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I have an excellent undergraduate student who is suddenly being put into math classes where she does not have access to a calculator. She is fine with all the new topics, but when running into something like $\sqrt{441}$, it takes her a significant amount of time to finish the calculation.
She wants to spend a couple weeks over the summer to learn to fend for herself without calculator assistance --- what kind of "syllabus" or "list of topics" could she use with a tutor to accomplish this goal?
Perhaps reading one or more of these books on the art of approximation (often: Fermi estimates) would be fun and helpful:
Weinstein, Lawrence, and John A. Adam. Guesstimation: Solving the world's problems on the back of a cocktail napkin. Princeton University Press, 2009.
Weinstein, Lawrence. Guesstimation 2.0: solving today's problems on the back of a napkin. Princeton University Press, 2012.
Mead, Carver A., and Sanjoy Mahajan. Street-Fighting Mathematics: The Art of Educated Guessing and Opportunistic Problem Solving. MIT Press, 2010.
This addresses the title of the question more than the situation. (So if, as commented, there are dyscalculia or time issues or something else, this advice won't be appropriate. Also, if precision is wanted, the following is less than optimal.)
I spent a lot of time before college doing mental calculation. A lot of it developed from memorization, some from pattern recognition, but practice to increase speed as well as check, check, check again, and then look at it to see what might have gone wrong are the main reasons why I can still do a lot of computation with only a paper and pencil.
An important part of this is estimation: what is the size and shape of the answer? Continuity of operations and experience speak to size, experience speaks to shape. For the given example, $\sqrt{441}$, I would say first the answer is north of (bigger than) 20 and, if it is an integer, has to be odd. Since I know tables of squares, memorization and experience gives me that the answer is the smallest odd number bigger than 20, but I would try $(20+ \epsilon)^2 = (400 + 2 \cdot 20 \cdot \epsilon + \epsilon \cdot \epsilon)$ as a check for enough values of $\epsilon$ to see if my first guess was correct.
Another important part is context: is an exact answer expected? How many significant digits are wanted? If it cannot be determined from the context, always, always, ALWAYS provide it in your answer, as whoever needs the answer also can use confirmation of the context to evaluate the answer. Some tools work well in some contexts, not so well in others. If you have picked the context and the tools you are using are not working well, check to see if you have the context right.
For personal performance, information representation can be important, as well as noting certain patterns. Rather than remembering half-angle formulae and other relations between trigonometric functions, I remember the trigonometric equivalent of the pythagorean theorem and a mnemonic (usually a picture) which help me remember and derive the desired relations. Once I have the relations derived, it is easier to use mnemonics and rederive than wonder whether I remembered the relation correctly or not. Similarly, noting cubes mod 10 (3 goes to 7 goes to 3 under cubing mod 10) helps in constructing answers and providing checks for calculations involving cubing integers, or even taking cube roots.
As a quick self-assessment, see if you can find some number sense tests, and perform the following experiment. They are usually 80 questions to be done in 10 minutes, and top solvers (in my day) rarely did more than 50; use this to set your benchmark (20, 30, or even 100 minutes). Then go through all questions once using a green pen. Those you feel good about (having solved in ten seconds or less correctly) answer in green pen. Any that don't, skip in five seconds or less. Then go through the list again focusing only on the unanswered questions, using a yellow pen, and this time allow 20 seconds to compute and be sure of your answer. Repeat with orange pen and 40 seconds, red pen and eighty seconds, black pen and 160 seconds.
The time and colors don't matter, but the process above serves as a quick assessment of levels of difficulty, and hopefully shows through color-coding some patterns where you know you want to improve in a certain type of calculation. Then you can focus on problems using those particular operations or combinations of operations.
If number sense tests aren't available, try some practice tests and other review tests in algebra texts and similar materials.
Gerhard "Still Uses A Calculator Occasionally" Paseman, 2015.04.06
