The following approach is a combination approach that I found helpful. It jumps around more than many educators would most likely choose, and it involves working on different exercises over a period of days. The advantage of it, however, is that the task of mastering the easiest-to-learn information is disposed of fairly quickly, leaving only a relatively few "trickier" pieces of information to be learned at the end of the process. Another advantage is that, rather than expecting the child to do a lot of memorization (and learning little else) all at once, this approach involves aiming for easier memorization, combined with exercises to help give the child a more solid understanding of multiplication, itself.

It may be easier for a child to learn to count by 2's, 3's, 4's, etc. (up to counting by 10's), without having to digest the concept of a multiplication table at the same time. Using a sing-song rhythm to teach counting by numbers others than 1 can make learning easier. If the child can easily progress from counting by 2's on up to counting by 9's, great. Not all children can easily and/or quickly get beyond counting by 5's. That's why aiming for the child to at least learn to county by 2's, 5's, and 10's make good first steps. Learning counting by 3's and 4's should follow. Counting by 6's, 7's, 8's and 9's can wait for a while.

Teaching a child to count by 2's until reaching 20 will familiarize him with everything up to 10 times 2.

Teaching him to count by 3's to 30 will, of course, familiarize him with up to 10 times 3.

Repeating this process for each number from 1 through 10, and telling the child, "You can learn how to count by numbers other than one" can make that much of this particular learning task seem less overwhelming. After all, he's only learning the "nifty" trick of counting by other numbers (and there are, after all, only numbers to remember each time a child learns to count by another number).

Not expecting the child to learn to count by different numbers all at once gives a child time to learn to count by one or two numbers well before trying to take in more information. Presenting counting by different numbers by comparing it to going upstairs two or three stairs at a time may help the child see counting by other numbers as "advanced" and particularly skilled.

Since counting "by 1's" is something the child learning multiplication already knows, that number is already taken of. Counting by 2's can be relatively easily for a child to learn, particularly if he keeps in mind that he's simply "skipping one step".

Showing a child how counting by 10's is easy will most likely result in his learning to do that fairly quickly. Showing him how counting by 10's is the same as counting by 1's and adding a zero should make him confidently master counting by 10's.

Counting by 5's can be easy if a child learns that all he has to do is think of how the numbers will start with 5 (of course) and then alternate when it comes to whether they end in 5 or 0. It is usually easy for a child to see that he simply counts from single digit numbers on up through the teens (with a 1 in front), twenties (with a 2 in front), etc. etc.

During the days when you're trying to teach the child to count by 2, 5, and 10, it may help to add a little time "playing with" 20 nickels. Pointing out how 5 nickels make 25 cents, and having the child count by 5's as he adds nickels will eventually lead to his becoming very familiar with multiplying by 5. Once he has become very comfortable with counting by 5's pointing out that he "happens" to now also know that 4 nickels (5's) make 20 cents AND that five 4's ALSO make 20.

Letting him practice by counting six nickels, seven nickels, etc. (up to the 20 nickels) should get him very comfortable not only with "The 5's", but with "how it all works". It can boost confidence when a child discover that he get "all the way up" to ten nickels and realize that 10 times 5 is just a matter of adding a 0 to the 5 he "already had".

Letting the child see what 8 nickels amounts to when he counts by 5, what 16 nickels amounts to when he counts by 5, can help. Having him divide up something like 16 nickels into two groups of 8 or four groups of 4, and then counting each pile by 5 should help him understand even more "how it all works".

Once a child has gotten this far, it may be a good time to draw up the lines for a multiplication table and letting him fill in what he already knows. Since he already know's 1 times any number, may have learned to count by 2's (and so knows 2 times any number), and probably knows 5's inside and out; filling in that much of the table should be relatively easy. Since he also probably learned 10's easily, that's another column that is easy to fill in.

Pointing out how easy filling in the 11's column is should take care of that very quickly. (Mentioning how 11 times any number can be figured out by thinking of 10 times the number and adding that number to it shows what multiplying by 11 means. Showing the child how to count by 11's will let him see the pattern to how 11's "work".)

It can help him see a pattern if you then point out how since he learned (through all that playing with nickles) that 6 nickels makes 30 cents, he also knows that 5 times 6 also makes 30. At this point he will have enough of his table filled in to perhaps start to see a pattern, as well as make the connection between moving nickels around and multiplying.

At this point, bringing out a pile of pennies and letting him see what 2, 3, 4, 5, 6, (etc) groups of 3, 4, 6, 7, 8, and 9 pennies look like may be helpful. Over the course of a few days he can probably learn to count by 3's, 4's, and 6's. Bringing out the table he started and having him fill in anything new he has learned will result in his seeing the table increasingly complete and, ideally, in understanding what it all means.

What is left to be learned, at this point, is a small portion of the table involving multiplying some 6's, 7's, 8', and 9's (by numbers higher than 5 and not including 10). Then, of course, the 12's must be learned. This is where using a separate sheet of paper to show, for example, "counting by 8's" and then "counting by 12's", and returning to the sing-song memorization can help.

Flash cards and playing with pennies are very useful as well.

Once the child has learned multiplication facts and has filled in the table he's been working on, encourage him to make smaller versions of the table just to keep him from forgetting. Having him "just make a table for the 4's" or "just make one for 6's through 9's" can help. Challenging the child to create one for "the 20's" or "the 100's" will further help him see how numbers work.

Inventing games to play, using coins (pennies, nickels, dimes, quarters), can reinforce what's he's learned. Coins, playing with them, rearranging them, and inventing games with them, can be useful for learning more about math later.

Like math, itself, describing the process of teaching it can make it seem far more complicated than it really is.

All it really requires is:

The simple memorization of counting by any number (rather than expecting memorization of all the multiplication facts of a child who is having difficulty memorizing them).

Hands-on experience creating and eventually filling in a multiplication table, which helps show how "it all works".

Hands-on experience with moving coins around, seeing what numbers represent, and seeing what multiplication actually means, how it amounts to counting by any number, and how it relates to adding.

Memorization of remaining facts with the help of counting by larger numbers or flash cards.

Practice exercises involving creating tables and playing with coins for the purpose of gaining a more solid knowledge and expanding knowledge of math in the future.

When children learn the "trick" of getting the easier material out of the way first, they will see how what they're left with is not as overwhelming as it once seemed to be.

When children can easily memorize math facts that's great and will probably result in their doing well on tests. When they learn "how it all works" and "what it all represents", however, they are more likely to math (not just memorization) interesting and easy to learn.