Actually, the fat underground parts of a dahlia are not tubers (storage material with growing eyes on them) but swollen roots. The actual eyes that will develop the aerial part appear along the base of the stem. To divide correctly the clump of roots you will have to wait until buds are visible along the stem, then with something really sharp make the cuts so each swollen root will have a new bud on a piece of the stem. In other words, a swollen root, and the piece of stem with a developing bud, all in one piece. Of course this is very simple if one has lifted the clump some time in advance and keep it in a box with dry sawdust, peat, etc. This process of dividing clumps does not work forever as old plants tend to become very bushy with less and less flowers. This is easily solved as 3-4 in. long buds that appear in the spring can be easily put to root and this will produce new plants with super vigor that will be covered with flowers (until of course, after several years they become bushy again).