At international conferences where planetary scientists discuss these proposals, there is general agreement that this would be a bad way to approach such a problem.
Forgetting for a moment the logistics of accomplishing it, what it creates is a great many more incoming chunks of asteroid, each of which would be at least slightly radioactive.
When this proposal was first put forward, studies showed that even if a significant amount of the original asteroid's mass missed Earth, the atmospheric heating effects of hundreds or thousands of relatively large (more than a few tons each) meteoroids entering the atmosphere together would be as bad as a single larger chunk.
While you'd think that having the smaller chunks more completely burn up before reaching the ground is a good thing, you must consider where all that heat ends up: in the atmosphere! That could be a very bad day, indeed.
There is no clear consensus as to how to approach these threats, other than that early identification is the primary goal. But there is general agreement that given enough time the best approach is some means of steering or nudging an incoming asteroid off course enough to ensure a miss.
You might find this interview with planetary scientist Jay Melosh interesting reading.