In our school district, we work with many elementary school teachers and their students who are studying systems within the context of the district curriculum's standards and benchmarks. Embedded in many of the standards and benchmarks are such systems concepts as change over time, interdependencies, accumulations, rates, boundaries, short/long term consequences, delays, and unintended consequences. A current third grade unit integrates curricular concepts on life cycles and ecosystems in science with patterns and data interpretation in math using systems thinking and dynamic modeling.

Since life cycles involve interdependencies among birth, growth, reproduction, and death, and ecosystems involve interdependencies among species, populations and communities, creating simple population models sets an experiential and visual basis for study. Operational Thinking skills, as explained by Barry Richmond, are needed to create and manipulate the interrelationships within a model; this pulls in several math benchmarks on number patterns and data interpretation.

Before working with a computer model, the students manipulate a hands-on model. They work in small groups rolling ping pong ball "animals" through a wrapping paper tube "birth" inflow into a cake pan "stock" of population, and out through a wrapping paper tube "death" outflow. As they experiment with this model, students predict, observe, and record resulting population trends on BOTGs(behavior over time graphs) and generate rules about how the birth and death rates affect the number of animals in the population.

The subsequent lesson involves whole class creation of a simple STELLA model based on the students' experience in using the previous physical model. As students help create and use the computer model, the nature of the variables included requires questions that draw on their prior knowledge and extend their thinking. (e.g., "If I have a population of elephants, can I put tigers through the birth flow?") Not only does this question insure the units are the same in the stock and the flows, it also leads to discussion of the science concept of species. Another question that links prior knowledge to new learnings is, "Who in the population is able to give birth?" Initially students answer, "Females," but when asked, "Can all females give birth?" students realize some animals are too young, too old, too sick, etc. This is an "Aha!" for many third graders.

When the class model is complete, students manipulate it by inputting birth and death fractions to achieve teacher-targeted results, such as, "What numbers could we use that would increase the population? What numbers would make it increase at a faster rate?" The model boundaries encompass births, population, and deaths; once the students input numbers that make the population grow excessively, we expand the boundaries through discussion. "Could a population grow indefinitely? What would happen?" Students experience another "Aha!" realizing that the growing population of one species will have a large effect on other populations and resources within a community. This also brings in the concepts of delays and short and long term consequences.

Debriefing this lesson includes questions that further extend student thinking about and understanding of life cycles. For example, students are asked whether this population model would work for all species and what elements are common among all life cycles.

We highly recommend the use of physical, hands-on models before creating computer models with young students. We believe this prepares students who are at different cognitive levels of development to understand the more abstract concepts associated with the computer model. Then, creating and manipulating a simple STELLA model will produce powerful "Aha!" learning moments for the students.

Julie Guerrero and Joan Scurran have been teachers in the Catalina Foothills School District in Tucson, Arizona for many years and are currently Waters Foundation Systems Thinking and Dynamic Modeling Mentors in the district.