Academic journal article The Mathematics Enthusiast

Mathematical Modeling Cycles as a Task Design Heuristic

Academic journal article The Mathematics Enthusiast

Mathematical Modeling Cycles as a Task Design Heuristic

Article excerpt

Since there are many theoretical perspectives on modeling (Sriraman, Kaiser, & Blomhoj, 2006) - and no consensus as to a definition of mathematical modeling (Cai et al., 2014) -assessments of mathematical modeling tend to be ad hoc or based on experience rather than theoretically grounded (Frejd, 2013). Therefore, the research presented here sought to take advantage of modeling task design principles in order to generate tasks that would evoke students' mathematical modeling processes. Typically in research reports, the smallest amount of space is accorded to instrument development despite the fact that all data, results, interpretations, and conclusions are derived from the instrument and therefore dependent upon its genesis. This paper attempts to make transparent the considerations and decisions that are present in a researcher's logs but that typically are cut from the dissemination of results. I then use these ideas to reflect on the domain-specific theory of modeling cycles and the methods used to design the tasks.

Theoretical Perspectives

Situating the Task Design Project

Kieran, Doorman, and Ohtani (2015) reviewed of sets of principles for task design in mathematics education studies. The review was organized around two dimensions. The first dimension was the scope of the theory informing the research objectives (grand, intermediate, or domain-specific) and the second was whether the design could be characterized as design as implementation or design as intention. Design as implementation studies focus attention on "the process by which a designed sequence is integrated into the classroom environment and subsequently is progressively refined" (Kieran et al., 2015, p. 28). This is consistent with how design research projects are understood(Cobb, Confrey, DiSessa, Lehrer, & Schauble, 2003). In contrast, design as intention studies "address the initial formulation of the design" and use well developed theoretical frames in order to provide clarity and coherence to the intention (Kieran et al., 2015, p. 28). Grand theoretical frames explain learning in general (e.g., cognitive constructivist or social constructivist theories). Intermediate-level frames can be applied across many mathematical areas and domains (e.g., theory of didactical situations). Finally, domain-specific frames specify reasoning processes (e.g., conjecturing or modeling) or content (e.g., place value, geometry).

Using Kieren et al.'s dimensions, I can situate this task design project as design as intention using a domain-specific theory. The objective of the research program was to study mathematical thinking from within a mathematical modeling paradigm. The participants were to be undergraduate engineering students in a course on differential equations. One sub goal of this project was to create an observational rubric that could be used to systematically observe students' modeling activity as it unfolded. Thus, domain-specific theories related to mathematical modeling processes were adopted. The creation of the rubric was to be guided by microanalysis of students' work on modeling tasks within a one-on-one interview setting. This required a "bootstrapping" approach that design studies are able to handle (DiSessa, Cobb, & Disessa,

2004, p. 85). The tasks and the rubric were developed through an iterative design process grounded in the students' mathematical modeling. The development of the rubric through qualitative analysis of the students' work on the tasks is presented elsewhere (Czocher, 2016).

The tasks designed needed to satisfy a particular intention, namely, evoking students' modeling processes so that those processes could be documented systematically. The goal in designing the tasks was not to create a direct measure of students' competencies nor to teach modeling. Instead, it was an attempt to study and explain in greater detail components of mathematical modeling identified by the literature as central to the mathematical modeling process or as difficult for students. …

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