Catalyzes the reversible interconversion of serine and glycine with tetrahydromethanopterin (H4MPT) serving as the one-carbon carrier. The use of tetrahydrofolate (THF or H4PteGlu) as the pteridine substrate is 450-fold less efficient than that of H4MPT. Also exhibits a pteridine-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. 5,10-methylenetetrahydromethanopterin + glycine + H2O = 5,6,7,8-tetrahydromethanopterin + L-serine L-allo-threonine = acetaldehyde + glycine kcat is 212 min(-1) for the L-serine hydroxymethyltransferase reaction, and 687 min(-1) for the L-allo-threonine cleavage, at 37 and 60 degrees Celsius, respectively. The presence of oxygen does not seem to affect significantly the kinetic parameters of the reactions. Optimum temperature is 70-80 degrees Celsius for the retro-aldol cleavage of L-allo-threonine. Amino-acid biosynthesis; glycine biosynthesis; glycine from L-serine: step 1/1. Homodimer. Belongs to the SHMT family.