How do we understand infinity and other abstract mathematical concepts?

Understanding Infinity and Abstract Mathematical Concepts

The concept of infinity and other abstract mathematical ideas are complex and often challenging to grasp. Research in this area explores various methods and perspectives to enhance understanding, particularly in educational settings.

Key Insights

Conceptual Metaphors and Tacit Models:
- Conceptual metaphors help ground the understanding of mathematical infinity in human experience, making abstract concepts more accessible .
- Tacit models, which are often unconscious, can lead to misunderstandings and paradoxes in interpreting infinity .
Educational Approaches:
- Pre-service teachers use different methods to conceptualize infinity, including mathematical examples, empirical measurements, and belief statements. These methods help bridge the gap between abstract and real-world applications of infinity.
- Integrating technology in teaching can foster new ways of thinking about infinity, engaging students in reflective and critical thinking.
Epistemological Perspectives:
- Both in mathematics and in Jewish tradition, infinity is treated as an inexhaustible source of new ideas and structures, rather than a concrete entity. This perspective emphasizes the ongoing nature of learning and discovery.

Conclusion

Understanding infinity and other abstract mathematical concepts involves a combination of conceptual metaphors, educational strategies, and epistemological perspectives. Conceptual metaphors and tacit models play a crucial role in making these concepts accessible, while educational approaches that integrate empirical and mathematical examples can enhance comprehension. Additionally, viewing infinity as an inexhaustible source of knowledge aligns with both mathematical and cultural traditions, highlighting the continuous nature of exploration and learning.

How do we understand infinity and other abstract mathematical concepts?

Summary

Understanding Infinity and Abstract Mathematical Concepts

Key Insights

Conclusion

Sources and full results

Conceptual Metaphors and Tacit Models in the Study of Mathematical Infinity

Conceptual metaphors help understand mathematical infinity by connecting conscious and unconscious models, improving teaching practice and fostering new ways of thinking about it.

Conceptualisations of infinity by primary pre-service teachers

Pre-service teachers conceptualize infinity as both an abstract and real concept using mathematical and empirical arguments, highlighting the need for connections between mathematical and empirical applications in teaching.

Ways of Infinity

Infinity is understood in mathematics and Judaism as inexhaustible sources of new thoughts, structures, ideas, and developments, rather than being theological or relating to an infinite being.

Tacit Models and Infinity

Tacit models, unconsciously influenced by mental models, can lead to erroneous interpretations and paradoxes in mathematical concepts related to infinity.

Walking Through Cantor's Paradise and Escher's Garden: Epistemological Reflections on the Mathematical Infinite (II)

Infinity is a complex concept with multiple meanings and value judgments, and this paper explores various aspects of its definition, including teleological and artistic aspects.

Panentheism in the Light of Mathematical Understandings of Infinity and Connectedness

A panentheistic conception of God can be supported by a mathematical understanding of infinity and can be seen as unifying regardless of the world's number of entities.

Mathematical Platonism and the Nature of Infinity

Mathematical Platonism supports the reality of abstractness and infinity, providing a plausible reason for the existence of something rather than nothing in the concrete universe.

A Simple Talk of the Thought about Infinity in Mathematics

Infinity is a key concept in mathematics and philosophy, with three crises in history caused by contradictions in understanding it, and the development of space concept has experiencedcd from finity to infinity.

Analysis of factors influencing the understanding of the concept of infinity

Primary teacher students' understanding of infinity varies depending on the task and context, with potential infinity being their primary method for dealing with infinity when unsure of themselves.

Student's Concept of Infinity in the Context of a Simple Geometrical Construct.

School mathematics does not generally develop students' ideas of infinity, highlighting the need for teachers to have a sound knowledge and communicative skills in discussing this concept.

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