Tiphanie & Richard,
I am a gifted program coordinator/teacher and also a past teacher of behavior management for kids, along with a series of parenting education courses. I agree 100% with your doctor on this one. It seems to definitely be an attention-getting mechanism that your daughter has developed. But I can also tell you (from personal experience) that it may take more than a month to break her of this.
My own daughter was much as you describe yours. She is extremely gifted (aces the AIMS test every year, etc.) and has always had trouble socially. She also started throwing tantrums around 5, and they occurred more often when she was having additional stress from social pressure. Now she still (age 13) occasionally blows up at us when she doesn't like what is going on.
Recent studies of "normal" kids have shown that the back part of their brain develops faster than the front part. The front part is where the reasoning center is. Their brain development doesn't catch up until they are 25. When you factor in the fact that your daughter is "gifted" you get a child whose knowledge is growing much faster than her ability to reason and analyze that knowledge. This is why they throw tantrums, and they are far more common for gifted children. Also, the social pressures for gifted girls are very different than for boys. A good read is "Gifted Girls, Gifted Women." You will be surprised how much of your daughter you recognize when you read it.
My daughter had problems making friends until we finally found a gifted program that worked well for her, and she found others in the same boat who thought on a higher level than their peers. It was not until she was able to make some real friends that the tantrums/anger seemed to slow.
I would follow your doctor's advice, and completely ignore the tantrums if possible. It will be extremely difficult since she is NOISY and you are worried that she will hurt herself, but it will be for the best in the long run. Also, she is old enough to have a talk with when she is calm. Let her know that you will not be talking to her when she is acting that way, but that you love her and are willing to talk to her once she has calmed down. Another thing that can help is showing her how to use her "cutie-pie smile" to get what she wants. Let her know that you are much more likely to listen to reason if she is smiling and happy rather than crying and screaming. Lastly, at one point I started a rule in our house that "whining means NO." If my kids whine/cry about something the answer is automatically no. If they calmly discuss something with me, I might say yes. This one rule made all the difference. It only took about 3 weeks to see the positive effects in my childrens' behavior.
Good luck. I hope this long post helps. Remember that what she needs now is your positive reinforcement for all the great things she does as she struggles to improve her behavior.