One of my private students is a 13year old girl who started school at age 5 (instead of the regular 6, in my country).
Testimony from parents show she had no sense of number at the time, would not grasp even quantities nor could she relate them to physical objects (like recognizing 3 or 4 pencils as being 3 or 4).
She was then entrusted to a person (private tutor) who decided to spend the paid time making small puppets and painting together. By the time the parents found out, she was on the 3rd grade struggling with addition and subtraction (unable of learning division and multiplication).
Nowadays, she has not appropriated multiplication, nor division, yet. 3 months ago, she could not even relate the concepts "double" and "half" to their real as well as to their mathematical expressions. One year ago, she could not read numbers with 3 or more digits, etc.
The max time I can get her attention on math work is precisely 1 hour and not 1 second more, even less if she is a bit ill or too tired from school that day.
I've spent the last 2,5 months creating specific exercises/problems based on the real world, using her personal "youtuber heros" as a motivational characters, to build some experience and familiarity with: patterns and regularities, addition, subtraction, equations, notion of variable, reference numbers, multiple representations of numbers, etc. (Not to mention the previous year, prior to hollidays, working on the very basics from primary school).
There was some progress, but meanwhile school started and the time we have (3 times a week) is only enough to go through what she is being taught at school.
It is obvious to me that she is smart and can reason very well, she has been progressing well in Probabilities, but there is always somewhat of a barrier when we deal with fractional/decimal numbers or expression manipulation (she still ads up instead of using multiplication, on most occasions, and she still doesn't really know how to subract if the result has to go into negative territory, or if she has to operate with negative numbers).
At this point I believe she will eventually fill up the gaps of what was left behind while we work on present day chores, I think I'm starting to see that happening (for example, the notion of probability, which she took in well, is helping to "understand" division, maybe working as "an entrance door" to it).
Has anyone dealt with such situations? What is your point of view on this matter? Possible approaches?