Students understanding of mathematics using from prototypical examples: Analyze in linear algebra

Mathematical objects are ʼnatural objects’ for the practical mind. Definitions and theories can only describe them, not create or construct them. A mathematical term is interpreted through its denotation as representing a collection of particular objects for the theoretical mind. So, theoretical and practical modes of thinking differ in the manner in which they constitute the meanings of words. Thinking in terms of prototypical examples, rather than definitions, became an obstacle to our students’ understanding the notion of linear transformation. In the course, linear transformations were defined as transformations of vector spaces which conserve linear combinations. The obstacle was revealed in the students’ attempts ‘linear extension problem’: given a transformation of a basis, to construct a linear transformation with those values on the basis. In the experimental course, the problem was not formulated in such general terms. It was restricted to two dimensions, and the students were not asked to ‘linearly extend a transformation from a basis to the whole plane’ but to assume the existence of such an extension and find some missing information about it. In this study we are discussing students understanding of mathematical concepts by means of linear transformation in linear algebra.