There's a lot of literature on this. One thing that people have tried to do is look at specific common flaws and then try to figure out what kids are thinking. One very studied area has been students confused over how .999...=1. See this summary section in Wikipedia which gives a pretty decent summary of that literature along with references. I do seem to vaguely recall also additional studies that have shown a correlation between not believing the .999... and likelyhood to make algebraic mistakes, but that isn't cited there, and I don't remember the authors names or any other relevant search terms. Is anyone familiar with this?
studies that have shown a correlation between not believing the .999... and likelyhood to make algebraic mistakes
People did studies on that? That just seems so frivolous!
I'm mostly asking this open question to those among us who are well-versed in developmental psychology (I'm mostly thinking of children) . Although, failing the actual scientific research on the topic, I guess some testable hypotheses would be great too.