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Tom Lehrer claims and the audience seems to agree with him that the 'old way', before new math to do subtraction was to say, for example, 'three from two is nine carry the one'. I never heard of this method. Does anyone still use it? What is it called? https://www.youtube.com/watch?v=UIKGV2cTgqA is the video showing Tom Lehrer's bit. Does it really not involve understanding what you are doing as the new math activists claimed (I presume).

Is it known what Lehrer really thought about this part of new math?

Lehrer's example is 342 - 173.

Edit: Also, when was the borrowing method of teaching subtraction invented?

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  • $\begingroup$ In elementary school, in the U.S., in the 1950's, I was taught "borrowing", not "carrying" for subtraction problems. $\endgroup$
    – Andreas Blass
    Apr 5, 2021 at 17:51
  • $\begingroup$ @AndreasBlass So it seems 'new math' was not entirely new. $\endgroup$
    – Matthew Christopher Bartsh
    Apr 6, 2021 at 6:31

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It sounds like a variation of subtracting that I learned in high school (1968). My instructor called it European subtraction.

\begin{array}{ccc} & 3 & 4 & 2 \\ - & 1 & 7 & 3 \\ \hline \end{array}

You start by saying $9$ plus $3$ is $12$. You write the $9$ as shown and "carry" the $1$, of the $12$, as a subscript of the $7$ as is shown below.

\begin{array}{ccc} & 3 & 4 & 2 \\ - & 1 & 7_{\color{red} 1} & 3 \\ \hline & & & \color{red} 9 \\ \end{array}

Then $7$ and $1$ is $8$ plus $6$ is $14$.

\begin{array}{ccc} & 3 & 4 & 2 \\ - & 1_{\color{red}1} & 7_1 & 3 \\ \hline & & \color{red}6 & 9 \\ \end{array}

Finally, $1$ and $1$ is $2$ plus $1$ is $3$.

\begin{array}{ccc} & 3 & 4 & 2 \\ - & 1_1 & 7_1 & 3 \\ \hline & \color{red}1 & 6 & 9 \\ \end{array}

The big advantage of this method is that you can do many subtractions at once.

\begin{array}{ccc} & 5 & 3 & 4 & 2 \\ - & 1 & 9 & 2 & 3 \\ - & 2 & 9 & 4 & 5 \\ \hline \end{array}

$5$ and $3$ is $8$, plus $4$ is $12$.

$5$ and $2$ is $7$, plus $7$ is $14$.

$10$ and $9$ is $19$, plus $4$ is $23$.

$4$ and $1$ is $5$, plus $0$ is $5$.

\begin{array}{llll} & 5 & 3 & 4 & 2 \\ - & 1 & 9 & 2 & 3 \\ - & 2_2 & 9_1 & 4_1 & 5 \\ \hline & & 4 & 7 & 4 \\ \end{array}

I think this is a wonderful way to subtract, but, I would not teach it to grade school students.

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  • $\begingroup$ This is the method and notation I learned in Finland: youtube.com/watch?v=HhldRs8Hkdw . I have no idea what it would be called or how it would be said in English, but it is certainly European. $\endgroup$
    – Tommi
    Apr 4, 2021 at 9:54
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    $\begingroup$ My recollection (Scotland in the 1940s/50s) is that the full-blown notation for beginners puts a little one at the top of the units as well as a little one at the foot of the tens (etc). I still do. The first subtraction then looks like 12 take away 3, so we write 9. Etc. $\endgroup$
    – ancient mathematician
    Apr 4, 2021 at 11:28
  • $\begingroup$ @ancientmathematician That method is awesome. It's like doing the same thing to each side of an equation so it remains true. $\endgroup$
    – Matthew Christopher Bartsh
    Apr 5, 2021 at 2:37
  • $\begingroup$ @MatthewChristopherBartsh we were all being trained up to be double-entry bookkeepers ;-) $\endgroup$
    – ancient mathematician
    Apr 5, 2021 at 6:59
  • $\begingroup$ If I may make one more comment. @Steven says "You start by saying 9 plus 3 is 12". That's not how I remember it: you start by saying "3 is bigger than 2, so borrow 1 " [here you write the little ones], "now 12 take away 3 is 9". In other words, we knew by heart two independent facts: (i) 3 add 9 is 12 (ii) 12 take away 3 is 9. In an addition sum you used the first, in a subtraction sum the second. It was years before I realised that it was more than a coincidence that these look so similar: we didn't waste time on semantics, we were being trained as efficient processors of strings of digits! $\endgroup$
    – ancient mathematician
    Apr 5, 2021 at 7:11

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