[sjk]

I'm showing electrolysis of water for my granddaughter using carbon electrodes (from a dead battery) and MgSO4 to add conductivity.

Very satisfying on the Hydrogen side but there's nothing like enough O2 (I guess proportions are about 1:10 rather than 1:2).  The anode is disintegrating too but it looks like mechanical breakdown not chemical.

Any ideas?

[Arkcon]

Should have worked as advertised.  Unless the components of the carbon rod are reacting with the oxygen.  So, proportions are off, but you're able to collect oxygen, and it has oxygen's properties?  That is colorless, odorless (careful with this assay) gas that reignites a glowing splint.  Try a mechanical pencil lead for an anode, its made of stronger stuff.

[Zyklonb]

Carbon anodes can be oxidized from the oxygen produced. Especially in the presents of chloride, but also with sulfate. The carbon dioxide bubbles off just like the oxygen so it looks like only mechanical degrading. 
I have tested this myself by bubbleing the gasess through a calcium hydroxide solution, and calcium carbonate percipitated within 20 minutes of running. The quantity was unknown, as the calcium carbonate could not be filtered and pH could not be tested as calcium hydroxide may still have been present.
 Like Arkcon said, pencil graphite degrades slower, both chemically and mechanically, to my knowledge.