In 1989, the National Council of Teachers of Mathematics (NCTM) developed new standards which were based on the belief that all students should learn higher-order thinking skills, and which recommended reduced emphasis on the teaching of traditional methods that relied on rote memorization, such as multiplication tables.
Yes, virtually all students can learn higher-order thinking skills. But automatic recall of multiplication facts is one of the ways that students can gain access to these higher-order skills.
Consider the following fraction:
If you have a strong command of your multiplication facts, you almost immediately simplified it to:
The difference between instantly spotting this relationship and having to think about it for several seconds (or longer) is the difference between success and frustration in every math course after fourth grade.
What about when you're faced with something like the rational expression below?
Again, if you are confident in your multiplication facts, the 12/84 will seem like an old friend. It will be less scary to simplify the rest of the expression. However, if you are still shaky on the arithmetic, the algebra will seem very daunting. You will labor over the coefficients and will have little energy to deal with the variables (which require still more multiplication).
The solution is to learn the multiplication facts. It's not enough just to be able to understand them - you need to be able to recall them as easily as you can recall your own birth date.
Is this learning or memorization? There is no difference. Once the facts are memorized, they are, in effect, learned.
Eventually, the NCTM figured out that "higher-order thinking skills" can't be accessed if you are laboring over recall of your times tables. In their 2006 "Curriculum Focal Points" document, they said that "[s]tudents [should] use understandings of multiplication to develop quick recall of the basic multiplication facts."
This is better than the 1989 document, where they called for "decreased attention" to "rote practice" and "written practice." In fact, such practice is exactly what it takes to acquire quick recall, whether or not you use your "understandings of multiplication" to get there.
I've been hard at work developing a bunch of emergency methods for helping students learn their math facts, which are working nicely. But the biggest benefit has been simply convincing the kids that:
- multiplication facts are important, and not knowing these facts is holding them back;
- everything should be as easy as 2 x 1, and will eventually be;
- facts that they don't know aren't harder, they're just less familiar.
You may have a number of memorization strategies you enjoy. Below is mine.
The McCann Method for Multiplication Memorization
I have tested this only with middle schoolers. It works. And it works better than anything else I've ever seen for students who have dyscalculia. If you want to know how to memorize times tables, read on! You'll need 12 half-length index cards.
- Buckle up! We're starting with the twelves. It's a huge confidence builder to learn "the hard ones."
- Create a flashcard for 12 x 1. Hold the front side up and state the answer. This will already be automatic.
- If 12 x 1 is 12, then what's 12 x 10? 120. Create a second card.
- Use your two flashcards to quiz yourself on the two facts you have so far. They should be automatic right from the start. You're training your brain to respond automatically to the visual input of the expression on the front of your card, without thinking. We already did the thinking when we created the card.
- Now, if 12 x 10 is 120, then 12 x 5 should be half that. What's half of 120? 60. Create a third card.
- Rotate through these three cards, quizzing yourself. This should still be effortless.
- What's 12 x 2? 12 + 12, or 2 tens plus 4 ones, or 24. Create a card.
- Drop 12 x 1. Cycle through your three remaining cards. If this is not effortless, start over with your first card only and build up again (skip the creating new cards part).
- The next card is 12 x 12. Using the distributive property, this is 12 x 10 plus 12 x 2. In other "words",144. Create the card and add it to your stack. Drop out 12 x 10 and cycle through your three remaining cards. Don't forget to continually change their order.
- Next is 12 x 6. This is 12 x 5 plus 12, or 72. Create this card, drop 12 x 5, and practice your three remaining cards. They should all be automatic. If you are not answering instantly, go back down to one card and start again.
- Next is 12 x 11. Think of it as 12 x 10 + 12, or 132. Create the card, drop 12 x 2, and practice your three remaining cards.
- We've arrived at 12 x 9. This is 12 x 10 - 12, or 108. Create the card, drop 12 x 12, and practice your three remaining cards.
- Now, 12 x 3. This is 12 x 2 + 12, or 12 + 12 + 12 (3 tens and 6 ones). 36! Create the card, drop 12 x 6, and practice the 3 cards remaining (they should be automatic).
- Time for 12 x 4. This is 12 x 3 + 12, or 12 + 12 + 12 + 12 (4 tens and 8 ones). 48. Create the card, drop 12 x 11, and practice the 3 cards remaining.
- We're almost there! Create a card for 12 x 7. Think of it as 12 x 6 + 12 (84). Drop 12 x 9 and practice your three remaining cards.
- Lastly, we have 12 x 8. This is 12 x 7 + 12, or 96. Drop 12 x 3 and practice the three cards remaining.
Note that it may take a few sessions to get through all of this. On the first time through, many students are fatigued by the addition of the sixth card and cannot continue.
With each practice session, start with the first card, and build up from there.
The order is not so critically important, but with this order, we have created a way of learning these facts without just counting up by twelves. We've given ourselves some "mental hooks" upon which to hang the information.
Once you have made it through the whole sequence, start over again. This time, don't drop a card until you're adding what would be the fifth card. Now, you'll have four cards in your hand until the end. Once you are comfortable with four cards, go through the sequence with a consistent five cards, and so on.
The end result is that you will be able to go through all twelve cards in any order, instantly.
Periodically test yourself to see how many problems you can do in one minute (I like this worksheet with only 12 as the top number and 3 - 12 as bottom numbers). Once you can get 40 or more in one minute, you're in good shape.
I hope I have convinced you of the importance of learning multiplication facts and also given you a good strategy for learning them. Let me know how it works for you!