How well do you know the multiplication table?

[Mark44]

The people who learn about FOIL either haven't been doing this multiplication for the last 8 to 10 years, or maybe aren't able to generalize from, say, 23 * 15 to (a + 3)(b + 5).

Does it require more thought to extend the multiplication in this way, than it does learning an mnemonic built around a very special case?

Yes, it requires much more thought to be able to extend multiplication from the arithmetic of multiplying integers to products of binomials. And the multiplication of binomials is by no means something that I would consider a special case. I would consider multiplying, say, a 7-term polynomial by a 4-term polynomial a special case.

And so what? Virtually any endeavor, if you want to get good at it, requires a lot of time on the basics. If you want to learn how to play the piano, you have to learn the names of the keys so that you can read music sheets. After that, there's a lot of time spent on practice, building "muscle memory" so that you can play a tune without having to think about each and every note. The same is true for sports of all kinds, with a lot of time spent hitting a ball, or throwing a ball into a basket, or whatever. The more you perform these actions, the more fluid and automatic they become. The same is true for arithmetic and mathematics at a higher level, I believe. If you aren't sure whether it's 6 x 9 = 63 or 54, it becomes much more difficult to do quick and dirty approximations as sanity checks on more difficult problems.

I don't consider memorizing tables without motivation a viable part of the basics. The basics would be working out the arithmetic , then as you become experienced you'll naturally develop a working memory for these calculations.

How do you "work out the arithmetic" if you don't have a firm grasp on multiplying two single-digit numbers? Earlier you said

solve simple equations like 5*6 + 4 = 34

First off, 5*6 + 4 is not an equation -- it's an expression, and there's really nothing to solve. If you haven't spent some time memorizing things such as 5 times 6, you'll be stuck right away.

And the "and so what" part I guess would be that the class time spent memorizing tables could be better spent working problems or motivating mathematics in general.?

This is like saying a football team would be more successful just playing football, and not spending valuable practice time doing running drills, blocking practice, running though tires, and all that other boring stuff. I don't think any successful coach would agree with you.

 

[Pepper Mint]

How do you "work out the arithmetic" if you don't have a firm grasp on multiplying two single-digit numbers?

True, and I think kids should learn 1-20 multiplication tables by heart.
For 23 x 15, I would think it is like 20 * 15 + 3*15
113*517 = 100*517 + 10*517+ 3*500+3*10+3*7

 

[Jonathan Scott]

At school we learned full multiplication tables up to 12, but I learned quite a few higher values up to 20. I was very good at fast mental arithmetic until I started using hexadecimal a lot, at which point all my short cuts got muddled with hexadecimal stuff!

I'm quite good at mental long multiplication and division, limited primarily by my ability to remember the intermediate results, and I often find myself estimating things including calculating square roots and similar in my head. I rely a lot on using rough figures and percentages, so for example if I'm trying to do a square root and my first guess is 10% too small, I increase my guess by about 5%.

 

[russ_watters]

I never saw the importance of memorizing multiplication tables, after enough practice using them simplifying other mathematics they become burned into your brain anyway.

I agree with Mark and think you are overstating the efficacy of memorizing-by-doing. The reason memorization is done by tables and done separately from solving problems is that they require different numbers of repetitions to learn, so it is more time efficient to separate them. If, for example, something gets burned into your memory after 10 tries, then you would need to work 1000 problems of the 2x4+3=?? to memorize the 100 combinations even though you could learn the method by doing 10 and separately (first) learn the multiplication table. It would require much, much more time with little extra value to work the other 990 problems.

 

[Ben Niehoff]

I think as early as 2nd or 3rd grade, we drilled multiplication tables up to 12x12, with loads of timed tests, etc. Today, they are all still more or less "programmed in", and I use them all the time.

 

[Student100]

Yes, it requires much more thought to be able to extend multiplication from the arithmetic of multiplying integers to products of binomials. And the multiplication of binomials is by no means something that I would consider a special case. I would consider multiplying, say, a 7-term polynomial by a 4-term polynomial a special case.

I actually think it's easier - to each their own.

Are the prolific cases of binomial multiplication a result of interesting mathematics, or simply a byproduct of what it's assumed students know how to do? I would also consider multiplying a 7-term polynomial by 4-term polynomial a special case. However, I would consider a method that allows students to multiply n-term polynomials by n-term polynomials a more useful method than one strictly restricted to binomials, especially when the extension of multiplication needed is almost trivial.

How do you "work out the arithmetic" if you don't have a firm grasp on multiplying two single-digit numbers?

The way we're taught to multiply at young ages? You read 5*6 as 5+5+5+5+5+5 or conversely, 6+6+6+6+6. It should be assumed students are familiar enough with addition to be able to work this out without needing to resort to tables.

First off, 5*6 + 4 is not an equation -- it's an expression, and there's really nothing to solve. If you haven't spent some time memorizing things such as 5 times 6, you'll be stuck right away.

The way in which I wrote it is what I would consider an equation, and it would be perfectly valid to give a student such a problem. They can learn to reduce the left hand side to right hand side and show they're truly equal. They also wouldn't be stuck without instantly knowing single digit multiplication, see above.

I agree with Mark and think you are overstating the efficacy of memorizing-by-doing. The reason memorization is done by tables and done separately from solving problems is that they require different numbers of repetitions to learn, so it is more time efficient to separate them. If, for example, something gets burned into your memory after 10 tries, then you would need to work 1000 problems of the 2x4+3=?? to memorize the 100 combinations even though you could learn the method by doing 10 and separately (first) learn the multiplication table. It would require much, much more time with little extra value to work the other 990 problems.

Mark knows more about mathematics, and mathematics education than I ever will. So I concede to his experience as an educator. I just know of my personal, subjective, experiences. My memory functions much better from working out problems than it ever did at memorizing from tables. In fact, I could probably transcribe various tables thousands of times and it would less efficient than working out problem sets that use the same information.

 

[Humanlimits]

Around 4th grade I used flash cards. Everything up to 12x12 comes pretty fast. For larger numbers I tend to add multiples together or just work from combinations I recall. I actually never learned to do arithmetic by hand very well, so I tended to do it mentally.

Now I'm studying the The Trachtenberg speed system of basic mathematics, since I saw it recommended here.

 

[vela]

Let me explain a bit better,when I was young we were giving laminated cards with both division and multiplication tables and told to memorize them, before we ever knew what multiplication or division was. I still don't see the benefit of this practice.

Division tables? I've never seen one of those.

When I was taught multiplication, we weren't given tables that we were simply told to memorize without knowing what multiplication meant. It was explained that, for example, 5×6 = 5+5+5+5+5+5. I'd guess that if kids weren't told to memorize the tables, a lot of them, when asked what 5×6 equaled would resort to adding six 5s every single time. It's more efficient pedagogically to simply tell students that they need to memorize the table. And the table reveal patterns which demonstrate, for example, that multiplication is commutative.

I get what you said about FOIL. The one I hate is SOHCAHTOA. I don't think they're complete garbage, but I think too many students use them as a crutch. I once tutored an engineering student who was in his junior year at the time, and he was using the mnemonic to recall how to calculate the sine of an angle. I pointed out to him that after the years of study he had already completed and the number of times he had already used trig functions, he should have already committed what sine, cosine, and tangent were to memory and that he shouldn't have to figure it out each time.

 

[Sophia]

We were taught MT up to 10 in the second grade and then squaring of numbers up to 20 in middle school (like 13x13) when we learned square roots. We weren't allowed to use a calculator until 9th grade.
The teacher explained meaning of multiplication before giving us the table to learn. I think we counted chocolate squares in whole chocolate :) I think it was easier to just memorize it because as was said above, it would take ages to remember the results only by doing exercises. But of course, solving as many problems as possible increases memory and deepens understanding of the operation. So both are important.
When I started to memorize the table I had problems with certain combinations, such as 7x8, 6x7 and 6x8 and I had to drill them so much that I eventually memorized them best. Nowadays, I know the result of these faster than the others, which didn't cause me problems when learning :)

 

[Mark44]

The way we're taught to multiply at young ages? You read 5*6 as 5+5+5+5+5+5 or conversely, 6+6+6+6+6. It should be assumed students are familiar enough with addition to be able to work this out without needing to resort to tables.

This is all fine for, say, third graders, but later in the fourth grade or so, when they have to do multiplication with decimal values, such as 5.4 * 6.3, this technique will be extremely cumbersome. They will need to do four multiplications with single digits -- 3 x 4, 3 x 5, 6 x 4, and 6 x 5, and line everything up and put the decimal point in the right place. Having to write 3 x 4 as 4 + 4 + 4, 3 x 5 as 5 + 5 + 5, and so on seems like a very inefficient use of their time. Slightly more difficult problems could take a student who doesn't know the times table an entire sheet of paper for a single problem.

The way in which I wrote it is what I would consider an equation, and it would be perfectly valid to give a student such a problem.

Here's what you wrote:

solve simple equations like 5*6 + 4 = 34

You're right -- this is an equation, but it's not something you would "solve," and that's what threw me off. You could ask the student to show that the left and right sides were equal, and I think this is what you had in mind, but it seems terribly artificial. A somewhat less artificial problem would be to ask the student to compute 5 * 6 + 4.

They can learn to reduce the left hand side to right hand side and show they're truly equal. They also wouldn't be stuck without instantly knowing single digit multiplication, see above.

And if the student in question doesn't know the "addition facts" to be able to add 5 + 5 + 5, I suppose he could count on his fingers.

My point is this: Having a few basic facts committed to memory, such as the pairwise sum of single-digit integers and the pairwise product of single-digit integers, isn't a hardship to most students, and is an important foundation for arithmetic and higher mathematics beyond that.

 

[Tosh5457]

I knew it in the first years of school, nowadays I just calculate them. It's easy if you make it into parts, 7*3 + 7*3 and it's easy.

 

[Charles Kottler]

If you're going to be doing arithmetic often it is worth learning tables to save time. At school I was taught by wrote - two twos are four, three twos are six etc. up to 12x12. These stuck with me and when I was about 30, while training for a half marathon I extended this up to 20x20 - not much else to do while bashing out the miles at night :).

 

[Kevin McHugh]

When I was 4th grade, my father drove me to school. He made me recite my 12 x 12s every morning. I also had to memorize perfect squares up to 20.

 

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[WWGD]

When I was 4th grade, my father drove me to school. He made me recite my 12 x 12s every morning. I also had to memorize perfect squares up to 20.

I knew my 1x1s table by 3!! :).

 

[Mark44]

I knew my 1x1s table by 3!! :).

That's quite old! 3! = 6, and 3!! = 6! = 720. I don't really believe you're that old.
It shouldn't take more than a year or two to learn all the products in that table...

 

[micromass]

That's quite old! 3! = 6, and 3!! = 6! = 720. I don't really believe you're that old.
It shouldn't take more than a year or two to learn all the products in that table...

Actually, the double factorial is defined differently than that https://en.wikipedia.org/wiki/Double_factorial
So $3!! = 3$, which is in line with his statement.

However, I think this notation for double factorials is pretty horrible.

 

[Mark44]

You're right, micromass. I was interpreting 3!! as (3!)!

 

[DaveC426913]

I did this in my head while (failing) to fall asleep. Can't bring a calculator to bed.
https://www.physicsforums.com/threads/joystick-geometry.885203/#post-5568515
I did it rounded to zero decimal places in my head, then followed up in the morning with pencil and paper to one decmial place.
I could do it to an arbitrary number of decimal places*, but there's no point.

*OK, I can't do square roots to more than one decimal place.

 

[WWGD]

You're right, micromass. I was interpreting 3!! as (3!)!

But the smiley takes care of any doubt :) (