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I recently started tutoring a $9^{\text{th}}$ grader who had managed to rotely memorize some processes (thats how he got this far).

In short, I have someone that is trying to learn basic concepts (that we use a positional number system for example and what that means) at the same time as learning things that rely on that (scientific notation).

Is it standard practice in these cases to jump right to expressing numbers as $\sum\limits^{n}_{i}{a_i10^i}$ (in words of course) and how that can be generalized, or should I in essence do the 3rd? grade part first, and then do scientific notation normally?

Note: This is just one example. There are many others, and I am asking for a general rule for what to do in these cases.

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I'd go for basic concepts, explained (somewhat) formally. Specially if they do know how to "do" problems, it might be enlightening to see "why" it works.

Be careful not to overdo it. What I find easy, even obvious, is often a deep mistery to students. Ditto, what I find interesting and fascinating in its possibilities is often seen as utterly pointless. Putting yourself into their shoes is hard.

Sites like Cut the knot have lots of discussion on serious mathematical problems in form of puzzles, many accompanied by interactive applets that allow you to experiment. This I can recommend highly, I'm sure others can give references to other (not so dry/serious) sites. Don't worry if they aren't learning why e.g. long division works, there is plenty of time for that later (if at all required).

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