While it is critical to support your child through a tantrum, it's important not to change your position if that is what triggered the outpouring of emotion. Giving in to her desires will teach the child that your limits are negotiable. Hold the limit while showing empathy for your child's feelings.
Concerning older children whose prefrontal cortices are developed (though not fully mature, this happens around age 4), it is still important to empathize while holding your limit as this sends the message "I hear you, and you matter." The simple act of not giving in to the demands prompting the tantrum will soon teach the child that behavior won't work to get what he wants.
If a school aged child tantrums in a way that is aggressive " hitting, kicking, or throwing things " this is a clue that the child needs emotions coaching, not punishment. During the tantrum, ensure everyone's safety. That may mean putting distance between the aggressive child and other family members.
You might say, "I see you are very angry. I won't let you hit. Take some time to calm down in your cool down zone."
Once the storm has passed, address the aggressive behaviors. Explain that his feelings are acceptable but his actions are not. Teach him how to manage his frustration and anger with techniques such as deep breathing, counting, going for a walk, or clapping to release energy. Punishing children for tantrums won't help them learn how to manage them. Until we give children better tools to deal with tough emotions, we can't expect them to do better.
Here are the other posts in the "Top 5 Behaviors That Cause Parents to Lose Their Cool" Series: