The following is assuming the child has selective mutism (can talk, but can't). This is all just my opinion, based on personal experience, my area of education, and on helping one child who also had this problem as a preschooler.
I had a diagnosis of elective mutism as a child (which is now called selective mutism.) I believe I was 6 when I was diagnosed. I never had any issues with talking at home, only at school/in social situations. I didn't speak at all outside the home. As an adult I'm still generally the shy, introverted type, but I definitely have "recovered" from mutism.
As a person who suffered through it I can tell you that any mention of the lack of speech or any bribes, praise, or punishments associated with speech are likely to make the problem worse. It is anxiety based and those methods are only going to increase anxiety. If a child has an extreme, debilitating phobia, would you bring up the item they were afraid of throughout the day and shame them for it? Would you constantly place demands on them to face their fear? I'm not talking about regular fears here, if you know someone with a phobia (which I also have), you know there is no logic or reason involved. I would jump out of a moving vehicle to avoid the thing I am afraid of... and likely die. And it's something relatively harmless. Fear and anxiety are not reasonable reactions and therefore can't be reasoned away.
Teachers and others whom the child has a hard time talking with should accept any response from the child. Pointing, nodding, writing, whispering, etc. If there is another child or person the kid is comfortable talking with that person can take whispered communications and relay them. The goal should be to reduce anxiety, which is the most likely way to see an increase in speech over time.
I remember when I had this condition and there was nothing that could have made me speak. I wanted to. I just couldn't. I was able to verbalize in my head what I wanted to say, but my body was frozen. I feared saying it wrong, my voice sounding funny, sounding stupid, being wrong, etc. At least at home I was comfortable enough to speak to my family. I do not remember them ever saying anything to me about the problem. At school I gradually got better when they switched my teacher to a much more accepting and kind person who never made an issue out of it.
My whole life I have been shy and reserved in public and that is not likely to change. But now I am able to talk to other parents and speak up at IEP meetings and even answer questions in a group setting. I still prefer to spend my time at home with my family, but I am a functional adult, able to hold a job and raise my 2E son. I even passed college speech class! I suspect the pressure of speech therapy and knowing that my parents were at their wits end with me would have pushed me in the opposite direction. If your DD has selective mutism she needs your support and understanding. Reduce her anxiety, reconnect as a family, treat her the same way you would if she was talking.
Obviously this only applies for selective mutism, if your DD is not talking for a different reason this will likely not be the best plan of action. I hope she's feeling better soon
