How Do Abstract Artists Counter the Claim That a Child Could Produce Similar Art?

A much oft repeated supposed witticism on observing a great piece of abstract art "My Child Could Do That." receives some very interesting answers as to Why No, your child could not do that.
 

Cy Twomby One of the most expensive selling artists in the Art World

                                                                  Cy Twomby
 

Anonymous Child Painting

Michelle Gaugy
Michelle Gaugy  The difference is that the child can do this only once or twice or three times. But no more. And they will never be able to get any further than this - designs, pretty, colorful ART requires a life source - experience, a story, an idea, a belief, a connection to something.

But, if you do wish to engage someone over the ignorant "my child could do this" statement rather than simply ignore it,  the way to do it, I think, is with compassion and some education. I usually say, "Yes, they might be able to copy it. But it's unlikely that they could have just come up with it out of their imagination without any prompting. And it's even more unlikely that they could then come up with another 100 or more other original works of the same quality,  in the next year or so, each one different, each of which have meaning to people that see them, and keep doing that for years and years. Because that's what professional artists do. They are deep and original - and consistent - over and over and over again - all the time. And children aren't. It's not just about one painting.
Michelle Gaugy

Painting on left is a child's painting the painting of the right is a world renowned artist

And as  Kathleen Grace explains 
Kathleen Grace

I'm so sick of fielding questions like this from people that I usually just shake my head and say to the person that they know very little about art if they believe that, and I often don't want to bother, a person will have their opinion no matter what I or anyone can explain.

My own mother stood in the halls of the Metropolitan Museum of Art and hollered at the top of her lungs, that she could scribble like that shit on the wall. So, I know very well how this kind of thinking works and how unwilling they may be to even embrace the concept of abstract.
From my own perspective of painting abstract, it is harder, much harder, you have to draw from nothing but what you feel, and work to create a sense in your work that emulates or evokes a mode of communication. I tell people that it's like communicating without having any language and trying to navigate enough through the work in order to say something.
Abstract is harder, you start from nothing and build a work to evoke a sense of something. That's not easy. And a child's work can't be compared to an adult's, a child is making random marks that may or may not be interpreted at having significance or meaning, when an artist has an intent.
So, I just ask that you don't insult the artist, ask if they can explain the work and what it is they are trying to convey. And forgive if the beginning of the question sounded like an insult, it wasn't meant to be but the question is what so very many people accuse abstract artists of, that they have no talent beyond what their child could do.
 Kathleen Grace
Kathleen Grace

And finally a video submitted by Annette McGuinness
Annette McGuinness explains and answer the question that refuses  to die an honorable death with this approachable and entertaining video by "The Art Assignment" which answers also answers the question quite well

Additional Reading
 Study Examines Difference Between Abstract Expressionist Masterpieces And Paintings By Children