Abstraction in mathematics

Area of Science:

• Mathematics
• Philosophy of Mathematics
• Mathematics Education

Background:

• Current interpretations of mathematical abstraction are often oversimplified.
• Existing views may reduce abstraction solely to generalization or decontextualization.
• These limited perspectives impact the understanding of mathematical object emergence and representation.

Purpose of the Study:

• To examine current interpretations of abstraction in mathematical settings.
• To explore the multifaceted nature of abstraction beyond generalization and decontextualization.
• To analyze the role of representations and linguistic aspects in mathematical abstraction.

Main Methods:

• Analysis of historical perspectives on mathematical abstraction.
• Examination of learning theories and cognitive processes in abstraction.
• Application of functional linguistics to understand mathematical notation.

Main Results:

• Abstraction is a complex cognitive and historical process, not reducible to single mechanisms.
• Abstraction is intrinsically linked to the creation and emergence of new mathematical objects.
• Representations play a crucial role, influenced by historical development and educational contexts.

Conclusions:

• One-dimensional interpretations of abstraction lead to ineffective mathematics teaching strategies.
• A comprehensive understanding of abstraction, including its link to new objects and representations, is vital.
• Future educational approaches must consider the complexity of abstraction for successful learning.