Elaborating reflective abstraction for instructional design in mathematics: Postulating a Second Type of Reflective Abstraction

ABSTRACT

The goal of our research program is to explicate the learning of mathematical concepts in ways that are useful for instructional design and to develop design principles based on those explications. I review one type of concept and our elaboration of reflective abstraction, coordination of actions (COA) that accounts for its construction. I then postulate a second type of concept and a second type of reflective abstraction that accounts for its construction, the abstraction of commonality (AOC).

KEYWORDS:

• Reflective abstraction
• instructional design
• mathematical concepts

Notes

1. I use “reflective abstraction” as a broad term that includes pseudo-empirical, reflecting, and reflected abstraction.

2. Simon (2006) introduced an expanded category, empirical learning processes, to include both empirical abstraction and empirical learning not related to properties of physical objects.

3. From here on, “CoA” refers to the CoA type of reflective abstraction. If it refers to the type of concept, “CoA concept” will be used.

4. Reflective abstraction does not always lead to the ability to justify.

5. The names of these two students are changed from the original text.

6. In partitive division (also called “sharing division”), one is trying to find the number in each group given the number of groups and the total quantity. In quotitive division (also called “measurement division”), one is trying to find the number of groups given the number in each group and the total quantity. The distinction is important, because they can be abstractions from different actions, dealing out or partitioning in the former and making groups of a given size or measuring a quantity of a certain size in the latter.

7. The MAKE button is an addition we made to Fraction Bars.

8. I used the same numbers as in the task above, so Figure 1 could be reused by the reader.

Additional information

Notes on contributors

Martin A. Simon

Martin A. Simon is Professor of Mathematics Education at New York University. His research focuses on explicating mathematics conceptual learning and how that learning process can be fostered. His recently completed NSF-funded project combined basic research in this area with research on a measurement-based approach to developing fraction and ratio concepts. A special section of the Journal of Mathematical Behavior (2018) is devoted to this work. Simon’s earlier research focused on the development of mathematics teachers as they learn to teach mathematics with a conceptual focus.