“Ways of looking" at quotient spaces in linear algebra. How to go beyond the modern definition? (English)

Tso, Tai-Yih (ed.), Proceedings of the 36th conference of the International Group for the Psychology of Mathematics Education “Opportunities to learn in mathematics education", PME 36, Taipei, Taiwan, July 18‒22, 2012, Vol. 4. Taipei: National Taiwan Normal University. 115-122 (2012).

Summary: What makes the concept of quotient vector space so hard to teach and learn? What kind of different representations and conceptions do teachers need to help students to make sense of this concept? We reflect on these questions from an epistemic point of view offering a panoramic of the advance and interactions of different conceptions of quotients, from school to last courses of a degree of mathematics, and the relations of these conceptions with the kind of representations used and with other notions needed for the development of the concept of quotient such as partition, equivalence relation or well-defined operation. We make this reflection thinking on linear algebra teachers interested in improving their students’ “cognitive flexibility" but also as a framework for further research.