We present a reduction procedure for gauge theories based on quotienting out the kernel of the presymplectic form in configuration-velocity space. Local expressions for a basis of this kernel are obtained using phase space procedures; the obstructions to the formulation of the dynamics in the reduced phase space are identified and circumvented. We show that this reduction procedure is equivalent to the standard Dirac method as long as the Dirac conjecture holds: that the Dirac Hamiltonian, containing the primary first class constraints, with their Lagrange multipliers, can be enlarged to an extended Dirac Hamiltonian which includes all first class constraints without any change of the dynamics. The quotienting procedure is always equivalent to the extended Dirac theory, even when it differs from the standard Dirac theory. The differences occur when there are ineffective constraints, and in these situations we conclude that the standard Dirac method is preferable --- at least for classical theories. An example is given to illustrate these features, as well as the possibility of having phase space formulations with an odd number of physical degrees of freedom.
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