The Standards& Testing Agency has just released their new sample materials for Key Stage 2 (upper primary). Among other things in Mr Gove's poisonous legacy is the insistence on everyone using formal methods of arithmetic, whether appropriate or not.
The examples given in the test are:
If you use a method other than long multiplication and division and get it wrong, you will receive no marks for either question - even if your alternative is perfectly reasonable.
I'm not for a moment suggesting that long multiplication and division are necessarily inappropriate ways to tackle these for everyone. If you know what you're doing, it'll give you the right answer with a little bit of work. It's just that they're not the only way to do them, and pretending that they are doesn't do anyone any favours - let alone the unfortunate teachers who'll need to teach their kids it1.
Especially in these cases, though, I can think of several ways of approaching the sums that are at least as good.
If someone asked me to figure out $2376 \times 15$ and I didn't happen to have a calculator handy, I'd say it was the same as $1188 \times 30$, which is $35,640$, with much less effort and scope for mistakes than long multiplication provides. I might recommend a student split it up as $2376 \times (10 + 5) = 23,760 + 11,880$ - effectively the same as long multiplication, but with more of a clue about why it works. In any case, I'd always suggest saying "it has to be about midway between 24,000 and 48,000, so 36,000 or so".
As for the division, $1652 \div 28$, my first instinct would be to cancel the fraction. There are several possible common factors I could take out, but four is the one that jumps out: it's $413 \div 7$, which is 59 (as it's seven less than $60 \times 7$). Again, I'd start with an estimate - I know $1500 \div 30$ is 50 and $1800 \div 30$ is 60, so somewhere in that ballpark is reasonable.
Long arithmetic is about as exciting and useful as copperplate handwriting: we don't do it any more because we've got machines that do it much more quickly and accurately than we can. I'm at a loss to figure out who is served by doing long division on paper (academically, it's moderately helpful in C2 if you do arithmetic division, but it's hard to think of a career2 where you'd work that out on paper rather than give a good guess and check it on the computer.) We'd be much better served encouraging our little humans to do the things humans are good at - interpreting real-life problems so that computers can solve them, developing models, making predictions, making decisions based on the results.
Insisting on any one method over any other seems to me like insisting that a tennis player only play drop shots, or a computer only run Excel. Different problems are solved by different mathematicians using different methods, and mandating one is - in my opinion - wrong-headed and harmful.
* Thanks to @robeastaway for pointing me towards this.