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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?

Thank you

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    $\begingroup$ Without being rude: why do you say that she is smart? You say he needs tutoring three times a week and in that time, you only can repeat the things she learnt at school, without adding anything else. Did you already consult her mathematics teacher about this? $\endgroup$ – Dominique Oct 3 '17 at 14:44
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    $\begingroup$ Not sure what country you are in, but in the US, you could ask that the child be evaluated to see if there are specific problems interefering with her learning. The student would then get an IEP which might allow calcuator use on tests, more time for tests, or specific strategies to help compensate for any disabilities. $\endgroup$ – Amy B Oct 3 '17 at 17:47
  • $\begingroup$ @Amy B: Many thanks for drawing my attention to these possibilities. I'm waiting on the results from her first test (statistics and probabilities) and will consider this. But I think you motivated me to be more influent. I'll talk to her parents and see if they'll allow me to go to school and see with whom I can talk to, get people aware of her specific situation. $\endgroup$ – Pietrov Oct 13 '17 at 16:50
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    $\begingroup$ @Dominique: I understand your question. Let me answer with a recent example. Some days ago, we were working on the #(AUB)=#A+#B - #(AintB), regarding random trials. The unknown was the #(AintB), the numbers were in the hundreds and she got to the result in seconds, only by some mind calculation she could not explain. I've seen on several occasions that although she can reason and calculate instinctively, she cannot make a link with the mathematical language. Personally, I see her performance in Math like the way we learn our own languages, even before we even can recognize letters or grammar. $\endgroup$ – Pietrov Oct 13 '17 at 16:58
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Your student's difficulties could be the result of early neglect of arithmetic.

But, if you don't see improvement, you may need to consult a specialist. There is something called "dyscalculia" which is (roughly) an inability to understand numbers and/or number symbols.

I believe that it is rare, but I have seen diagnosed in college students it a couple of times over the years.

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  • $\begingroup$ I find it very difficult to believe that Picasso never dealt with a written number in his life. Most likely he was being facetious, if the quote is even true. $\endgroup$ – Javier Oct 3 '17 at 3:19
  • $\begingroup$ @Jum H: I see it frequently with her, she prefers to add (the only operation she "masters") instead of multiplying. The weirdest is that mentally, she can do some calculations mostly instinctively, but she finds it very difficult to translate them into mathematical language. $\endgroup$ – Pietrov Oct 13 '17 at 17:04
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In my view, the worst thing you can do is go back and only focus on elementary topics. I've taught students grades 7-12 for the past several years, and many 7th graders come in not knowing their multiplication facts, struggling to add and subtract negative integers, and struggling with operations on fractions. However, I've found that students will eventually learn these elementary concepts if you move on with learning at their appropriate grade level.

Interesting side note: The use of calculators has been found to have no negative effects on student computational skills. In fact, it generally aids in learning computational skills. See article here: http://www.nctm.org/uploadedFiles/Research_and_Advocacy/research_brief_and_clips/2011-Research_brief_18-calculator.pdf. So, I would allow her to use a calculator which may catch her up to speed on some elementary topics.

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  • $\begingroup$ Currently I am shifting my beliefs towards exactly what you state here, I'm seeing that happening little by little. I will take your advice about the calculator and implement it whenever possible. $\endgroup$ – Pietrov Oct 13 '17 at 17:06

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