How to learn and teach multiplication

The standard method in western countries is to insist that you stare at some page filled with "the multiplication table" and "Just shut up and memorize it!"

Forget that, entirely! First of all, we are usually not taught any meaningful way to memorize things. The default method that most people have is untrained, and pretty shakey, at best.

My Idea is to use an active method that forces you to use your brain a bit. When you do this, your brain (which is smarter than your teacher) creates new mental pathways. When it does it often enough, those pathways are "etched" deeper into your brain, so they become "highways." Remember, though, that this is just a metaphor, and isn't quite the way your neurology works, but it makes sense to us as a metaphor.

If you practice something enough, it becomes easier. If you stare at something passively long enough and often enough, it will create pathways. But if you actively create the pathways, they will be made sooner and deeper.
And if you use a bunch of different ways, you will create wider, deeper paths.

So if you find ways to calculate the multiplications, instead of using passive memorization, you will be more successful.

The simplest (not necessarily the easiest or the best) way to calculate your multiplications of whole numbers {1,2,3...}, is to perform repeated addition on it. In other words 6*4 becomes 4+4+4+4+4+4.

(It is important to know that although multiplication of whole numbers can be accomplished by repeated addition, repeated addition is not the definition of multiplication. Like most things, the more you learn, the more you can understand, and once you learn basic multiplication and arithmetic you can go on to learn very interesting properties about multiplication that go beyond simple repeated addition. But when you are starting out learning with whole numbers, "multiplication can be accomplished by repeated addition" is a good rule of thumb.)

If you do your multiplications that way often enough, your brain starts thinking, "Hey, this is a pain in the neck. I can do it, of course, but there must be a better way. So the next time we do this, I'm going to store the answer in my memory, so that after that I won't have to go through this tedious process anymore."

That all happens on a subconcious level, you understand. It's a great talent, that brain of yours.

If you need an illustration of how this works in other fields, take someone who is learning to shoot baskets. If he has a good coach, the coach tells him all the things he needs to know, like how to bend at the knee, use the wrists, etc.

Now, the player can just listen to the coach, make notes of what he's saying, and repeat those instructions to himself over and over, with no physical effort.

But you and I know that at some time he's going to have to go out and shoot some hoops. And only by repeated trial, does his body memorize the actions, and streamline the process. That's because the body and the brain don't want to work too hard forever.

Somehow the term "multiplication tables" gives one the impression that there are some "magical tablets brought down from the mountain," like they are the only way to learn.

The "tables" are simply a list of what happens when you multiply a set of numbers by each other, they are not the multiplications themselves. (Give that a chance to sink in - it's deeper than you may think.)

The "tables" are like a roadmap, but they are not the road. If you teach a kid that the tables are the only way to learn, you are depriving him from experiencing real multiplication. Multiplication is about manipulating numbers, or amounts. Deep understanding of simple multiplication is a necessary part of developing numeracy (the ability to understand and work with numbers).

The term "multiplication facts" is also a really bad Idea to inculcate young minds with. Not only does it suffer from the problem mentioned above, but it also gives the impression that they just happen to be "facts" that exist outside of you. Yes, they are facts, but they exist inside of you. You can reconstruct them simply by using your mind. They are not like facts like, say, historical facts like "The Magna Carta was signed in 1215." That is something you cannot directly experience, and must simply learn as a "fact." (Actually, that is not strictly true either, just ask a real historian.)

See, learning something strictly as a "fact" makes you feel like a victim of that fact, rather than a participant. Direct, or indirect experience with something makes it so much more alive, real, and actually fun.

So the best ways to learn math are not with flash cards, silly rhymes, obnoxious cartoon characters or other methods that distance a child from direct experience of numbers and how to use them. (Don't even get me started about the evils of calculators...)

The best ways to learn math are ways that have you use actual math. Imagine that! Learning how numbers relate to each other is the way to go. I'm biased, of course (because I wrote it) but I think "Numbers Juggling - Times without the Tables" is the best explanation of how to learn basic multiplication that works for most people, especially the ones who think the "tables" are a pain.

I wrote this e-book because I was disgusted with the way we are normally taught how to multiply. This booklet will open your eyes to a new world. You will learn a method to teach any child basic multiplication of single-digit numbers (what we normally call "the times tables" or "multiplication facts") in about two minutes.

The booklet then goes on to show how to "lock this knowledge in."
It also comes with seven e-mail lessons that will show you the math behind the method, so you can actually understand and show your child why it works!

And here's the big news: I've just added over a dozen videos to help you super-charge your learning and practicing. They are easy to follow and fun. I "hold your hand" as you practice, until you can do over thirty multiplications in a minute.

I truly believe that every parent and teacher should know what is taught in this booklet, so every child can have a meaningful, helpful method for dealing with this important subject.

You can order "Numbers Juggling - (Times without the Tables)" here.

Just as a refresher, a whole number is zero and all the counting numbers from 1 on, forever. The whole numbers are 0,1,2,3,4,5... (The three dots mean "and so on).

Now, there is a really good interactive lesson on how to multiply any whole number by 11, mentally, at this page at MathMojo.com. You should learn that before reading further.

1. 
   Start at the left of the multiplicand (that's the 674) and pretend there are two zeros behind it and in front of it, making it 0067400. Now add the final three digits, (in 0067400 that would be 4+0+0) and write the sum (4) beneath the 4 in the 0067400 . That's the digit in the units column of the product. (The product is the answer.)
   


2. 
   Next, add the three digits that start second from the end of the multiplicand (in 0067400 that would be 7+4+0 = 11) and write the units digit(1) of that number to the left of the last number you wrote, giving you 14 so far. Keep the tens digit of that number in your head - you are going to carry it to the next addition.
   


3. 
   Now add the three digits (plus the carry) that start third from the end of the multiplicand (in 0067400 that would be 6+7+4+1 = 18) and write the units digit of that number (8) to the left of the last number you wrote, giving you 814 so far. Keep the tens digit of that number in your head - you are going to carry it to the next addition.
   


4. 
   Now add the three digits (plus the carry) that start fourth from the end of the multiplicand (in 0067400 that would be 0+6+7+1 = 14) and write the units digit of that number (4) to the left of the last number you wrote, giving you 4814 so far. Keep the tens digit of that number in your head - you are going to carry it to the next addition.
   


5. 
   Now add the three digits (plus the carry) that start fifth from the end of the multiplicand (in 0067400 that would be 0+0+6+1 = 7) and write the units digit of that number (7) to the left of the last number you wrote, giving you 74814 so far. There is no tens digit to carry this time. You are finished.
   

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If the multiplicand had been longer, say, 876,346,974, it would still work the same way. You would just continue on until you ran out of digits.

If you learned this, and the original lesson for multiplication by 11 at MathMojo.com, you must see a pattern by now:

If your repunit has four digits, you put three imaginary zeros in front and behind, then, starting at the end, add four digits each time.

If your repunit has five digits, you put four imaginary zeros in front and behind, then, starting at the end, add five digits each time.

and so forth.

P.S. Are you aware that squaring any repunit (at least up to nine digits) will give you a palindrome? Try it.

Some people like to complain that math is so uninteresting because "all you do is the same thing over and over, and get the same answers, etc." I feel bad for them. They don't get it that although, in arithmetic at least, any problem has only one answer, there are lots of ways to get there, and only the one you learned in school is boring.