Mathematics has often been likened to a language.
Idiot savants (often left-brain impaired) occasionally take to lightning calculating naturally with no formal training in mathematics at all.
Your left brain hemisphere processes information sequentially (one part after another).
Your right hemisphere uses simultaneous processing.
During World War II, plane spotters were trained to count at a glance not only the number of planes that were passing by, but by their silhouettes alone, how many of each type! They didn't count 1, 2, 3, 4 or 3, 6, 9, 12, but they acknowledged them all as a group! Small children learn to count this way very easily, that is before they are discouraged from doing so.
This right brain faculty is also demonstrated when carpenters size a board without using a ruler or mechanics assess a certain size wrench for a particular bolt just by looking at it.
To initiate the right brain's innate ability to calculate instantaneously, count your footsteps from 1 to 100 rhythmically as you walk or run, and observe simultaneously the sights and sounds with your 'conscious awareness' while still automatically counting.
When you reach 100, start all over again.
When the process becomes automatic, this simple drill can be expanded to reciting the 2's table -- until that too becomes automatic.
While taking a walk, estimate the number of strides that it will take to reach a certain street, tree or corner.
Count the steps and check yourself.
Look at a building and guess the number of floors or windows it contains.
Count and verify.
When you get on a bus, flash to yourself the number of people already on board, then count to double check.
Look at a parking lot and guesstimate the number of autos.
In your home, count the number of tiles on a wall or the number of boards in the ceiling or the number of books on a shelf -- but always at a glance, then check yourself.
Open a book and glance at a particular paragraph, then close your eyes and relate the number of lines in the paragraph.
With enough practice, you'll eventually be able to look at a flock of birds and instantly give the number.
As a further drill, take a few dominoes and glance at them giving the total number of spots instantly.
With each day of progression, add more dominoes.
In lieu of dominoes, practice with a small handful of peas or beans thrown on the table, each time varying the number.
With 15-30 minutes of practice per day, in a week you'll suddenly be counting instantaneously and amazing yourself.
For another drill, estimate the length, width and thickness of a board from a 4-foot distance and then again from a 20-foot distance.
Estimate the height and weight of a person the same way.
Carnival show men do it.
So can you! For the auditory minded, practice with a tape recorder and a friend.
Have the friend prearrange a series of calculations on paper to be read into the tape recorder for playback by you later.
For instance, have him read at a certain tempo 40 x 3 + 20 ÷ 7 ÷ 5 x 4 x 3 ÷ 12 = ?
Design your own variety of problems with varying complexity and press yourself for speed by having the tempo increased progressively.
With practice, you'll find a 'letting go' procedure takes place and the answer just pops up! It might also help to disengage the left brain from participating in the calculations, so say the ABC's while mentally doing each mathematical problem.
As another drill, have a friend read words to you from a book and you give the number of letters in them.
The left brain is comfortable with counting by 2's, 3's, 4's, 5's, 6's and even by 7-item groupings before it has to slide over to the right brain for assistance in higher numbers.
To acquaint your brain's acknowledgement of number groupings, practice with the following for 1 second glances: After completing the above, take a 1" x 1" window square cut out from a piece of paper and slide it over the dots below.
Determine the number of each random dot grouping within the window as quickly as possible; then confirm your count.
Afterwards, do the same thing with a 2" x 2" window square.
To determine the dots more effectively, flash your focus over the dots in a series of saccadic jumps.
Also to disengage your left brain counting tendency, say the ABC's simultaneously to yourself while you do this flash counting in your right field of vision.