The product of two Heisenberg-Weil algebras contains the Jordan-Schwinger representation of su(2). This algebra is quotiented by the square-root of the Casimir to produce a nonassociative algebra denoted by Psi. This algebra may be viewed as the right-module over one of its associative subalgebras which corresponds to the algebra of scalar fields on the noncommutative sphere. It is now possible to interpret other subspaces as the space of spinor or vector fields on the noncommutative sphere. A natural basis of Psi is given which may be interpreted as the deformed entries in the rotation matrices of SU(2).