  J1044+0353 is considered a local analog of the young galaxies that ionized
the intergalactic medium at high-redshift due to its low mass, low metallicity,
high specific star formation rate, and strong high-ionization emission lines.
We use integral field spectroscopy to trace the propagation of the starburst
across this small galaxy using Balmer emission- and absorption-line equivalent
widths and find a post-starburst population (~ 15 - 20 Myr) roughly one kpc
east of the much younger, compact starburst (~ 3 - 4 Myr). Using the direct
electron temperature method to map the O/H abundance ratio, we find similar
metallicity (1 to 3 sigma) between the starburst and post-starburst regions but
with a significant dispersion of about 0.3 dex within the latter. We also map
the Doppler shift and width of the strong emission lines. Over scales several
times the size of the galaxy, we discover a velocity gradient parallel to the
galaxy's minor axis. The steepest gradients (~ 30 $\mathrm{km \ s^{-1} \
kpc^{-1}}$) appear to emanate from the oldest stellar association. We identify
the velocity gradient as an outflow viewed edge-on based on the increased line
width and skew in a biconical region. We discuss how this outflow and the gas
inflow necessary to trigger the starburst affect the chemical evolution of
J1044+0353. We conclude that the stellar associations driving the galactic
outflow are spatially offset from the youngest association, and a chemical
evolution model with a metal-enriched wind requires a more realistic inflow
rate than a homogeneous chemical evolution model.
