A new finite element, which is continuously differentiable,
but only piecewise quadratic
polynomials on a type of uniform triangulations, is introduced.
We construct a local basis which
does not involve nodal values nor derivatives.
Different from the traditional finite elements, we have to
construct a special, averaging operator
which is stable and preserves quadratic polynomials.
We show the optimal order of approximation
of the finite element in interpolation, and in solving
the biharmonic equation.
Numerical results are provided confirming the analysis.