My answer to a question on Quora: What is a good way to help a student to remember to carry numbers in multiplication?
The answer was already given in this thread by Mark McLenaghan: place value. As soon as children understand place value, carries will be easier to understand. And another point: carries should appear first in addition.
Next come comes multiplication of a number by a single digit number – first without carries ( such as \(23 \times 3\)) and then with carries, first occurring only once and going to the empty position, (\(43 \times 3\)), then once with addition in the next place (\(23 \times 4\)), and several times (such as \(23 \times 9\) and \(123\times 9\)).
Finally, multiplication of arbitrary numbers is bolted together from multiplication by single digit numbers — of course, starting with multiplication by two-digit numbers.
I think experienced teachers can subdivide the learning process in even larger number of steps, avoiding sudden jumps in difficulty and making the progress smooth.
For learning place value, an useful exercise is playing with casino-style tokens of value 1, 10, 100, maybe even 1000 and counting pods with holes for 10 and 100 tokens (of the kind that were used in banks in pre-machine age for quick counting of coins); it could be fun.
Please notice that I am talking about the pre-machine age. This is the general principle of development of an individual: ontogeny recapitulates phylogeny (as formulated by Ernst Haeckel in case of evolution: a human embryo at early stages of development looks like a fish). So the use of archaic tools in learning is natural and perhaps unavoidable.
From the mathematical point of view, the long multiplication (and, actually, long addition) are immensely deep; I apologise for plugging in my old blog posts, but this one is illumination: The secrets of long multiplication. And this one, about long division: A tale about long division. A good discussion of place value can be found here: R. Howe, Three pillars of first grade mathematics.