  We report a measurement of the flux-averaged neutral-current elastic
differential cross section for neutrinos scattering on mineral oil (CH$_2$) as
a function of four-momentum transferred squared. It is obtained by measuring
the kinematics of recoiling nucleons with kinetic energy greater than 50~MeV
which are readily detected in MiniBooNE. This differential cross-section
distribution is fit with fixed nucleon form factors apart from an axial mass,
$M_{A}$, that provides a best fit for $M_A= 1.39\pm0.11$~GeV. Additionally,
single protons with kinetic energies above 350 MeV can be distinguished from
neutrons and multiple nucleon events. Using this marker, the strange quark
contribution to the neutral-current axial vector form factor at $Q^2 = 0$,
$\Delta s$, is found to be $\Delta s=0.08\pm0.26$.
