A cube-tower instructional sequence instantiated concreteness fading in an elaborated form for two preservice teachers' combinatorial reasoning¶

Claim · All claims
Evidence · 1 study · theoretical r3 · q2
Currency · newest study 2022 · not yet checked against its sources · not reviewed by a person
Evidence quality · No study behind this claim is a peer-reviewed article with a DOI: it rests on an ERIC document (a report, conference paper or thesis). It is kept because 4 design pages cite it; treat it as a lead to check, not an established finding.

Subclaims¶

q2 i? Two preservice teachers progressed from enactive one-by-one tower construction, to iconic mental imaging of towers, to symbolic multiplicative operations conceptually grounded in the cube-tower context, consistent with the enactive-iconic-symbolic stages of CF. → Joseph Antonides 2022

Evidence¶

Joseph Antonides 2022¶

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

q2 · i? · theoretical · r3

One-on-one constructivist teaching experiments with two preservice middle school teachers (DC and NK) enumerating permutations as cube towers. The authors report that students' reasoning "shifted from operating on towers to operating primarily on symbolic representations", matching CF's enactive-iconic-symbolic stages.

"Gradually, the students' reasoning shifted from operating on towers to operating primarily on symbolic representations, which they could later use to reason in novel situations."

Discussion¶

Related Claims¶

• Students transferred cube-based permutation reasoning to permutation tasks not involving cubes — related
• A systematic review argues that concrete materials aid mathematics and science learning when they are explicitly and gradually connected to the abstract symbols they represent — a broader claim this one bears on
• Combinatorial enumeration yields 216 scenarios, filtered to 24 priority scenarios across three developmental tiers — related
• Concreteness fading enables learners to build a store of memorable images usable when abstract symbols lose meaning — related
• Manipulatives Improve Math Learning — a broader claim this one bears on
• Collaboration aids learning of mathematical elaboration skills but not procedural skills — related
• Concreteness fading may not be as broadly generalizable across mathematics and natural science domains as has been suggested — related