Elaborated theory of levels of abstraction with a parallel set of levels for computational operation schemes¶

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

Description¶

The article elaborates Battista's (2007) theory of levels of abstraction—perceptual/recognition, internalized, and interiorized levels—by adding an empirically supported parallel set of levels for computational operation schemes (COS), from perceptual abstraction through internalization, interiorization, second interiorization (operating on the COS as a conceptual symbol), third interiorization (algebraic generalization), and fourth interiorization (symbols as pointers to generalized computations). The authors state the new set "served to explain students' abstractions of computational operation schemes", occurring in parallel with the original set. This elaborated theory is used to explicate the fading mechanism central to CF.

Design Implications¶

Context¶

Requirements¶

• Analysis requires framing students' conceptual progressions in a specific mathematical domain, here permutation enumeration with spatial material

Constraints¶

• The additional levels were developed from data analysis of two students' combinatorial reasoning in one-on-one teaching experiments

Target Learners¶

• undergraduate preservice middle school teachers

Target Learning Objectives¶

• combinatorial reasoning and permutation enumeration
• transition from concrete manipulatives to symbolic reasoning

Claims¶

• Cube Tower Sequence Elaborated Cf Instantiation [+M]
• Students transferred cube-based permutation reasoning to permutation tasks not involving cubes [+W]

Related Theories¶

• S*NLS as the linking mechanism between spatial structuring and numerical structuring

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