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 $M_A= 1.39pm0.11$~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 $Delta s=0.08pm0.26$.
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