Cube-tower permutation enumeration task sequence¶

Element · All elements
Evidence · 2 claims (2 for) · 1 study (1 theoretical), q2 · 0 of 1 report an effect size · 2 claims rest on one study

Description¶

An instructional sequence representing permutations of n objects as n-cube towers built from multi-colored connecting cubes in vertical orientation. Students constructed towers one-by-one for 3- and 4-cube tasks, then reasoned multiplicatively for 5-cube and larger tasks (up to 100-cube and general n-cube towers), with numerical symbols and computational expressions explicitly linked to their tower constructions. The article's Table 1 summarizes the eight-task sequence and each student's reasoning, showing how students progressed to describing formulas such as n × (n−1) × … × 2 × 1.

Design Implications¶

Context¶

Requirements¶

• Physical multi-colored connecting cubes; more than enough cubes to construct all 3- and 4-cube towers

Constraints¶

• Not enough cubes were available to construct all 120 possible 5-cube towers, which supported the shift to multiplicative reasoning

Target Learners¶

• undergraduate preservice middle school teachers with no prior combinatorics study

Target Learning Goals¶

• enumerate permutations
• construct generalized counting formulas

Claims¶

• A cube-tower instructional sequence instantiated concreteness fading in an elaborated form for two preservice teachers' combinatorial reasoning [+W]
• Students transferred cube-based permutation reasoning to permutation tasks not involving cubes [+W]

Related Elements¶

Examples¶

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

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