Critical Analysis of Infinity: Historical Development and Student Response Schemes in Mathematical Concept Construction
Overview
This paper examines the historical and epistemological development of the concept of infinity through critical analysis, comparing how mathematicians throughout history have grappled with infinity as a mathematical object with how contemporary students construct understanding of formal definitions. The study highlights the gap between historical conceptual evolution and current instructional approaches in mathematics education.
What this research paper covers
- Paragraph 1-2: Identifies critical analysis approach and historical stages in infinity's development across multiple scientists
- Paragraph 3-5: Analyzes parallels between student response schemes and historical mathematician approaches; examines individual and historical perspectives
- Paragraph 6-8: Traces emergence of infinity from Aristotelian conception through Bolzano's introduction of actual infinity, including grammatical and linguistic analysis
Preview
Paragraph 1
The moves in paragraph one is to identify how critical analysis of concepts can throw light on knowledge construction. One of the moves is show the existence of well-differentiated stages in the historical development of infinity. Another move is the purpose of the study, and what it sets out to do.
Paragraph 2
In paragraph two, we can see how history presents us with general view of the evolution concept, where evolutions are made by many scientists. The paragraph also shows how different aspects of the epistemic subject have been explored by different authors, leading to a kind of homology.
Paragraph 3
The move is showing situations involving infinity, the student response schemes are similar to response schemes offered by mathematicians throughout history. The move is revealed by showing examples of mathematicians who were faced with similar questions.
Paragraph 4
The move shows how current school instruction does not assist the evolution of student response schemes towards construction of formal Cantorian definitions.
Paragraph 5
The move focuses on the evolution of the concept from the individual as well as historical perspective. The question focuses on carrying out and taking into account the diverse forms of existence of the concept and the associated domains of operation.
Paragraph 6
The moves in paragraph six center on the historical development of actual infinity. They are highlighted through the work of Bolzano, who is attributed to have introduced actual infinity.
Paragraph 7
The move focuses on the emergence of infinity in Aristotle’s work, but not yet as a mathematical object.