Abstract
In the acquisition and formalization of mathematical concepts, the transition between algebraic and geometric representations and the use of different modes of representation contextualizes abstract algebra. Regrettably, the role of geometry is often limited to the visualization of algebraic facts and figurative memory aids. Such visualizations are blind for the underlying concepts, since transitions between concepts in different representations assume the existence of symbols, language, rules and operations in both systems. The history of mathematics offers contexts to develop geometrical language and intuition in areas currently being taught in school in a purely algebraic fashion. The example of the determination of zeros of polynomials shows how reflecting on posing a problem in ancient Greek mathematics, engineering mathematics (19th century) and paper folding (beginning of the 20th century) can help to develop geometrical concepts, language and intuition stemming from an algebraic context.
The title is an allusion to Immanuel Kant’s famous quote: “Gedanken ohne Inhalt sind leer, Anschauungen ohne Begriffe sind blind”—Thoughts, without the content are empty, perception without a concept is blind.
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Acknowledgements
The authors thank Carl-Peter Fitting for alerting them to the method of Captain Lill that turned out to be such a fruitful topic when using history to foster concept development. They are grateful for the elaborate and constructive comments of the referees that helped to improve this chapter considerably.
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Kaenders, R., Weiss, Y. (2018). Algebra Without Context Is Empty, Visualizations Without Concepts Are Blind. In: Clark, K., Kjeldsen, T., Schorcht, S., Tzanakis, C. (eds) Mathematics, Education and History . ICME-13 Monographs. Springer, Cham. https://doi.org/10.1007/978-3-319-73924-3_7
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