Catalyzes the reversible interconversion of serine and glycine with tetrahydrofolate (THF) serving as the one-carbon carrier. This reaction serves as the major source of one-carbon groups required for the biosynthesis of purines, thymidylate, methionine, and other important biomolecules. 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. (6R)-5,10-methylene-5,6,7,8-tetrahydrofolate + glycine + H2O = (6S)-5,6,7,8-tetrahydrofolate + L-serine Binds 1 pyridoxal phosphate per subunit. Optimum pH is about 7.8. Is completely inactive at pH below 4.0 and above 10.0. One-carbon metabolism; tetrahydrofolate interconversion. Amino-acid biosynthesis; glycine biosynthesis; glycine from L-serine: step 1/1. Homodimer. Belongs to the SHMT family.