As a group, the artists educated near the turn of the 19th and 20th centuries possessed greater mathematical knowledge than expected of artists today, especially regarding constructive skills in Euclidean geometry. Educational theory of the time stressed such skills for students in general, who needed these to enter the workplace of the time. Mathematics teaching then stressed the use of manipulatives, i.e., visual and interactive aids thought to better fix the student’s acquisition of mathematical skills. This visual training, especially in geometry, significantly affected the early development of abstraction in art. This paper presents examples of this visual mathematics education and samples its effects on the development of abstract art in the first decades of the 20th century.

中文翻译：

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几何视觉教学与20世纪抽象艺术的起源

作为一个整体，受教育的艺术家在19世纪和20世纪之交时拥有比当今艺术家所期望的更多的数学知识，尤其是在欧几里得几何学中的构造技巧方面。当时的教育理论为一般学生强调了这类技能，他们需要这些技能才能进入当时的工作场所。然后，数学教学强调使用手法，即视觉和互动辅助手段，以更好地固定学生对数学技能的掌握。这种视觉训练，尤其是几何学的视觉训练，极大地影响了艺术抽象的早期发展。本文介绍了这种视觉数学教育的实例，并对其在20世纪前几十年对抽象艺术发展的影响进行了采样。