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Spectral variability of radio sources encodes information about the conditions of intervening media, source structure, and emission processes. With new low-frequency radio interferometers observing over wide fractional bandwidths, studies of spectral variability for a large population of extragalactic radio sources are now possible. Using two epochs of observations from the GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) survey that were taken one year apart, we search for spectral variability across 100--230 MHz for 21,558 sources. We present methodologies for detecting variability in the spectrum between epochs and for classifying the type of variability: either as a change in spectral shape or as a uniform change in flux density across the bandwidth. We identify 323 sources with significant spectral variability over a year-long timescale. Of the 323 variable sources, we classify 51 of these as showing a significant change in spectral shape. Variability is more prevalent in peaked-spectrum sources, analogous to gigahertz-peaked spectrum and compact steep-spectrum sources, compared to typical radio galaxies. We discuss the viability of several potential explanations of the observed spectral variability, such as interstellar scintillation and jet evolution. Our results suggest that the radio sky in the megahertz regime is more dynamic than previously suggested.