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<p>To effectively teach modeling, instructors need to select tasks that allow students to learn which mathematical operations are appropriate for modeling different scenarios. Naturally, instructors might select tasks using a priori task classification systems—ones that group real-world problems for a given operation based on mathematical formalisms (e.g., as reported by Vergnaud (in: Hiebert (ed) Number concepts and operations in the middle grades, National Council of Teachers of Mathematics, 1988). In this paper, we critique the robustness of a priori task classifications systems for guiding the selection of modeling tasks. We investigated the meanings undergraduate STEM majors attributed to multiplication while modeling a predator–prey system using differential equations. Through analysis of task-based clinical interviews with 23 participants, six distinct justifications for <i>why</i> multiplication was an appropriate operation for modeling the scenario were identified. These six justifications confirm that learners’ assimilation of scenarios to operations may differ from how educators classify problems using a priori classification schemes. Our findings challenge the use of a priori task classification systems for guiding the pedagogical selection of real-world scenarios to model because classifying real-world scenarios using a priori systems can overlook nuances in modelers’ <i>structuring</i> and <i>validating</i>. We highlight the importance of these nuances for generating task trajectories that would leverage learners existing meanings for mathematical operations to build new associations between mathematical operations and novel problems. We end by suggesting a shift towards reasoning-based classification systems for selecting real-world scenarios to model—ones that are based on students reasoning about and within a scenario.</p>

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Reconsidering task classification systems for situations that can be modelled with multiplication

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年1月31日

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  • Elizabeth Roan,
  • Sindura Kularajan,
  • Jennifer A. Czocher

摘要

To effectively teach modeling, instructors need to select tasks that allow students to learn which mathematical operations are appropriate for modeling different scenarios. Naturally, instructors might select tasks using a priori task classification systems—ones that group real-world problems for a given operation based on mathematical formalisms (e.g., as reported by Vergnaud (in: Hiebert (ed) Number concepts and operations in the middle grades, National Council of Teachers of Mathematics, 1988). In this paper, we critique the robustness of a priori task classifications systems for guiding the selection of modeling tasks. We investigated the meanings undergraduate STEM majors attributed to multiplication while modeling a predator–prey system using differential equations. Through analysis of task-based clinical interviews with 23 participants, six distinct justifications for why multiplication was an appropriate operation for modeling the scenario were identified. These six justifications confirm that learners’ assimilation of scenarios to operations may differ from how educators classify problems using a priori classification schemes. Our findings challenge the use of a priori task classification systems for guiding the pedagogical selection of real-world scenarios to model because classifying real-world scenarios using a priori systems can overlook nuances in modelers’ structuring and validating. We highlight the importance of these nuances for generating task trajectories that would leverage learners existing meanings for mathematical operations to build new associations between mathematical operations and novel problems. We end by suggesting a shift towards reasoning-based classification systems for selecting real-world scenarios to model—ones that are based on students reasoning about and within a scenario.