从教育数学的角度探讨行列式教学

On Teaching of Determinant in View of Educational Mathematics

本文根据教育数学思想,讨论大学《线性代数》公共课中行列式部分的教学,通过设计几个教学场景,帮助学生以更直观的方法掌握行列式本质.所设计的场景包括:从行列式定义的意图出发合情推理行列式的可能表达式;从低阶行列式性质类比证明行列式定义的必然形式;通过矩阵初等变换与等底单形的体积之间的关系建立行列式与单形体积的关联;通过仿射变换保持单形体积比的性质导出Cramer法则的直观证明;以及分析行列式的不同计算方法所对应的计算复杂度.最后,文章列出行列式知识产生和发展的部分数学史材料,供教师在教学中穿插使用,达到更好引导学生理解和应用行列式知识.

This paper discusses the teaching of the determinant in the common course of Linear Algebra, and designs several teaching scenarios to help students grasp the essence of the determinant in a more intuitive way. The necessary form of the determinant is compared with the properties of lower order determinants. The relation between the determinant and the volume of a simplex is established by the relation between the matrix of an elementary transformation and the volume of the simplex. An intuitive proof of Cramer’s rule is derived by the fact that an affine transformation preserves the ratio of volumes of simplexes. Finally, the paper lists some historical mathematical materials which can be used in teaching and learning.