Tai-chi Dantian roll (丹田内转) is one of those intermediate/advanced tai-chi techniques that boggled the mind of many practitioners. It has been taught in different ways by different masters, some teach moving one's waist in circular motions, some teach rubbing one's abdomen in circular motions, some teach using ones hands holding like a ball and move them in circular motions, some teach holding a one-foot tai-chi stick and move in various circular motions... It is difficult to tell who is doing it correctly from the outside. Like all internal disciplines, everything happens inside. The best way to approach it is to analyze and understand the logic behind, and leave the final judgment to each individual practitioner.
What is the logic of Dantian roll? Simply put, a practitioner is to visualize a sphere inside his Dantian, and further more he should be able to guide his chi to move around the sphere in circle in all directions. It is easier said than done, assuming we're talking about doing it in more or less perfect movement in circles and with power. And in its more advanced form, the size as well as the position of the sphere are variables in the training process too! Before we go further, you might ask why practice Dantian roll? A good foundation in tai-chi roll can greatly promote structural power and balance as well as greatly promote the speed and amount of chi-generation in form practice. Besides, it is also an intermediate level for Taoist meditation (Neidan), to be practiced only after a successful milestone in doing microcosmic circulation (小周天).
So, how do we go about doing the Dantian roll? As a prerequisite, a good foundation in chi generation has to be built first. It can be achieved through standing meditation (zhan zhuang), or, better still, accomplishing the microcosmic circulation (both I discussed in various posts in some depth, interested readers can search this blog for the posts). The classic way of doing Dantian roll in Neidan is without any movement, or more correctly put without any ostensible movement. In standing or seated meditation, with stretched hands (or in one's preferred mudra), and with one's mind focused on one's three eyes (i.e. third eye [point between one's eyebrows] included). The way to go around the sphere is to be guided by the movement of one's two physical eyes (which in turns being guided by one's third eye). Three planes of movements have to be done, and in both directions. The three planes are the x, y and z axis in math.
If a practitioner can understand the inner logic of Dantian roll as explained above, he will appreciate that any outward movement of hands or waist are just incidental. The visible movement serves good training purpose, but only those who can understand the inner logic can truly benefit from the practice. Did your master explain to you this inner logic? Perhaps not...