Accomplish concreteness fading by incrementally increasing task size until physical manipulation becomes impractical¶

Strategy · All strategies
Evidence · no claims cited

Description¶

The article's fading was accomplished by starting with small-number permutations of physical cubes and incrementally increasing the number of cubes so that actual manipulation becomes impractical, creating pressure for symbolic representations to be abstracted and come to the fore. The authors report that "actual manipulation of cubes becomes impractical and necessarily needs to fade into the background as symbolic representations are abstracted". Throughout, students were supported via S*NLS to connect each new problem's reasoning to the previous one using recursion.

Design Implications¶

Context¶

Requirements¶

• Physical manipulatives must be available for initial small-number tasks
• Tasks must increase in size so full physical modeling becomes impractical

Constraints¶

• Studied in one-on-one instruction with undergraduate preservice teachers; the authors hypothesize, as future research, that the enactive stage may be more effective when complete modeling of the task situation with concrete materials is impractical or impossible

Target Learners¶

• undergraduate preservice middle school teachers

Target Learning Goals¶

• generalizing permutation counting from concrete enumeration to symbolic formulas

Related Strategies¶

Examples¶

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Key Sources¶

• Joseph Antonides, Michael T. Battista. (2022). Towards an Elaboration of Concreteness Fading: Reflections on a Constructivist Teaching Experiment. Proceedings of the forty-fourth annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. https://eric.ed.gov/?id=ED630441