Trachtenberg Speed System of Arithmetic

In summary, the Trachtenberg Speed System of Arithmetic is a very old, but extremely effective, way to teach basic arithmetic. It is a system that is very similar to the one that was taught in Switzerland up until a few decades ago, and it is claimed that this system results in students being able to understand arithmetic to an astonishing degree, to the point where it is no longer necessary for them to think about the process. However, there are those who argue that this system is a waste of time, as it is a system that is only effective if the student is able to master it on their own.
  • #1
Ackbach
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I was very excited to learn that you can download the entire Trachtenberg Speed System of Arithmetic book for free from this website. (https://archive.org/details/TheTrachtenbergSpeedSystemOfBasicMathematics_201803)

Why spend such an inordinate amount of time on arithmetic? Don't we have calculators?

This is a very old discussion, but I think there can be a definitive answer.

1. Computers make fast, very accurate, mistakes. They are only as good as the operator, and the operator needs to know if the answer coming out is correct. The only way to check an answer is by having a different way to estimate or arrive at the answer. Mental arithmetic is thus a check on the calculator result, but only if you can actually do it.

2. According to the excellent book Why Don't Students Like School, by Daniel Willingham, the only way students can possibly understand all the abstractions we pile onto arithmetic in advanced math courses is if the fundamental processes are understood so well that the student does NOT have to think about them. Thus, Trachtenberg allows the student to master arithmetic to an astonishing degree (approaching Gauss, possibly!) - to the degree that it's automatic.

So, imagine you're solving an algebra problem, and $9\times 7$ comes up. If you don't know instantly, without thinking, that it's $63$, you're going to have to use working memory slots to do that computation. Those are now working memory slots that are unavailable to think about the algebra problem. Moreover, once you've actually computed $9\times 7$, you must reacquaint yourself with the algebra problem. You take a double-hit in time and efficiency. It also makes the algebra problem seem unnecessarily difficult.

For these reasons, I am a big believer in drilling the basics so that they are automatic, and the students do not have to think about them. Then those mental resources are available for the higher-level problem they are working on. And Trachtenberg allows the students to master arithmetic to a degree unheard-of outside Switzerland (apparently, the Swiss teach Trachtenberg everywhere).

Highly recommended!
 
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I am of a different opinion. E.g. on page 130(270): ##302\cdot 114=34428.##

I agree that basic calculations should be done in mind. But reading 270 pages in order to excellent what most of us have learned at elementary school is in my opinion a waste of time.

My mentor at the university told me about a dialogue he had with his mentor:
"How old are you?"
"40."
"How many books do you read per year? Say you read 3 or 4. So until the end of your life, you will be able to read about 200 more books. Don't you think it's time to make a qualified choice?"

I don't remember the actual numbers in this anecdote, but the message is clear: read what promises a gain. Being better in multiplications when calculators are all over the place seems to me a bad choice. And I have read Artin's Galois Theory which many would also consider a waste of time!
 
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  • #4
fresh_42 said:
I am of a different opinion. E.g. on page 130(270): ##302\cdot 114=34428.##

I agree that basic calculations should be done in mind. But reading 270 pages in order to excellent what most of us have learned at elementary school is in my opinion a waste of time.

My mentor at the university told me about a dialogue he had with his mentor:
"How old are you?"
"40."
"How many books do you read per year? Say you read 3 or 4. So until the end of your life, you will be able to read about 200 more books. Don't you think it's time to make a qualified choice?"

I don't remember the actual numbers in this anecdote, but the message is clear: read what promises a gain. Being better in multiplications when calculators are all over the place seems to me a bad choice. And I have read Artin's Galois Theory which many would also consider a waste of time!
I think you're entirely missing the point. The point is that I would advocate that the Trachtenberg System be taught in elementary school! That is what Switzerland does, with very good results. The book to which I have linked is admittedly not an elementary-level text. Someone should write that. I don't know what they have in Switzerland, but it works, and it works very well.
 
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  • #5
Ackbach said:
I think you're entirely missing the point. The point is that I would advocate that the Trachtenberg System be taught in elementary school!
Thanks for the clarification. I moved it to the "education" forum.
 
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  • #6
Ackbach said:
That is what Switzerland does, with very good results.
That depends on how you define a good result:
In no other subject are the students as bad as in mathematics.
https://www.srf.ch/wissen/lernen-gewusst-wie/die-schweiz-steckt-in-einer-mathe-misere

Admittedly, mathematics at the final level is different from the beginner's level. But can we speak of success it finally fails? I tried to find more details. Not easy, if you run in general statements and social-political studies on one side, and dyscalculia on the other.
 
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  • #7
Ackbach said:
So, imagine you're solving an algebra problem, and 9×7 comes up. If you don't know instantly, without thinking, that it's 63, you're going to have to use working memory slots to do that computation. Those are now working memory slots that are unavailable to think about the algebra problem.
The Trachtenberg system doesn't make that easier to memorize; it only makes more involved arithmetic easier, at which point a calculator could be used. It might make sense to use them in place of the standard algorithms (at the risk of even further removing students from the underlying logic behind the operations), and it's certainly interesting to see a historical perspective of arithmetic (newer isn't necessarily better)
 
  • #8
Muu9 said:
The Trachtenberg system doesn't make that easier to memorize; it only makes more involved arithmetic easier, at which point a calculator could be used. It might make sense to use them in place of the standard algorithms (at the risk of even further removing students from the underlying logic behind the operations), and it's certainly interesting to see a historical perspective of arithmetic (newer isn't necessarily better)
And how do you know that the calculator answer is correct? How do you know you didn't fat-finger a number, or hit the wrong operation? As John Wheeler was fond of saying, "Never make a calculation until you know the answer." He meant that you need to have a second, completely independent method of computing the answer as a means of checking your work. The calculator is one way. What's your other way?
 
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  • #9
Ackbach said:
And how do you know that the calculator answer is correct? How do you know you didn't fat-finger a number, or hit the wrong operation? As John Wheeler was fond of saying, "Never make a calculation until you know the answer." He meant that you need to have a second, completely independent method of computing the answer as a means of checking your work.
I'm not convinced that completely independent method is what he meant. I think he means you should have an estimate of the answer ahead of doing detailed calculation. If the detailed result is way off from the estimate, then you have made at least one mistake somewhere.
The calculator is one way. What's your other way?
My other way is certainly not to do the digit by digit arithmetic in my head.

This is not to say the link you provided isn't interesting. It is. But after skimming through it, I tend to agree with @Muu9

Muu9 said:
It might make sense to use them in place of the standard algorithms (at the risk of even further removing students from the underlying logic behind the operations)
 
  • #10
I've not looked at the link but 302 x 114 is certainly the sort of calculation I'd be likely to do in my head. The only complication is that my head works left-to-right and my working memory is not what it was so for example I read 3 x 114 as 3 x "1" "1" "4" = "3" "3" "12" and then resolve the carry on the next pass to "342".
 
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  • #11
Jonathan Scott said:
I've not looked at the link but 302 x 114 is certainly the sort of calculation I'd be likely to do in my head. The only complication is that my head works left-to-right and my working memory is not what it was so for example I read 3 x 114 as 3 x "1" "1" "4" = "3" "3" "12" and then resolve the carry on the next pass to "342".
What does do in my head mean? Do you write down the digits as you compute them or store them first then write the whole answer down? I can do these computations in my head but I have to write down the digits as I get them or I'll lose track. For example, it's easy to do 10121X30411 in mind but I have to write down the digits from right to left as I go. I also have to arrange the numbers one above the other to see the pattern.
 
  • #12
bob012345 said:
What does do in my head mean?
I didn't write anything down. I read 302 x 114, and calculated the result as follows:
3 0 2 x 1 1 4 = 3 x 1 1 4 0 0 + 2 x 1 1 4 = 3 3 12 0 0 + 2 2 8 = 3 3 14 2 8 = 3 4 4 2 8.
I repeat the current version to myself in my head, as written, while I transform it.

I don't think I'd normally try that for 10121x30411 as there are too many intermediate results to remember.
 
  • #13
Jonathan Scott said:
I didn't write anything down. I read 302 x 114, and calculated the result as follows:
3 0 2 x 1 1 4 = 3 x 1 1 4 0 0 + 2 x 1 1 4 = 3 3 12 0 0 + 2 2 8 = 3 3 14 2 8 = 3 4 4 2 8.
I repeat the current version to myself in my head, as written, while I transform it.

I don't think I'd normally try that for 10121x30411 as there are too many intermediate results to remember.
You're like a mentat! What I do is more like matrix multiplication. I do the right most numbers then the cross product of the two right most numbers of each then three, ect till the left most two.

For example ab X cd = ac, ad+cd, bd where these are digit locations. I do the cross product in my head first (I call that the kernel) so 34x41= 12,19,4 or 13,9,4 or 1394. 341x411 would be 12, 19,11,5,1 or 140151 when added as I go from right to left. I don't need to put number in the correct positions with zero's with this method.
 
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  • #14
bob012345 said:
You're like a mentat!
I grew up before calculators, so I guess that gave me more practice. My working memory (at 66) is much smaller than it was when I was young, but I still find that if I can represent the current state of a calculation as something I can say to myself and gradually transform it, I can do that in my head.

I often do calculations in my head when I'm curious to estimate something, but those calculations are generally approximate and quick. For example, if I want to know the square root of some number, I find a nearby number for which I know the square root, estimate how far out it is as a percentage or similar, halve that and adjust the square root accordingly. I then square the new estimate and find out how far out that is, until it is close enough for my purposes. For example, I recently needed to know roughly the square root of 750, which is 20% more than 625 which is 25 squared, so I increased 25 by 10%, giving 27.5 as the next estimate. The square is then equal to (25+2.5)^2 which is 625 + (2*2.5*25) + 2.5*2.5 = 756.25 which was close enough in that context.
 
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If one looks at the old method they taught in grammar school in the U.S. it looks like this using two two digit numbers ##(a b)## and ##(c d)##;

\begin{matrix}
&a&b\\
&c&d\\
&\\
&da&db\\
ca&cb&0\\
&&\\
ca,&da+cb,&db
\end{matrix}but one can skip the intermediate steps and just write down the answer from right to left if one sees the multiplication pattern is always the same simple pattern.

For two three digit numbers;
\begin{matrix}
&&a&b&c\\
&&d&e&f\\
&\\
&&fa&fb&fc\\
&ea&eb&ec&0\\
da&db&dc&0&0\\
&&\\
da,&ea+db,&fa+dc+eb,&fb+ec,&fc
\end{matrix}

For two four digit numbers;
\begin{matrix}
&&&a&b&c&d\\
&&&e&f&g&h\\
&\\
&&&ha&hb&hc&hd\\
&&ga&gb&gc&gd&0\\
&fa&fb&fc&fd&0&0\\
ea&eb&ec&ed&0&0&0\\
&&\\
ea,& fa+eb,& ga+ec+fb,& ha+ed +gb+fc&hb+fd+gc&hc+gd,&hd
\end{matrix}

It's the same pattern when the number of digits is different just some terms zero out.
 
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  • #16
Jonathan Scott said:
I've not looked at the link but 302 x 114 is certainly the sort of calculation I'd be likely to do in my head. The only complication is that my head works left-to-right and my working memory is not what it was so for example I read 3 x 114 as 3 x "1" "1" "4" = "3" "3" "12" and then resolve the carry on the next pass to "342".
Nice as might be to be able to do that example mentally, many people will not want to be required to.

Looking at that example, I know already two things:
  • Product probably be a six-digit number.
  • An ESTIMATION is that the product is near 300 x 100 = 30000; well obviously it will not be big enough for six; more realistically, guess five digits.

next, I could write the long-multiplication process or algorithm on paper using the meaning of place-value:
300*100 + 300*10 + 300*4 + 0*100 + 0*10 + 0*4 + 2*100 + 2*10 +2*4
And arrange in columnar form and perform the addition.

Were we all not taught this kind of thinking when we attended elementary school?
 

FAQ: Trachtenberg Speed System of Arithmetic

1. What is the Trachtenberg Speed System of Arithmetic?

The Trachtenberg Speed System of Arithmetic is a mental math system developed by the Russian engineer Jakow Trachtenberg. It is a set of techniques and methods that allow for quick and accurate calculations without the use of traditional methods such as multiplication tables or long division.

2. How does the Trachtenberg Speed System work?

The Trachtenberg Speed System utilizes a series of simple and efficient methods to break down complex calculations into smaller, easier steps. These methods include techniques for multiplication, division, addition, and subtraction, as well as shortcuts for working with fractions, decimals, and square roots.

3. Who can benefit from using the Trachtenberg Speed System?

The Trachtenberg Speed System can benefit anyone looking to improve their mental math skills, but it is especially useful for students, teachers, engineers, and anyone in a profession that requires quick and accurate calculations. It can also be helpful for individuals who struggle with traditional math methods or have learning disabilities.

4. Can the Trachtenberg Speed System be learned by anyone?

Yes, the Trachtenberg Speed System can be learned by anyone, regardless of their age or mathematical ability. It does not require any prior knowledge of advanced math concepts, making it accessible to everyone.

5. Are there any drawbacks to using the Trachtenberg Speed System?

One potential drawback of using the Trachtenberg Speed System is that it may not be as useful for complex calculations or in situations where precise calculations are required. It is also important to note that while the system can improve mental math skills, it is not a replacement for traditional math methods and should be used in conjunction with them.

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