Expanding the Disciplinary Expertise of a Middle School Mathematics Classroom: Re-Contextualizing Student Models in Conversations With Visiting Specialists

Abstract

This article examines how conversations during design reviews in which 8th-grade mathematics students shared population models with visiting specialists expanded the disciplinary expertise of the classroom. Re-contextualizing is a conversational exchange that visiting specialists initiated to invite groups to consider their models in novel contexts. Analysis of 14 design reviews in 2 classrooms showed that re-contextualizing resulted in both the elaboration of ideas students already understood and new contributions to students' understandings of mathematical aspects of population modeling. This article presents case studies of 2 groups that differed in terms of their interest in the curricular task and the level of conceptual integrity of their population models. Despite these differences, the re-contextualizing exchanges that emerged in their design reviews led to new insights for both groups and provided them with opportunities to try on ways of thinking and acting like population biologists.

Notes

¹The Math at Work project was a 4-year ethnographic study that compared the organization of mathematics in school and in workplaces where design was a leading activity (Hall, 1999).

²The three classroom researchers (Rogers Hall, Susan Jurow, and Tony Torralba) focused on three focal student groups in each class. On occasions such as the design review or other presentations to the class, we decided ahead of time who would focus on what aspects of the interaction (e.g., whole-class interactions, teacher interactions, the activities of a particular student group).

³Our extension scenarios were added to the original MMAP units in order to give students a chance to use what they had learned about modeling fish populations in substantially more challenging, open-ended problems. We also wanted student projects to be complex enough to interest visiting specialists (in this case biologists) whom we had been following in parallel studies of using and learning mathematics at work (Hall, 1999).

⁴We also studied teaching and learning in MMAP curriculum units with a substantive design focus both in architecture and in cartography. These MMAP curriculum units also were designed to have students build, evaluate, and refine computer simulations of objects typical in professional practice (e.g., a live/work structure for scientists wintering over in Antarctica, a multilayer Geographic Information Systems (GIS) map tailored to the particular interests of a client; see Jurow, 2005, and Stevens, 2000, for analyses of these classroom environments). For another curriculum effort using design as a context for teaching, see Kolodner et al.'s (2003) Learning by Design curriculum.

⁵ Carrying capacity is conventionally defined as the maximum number of individuals an environment can support. The teacher and the classroom researchers presented a mini-lecture to the students on the concept of carrying capacity, showed a video (a Bill Nye the Science Guy episode) that addressed the idea of carrying capacity, and presented the students with a formula that they could implement in the Habitech^© software for incorporating a carrying capacity into their population model (e.g., reducing the birth rate when total population approached the theoretical maximum).

⁶We follow transcript conventions typical in conversation analysis (Jordan & Henderson, 1995; for other examples, see C. Goodwin, 2007; M. H. Goodwin, 1990; Hall et al., 2002), but we pay particularly close attention to turn boundaries and the coordination of talk, gesture, and action with representational forms. Our transcripts number turns for identified speakers, breaking lines at thematic boundaries and inserting index numbers for action or gestural sequences that are important for our analysis. Numbered action descriptions appear, in italics, below utterances in which they occur. Unless otherwise noted, EMPHATIC speech is shown in uppercase, stre:::tched enunciation is shown with repeated colons, brief ((action descriptions)) are show in italics within double parentheses, and [overlapping talk is marked with [matching square brackets across speaking turns.

^aIn the design review for Group 4, the student responsible for putting finishing touches on the poster was absent the day Mark visited.

⁷The questioning during the design reviews was led by the visitors (Mark and Jane); the classroom researchers (Tony and Rogers) were asked questions much less frequently in comparison.