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Ackbach

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MHB

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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

2. According to the excellent book

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

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!

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*Thus, Trachtenberg allows the student to master arithmetic to an astonishing degree (approaching Gauss, possibly!) - to the degree that it's automatic.**have to think about them.**__NOT__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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