You are not wasting your time but I must also add that there is no guarantee that any cutting will root so I always recommend taking as many cuttings as you can. This will increase the odds of a plant forming roots.  As for the time line, bougainvillea can take from 3 to 6 months to grow roots when the cutting is placed into moist soil and even longer when placed in water. The white circles are a good sign as it means a callus for root growth is forming. To really speed up rooting you should dip each cutting into a rooting hormone powder then stick into a pot of moist potting soil. Cover with a plastic bag to increase humidity. One more tip. Dirt cheap gardeners avoid purchasing the rooting hormone by using willow water instead. This means you cut a handful of stems from a true willow tree (Salix family) and place these cut stems into a glass jar of water for at least two days. Place your cuttings into the willow water before you root them and water your potted cuttings with the willow water as well. The rooting hormone in the willow stems will leach into the water and speed up any cutting to take root. You can add willow water to your cuttings that are now in plain water. Good luck and I see some new plants in your future. Keep growing, Marianne Binetti