The example in the link - well, that's how I was taught to do it more about 60 years ago in our C of E Junior School. However, I've never seen the 'Rules' for it written down anywhere ever before. Interesting stuff!
In those days the teachers were very, very good and the one who taught me how to do arithmetic was called Miss Mabel Woollfitt. She was a cousin of Donald Wolfit (note how he changed his name), the famous actor and film star. In fact, she helped me get through '11+'. Actually, I was the only one in our whole small village school that year to pass and get a scholarship to go to the Grammar School. She was one of the old school of so called, 'Dedicated Teachers', which the country had plenty of at that time. They were extremely well-trained and the School Inspectors were also very strict. I'm talking of the '1950's here.
Discipline and dress and cleanliness were all extremely important. Mess about in class and you would get corporal punishment with a ruler on your bare hand, a slipper on your trousered bottom or even the cane on it instead - even if you were a girl!
This was the 'post war era'. We had tons of 'parrot fashion' learning and used to stand in a circle in front of the blackboard and easel and take it in turns to recite our 'tables'. As kids, we used to compete with each other to see who could rattle them off the quickest - not that anyone won anything for the doing of it, and whoever seemed to be the quickest was actually a matter of personal estimation or guesswork. We were all of us in the class quite utterly brilliant at it and my 'Tables' are absolutely engrained in my memory and will be so until I die.
Miss Woollffitt also gave us 20 Mental Arithmetic Questions every morning to start off the day after 'Assembly and Prayers'.
These were the days of:
Pounds, Shillings (20), Pence (12), Half-pennies, Farthings (4)
Miles, Furlongs (8), Chains (10), Yards (22), Feet (3), Inches (12)
Tons, Hundredweights (20), Stones (8), Pounds (lbs) (14), Ounces (16)
...and so I'm sure you can well imagine that questions on this stuff were quite hard according to 'modern day 2011 standards'.
Indeed, to answer mental arithmetic questions on any of the above topics, the amount of stuff you had to commit to and recall from memory was incredible - and the age of the kids in the class was from 9 - 11 years old.
eg, How much would 15 articles cost at 3 3/4p (threepence three farthings) each?
The answer is extremely easy to work out if you 1. Know your tables and 2. Know the quick ways to use to work it out in your head.
So for the example used of 425 divided by 25, I would have been able to write '17' down virtually instantly at 11 years old - and having worked it out using mental arithmetic. And I was not unique in that ability, such was the standard of the teaching in those days.
And if you don't believe me, ask your grand parents. That's why some of us 'oldies' fundamentally know and believe that today's teaching and teachers are 'total crap', when compared with the 'old days'!
And we also read that today's kids are encouraged to use calculators in class too!
(Stands by mi' bed ready to get shot down)