Memorizing multiplication facts

[Paul C. Anagnostopoulos]

My son Luke hasn't managed to memorize his multiplication facts despite our having gone through them from beginning to end five or six times in two years. We're using flash cards. He might know about half of them now. Does anyone have any suggestions for other approaches to memorizing them?

Observation: He doesn't know his addition facts, either, and still uses his fingers to add. However, I'm suspicious that he does really know them, but doesn't trust himself. I've watched him add 6 + 4 and his finger movements are almost perfunctory.

He has a reading disability, but seems to have a perfectly good memory.

~~ Paul

[The Don]

I take it that they're the same as Multiplication Tables. I was just forced to learn them by rote (although I still have some blind spots) and was then quizzed over dinner by my parents.

For each table rote learned I got some stickers, for each set of ten questions right I got some stickers. Good "old fashioned" teaching methods.

[TeaBag420]

How old is he? Two years is an awfully long time, but if he's only five now.... I was nine when I learned my times table.

What is his reading disability?

Since the flashcards aren't working, try recitation. Take one multiplier at a time and have him say out loud "Two times two is four...... two times twelve is twenty-four."

You might want to revisit addition first using the same method.

[Diogenes]

Does he spend anytime playing video games?

How important is it for him to be allowed to play?



( hint: Learning the factors of 8 ( 7, 6, whatever.. ) earns him one ( or whatever ) hour of playing time... )

[Brown]

Multiplication of a single digit by 9 is easy. The digits always add up to 9. The movie "Stand and Deliver" shows a cute little finger mnemonic for "remembering your 9's."

Some multiplications are pretty neat, like:
12=3*4, 56=7*8
which lists the first eight integers in order.

Six times every even single digit is easy to remember. The last digit is the same as the number you multiplied by, and the first digit is half of that. So 6*2=12, 6*4=24, 6*6=36, etc. There are other patterns in the times tables, too.

A friend of mine had trouble with her multiplication, but she was able to get a better grasp of it by relating it to money. She knew, for example, that eight nickels was forty cents, so she knew 8*5=40. She was able to multiply by seven by multiplying the nickels first, then multiplying the two pennies, then adding the products together.

[CFLarsen]

Paul,

What would you say if he could multiply a 30-digit number with 11 in 30 seconds - a week from now?

[Diogenes]

Originally posted by CFLarsen
Paul,

What would you say if he could multiply a 30-digit number with 11 in 30 seconds - a week from now?

I'm sure Paul will have an answer also, but I would wonder why anyone would want to do that?

[CFLarsen]

Originally posted by Diogenes
I'm sure Paul will have an answer also, but I would wonder why anyone would want to do that?

It kicks ass - especially from a kid, who has a rep for being bad at math... ;)

[pgwenthold]

Go out and get Schoolhouse Rock so he can sing them.

3 - 6- 9
12- 15- 18
21- 24 - 27
30

A man and woman had a little baby, yes they did. There were three-ee-ee in the family. Its a magic number.

[Diogenes]

Originally posted by CFLarsen
It kicks ass - especially from a kid, who has a rep for being bad at math... ;)

Point taken..

Is Paul going to have to answer before you share this with us?


I see an easy way to do this on paper .. Is your method done without pen and paper? ( assuming one can keep a 30 digit number in their head ) :)

[CFLarsen]

The kid has to learn one rule:

Add the neighbour (if any), add the ten (if any).

A number's neigbour is the number to the right.

Here's how it works:


987654321 x 11
--------------


We start from the right:

1. There is no neighbour, so we simply put down 1:

987654321 x 11
--------------
1

Next number is 2. We add the neighbour (1), and get 3:

987654321 x 11
--------------
31

Next number is 3. We add the neighbour (2), and get 5:

987654321 x 11
--------------
531

Next is 4. Add the neighbour (3), and get 7:

987654321 x 11
--------------
7531

Next is 5. Add (4) and get 9:

987654321 x 11
--------------
97531

Next is 6. Add (5) and get 11. But hey, that is bigger than 9, so we put down 1, and carry the ten (1):

1
987654321 x 11
--------------
197531

Next is 7. Add (6) and get 13 - but don't forget the "ten": 14. Again, put down 4, carry the ten (1):

1
987654321 x 11
--------------
4197531

Next is 8. Add (7) and get 15 - plus the "ten": 16. Put down 6, and carry the ten:

1
987654321 x 11
--------------
64197531

Next is 9. Add (8) and get 17 - plus the "ten": 18. Put down 8 and carry the ten:

1
987654321 x 11
--------------
864197531

Last thing to do is get the (nothing) and add 9 - plus the 10: Put down 10 - done!

987654321 x 11
--------------
10864197531

In about an hour, he will learn the principle.

After the first evening, he can do 3-digit numbers below 500 (no tens, to make it easier) - blindingly fast.

In a week, he can do the 30-digit number in 30 seconds flat. That will amaze his classmates, as well as his teacher. ;)

This was invented by Jakov Trachtenberg while in concentration camp during WWII. I am not saying that you should put your kid through the whole thingamabob, but it will certainly give him a boost of confidence to multiply a 30-digit number with 11 in 30 seconds!

Web sites
http://hucellbiol.mdc-berlin.de/~mp01mg/math.htm
Explanations and examples.

http://www.shermankeene.com/tracten.html
Free Trachtenberg Mathematics Software.

http://www.ecis.org/profdev/newsletter/aug96/maths/maths6.htm
Simple school project on Trachtenberg.

http://forum.swarthmore.edu/dr.math/faq/faq.trachten.html
Dr. Math's FAQ on Trachtenberg (rules and examples).

[69dodge]

Binary. Fewer facts to remember. :D

Balanced ternary is nice too.

[Diogenes]

Originally posted by 69dodge
Binary. Fewer facts to remember. :D

But you can use your weight in check book registers real fast!
:D

[epepke]

Originally posted by Paul C. Anagnostopoulos
My son Luke hasn't managed to memorize his multiplication facts despite our having gone through them from beginning to end five or six times in two years. We're using flash cards. He might know about half of them now. Does anyone have any suggestions for other approaches to memorizing them?

Observation: He doesn't know his addition facts, either, and still uses his fingers to add. However, I'm suspicious that he does really know them, but doesn't trust himself. I've watched him add 6 + 4 and his finger movements are almost perfunctory.

He has a reading disability, but seems to have a perfectly good memory.

~~ Paul

Maybe he's a genius. Einstein said that when he thought about problems, he thought in terms of muscle sensations rather than vision or symbols.

I never learned my times tables as a kid. Instead, I learned the patterns for multiplication by various digits. This left all but a few combinations, which I got in time.

[Hellbound]

Have to agree, I never memorized tables.

My brother expalined multiplication to me when I was in 1st grade, and although I was slow at first I gained a lot of speed with practice. Early understanding helped, I think. He related it to terms of counting:

"Five times four just means you count to five, four times, adding it all together." That was a pretty easy concept for me at that age.

Of course, I picked up certain patterns as I used it. Like any number times five equals half that number times ten (add an ending 0 and divide by 2). I could figure out the answers for those multiplication charts tests without memorizing the tables. Personally, I think memorization of the tables is rather silly...I'd rather know why certain numbers go at certain places in the table rather than just what the numbers are.

[wipeout]

I have an interest in mnemonic tricks, and they can be used for remembering huge amounts of data and manipulating it in your mind to do feats like vast mental calculations, memorizing numbers that are tens of thousands of digits long and... er... card-counting at blackjack. :D

Visual associations are the strongest form of mnemonics and that's what the books recommend but they are ludicrously time-consuming to create, so I generally ignore the books and go with verbal mnemonics if I have to use any.

Here's how I would do the times tables:

7 x 7 = 49
7 x 8 = 56

Use the first syllables of each word to create a new "word":

"sev-sev-for-nine"
"sev-eh-fif-six"

You then have a new "word" that is short enough to remember without too much practice.

Hope this helps. :)

[Earthborn]

If he likes to count on with his fingers, there are a few tricks to get him interested in multiplication.

The first is a trick that only works for the table of 9. Use both hands and spread the fingers. The fingers are numbered from 1 to 10, left to right. If necessary, use a pen to actually number the fingers. You can easily calculate any number from 1 to 10 multiplied by 9. Just fold the finger with the number you want to mulitply by 9, put a ring on it or remember which one it is, whatever appeals to your kid the most. Then count the fingers to the left of that finger: those are the tens. If you chose the first finger, there are 0 tens, if you chose the third there are 2 tens, the seventh there are 6 tens. Then count the fingers to the left of the special finger. Those are the ones: for a multiplication with 1, that's 9, with 3 that's 7 and for 7 that's 3. The answer is easy to derive: stick the tens and the ones together: 09, 36, 63, etc...

Also point out that if you have a table of nine from 1 to 10, the 6 through 10 is the exact mirror image of 1 through 5

Like this:
09 90
18 81
27 72
36 63
45 54

That should reduce the time spent filling in a complete table almost in half.

There is another trick with fingers that is more complex, and requires some memorization of the tables 0 through 4 and some addition. It helps to solve the more complicated tables from 6 through 10. The fingers of both hands are number differently from the other trick, so don't get confused. The fingers of both hands are numbered the same. The pinkies are 6s, ring fingers 7s etc... and the thumbs are 10s. Chose which numbers you want to multiply together and hold the corresponding fingers together. Now count how many fingers you hold together (that's always 2) and how many are below them. Together that makes your tens. To get the right answer, you must further take the number of fingers that are above the touching fingers from the right hand and multiply that to the number of fingers above the touching fingers from the left hand, add the answer to the tens you derived earlier. It's tricky to explain. Here's an example:

10 10
9 9
8 8
7 7
6 - 6

Here the pinkies (sixes) touch. There are only 2 fingers touching and below the touching ones, so that makes 20 total. There are 4 fingers above the touching fingers on the right hand, and 4 on the left hand, multiply them and you get 16. 20+16 = 36. and that's exactly the answer to 6 x 6.

10
9 10
8 9
7 - 8
6 7
6

Here the 7 and 8 touch, so we try to find the answer to 7 x 8. There are 5 fingers touching and below the touching ones. That makes 50. Above the touching finger on the left hand there are 3 fingers, above the right hand there are 2. 3 x 2 = 6, add that to 50: 50+6=56. And again that is the correct answer to 7 x 8.

You can also introduce your kid to Russian Peasant Multiplication (http://mathforum.org/dr.math/faq/faq.peasant.html), that requires no memorization, just doubling and halving and addition, or the similar Egyption Multiplication (http://mathforum.org/library/drmath/view/57542.html). I think it will boost his confidence if he can do the same thing in many different ways, provided you manage to introduce them in a playful manner.

[American]

Try my new super-memory-recall-prodigy system, I'm a Genius!, for 30 days.

Call now and you will receive my patented super-speed-reading-prodigy training system at no extra cost.

[Zep]

Make it about money, Paul, and see what happens. Zeplette had a similar issue. She was stymied by "What is 20 times 5?", but ask her how much money was twenty 5-cent pieces (OK, nickels), and the correct answer came back in a flash.

Or maybe that's just a "girl thing...

:bricks:

[TeaBag420]

Originally posted by CFLarsen
The kid has to learn one rule:

Add the neighbour (if any), add the ten (if any).

..................

Wouldn't it be simpler to stick a zero on the end and add the original number to that number? Fewer steps to remember, and it teaches more clearly about multiplying by 11.

[T'ai Chi]

Make sure your child understands that a*b = b*a, and then teach him only half of the multiplication table.

I'd also suggest making a poster of this table and putting it on a wall in his room, the kitchen, or some place in the house.

[CFLarsen]

Originally posted by TeaBag420
Wouldn't it be simpler to stick a zero on the end and add the original number to that number? Fewer steps to remember, and it teaches more clearly about multiplying by 11.

Yeah, but you see, you are going to reuse the rule (with one modification) when you want to multiply with 12.... ;)

This way, there is no middle calculations. You get the result in one swipe.

[SezMe]

Originally posted by Paul C. Anagnostopoulos
My son Luke hasn't managed to memorize his multiplication facts despite our having gone through them from beginning to end five or six times in two years. We're using flash cards.
I did my daughter "five or six times" per month for months on end. Your doing it in two years is NOTHING. My daughter at the time HATED it, but now (10 years later) we both chuckle about it. I still ask her what 8x7 is (her toughest one) and she still pauses before answering and then we both laugh.

Get tough. Rote is not wrong.

ETA: She is now in her second year of grad school.

[Paul C. Anagnostopoulos]

Luke is 12 years old. His reading disability is basically a complete lack of phonemic awareness. I think he reads mostly by sight, and is improving slowly. We have had someone tutoring him in phonics for years. [They are now teaching spelling rules in Language Arts; what a laugh for Luke.]

He knows his 2s by heart, his 5s fairly well, and his 9s by the finger trick. He can do his 4s fairly well by doubling twice. He knows songs for some of the others facts.

Wipeout, the "new word" trick won't work because he cannot pronounce nonsense words.

SezMe, you're probably right. At the very least, we'll just keep practicing. Part of the problem, of course, is that extra time is often better spent reviewing the current topic in his math class.

~~ Paul

[TeaBag420]

If he's 12 now, he's already at least two years behind. As for reviewing current topics, presumably sometime before graduation from high school he'll be expected to do division. Difficult if he can't multiply.

Presumably you've had him evaluated for non-reading related disabilities.

If he knows his 5's "fairly well" there's no reason (<-- my opinion) that he can't know them perfectly by memorizing. Perhaps you could start by simply having him recite (on demand) the products of 5 multiplied by 1 through 12:

5, 10, 15, 20, 25, 30, etc.

Give him 12 nickels if you think that will help.

So that's 5 and 2 taken care of. 10 is trivial, but make sure he can recite on demand.

My point is, he shouldn't have to think or figure things out. He has to MEMORIZE 72 products (because of the a*b thing) of which 12 (1 and 10) are trivial. He already knows his 2's and will shortly know his 5's, so you're down to 48 products.

You might want to look at this link, which endorses the finger method for 9 (I never learned it until today).

http://www.col-ed.org/cur/math/math02.txt

[JSFolk]

My stepkids were having trouble memorizing them, and what worked for us was having them do 20 minutes of basic multiplication before they could play any computer games. There's a site that will produce random sheets of multiplication (or division, addition, subtraction) online, so I would just tell them to keep doing new sheets until they ran out of time. They were slow at first since their school used the singing method of table memorization, but as they did more of it, they sped up. After they were doing a bit better, I had them start doing division too, and now they are both faster than most of the other kids in the class.

$.02

[CFLarsen]

Luke's "problem" is that he cannot slow down. His brain simply runs too fast. He is way too human to grasp the extremely simple concepts of math.

Humans are extremely good at processing large amounts of data, without really "thinking" about it. When we see something, we rarely are aware that we are processing data: It comes to us so fast, that it seems almost like magic. And we do it so freakin' effectively that we don't even know what we are doing.

But math doesn't work that way. If you need to understand math, you have to understand a very limited number of, but also very simple, concepts.

1+1=2.

7-2=5.

2*5=10.

25/5=5.

This is way too slow for the human brain. We are wired to determine if the animal tracks are leading to the prey, or away from it. We are genetically determined to decide whether or not the clouds will give us rain or not. If we are not capable of shifting through this massive body of data, we die.

So, catching on to basic math is a very straining task for us. It requires mental strength to slow down, because the nature of pure math is so simple. It follows very simple rules, that are used over and over again, at every step of the way. But we are not geared to a life of math. We are geared to a highly complex life, where we have to process a huge amount of information in a very short time, because if we don't, we will overlook the subtle hints, and the lion will eat us. Or we will overlook the signs that tells us where the prey is, so we won't eat. And starve to death.

Luke needs to slow down his thought processes. He is not "slow" in learning, he is too fast.

Just my 5 cents... :)

[American]

Originally posted by CFLarsen

Luke needs to slow down his thought processes. He is not "slow" in learning, he is too fast.

Sounds like when a restless person takes on a new task- they need to slow down and learn the rules before jumping into the job.

You might not totally believe this, but I wasn't always a great student. I fell way behind in the 2nd grade, and never really did well until middle school.

What changed? I felt a life-threatening, paralyzing, never-before felt fear of failure. I have no idea what I was scared of, except the teachers were terribly stern figures. And so to deal with that, I overcompensated academically and became this sick study-machine, a condition which remained until my 1st day of college when I discovered sweet, cold beer.

My advice is not always sage on matters with children, but my suggestion is to stay assignment-oriented. I believe "learning the fundamentals" is mostly a diversion that lets you avoid diving into the difficult work- a form of regression where you keep going back to easy lessons and what you know, dodging what needs to be learned. (People who sell study aids would surely disagree with me.)

[Paul C. Anagnostopoulos]

TeaBag said:
My point is, he shouldn't have to think or figure things out. He has to MEMORIZE 72 products (because of the a*b thing) of which 12 (1 and 10) are trivial. He already knows his 2's and will shortly know his 5's, so you're down to 48 products.
And we'll keep on chugging.

He has certainly gotten to the point with 6s, say, where he can do the flash cards in random order quite well. But then try it again a few months later and he doesn't know some of them. Has he forgotten, or is he just unsure of himself? Sometimes I think it's the latter.

~~ Paul

[American]

Originally posted by Paul C. Anagnostopoulos
And we'll keep on chugging.

He has certainly gotten to the point with 6s, say, where he can do the flash cards in random order quite well. But then try it again a few months later and he doesn't know some of them. Has he forgotten, or is he just unsure of himself? Sometimes I think it's the latter.


Is it too soon to rely on patterns? Example, 6x6 is actually "6x5 plus 6". It's also the same as "6x4 plus 12", etc.

[TeaBag420]

Originally posted by Paul C. Anagnostopoulos
And we'll keep on chugging.

He has certainly gotten to the point with 6s, say, where he can do the flash cards in random order quite well. But then try it again a few months later and he doesn't know some of them. Has he forgotten, or is he just unsure of himself? Sometimes I think it's the latter.

~~ Paul

What do you mean "a few months later"? Do you not have regular access to this child? What about "the next day" and "the day after that"? The flashcards, as you are applying them, clearly are not working. Deepsix that method for now. Repetitio mater studiorum est. Memorization on a daily basis won't kill him.

[Art Vandelay]

First of all, I don’t think you should be worrying about multiplication until your son has addition down. Also, I don’t think that flash cards are the way to go. They will teach your son to think of the products as separate facts, rather than different parts to one coherent whole. Thirdly, the best way to learn products is just to use them a lot.

Here are some games you can teach your son:
Get an empty multiplication table. Both players have a different color pen, and take turns writing the appropriate product in one of the boxes. First players to get five boxes in a row wins. This will teach him the relationships between different products.

Pick a number. Get some checkers, and start a stack of checkers. If your number is even, put down a white checker, and halve your number. If it is odd, subtract one, halve the result, and then put down a red checker. Then do the same with the new number, continuing until you reach zero.

Take a stack of checkers. Pick another number. Take the top checker off the stack. If it's white, do nothing. If it's red, add the original number. Before taking the next checker, double the current number. Continue until the stack is gone. When you're done, you've multiplied the two numbers!

Here's an example. Suppose the first number is 34.
34 is even, so a white checker goes down, then 34 is divided by 2.
17 is odd, so red checker goes down, subtract one, divide by two.
8 is even, so white checker goes down.
4 is even, white checker.
2 even, white.
1 odd, red.
0 means stop.

So you now have, from top, rwwwrw.

Now say the second number is 57.
Pick up the first checker. It's red, so add the original number (57). Now double it (114). Now pick up the next checker. White; do nothing. Now double (228), and pick up the next checker. White; do nothing. Double (456), and pick next. White; do nothing. Double (912), then pick up the next checker. It’s red, so add the original number (969). Double the number (1938), then pick up the next checker. It’s white, so do nothing. You’re out of checkers now, so stop.

Now go to a calculator. 34*57=1938. Isn’t that cool? Obviously, you’d want to start your son out with smaller numbers, but this method allows large numbers to be easily multiplied. And the second number can be represented with checkers, too, making the addition and multiplication even easier. To multiply a checker number by two, simply put a white checker at the bottom. To add two checker numbers, first flip them over. Take the top checkers from both stacks. If they’re both white, put one of them on the top of the sum stack. If one of them is red, put it on the top of the sum stack. If both of them are red, put a white checker on the top of the sum stack, and look through the summand stacks for the first white checker. Change that checker to red, and all the checkers above it to white. If none of the stacks have any white checkers, put a red checker at the bottom of the tallest stack (if they are the same size, pick either), and change the rest of the checkers in that stack to white. Once one stack runs out, transfer the remaining checkers to the sum stack, top first.

With a few dozen checkers and ample table space, your son should be able to multiply two, even three, digit numbers with minimal thought.

[davefoc]

I have worked with my first daughter on her multipiclation tables by getting her to learn to count in the different numbers.

This did a few things, I thought:
It reinforced her adding skills
It helped to make it clear exactly what multiplication was
It gave her an easy way of calculating the answer if she forgot a particular multiplication fact
I also went over some of the tricks that have been discussed for various numbers above. Apparently they don't always teach those.

[RamblingOnwards]

I'm clearly being a heretic here, but...

If he understands the principles involved, why not simply buy him a calculator?

In my final physics exam I used a calculator for 2*4, because the risks of getting it wrong (answer of 6, say) under that kind of stress was higher than I was willing to take. I never bothered learning the 8 * table (although I can quote pi to 12 decimal places - I read 'Starman Jones' by Heinlein and it sounded cool). It really hasn't affected my adult life.

What exactly is your end aim here?

[wipeout]

Originally posted by Paul C. Anagnostopoulos
Wipeout, the "new word" trick won't work because he cannot pronounce nonsense words.

I see. Well, the standard way to memorize numbers is to make visual associations (like "7" is a golf club, "8" is a snowman, etc.) and imagine a little mental story involving them in the right sequence but I don't think it's worthwhile for learning times tables.

On a different point, the recommended way to memorize for people studying any subject at any level is to review what you just studied straight away, then a hour later, then a day later, then a week later, and so on and to have a regular schedule.

[steenkh]

I had big troubles learning to calculate when I was a child. I was forced to do extra courses of arithmetic during my summer holidays, but it just kept me a little bit above the lowest possible grades.

I started on a Trachtenberg book on my own, and though it never really helped my grades, for the first time I had a feeling that it might be possible to have fun with arithmetic. Like Claus, I also found it fascinating to be able to multiply figures of any length with 11. come to think of it, I tried to learn multiplication, but without noticing it, I learned addition!

The other tip about imagining you are using coins was also a great help.

Eventually, when I got to the university, the calculator had become common, and I never had to look back. Life has been so much easier with this tool. What has been important at school was to be able to understand the mechanism and concept of calculations, so that you can spot when there has been a typing error on the calculator.

It is interesting that at 50 years age, I am actually much better at using my head for calculating than when at school, although I mostly use a calculator. When the toil and sweat has been taken out of calculation, it becomes much easier to absorb some of the knowledge too.

[Paul C. Anagnostopoulos]

TeaBag said:
What do you mean "a few months later"? Do you not have regular access to this child? What about "the next day" and "the day after that"? The flashcards, as you are applying them, clearly are not working. Deepsix that method for now. Repetitio mater studiorum est. Memorization on a daily basis won't kill him.
So I find out he has forgotten them 14.5 days later, or 26.4 days later, what difference does that make? When I said we have gone over his facts five or six times in two years, I did not mean that we went through the 6s once each of those five or six times. I meant we worked on them for a week or two each of those five or six times, until he had them down. What it means that he forgets some of them in between each pass, I do not know. Hell, it might be related to his reading disability and nothing is going to fix it.

The suggestions that it is time for a different approach are certainly reasonable. I'll put together some other way to approach them.

Rambling said:
If he understands the principles involved, why not simply buy him a calculator?
He has several and uses them. The problem comes when he has to figure out, say, the prime factors of a number. It's quicker if you know by heart which numbers divide that number. But, it's only quicker. A prime factor sheet does the trick, too.

Interesting sidelight: We had a meeting with all his teachers yesterday to talk about ways to help him. Fantastic meeting; everyone is pulling together and coming up with extra things they can do to help. I suggested to his language arts teacher that we simply give up on spelling. She was a bit taken aback by that suggestion.

I also found out that is science teacher is an immunologist. Pretty cool for a sixth grade science teacher.

~~ Paul

[davefoc]

Correction here it was actually RamblingOnwards that made the "why not buy him a calculator" comment.

I tend to disagree with it.

I think a solid foundation in arithmetic is probably next to reading is the most important academic base skill learned in elementary school. Learning a few rote facts is not that much of a burden usually and it is in my mind an important part of the basic arithmetic skill set.

I recently tutored a girl in algebra that had not mastered basic arithmetic skills and the difficulty she was having was profound. But skip algebra or other advanced math endeavors which are arguably only important for technical careers. Basic arithmetic also is a requirement for almost every career in the modern world and for that matter it is just pretty much a basic required life skill.

[flume]

I wonder if it would help if he did a lot of practice problems or flash cards using the calculator. He wouldn't have to struggle to remember, but he'd see the same answers over and over, and he'd have to write them down which would reinforce the memory.

When I was a kid my dad played cribbage with me, and I think that helped my math skills. Cribbage doesn't involve multiplication but it does reinforce addition. And it looks at a set of numbers in a different way. But it might not be enjoyable to a kid who has trouble with math.

[RamblingOnwards]

Originally posted by davefoc
I think a solid foundation in arithmetic is probably next to reading is the most important academic base skill learned in elementary school.

I agree with this bit (which is why I added 'if he understands the principles')

Learning a few rote facts ... is in my mind an important part of the basic arithmetic skill set.

but not neccesarily with this.

I agree with steenkh that a vital part (perhaps even the most vital part) of arithmetic skills is developing an idea of what the end result needs to be. This may require some memorisation. But all the memorisation in the world is completely useless if you do not understand the principles - and it may serve to hide the fact that you don't.

I tutored a girl in maths who was completely awed that I could predict fairly acurately whether a sequence converged or diverged before working it out rigourously. We took some time out at that point to discuss guestimating answers, which I was horrified to discover was a new concept to her. She'd been under the impression that I'd memorised all those sequences. I'd been under the impression she understood the various indicators.

If Paul's son knows that 11*13 is larger than 100, but smaller than 300, and that 116*7889 is somewhere in the order of 1 000 000, then I wouldn't be too worried if he can't recite what 6*8 is.

[roger]

Originally posted by davefoc
I think a solid foundation in arithmetic is probably next to reading is the most important academic base skill learned in elementary school. Learning a few rote facts is not that much of a burden usually and it is in my mind an important part of the basic arithmetic skill set. I know what you are saying, and mostly agree, but let me offer a counterexample. One of the most brilliant mathematical minds I ever came across (fair to point out - I haven't rubbed shoulders with Harvard Illuminati, or such) was a math professor in college. You could ask him a question to a problem he didn't know the answer to, he'd gaze into the air, write down the answer, and then say "now, how would I prove that... hmm, theorem X seems to apply, but how would we get to that..." and he'd weave a proof together from both ends, meeting in the middle. But if somewhere along the way he had to do arithmetic, he'd have to helplessly ask the class the for the answer. He just could not do arithmatic. 15 - 8, he'd have to ask, or stand there and count it out. I'm serious, not exagerating.

I don't know Paul's son, but some people just aren't wired for this kind of memorization, but that doesn't preclude great achievements, even in mathmatics.

[davefoc]

roger and RamblingOnwards,
I think there is something to what you are saying.

Still, when doing straightforward algebra problems that are usually made up to involve integers or simple fractions a failure to have simple math facts memorized can make doing them surprisingly difficult, even if one has got the principles nailed.

But in this case, the girl was significantly below her grade level with her grasp of basic arithmetic principles on such things as fractions and negative numbers. I suppose this supports your view a bit in that if people had focused on basic principles excluding math facts she might have been better off.

But notice to acquire the next step up in the understanding of basic arithmetic skills students are normally required to problems that require a knowledge of basic math facts. Now I suppose you could still argue that the student should just rely endlessly on the calculator for the basic math facts. But to me, this seems like one is needlessly complicating the next step of learning because one just hasn't made the effort to learn the basic math facts.

[Paul C. Anagnostopoulos]

Oops, sorry Davefoc. Attribution corrected.

~~ Paul

[Earthborn]

Originally posted by Paul C. Anagnostopoulos
His reading disability is basically a complete lack of phonemic awareness. I think he reads mostly by sightThis could also mean that he has difficulty reading sums that are written down because he cannot internalize the symbolic meaning of the sequence of mathematical symbols. Perhaps it will help him if he can visualize the calculation. For that he'll probably benefit from a set of Cuisenaire rods (http://www.homespunkids.com/browse.asp?cat=512&path=500,512). They can also help with addition, subtraction, division, understanding the metric system and lots more. They are fun and colourful too, I still love mine. :cool:

[TeaBag420]

"So I find out he has forgotten them 14.5 days later, or 26.4 days later, what difference does that make? When I said we have gone over his facts five or six times in two years, I did not mean that we went through the 6s once each of those five or six times. I meant we worked on them for a week or two each of those five or six times, until he had them down. What it means that he forgets some of them in between each pass, I do not know. Hell, it might be related to his reading disability and nothing is going to fix it."

I say again, have you had him evaluated for NON-READING RELATED disabilities? Because it doesn't take a normal person a week to learn to multiply by six. There is clearly something not usual about your son. Don't waste another two years. You need to bring in professional help that you aren't finding at the school or here.

And again, no paper, no pencil. You recite to him, he recites to you. No reading involved. Yes, he will have to read the problems in school, but if he doesn't know the answer it doesn't matter if he can read the problem or not. No tricks, no fingers, no multiplying by 11, no flashcards. And get a qualified professional to evaluate him for non-reading related disabilities.

I think you indicated he was also having trouble with addition. Can he count? Aloud, with his eyes closed?

What grade level is he reading at?

[epepke]

John Allen wrote a book called Innumeracy. Much of it is polemic, but it has some good math tricks.

Martin Gardner wrote a kid's book called Science Puzzlers, which has some good ones, too.

[Paul C. Anagnostopoulos]

TeaBag said:
I say again, have you had him evaluated for NON-READING RELATED disabilities? Because it doesn't take a normal person a week to learn to multiply by six. There is clearly something not usual about your son. Don't waste another two years. You need to bring in professional help that you aren't finding at the school or here.
Teabag, you appear to be assuming that we are idiots. We've had him evaluated up the yin-yang. Latest evaluation is "non-verbal learning disability." No kidding. We have listened to countless suggestions from all sort of professional and nonprofessional people. We implement many of the suggestions, including lots of tutoring. I doubt there is a silver bullet.

He can count fine. I'll try it with his eyes closed.

He is reading at about 3rd-4th grade level. Comprehension is fine.

I think there is a psychological factor at work, too.

~~ Paul

[marting]

Originally posted by RamblingOnwards
I agree with steenkh that a vital part (perhaps even the most vital part) of arithmetic skills is developing an idea of what the end result needs to be. This may require some memorisation. But all the memorisation in the world is completely useless if you do not understand the principles - and it may serve to hide the fact that you don't.

That's a very key point. Understanding approximations may well have declined with the advent of cheap calculators. Slide rules and logarithms used to be regularly used tools and one side effect was developing a good sense of estimation.

A great many things can be estimated far quicker than getting out a calculator and it really helps prevent gross errors.

This doesn't apply to all professions though. An accountant needs to be able to answer the question "What is two plus two?" correctly. The proper answer being "What do you want it to be, boss?"

-Just kidding, just kidding.....

[Zombified]

Originally posted by marting
That's a very key point. Understanding approximations may well have declined with the advent of cheap calculators. Slide rules and logarithms used to be regularly used tools and one side effect was developing a good sense of estimation.

A great many things can be estimated far quicker than getting out a calculator and it really helps prevent gross errors.
Understanding approximations, estimating magnitude, and particularly being aware of precision are still extremely important concepts. Another key skill is dimensional analysis - very useful for both estimation and checking your algebra. This stuff still gets taught, and is still powerful and useful even for advanced subjects.

Of course, Paul won't have to teach some of that stuff for a while yet...

[marting]

Originally posted by Zombified
Understanding approximations, estimating magnitude, and particularly being aware of precision are still extremely important concepts. Another key skill is dimensional analysis - very useful for both estimation and checking your algebra. This stuff still gets taught, and is still powerful and useful even for advanced subjects.

Of course, Paul won't have to teach some of that stuff for a while yet...

Yup, one of the few benefits of English units is it necessitates a grounding in dimensional analysis. Too bad too many K-12 teachers outside of the sciences don't get it across adequately. Teaching it earlier would help understanding.

[TeaBag420]

Originally posted by Paul C. Anagnostopoulos
Teabag, you appear to be assuming that we are idiots. We've had him evaluated up the yin-yang. Latest evaluation is "non-verbal learning disability." No kidding. We have listened to countless suggestions from all sort of professional and nonprofessional people. We implement many of the suggestions, including lots of tutoring. I doubt there is a silver bullet.

He can count fine. I'll try it with his eyes closed.

He is reading at about 3rd-4th grade level. Comprehension is fine.

I think there is a psychological factor at work, too.

~~ Paul No, I'm not assuming that you (I suppose "we" means you and Luke's mother) are idiots, I am merely reposing a question you didn't answer the first time I asked it.... have you had him evaluated for non-reading (not "non-verbal") learning disabilities? There, third time.

What did the evaluator tell you "non-verbal learning disability" means? Please feel free to provide a link if it's a long explanation. If he didn't explain it to you so you could explain it to another person you got cheated.

Can he count backwards from 10? From 100? By all means have him do it with his eyes closed.

Memorization. Recitation. No flash cards, no reading, no writing. Have you tried it yet?

I'm sorry if some of this sounds harsh, but your coming here for advice indicates that you're at the end of your tether and you're ready to think outside the box, and I'm about as far outside the box as anyone you've never met.

[Paul C. Anagnostopoulos]

I do not know how to answer your question about the evaluations. He has been evaluated on many dimensions. The last neuropsychologist said "non-verbal learning disorder," but our research led us to disagree. Unforunately, her husband passed away and she has effectively disappeared. He certainly has some issues other than phonemic ones.

If you have a specific suggestion for evaluation, by all means tell me.

How are you proposing he memorize his multiplication facts? By going over them verbally with no flash cards? Are you trying to eliminate the visual cues?

~~ Paul

[Paul C. Anagnostopoulos]

Aha, here we go. From http://www.nldline.com/

What is NLD? Nonverbal learning disorders (NLD) is a neurological syndrome consisting of specific assets and deficits. The assets include early speech and vocabulary development,
A little early, but nothing extraordinary.

remarkable rote memory skills,
No.

attention to detail,
No.

early reading skills development and excellent spelling skills.
No.

In addition, these individuals have the verbal ability to express themselves eloquently.
Fine, but I wouldn't say eloquent.

Moreover, persons with NLD have strong auditory retention.
Not sure, really.

Four major categories of deficits and dysfunction also present themselves:

motoric (lack of coordination, severe balance problems, and difficulties with graphomotor skills).
He certainly has some fine motor deficit, but not this bad.

visual-spatial-organizational (lack of image, poor visual recall, faulty spatial perceptions, difficulties with executive functioning* and problems with spatial relations).
No.

social (lack of ability to comprehend nonverbal communication, difficulties adjusting to transitions and novel situations, and deficits in social judgment and social interaction).
To some degree, but not severe. You'd just think he is a bit of a loner.

sensory (sensitivity in any of the sensory modes: visual, auditory, tactile, taste or olfactory)
What is "sensitivity"?

~~ Paul

[Soapy Sam]

Paul- Perhaps you could concentrate on something he is able to remember (and recall) well. Then associate the numbers with that.Cars, sports, whatever.

Rhyming and chanting may help too- the mnemonic /poetic approach.

In primary school, we group-chanted our tables up to twelve times, with each table itself reaching twelve values.

"Six ones are six, six twos are twelve, six threes are eighteen...
six twelves are seventy two." Heavy stress on the "are" (stamp feet).. The monotonous rythm fixed it in the old bonce, to the extent that I occasionally find myself mentally chanting when counting. This suited me, because I'm a verbal thinker. Frankly, I still have no idea why six threes are eighteen, but I recall the fact, just as I do dates.

On the other hand, if he is a visualiser, maybe substituting dot patterns (like playing cards) for the Arabic numerals would help?

[TeaBag420]

Yes, verbal repetition, no visual cues. What Soapy Sam said. It sounds like you haven't tried it, so what do you have to lose after two years? Try it for a week maybe.

Although I prefer "Six times one is six." I think either phrasing would work.

[Paul C. Anagnostopoulos]

Okay, tomorrow we'll start working on 6s with no flash cards. I'm game.

Teabag: Counting backwards with eyes closed was no problem. Of course, after a couple of tests, Luke basically said "Hey, screw this, this is a test." That 'bout summarizes his opinion of all tests.

~~ Paul

[bpesta22]

Hey Paul

Outta lurk mode here.

It's odd but I may be one of the world's leading experts on memory for solved multiplication answers.

o Pesta, B., Sanders, R., & Murphy, M. (2001). Misguided multiplication: Creating false memories with numbers rather than words. Memory & Cognition, 29, 478-483.

o Pesta, B., Sanders, R., & Murphy, M. (1999). A beautiful day in the neighborhood: What factors determine the generation effect for simple multiplication problems? Memory & Cognition, 27, 106-115.

o Pesta, B., Sanders, R., & Nemec, B. (1996). Older adults’ strategic superiority with mental multiplication: A generation effect assessment. Experimental Aging Research, 93, 155-169.


There's actually a fairly large literature on "mental math"-- most all of it focuses on how people learn multiplication facts.

In fact, there's a guy at my school-- CSU-- who sorta started research in this area.

I didn't read the suggestions you got here, but there is an article by Siegler, "development is destiny" that really offers a cool theory on how kids learn simple multiplication problems.

I'll see if I can find a link, but in the mean time, you're son is not forgetting the answers (assuming he indeed learned them at one time), he's probably just having trouble retrieving them.

Either the strength between the operands and the answers is not strong enough, or he's getting interference from other (probably table related) wrong answers.

For example, consider these two lists...the goal is to remember the answers:

7 x 3 = ?
2 x 4 = 8
4 x 7 = 28
6 x 5 = ??

There's a pretty large effect on memory for the solved answers above (21 and 30) versus the ones with the answers supplied (8, 28).

So, flash cards is a real good way to learn math facts (it capitalizes on the effect above-- versus drill and practice).

Interestingly, the memory effect for solving is even bigger when the problems are relatively difficult (e.g., 8 x 7 = ?? versus 2 x 3 = ??)

And, solving any one problem results in strenghtening of associations between the problem and the answer produced (whether right or wrong) as well as associations between other table related answers!

So look at this list and compare it with the one above

3 x 5 = 15
2 x 4 = 8
4 x 7 = 28
6 x 5 = ??

The only difference is the first problem. In one of the papers above, though, I show that memory for the third problem's answer-- 28-- is better in the first list than in the second list.

This is because in the first list, you solve 7 x 3 = ??, for 21, which is a table-related near neighbor of 28. So, as you solve 21, activation spreads from the answer to it's neighbors and partially lights up 28. By the time you get to problem 3, the answer--28-- is already partially activated which leads to better memory for it, then compared with people in using the control list.

Ah, I'm starting to ramble, but before I forget, I don't think memory tricks are a good idea for math facts (unlike any other fact we need to know). Math facts are inter-related, and the need to be "hard wired" in the brain.

Brutal drill and practice and flash cards are the way to go.

Overlearn them if you have to.

Finally, in the siegler article I'll try to find, getting the answer wrong is really bad for learning because a memory trace is formed that links the problem to the wrong answer.

When you see the problem again, this wrong answer memory trace will compete with retrieval for memory of the right answer.

The key practical application is let you kid do things like verify, count with fingers, use paper for intermediate steps, because getting a wrong answer begets getting more wrong answers.

Best

B

[alfaniner]

Ever tried using Legos? I always thought that would be a good aid for teaching fractions, anyway. Might work for multiplication, too.

[Brown]

Here's a pretty neat multiplication "trick."

Learning the "squares" helps you learn some nearby products, too.

Take the square of 5. 5 times 5 is 25. Now look at the numbers on either side of 5, 4 and 6. 4 times 6 is 24, one less than 5 times 5.

7 times 7 is 49. 6 times 8 is one less, 48.
8 times 8 is 64. 7 times 9 is one less, 63.

Algebraically, it is easy to prove that this always works.

This can be a confidence-builder. You can tell the child, "14 times 14 is 196," then ask "What is 13 times 15?" and the child will know right away (even though it would take most adults quite a long time to compute the product).

[TeaBag420]

Originally posted by Brown
Here's a pretty neat multiplication "trick."

Learning the "squares" helps you learn some nearby products, too.

Take the square of 5. 5 times 5 is 25. Now look at the numbers on either side of 5, 4 and 6. 4 times 6 is 24, one less than 5 times 5.

7 times 7 is 49. 6 times 8 is one less, 48.
8 times 8 is 64. 7 times 9 is one less, 63.

Algebraically, it is easy to prove that this always works.

This can be a confidence-builder. You can tell the child, "14 times 14 is 196," then ask "What is 13 times 15?" and the child will know right away (even though it would take most adults quite a long time to compute the product). If you're going to tell them what 14 times 14 is, why not just cut to the chase and tell them what 13 times 15 is? The problem is that the kid can't even do simple multiplication. This trick only builds confidence in the trick... it doesn't help the kid do multiplication.

13 times 15 is 150 + 45. It took longer to type that than to calculate it. THAT's multiplication, and a more fungible method.

To build confidence, have Luke memorize his 15, 20, and 25 times tables (15 up to 75, 20 and 25 up to 100). It's only 11 products, and might give him a framework to hang the rest of it on. Quick, what's 2 times 50 ? There's another piece of the framework.

You're trying to help, but all the checkers, legos, and tricks in the world aren't going to help this kid learn his times tables. That checkers idea actually made my brain hurt, and I already know my times tables.

[Paul C. Anagnostopoulos]

But Teabag, we don't know why he can't seem to memorize his multiplication facts. Therefore, saying that tricks won't help is not necessarily true.

For example, if he isn't memorizing them because he doesn't want to, then tricks might or might not help by making it more interesting.

If he isn't memorizing them because there is a link to his phonemic problem, then tricks might be just the trick.

If he needs to memorize them "by sight," as with words, then flash cards might be absolutely necessary.

Interesting observation: First, remember that he doesn't seem to know some of his addition facts, either. So this afternoon he had to add:

9...
8...
1...
+ 8...

He did it in a few seconds. I asked him how. He said "9+1 is 10. 2 x 8 is 16. 16 + 10 is 26."

Figure that one out, boys and girls.

~~ Paul

[TeaBag420]

Paul,

I see nothing wrong with that approach, and it doesn't show that he doesn't know some of his addition, rather it shows that he does know how to chunk a problem into pieces.

Instead of 9 plus 8 is 17, 17 plus 1 is 18, 18 plus 8 is 26, he just said, "9 plus one is trivial, and at least I can multiply by 2, so I just have to add 10 and 16." It's not how I would have done it, or maybe it is; the important thing is he solved the problem just by looking at it, and in a way that makes perfect sense. You seem to be implying that he has a long term memory problem, at least for part of math. I'm out of suggestions at this point.

As far as tricks go, the trick presented boiled down to "I'm going to say a number and I want you to subtract one from it." That trick doesn't teach multiplication, unless the student can memorize a table of squares, which means memorizing a multiplication table. So you're back to square one. The trick is useless.

[Cabbage]

Paul, I haven't read through this entire thread, so maybe this has already been mentioned, but I thought I would mention this interesting way of multiplying using your fingers. Hopefully it may be of some help:

http://www.geocities.com/Heartland/Lake/3262/maths/multiplying_fingers.htm

[epepke]

Originally posted by Paul C. Anagnostopoulos
He did it in a few seconds. I asked him how. He said "9+1 is 10. 2 x 8 is 16. 16 + 10 is 26."

Figure that one out, boys and girls.

~~ Paul

I'm not sure what you mean by "figure that one out," but that's exactly how I solved the problem.

[TeaBag420]

Originally posted by Paul C. Anagnostopoulos
But Teabag, we don't know why he can't seem to memorize his multiplication facts. Therefore, saying that tricks won't help is not necessarily true.

For example, if he isn't memorizing them because he doesn't want to, then tricks might or might not help by making it more interesting.

If he isn't memorizing them because there is a link to his phonemic problem, then tricks might be just the trick.

If he needs to memorize them "by sight," as with words, then flash cards might be absolutely necessary.

Interesting observation: First, remember that he doesn't seem to know some of his addition facts, either. So this afternoon he had to add:

9...
8...
1...
+ 8...

He did it in a few seconds. I asked him how. He said "9+1 is 10. 2 x 8 is 16. 16 + 10 is 26."

Figure that one out, boys and girls.

~~ Paul


If, if, if. You don't address recitation, which makes me think you haven't tried it. If his teachers and you haven't succeeded, you may have to look into outsourcing (which term is currently being misused as a synonym for "offshoring") the project. Get back to me when you've worked on recitation for a couple days. Show me you're serious. Chin up, soldier. Denny Crane.

[T'ai Chi]

I'd also suggest playing many games that involve math or math-like thinking.

Did I already mention Hog (http://www.amstat.org/publications/jse/v11n2/feldman.html)? It is based on adding, but it could probably be modified to be based on multiplication.

I did a search and found some multiplication games (http://www.multiplication.com/interactive_games.htm) that are pretty fun. :)