Interfaces between mathematical practices and mathematical education Introduction 1.Foundations of mathematics buried in school garbage (Southern Mesopotamia early second millennium BCE) 2. From the practice of explanation to the ideology of demonstration: an informal essay 3.Catholicism and Mathematics in the Early Modernity 4.“Je n’ai point ambitionnée d’être neuf”: Modern Geometry in Early Nineteenth Century French Textbooks 5.The impact of teaching mathematics upon the development of mathematical practices 6.A Multiple Perspective Approach to History of Mathematics: Mathematical programming and Rashevsky’s early development of mathematical biology in the 20th century 7.Teaching as an indicator of mathematical practices This contributed volume investigates the active role of the different contexts of mathematics teaching on the evolution of the practices of mathematical concepts with particular focus on their foundations. The book aims to deconstruct the strong and generally wide held conviction that research in mathematics constitutes the only driving force for any progress in the development of mathematics as a field. In compelling and convincing contrast these chapters aim to show the productive function of teaching showcasing investigations from countries and regions throughout various eras from Old Babylonia through the 20th Century. In so doing they provide a critical reflection on the foundations of mathematics as well as instigate new research questions and explore the interfaces between teaching and research