I think teenagers and children today would stand more of a chance of grasping maths - sorry, numeracy, if they didn't need a whole sheet of paper to work out one multiplication or division sum - chunking, number lines etc - what was wrong with good old fashioned long multiplication or division
You don't need a blind knowledge of how to multiply on paper: you either need to do it in your head (perhaps as an estimate to cross-check a calculator) or you do it on a calculator. Very few people indeed could explain how long multiplication (and, particularly, long division) actually works: it was a rote process learnt because learning it by rote was the least worst option. Now there's no point in teaching a rote method to do multiplication, because it's just as much a black box with no checking for errors as using a calculator. So you either teach a method that actually shows what's happening (the distinction between maths and numeracy), or if you're going to use a black box, you may as well use a less error prone calculator rather than an error-prone rote method.
Long multiplication of reals using involves some rule of thumb about counting the numbers to the right of the decimal point, adding the results together and then putting the decimal point in the answer there. Explain how that works. Now, explain how to locate the decimal point in long division, and why that works.
We had someone on this very forum a few months ago puzzled as to why division by 0.1 made the answer bigger. Chunking explains that. Long division doesn't, and is astoundingly error prone, which is why, outside of school exercises, no-one ever did it, even prior to calculators: they used slide rules, including ones the size of a door with a magnifying lens on the scale, ready reckoners, Brunsviga mechanical calculators and log tables.
Ah, log tables. A lot of people in their late forties and older will have been taught logarithms as a means to multiply and divide large numbers. You know: look up the two numbers in the book, add or subtract the results (remember the mean differences!) and then look up the result on another page to get the rough answer. If you were flash you'd have a table of logarithmic sines and so on in order to cut out the middle-man while doing trigonometry.
Now then: explain how it works. Extra credit for the business about putting the minus sign on top of the integer part, not to the left.
_That's_ why logarithms have moved from pre-O Level to A Level: instead of being a black-box tool to perform a task that there's no other way of doing, they arise (as natural logs, rather than base ten, too) as part of calculus and not before. You're going to need a calculator anyway (as you can't buy log tables any more) so using logs as a means to do arithmetic as a black box is entirely pointless.
Long multiplication and division were rubbish as means to actually do work. That's why so many alternative methods were used to avoid using them. If they were so easy, why did people use slide rules? Why were ready reckoners on the counter in every shop? Why did Brunsviga (http://www.columbia.edu/cu/computinghis ... sviga.html
) make so much money? Why was the Busicom 141-PF (http://www.vintagecalculators.com/html/ ... 41-pf.html
) something companies would mortgage their souls to buy?