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The ideas in this drill guide are taken from Multiplication 1 ebook.

How to memorize multiplication tables using a structured drill

I have prepared a video where you can see this structured drill method first hand.

This drilling is aimed at memorizing a certain times table. It should be used only after the child already understands the concept of multiplication.

When you are doing drills to memorize, explain to the child that the goal is to memorize the facts, to recall from memory, and not to get the answers by counting or some other method.  Just like your child probably has memorized your address and phone number, now she/he is going to memorize some math facts.  You can easily see if the student is trying to count because producing the answer takes much more time.  You should expect the answers from the child immediately when you are drilling.  If he/she doesn't know the answer by heart (from memory), then tell him/her the right answer.

Usually short drill sessions are best. You can drill for 5-10 minutes at a time, depending on the child.

Try to have at least two sessions within a day though, as your schedule permits. Brain research shows us that forgetting happens fast, and that new information is retained far better if the first reviewing session is done within 4-6 hours of the first time learning. (This principle applies to anything new you are learning.)

Paper-pencil activities where the child is left alone, do not work really well for memorizing the facts - the child may get the answers by counting and not from memory.  So it will take time from the teacher/parent.  If you can, utilize older siblings in the drilling task too.  Computers are great drillers since they won't get tired and you can usually choose a timed session where the child is then forced to produce the answers quickly.  Children can actually enjoy the memorization process when they notice they are truly learning the facts and are able to go through the drills successfully.  Computer programs and computer-based drilling can be very rewarding to children and let them enjoy memorizing times tables.  See a list of free online multiplication activities. 

The method below has several steps from 1 to 5. You can work on only a few of the steps in one session, again, depending on the child's concentration and ability.  

Memorizing the table of 3 - in steps

Have a table to be worked on all ready written on paper.  We will use here the table of three as an example.

1. The first task is to memorize the list of answers, so to speak. Study first the skip-counting list up until the midpoint (3, 6, 9, 12, 15, 18).  Have your child say it alound while pointing to the answers one by one with a finger or pen - thereby using many of his senses simultaneously. After he has gone through if a few times, ask him to repeat the list from memory.
    
   Try require the answers from your child, and not give them to her too easily, because ONLY by straining her mind will she make the effort to eventually memorize these facts.  The mind is like muscles: it needs exercise to become stronger.  
   
   Require her to memorize this list both upwards and downwards.  Continue this way until she can 'rattle off' the first list of 3, 6, 9, 12, 15, 18.
   With some tables, like table of 2, table of 5, or table of 10, point out the pattern in them.  The pattern in table of 9 is more subtle but still usable.
   
   
2. Then tackle the last part of the list: 21, 24, 27, 30, 33, 36.  Do the same things you did with the first part of the list.
    
3. Lastly, work with the whole list of answers.  Practice the list UP AND DOWN until it goes smooth and easy.
   
   This part may be enough for one day. But review it later in the day.
    
4. Next, practice individual problems randomly.  You can ask orally ("What is 5 times 3?") or point to the problems on the paper, or use flashcards.  However, I would recommend saying a question aloud and simultaneously pointing to the problem the child can see, because again, using multiple senses should help fix them in the mind better. 
   
   The goal at this stage is to associate each answer 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, with a certain multiplication fact (such as 7 x 3). 
    
   You can also mix earlier tables that she already knows with these new problems, and drill both with flashcards.
    
5. The last step is to do this the other way round so that YOU say the answer, say 21, and the student has to produce the problem (from table of 3).  Have the table handy, hide the problems, and point to the answers in random order.
   This one you can work the other way around: the student says answers, and you produce problems.  Answer wrong sometimes, too, to check her out.
   
   As an extension, you can say answers from several tables that you've studied, and the student gives the corresponding problem.  Sometimes there are several answers: for example 36, 30, 24, and 20 are in several different times tables.  This is an especially good exercise as it prepares to division concept and factoring.

The memorization won't probably happen overnight.  On subsequent days, you can mix these drills 1-5 (and hopefully you don't need to concentrate on steps 1 and 2).  This kind of drilling takes a little time and effort from the teacher, but it can be very effective.  And, homeschoolers can obviously do some of it while going about other tasks, or while traveling in the car, etc.

While you are doing this table by table, you can also try to teach the process to your child, so that she will learn how to do memorization herself.  She can hide the answers and try to produce the list in her mind.
 

Other helpful ideas

• Hang a poster with the 12x12 or 10x10 grid on the wall. Remind your child to glance at it a few times a day. It can work wonders for visual learners!

• Hang beside it another, initially empty, poster, to which the child fills in those facts he has mastered.

• Recite the skip-counting lists or multiplication facts aloud just before going to bed. This can turn them into mastered facts by the next morning.

In my book Math Mammoth Multiplication 1, the tables are not studied in just ascending number order, but in a specific order: first table of 2, then table of 4, table of 10 (very easy), table of 5 (easy), then some review. Then, table of 3, table of 6, table of 11 (very easy), and table of 9 (which has a pattern) - and then some review. Lastly are studied the 'hard ones', table of 7, table of 8, and table of 12.

Along with that, in each lesson there is a 12x12 grid to fill where all the facts not yet studied are colored full. Each time a new table is studied, those still colored places get less and less, which can encourage the student a lot. And, with each new table, the facts in it that the student already knows, are explicitly pointed out. With the last tables there are but a few new facts to memorize.