at three different diagrams from other groups, ensuring that the members of a group have sampled multiple other diagrams.


A suggested rubric for assessment of student essays is provided here (Writing About Complexity Rubric PDF (Acrobat (PDF) 66kB Oct17 16) or Word doc (Microsoft Word 2007 (.docx) 16kB Oct17 16)), and is provided for the students on the homework assignment handout that is given in Part 4. The homework assignment provides an opportunity to assess whether students are appropriately using terminology, identifying measurable quantities in a system diagram, and thinking about how to obtain the quantitative information necessary to transform a diagram into a model. Students who meet the "Acceptable" or "Exemplary" criteria in most of the categories demonstrate good mastery of the ability to think about how to take a system diagram to a more quantitative level. Students can be asked to self-evaluate their work using the rubric or can be organized into peer-review teams to evaluate each others' work using the rubric, in addition to the instructor grading the work.

References and Resources

Carbon Cycle Example:

Data for the quantitative relationships in the carbon cycle can be found in the Chapter 6 of Climate Change 2013: The Physical Science Basis, from the Intergovernmental Panel on Climate Change (IPCC):

Ciais, P., C. Sabine, G. Bala, L. Bopp, V. Brovkin, J. Canadell, A. Chhabra, R. DeFries, J. Galloway, M. Heimann, C. Jones, C. Le Quéré, R.B. Myneni, S. Piao and P. Thornton, 2013: Carbon and Other Biogeochemical Cycles. Climate Change 2013: The Physical Science Basis. Contribution of Working Group I to the Fifth Assessment Report of the Intergovernmental Panel on Climate Change [Stocker, T.F., D. Qin, G.-K. Plattner, M. Tignor, S.K. Allen, J. Boschung, A. Nauels, Y. Xia, V. Bex and P.M. Midgley (eds.)]. Cambridge University Press, Cambridge, United Kingdom and New York, NY, USA. (Link to the full report is here:

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These materials are part of a collection of classroom-tested modules and courses developed by InTeGrate. The materials engage students in understanding the earth system as it intertwines with key societal issues. The collection is freely available and ready to be adapted by undergraduate educators across a range of courses including: general education or majors courses in Earth-focused disciplines such as geoscience or environmental science, social science, engineering, and other sciences, as well as courses for interdisciplinary programs.
Explore the Collection »