My son went to a full day session to meet up with some other children around his age who are being homeschooled. During this time he had a class on engineering with Lego, and a lot of free time with which to play and socialize with the other children.
Interestingly enough, the kids decide to play the Game of School. They found a chalkboard and took turns pretending to be the teacher and explaining things to the other students. My son decided to be the math teacher, and according to him, this is what he tried to explain.
“9 x 10 = 90” and “90 x 10 = ?” and “? x 10 = a different ?” and “a different ? x 10 = another ?” and so on.
In other words, he was trying to show a pattern with multiplication by 10, and he decided to use a place-holder to show that the result of his multiplication from the first times 10 would be used in the next calculation.
We have another word we use commonly in mathematics for these place-holders. We call them variables. My son is 7. He has had no formal instruction on variables, and I have certainly never talked about the idea in our discussions about math. This is him inventing a new-to-him mathematical idea to help describe a process to another child.
Why is my son using variables in this explanation? I think it’s because he had a need for them. Does he understand the concept of variable completely? I doubt it, but this is a good start.
My child certainly has a stronger-than-usual background in mathematics (he has me as his father, and we talk about math ideas and numbers a lot) but if he can invent this concept, other children can too, given the necessary experiences.