We introduce generalized functions and generalized derivatives.  By a finite dimensional approximation, the generalized derivatives are the so-called weak gradients.  Consequently we introduce the weak Galerkin finite element methods.  We discuss the equivalence and differences between WG and other finite element methods.
As an example showing the flexibility and the advantage of WG,  we construct a special P2 WG finite element which is order 3 convergent in L2 for solving biharmonic equations, while it is not possible for all other P2 finite method methods.  Jun Hu proved an order 2 lower bound in L2 for conforming and nonconforming P2 finite elements when solving biharmonic equations.