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<p>For decades, in German-speaking areas, well-founded recommendations for practice formats in the mathematics classroom have been available, and instructional models of practice are also examined in empirical research on learning and instruction. While empirical studies often focus on consolidating factual knowledge and skills, suggestions typically adopt a&#xa0;broader view of goals for practice: Consolidation then refers to both skills and conceptual understanding, often in connection with each other. This article discusses the research available as well as the possible implications for practice in mathematics classrooms and for desirable further research on practicing.</p><p>For the design of practice phases, it is first necessary to specify the goals (especially mathematical concepts and mathematical procedures and strategies) and then identify the associated psychological types of knowledge (factual knowledge, skills, and understanding) and the function of the practicing (securing, flexibilizing, or deepening). These determinations then allow for the selection and design of suitable practice formats, of which the following five types—each with their theoretical foundation, empirical evidence, and considerations for practical implementation—are presented: (1)&#xa0;Retrieval practice, (2)&#xa0;distributed practice (primarily for declarative factual knowledge or simple skills), (3) interleaved practice with the implicit possibility for comparisons (especially for easily confusable facts or skills) or with explicit comparison prompts (also for more complex types of knowledge, such as procedural flexibility or understanding), (4) elaborative practice (with a&#xa0;focus on understanding), and (5) combinatory practice as an “intelligent” integration of multiple practice goals. While the first three practice formats have been systematically researched, the last two are more commonly found under the umbrella term “productive practice” in theoretically justifiable but less extensively studied practical recommendations.</p>

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Üben im Mathematikunterricht – Konsolidierende Übungsformate und ihre lerntheoretische und empirische Fundierung

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  • 发表日期：

    2025年6月3日

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  • Timo Leuders,
  • Katharina Loibl

摘要

For decades, in German-speaking areas, well-founded recommendations for practice formats in the mathematics classroom have been available, and instructional models of practice are also examined in empirical research on learning and instruction. While empirical studies often focus on consolidating factual knowledge and skills, suggestions typically adopt a broader view of goals for practice: Consolidation then refers to both skills and conceptual understanding, often in connection with each other. This article discusses the research available as well as the possible implications for practice in mathematics classrooms and for desirable further research on practicing.

For the design of practice phases, it is first necessary to specify the goals (especially mathematical concepts and mathematical procedures and strategies) and then identify the associated psychological types of knowledge (factual knowledge, skills, and understanding) and the function of the practicing (securing, flexibilizing, or deepening). These determinations then allow for the selection and design of suitable practice formats, of which the following five types—each with their theoretical foundation, empirical evidence, and considerations for practical implementation—are presented: (1) Retrieval practice, (2) distributed practice (primarily for declarative factual knowledge or simple skills), (3) interleaved practice with the implicit possibility for comparisons (especially for easily confusable facts or skills) or with explicit comparison prompts (also for more complex types of knowledge, such as procedural flexibility or understanding), (4) elaborative practice (with a focus on understanding), and (5) combinatory practice as an “intelligent” integration of multiple practice goals. While the first three practice formats have been systematically researched, the last two are more commonly found under the umbrella term “productive practice” in theoretically justifiable but less extensively studied practical recommendations.