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<p>The acquisition of mathematical knowledge frequently poses challenges for students. It can be described in terms of Sfard’s theory of reification, which encompasses the acquisition of operational and structural knowledge through the processes of interiorization, condensation, and reification. Applying this theory to mathematical concepts, such as the derivative, allows a theoretical account for identifying relevant learning phases in concept acquisition and identifying risks such as the development of pseudostructural knowledge, that is, structural knowledge without a sufficient operational foundation, which may hinder sustainable further learning Data from digital learning environments enable tracing individual learning trajectories during concept acquisition and identifying learners who may need additional support. This study examines whether individual learning trajectories derived from a digital learning environment implemented in real classrooms can support early, low-stakes identification of students at risk of developing pseudostructural knowledge of the derivative, and which theoretically derived learning phases are most predictive. Using data from 365 students from 15 upper-secondary classes, we aggregated students’ performance into indicators aligned with the subconcepts of the derivative and Sfard’s phases of concept acquisition. Machine-learning classifiers achieved modest but systematic discrimination based on these theory-grounded trajectory indicators. Adding covariates, including prior knowledge and cognitive abilities, yielded selective improvements and increased robustness under conservative operating rules, illustrating the recall–precision trade-off inherent in screening applications. Across models, early difference-quotient indicators, particularly those targeting condensation, contributed most to screening performance, whereas later indicators showed smaller incremental contributions. Overall, theory-based process indicators, complemented by learner prerequisites, provide a substantively interpretable basis for low-stakes screening of learners at risk of acquiring pseudostructural knowledge of the derivative. </p>

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Tracing learning in a digital learning environment: combining process-object theories, individual learning trajectories, and machine learning to predict concept acquisition in mathematics

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  • 出版模式：

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  • 发表日期：

    2026年6月17日

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  • David Bednorz,
  • Daniel Sommerhoff,
  • Aiso Heinze

摘要

The acquisition of mathematical knowledge frequently poses challenges for students. It can be described in terms of Sfard’s theory of reification, which encompasses the acquisition of operational and structural knowledge through the processes of interiorization, condensation, and reification. Applying this theory to mathematical concepts, such as the derivative, allows a theoretical account for identifying relevant learning phases in concept acquisition and identifying risks such as the development of pseudostructural knowledge, that is, structural knowledge without a sufficient operational foundation, which may hinder sustainable further learning Data from digital learning environments enable tracing individual learning trajectories during concept acquisition and identifying learners who may need additional support. This study examines whether individual learning trajectories derived from a digital learning environment implemented in real classrooms can support early, low-stakes identification of students at risk of developing pseudostructural knowledge of the derivative, and which theoretically derived learning phases are most predictive. Using data from 365 students from 15 upper-secondary classes, we aggregated students’ performance into indicators aligned with the subconcepts of the derivative and Sfard’s phases of concept acquisition. Machine-learning classifiers achieved modest but systematic discrimination based on these theory-grounded trajectory indicators. Adding covariates, including prior knowledge and cognitive abilities, yielded selective improvements and increased robustness under conservative operating rules, illustrating the recall–precision trade-off inherent in screening applications. Across models, early difference-quotient indicators, particularly those targeting condensation, contributed most to screening performance, whereas later indicators showed smaller incremental contributions. Overall, theory-based process indicators, complemented by learner prerequisites, provide a substantively interpretable basis for low-stakes screening of learners at risk of acquiring pseudostructural knowledge of the derivative.