How PSTs Engage in the Modeling Cycle

Background

As we reviewed audio transcripts of PSMTs working on the Having Kittens modeling task from MARS we noticed that there deviations between the paths students took to obtain their solutions and the hypothetical paths available in published modeling cycles. To explore these deviations we created a modified Modeling Activity Diagram (MAD) for each group of students, which enable the viewer to visualize the way groups shifted between modeling phases as they found their solution to the task. These MADs were based on the work of Ärlebäck and Albarracin (2019), but were modified to align with the Bliss et al. (2014) modeling cycle and phase descriptions.

MAD Framework from Ärlebäck and Albarracin (2019)

Gaps in Research

- No current MAD analysis for practitioner-based modeling cycles
- Need a framework that is more general and inclusive for all modeling tasks

Research Question: How does pre-service math teachers’ (PSMTs) engagement in a modeling task align with the Bliss et al. modeling cycle?

Methods

Two sessions on mathematical modeling were conducted with PSMTs in a MSU math methods course
During the second session the PSMTs engaged in the Having Kittens modeling task from MARS
Audio transcripts were analyzed for each small group
A modified MAD was created for each group using modeling phases that aligned with the Bliss et al. (2014) modeling cycle and phase descriptions
The resulting MAD enables the viewer to visualize the way groups shifted between modeling phases as they found their solution to the task

Modeling Cycle & Phase Classifications

When analyzing the audio transcriptions we used the categories below to classify each group's activity throughout thier time working on the modeling task. These classifications were developed using the Bliss et al. (2014) phase descriptions.

Defining the Problem Statement

Understanding the situation
Unpacking the task
Referring back to given information

Making Assumptions

Breaking the problem down into manageable pieces
Reducing the number of factors

Defining Variables

Identifying and considering additional variables
Mathematizing

Getting a Solution

Mathematical work to produce a solution
Discussing next math steps
Stating intermediate results to group members

Analysis & Model Assessment

Analyzing the quality of the model, including: strengths and weaknesses, model sensitivity if assumptions are altered
Reflecting on and validating results
Questioning and revising the solution

Reporting the Results

Providing and sharing results
Reporting results to activity facilitators or others outside of the group

Miscellaneous

Off-task behavior
Talking with activity facilitators

Findings

All groups started off defining the problem statement, which is not explicitly listed as the first step in the Bliss et al. (2014) modeling cycle
Participants’ modeling behaviors deviated from Bliss et al.’s modeling cycle
All groups started off defining the problem statement, which is not explicitly listed as the first step in the Bliss et al. modeling cycle
Defining the problem statement was revisited less and less as time went on
Two groups did not define variables according to our coding, they may have not felt the need to consider additional variables
Reporting results may have been less frequent because students’ presentations of solutions were not included in the recording
Students worked in ways that were not represented by the arrows in the Bliss et al. modeling cycle

Limitations

Some parts of the transcripts were difficult to code, multiple group conversations took place to come to a consensus
Each time increment in the chart is 5 seconds in length, any code lasting less than 3 seconds was not represented

Future Research

Comparing what we observed to different modeling cycles
Applying the MAD analysis to a different modeling task
Repeating the exercise with in-service teachers

References

Ärlebäck, J. B., & Albarracín, L. (2019, February). An extension of the MAD framework and its possible implication for research. In Eleventh Congress of the European Society for Research in Mathematics Education (No. 4). Freudenthal Group; Freudenthal Institute; ERME.

Bliss, K. M., Fowler, K. R., & Galluzo, B. J. (2014). Math modeling: Getting started and getting solutions. Society for Industrial and Applied Mathematics (SIAM).

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