Catalyzes the reversible interconversion of serine and glycine with tetrahydrofolate (THF) serving as the one-carbon carrier. Appears to be specific for THF as the pteridine substrate, since the use of tetrahydromethanopterin (H4MPT) is much less efficient. Also exhibits THF-independent aldolase activity toward beta-hydroxyamino acids, producing glycine and aldehydes, via a retro-aldol mechanism. Thus, is able to catalyze the cleavage of L-allo-threonine and L-threo-beta-phenylserine. (6R)-5,10-methylene-5,6,7,8-tetrahydrofolate + glycine + H2O = (6S)-5,6,7,8-tetrahydrofolate + L-serine The apparent KM for tetrahydromethanopterin (H4MPT) is superior to 0.3 mM (at pH 7.2 and 37 degrees Celsius). One-carbon metabolism; tetrahydrofolate interconversion. Amino-acid biosynthesis; glycine biosynthesis; glycine from L-serine: step 1/1. Homodimer. Belongs to the SHMT family.