We perform a 'fossil record' analysis for ≍800 low-redshift spiral galaxies, using STARLIGHT applied to integral field spectroscopic observations from the SDSS-IV MaNGA survey to obtain fully spatially resolved high-resolution star formation histories (SFHs). From the SFHs, we are able to build maps indicating the present-day distribution of stellar populations of different ages in each galaxy. We find small negative mean age gradients in most spiral galaxies, especially at high stellar mass, which reflects the formation times of stellar populations at different galactocentric radii. We show that the youngest (<108.5 yr) populations exhibit significantly more extended distributions than the oldest (>109.5 yr), again with a strong dependence on stellar mass. By interpreting the radial profiles of 'time slices' as indicative of the size of the galaxy at the time those populations had formed, we are able to trace the simultaneous growth in mass and size of the spiral galaxies over the last 10 Gyr. Despite finding that the evolution of the measured light-weighted radius is consistent with inside-out growth in the majority of spiral galaxies, the evolution of an equivalent mass-weighted radius has changed little over the same time period. Since radial migration effects are likely to be small, we conclude that the growth of discs in spiral galaxies has occurred predominantly through an inside-out mode (with the effect greatest in high-mass galaxies), but this has not had anywhere near as much impact on the distribution of mass within spiral galaxies.