In mathematics, transformation geometry is a name for a pedagogic theory for teaching Euclidean geometry, based on the Erlangen programme. Felix Klein, who pioneered this point of view, was himself interested in mathematical education. It took many years, though, for his "modern" point of view to have much effect, with the synthetic geometry remaining dominant.
In the end, the reform of geometry teaching came simultaneously with the New Math movement. To do real transformation geometry requires some work with symmetry groups; if not group theory, matrix computations are required. The tide of transformation geometry retreated, leaving behind some vector methods.