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).