The two principal approaches to the quantum-mechanical calculation of scattering by flux tubes in two spatial dimensions are reviewed and corrections made where needed. It is noted that even when appropriate care is taken the standard phase-shift method fails to agree with the (corrected) Aharonov-Bohm result. The discrepancy is resolved by noting that the former approach requires an improper interchange of a limit and a summation. That interchange is justified only when the phase shifts go to zero sufficiently rapidly as the angular momentum becomes arbitrarily large, a condition which fails to hold in this particular scattering problem.