The modeling cycle has two stages, involving the two general classes of modeling activities: **model development and model deployment.** Roughly speaking, model development encompasses the exploration and invention stages of the learning cycle, while model deployment corresponds to the discovery stage. It will be noted that the "modeling terminology" is more descriptive of what the students actually do in the cycle.

The two stage modeling cycle has a generic and flexible format which can be adapted to any physics topic. In its high school physics implementation, the cycle is two or three weeks long, with at least a week devoted to each stage, and there are six cycles in a semester, each devoted to a major topic. Each topic is centered on the development and deployment of a well-defined mathematical model, including investigations of empirical implications and general physical principles involved.

Throughout the modeling cycle the teacher has a definite agenda and specific objectives for every class activity, including concepts and terminology to be introduced, conclusions to be reached, issues to be raised and misconceptions to be addressed. Though the teacher sets the goals of instruction and controls the agenda, this is done unobtrusively. The teacher assumes the roles of activity facilitator, Socratic inquisitor, and arbiter (more the role of a physics coach than a traditional teacher). To the students, the skilled teacher is transparent, appearing primarily as a facilitator of student goals and agendas.

To make the present discussion of details in the modeling cycle more concrete, we choose a specific topic which appears in both high school and university physics courses. Accordingly, as major objectives for the instructional agenda in the cycle, we aim to develop student conceptual understanding of the following :

**Target model: **Motion of a material particle subject to a constant force.

**Physical principle: **Newton's second law of motion.

**Experimental context:** Modified Atwood's machine.

**Prerequisite:** Before beginning this cycle, the students should have previous experience with kinematic models (two cycles in the high school course), so they have fairly clear concepts of velocity and acceleration. Many students still have only a shaky grasp of these concepts at this point, and more experience with the concepts in a variety of contexts is necessary to consolidate them. Conceptual development takes time, and it will be haphazard unless instruction is carefully designed to promote it systematically.

Stage I begins with the presentation of, for example, the modified Atwood machine for the class to consider. Eventually they will realize that a scientific understanding of the system requires

- the specification of a model to represent it conceptually, and
- an evaluation of the fidelity of the representation

At the conclusion of the descriptive stage, the students are directed, collectively, to identify quantitatively measurable parameters that might be expected to exhibit some cause-effect relationship. A variable under direct control by the experimenters is identified as the independent variable, while the effect is identified as the dependent variable. This is a critical step in the modeling process. It is at this point that the students learn to differentiate aspects of the phenomenon to which they must attend from those which are distracters. While this issue of identifying and controlling variables is critical to modeling, it is scarcely addressed in traditional instruction, where a lab manual typically provides students with the lab purpose, procedure, evaluation of data and even questions suggesting appropriate conclusions. This critical issue is also missed in conventional homework and test problems, which typically provide only that information necessary to accommodate the author's choice of solutions.

Having completed the descriptive phase of modeling by settling on a suitable set of descriptive variables, the instructor guides the class into the formulation phase by raising the central problem: to develop a functional relationship between the specified variables. A brief class discussion of the essential elements of the experimental design (which parameters will be held constant and which will be varied) is pursued at this time. The class then divides into teams of two or three to devise and perform experiments of their own.

Before starting data acquisition, each team must develop a detailed experimental design. Except where the design might pose risk of injury to persons or equipment, the teams are permitted to pursue their own experimental procedures without intrusion by the instructor. For a post-lab presentation to the class, the instructor selects a group which is likely to raise significant issues for class discussion ÜÜ often a group that has taken an inappropriate approach. At that time, the group members are expected to present a detailed explanation and defense of their experimental design and conclusions.

Each lab team performs its own data analysis cooperatively, using computers and striving to construct graphical and mathematical representations of the functional relationships previously posited. The principal goal of the laboratory activities is to lead students to develop a conceptual correspondence between targeted aspects of the real world phenomenon and corresponding symbolic representations.

Every lab activity is concluded by each lab team preparing, on a whiteboard, a detailed post-lab analysis of the activity and reasoning that led to the proposed model(s). The teacher then selects one or more of the lab groups to make presentations before the class, explaining and defending their experimental design, analysis of data and proposed model.

Laboratory reports for each activity are written up in a laboratory notebook according to a given format. It is stressed that the purpose of the laboratory report is to articulate a coherent argument in support of their model construction. While each student must prepare and submit a lab notebook, most of the work is done in class in their cooperative study groups. Grading is done by selecting one report at random from each group and selecting different members of the group to defend different aspects of the report. This induces students, during the preparation of reports by the groups, to ensure that every member of the group understands all aspects of the model that they have developed, thus instilling a sense of shared responsibility for the knowledge. This concludes **Stage I**.

The end product of Stage I is a mathematical model together with evidence for a claim that accurately represents the behavior (or structure) of some physical system, in this case the Modified Atwood's Machine. Students have verified that the equation a = F/m accurately describes the acceleration when F and m are varied independently. They are encouraged to consider the possibility that this equation represents a general law of nature, but they should be led to realize that there is no such thing as an experimental proof of a general law. At best, experiment can validate specific models which conform to the law, as in the present case.

Each study group develops solutions for each problem in the study set. Each group is then assigned one of the problems in the set to prepare, on the white boards, for class presentation. One member of the group is then selected to make the presentation. The same recitation grade is given to the entire group, and it depends on the quality of the presentation. During the presentation, if questions are asked by fellow students that the selected presenter can not answer, other members of the group may offer assistance. If however any assistance from other members of the group is required to satisfy the questioner, the recitation grade awarded the group may be reduced. The recitation scores of the groups are enhanced if the members ask valid, well thought out questions during the presentations (shared responsibility).

On each pass through the modeling cycle the students' understanding of models and modeling is progressively deepened; students become more independent in formulating and executing tasks and more articulate in presenting and defending their points of view. The ultimate objective is, of course, to have them become autonomous scientific thinkers, fluent in the vicissitudes of mathematical modeling.

Return to Synopsis

or

*last modified on 1/18/09*