The dimension formula of quotient spaces is a core proposition in advanced algebra that describes the structure of vector spaces and the correlation between subspaces. Its high abstraction and rigorous logic often lead students to a learning misunderstanding of “mechanically memorizing formulas and blindly applying question types”, making it difficult for them to grasp the generative logic and application value of the formula. The concept of “knowledge rediscovery” emphasizes guiding students to revisit the trajectory of knowledge construction as “quasi-researchers” through a progressive process of “problem-driven guidance – concrete instantiation – conjecture and deduction – rigorous proof – multi-dimensional application”, which precisely adapts to the teaching characteristics of the dimension formula of quotient spaces. Based on this concept, this paper reconstructs the teaching framework of the dimension formula of quotient spaces: starting from the geometric intuition of Euclidean space, it guides students to independently analyze the essence of equivalence classes of quotient spaces, completes the strict proof of the formula through basis extension and conjecture of dimensional relationships, and then designs application tasks combining multiple scenarios such as the derivation of the dimension formula for the sum of subspaces and the verification of the rank-nullity theorem of linear transformations. The research aims to help students break through the learning barriers of abstract knowledge, deepen their essential understanding of the structure of vector spaces, and cultivate core competencies such as algebraic abstraction, logical reasoning, and knowledge transfer. Meanwhile, it provides a new practical path and theoretical reference for the teaching innovation of abstract formula modules in advanced algebra.
Wang Qingwen (Thu,) studied this question.