# Trachtenberg Speed System of Arithmetic

• MHB
Gold Member
MHB
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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• symbolipoint and jim mcnamara

## Answers and Replies

Gold Member
Thanks for sharing with us!

Mentor
2021 Award
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!

• vanhees71, pbuk and Wrichik Basu
Gold Member
MHB
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.

• symbolipoint
Mentor
2021 Award
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.

• Ackbach
Mentor
2021 Award
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.

• vanhees71
Muu9
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)

Gold Member
MHB
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?

• symbolipoint
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

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)

Gold Member
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".

• Ackbach
Gold Member
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.

Gold Member
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.

Gold Member
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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Gold Member
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.

• symbolipoint and bob012345
Gold Member
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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• symbolipoint
Homework Helper
Gold Member
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?