We investigate the composition dependence of glass-forming ability (GFA) of a series of iron-containing soda lime borate liquids by substituting Na2O for B2O3. We have characterized GFA by measuring the glass stability against crystallization using a differential scanning calorimeter (DSC). The results show that the GFA decreases when substituting Na2O for B2O3. Moreover, we find that there is no direct link between the kinetic fragility and GFA for the soda lime borate series studied herein. We have also discovered and clarified a striking thermal history dependence of the glass stability against crystallization. In particular, the two glasses containing 20 and 25 Na2O mol% do not exhibit crystallization exotherms during the second DSC upscan at 10 and 20 K/min following prior slow (10 and 20 K/min) downscans. This indicates that the glass stability of these compositions can be enhanced by slowly cooling their melts to the glassy state, before any reheating. We explain this phenomenon in terms of the thermal history dependence of boron speciation.