Sunday 16 October 2022

#21 Subtraction....from old Math Terms notes

Subtract joins two easy to understand roots, the sub which commonly means under or below, and the tract from words like tractor and traction meaning to pull or carry away. Subtraction then, literally means to carry away the bottom part. The "-" symbol for subtraction was first used as markings on barrels to indicate those that were underfilled. Around the 1500's it began to be used as an operational symbol and it became common in English after it was used by Robert Recorde in The Whetstone of Witte in 1557.

In a subtraction relationship, a-b=c, all three numbers have a special name. The first number, a, is called the minuend, from the same root as minus, and literally means that which is to be made smaller. The part to be removed, b, is called the subtrahend and means that which is to be pulled from below. The answer, c, is most often called the difference or result, but in many applied statistical uses it is also called the residueor residual, that which remains. In statistical uses it may also be called a deviation.

The term subduction was often used in older English books up until about 1800. John Wallis uses the term in his "Treatise on Arithmetic", 1685 in describing subtraction... "Supposing a man to have advanced or moved forward, (from A to B,) 5 yards; and then to retreat (from B to C) 2 yards: If it be asked, how much he had advanced (upon the whole march) when at C? Or how many yards he is now forwarder than when he was at A? I find (by subducting 2 from 5,) that he is advanced 3 yards. Samuel Johnson's 1768 dictionary defines both terms, but includes "substraction" as part of the definition of subduction.

In the same dictionary, Johnson defines both subtract and substract, but for subtraction, the reader is referred to "see substraction" so I assume that was the more common term.


The method of subtraction commonly called Borrowing or decomposition seems to go back at least to the 1200s. In The Art of Nombrynge by John of Hollywood (Sacrobosco) subtraction was taught "entirley like the method of today, 'borrowing' and all." [E. R. Slight from "The Craft of Nombrynge, Mathematics Teacher, Oct. 1939]. The word, "borrow" may not have been used until around 1600 as the earliest listing in the OED associated with subtraction is "[1594 BLUNDEVIL Exerc. I. (ed. 7) 91] Take 6 out of nothing, which will not bee, wherefore you must borrow 60." The borrowing of 60 suggests the exercise may have been about time. Here is a link with an image of a page from the Arithmetic of John Ayres in 1695 in which the word borrow is used in subtraction, although the method is more like what Ross and Pratt-Cotter (below) call the "equal additions" method.

Susan Ross and Mary Pratt-Cotter [Subtraction in the United States: An Historical Perspective, from The Mathematics Educator] show that prior to about 1940 in the US there were three common approaches to subtraction in arithmetic texts. The Borrow method, a method they call "equal additions" which also seems to date back to the 15th century (and is probably a logic based alternative to the borrowing approach), and a method of subtracting by adding on from the subtrahend, which is sometimes called the Austrian algorithm. (images of all three types can be found in the link above to Ross and Pratt-Cotter's document)

The research by Ross and Pratt-Cotter indicated that before 1937 there were few illustrations in American textbooks that show any physical "marking through or numbers being rewritten". Their work states that almost overnight, after a study by William Brownell, "most textbooks used the decomposition (borrow) method for describing borrowing in subtraction, and the use of the crutch described by Brownell became very popular. Today this method of subtraction is used in most textbooks that teach subtraction." The study also states,

Only one example was found, from a text published in 1857(Ray's Practical Arithmetic), where markings were used to keep track of the renaming process. This was done in only one problem in the text, with all other problems worked without any markings. Brownell was not aware, however, of any textbook employing this technique.

This statement, which I assume to be true, and the existence of a clear example of the "borrow" with markings in an 1898 copy of Gill's Oxford and Cambridge Practical Arithmetic , shown at right, make me suspect that the borrowing crutch first appeared in England and then made it's way to America. {if you have knowledge of any earlier appearance in textbooks in ANY country, please write)

I posted a request for information about texts or other sources of the use of the "crutch" and received the following from Ralph Raimi of the University of Rochester:

" I entered kindergarten in 1929, ten years before the Brownell article, and while I don't remember distinctly just which grade introduced me to the "borrowing" scheme for subtractions, it was surely in my schoolwork by 1933, probably 1932, in the Ferry School, Detroit, Michigan; and the crutch pictured in Ballew's example was standard procedure for us. We did (and every grocer did, too) the corresponding thing when adding a column of figures, as a grocer would do on his brown paper bag, listing and summing the item prices in a vertical column on the side of the bag before filling it with the items themselves. If the sum of the right-hand column (in cents) was, say 126, he would enter the 6 below the line, as part of the ultimate sum, and enter the 2 above the top of the tens column and the 1 above the top of the hundreds column, etc. In practical commercial sums and differences the place of the decimal point was implicit (dollars and cents) and disregarded in grocery stores until the end. In school we were careful to make vertical things *beautifully* vertical, and to preserve the decimal points throughout. Of course this made no difference, except to our understanding. Later, probably by the fifth grade, we were encouraged to *imagine* the crutch in a subtraction problem; writing it down was a sign of weakness, akin to moving the lips while reading. I'm sorry I have no documents from that era in my education, but I do know that my work was always supervised by my older brother, who was five years older than me, and he never showed any surprise at anything I did in arithmetic, so I imagine that by 1925 or 1926 he had also been learning to write subtractions (also in Ferry School) in the same way. Some years ago I was studying a facet of the history of the Detroit Central High School in the period 1898 - 1950 (it was debating clubs that interested me, not math lessons, but no matter), and I found that the Detroit Public Library had several archives of random materials of no particular importance that they had filed under the names of certain (dead)teachers, but cross-referenced so as to make it easy to know what they were. How those particular teachers, or their heirs, got these memorabilia into the library I can't imagine, for the files contained only old school newspapers, club meeting minutes, letters and so on, and the teachers themselves had not been notable; but I believe that if you go to any big city public library and ask for archives of local school teachers of a certain era (1900-1940, say), you might find a sheet of homework or a set of exams or answers, written out in that teacher's hand, or a student's, in some one of them. That might tell you more about the arithmetic style of the time than even the popular textbooks would. "

I have not yet followed up on Mr Raimi's suggestion in my brief visits to the US, but if anyone else finds information on the use of this "crutch" before 1937, I would appreciate a note.

It may be that the use of the "crutch" markings were commonly taught, but not found in books because disagreement about whether they should be used. In The Teaching of Arithmetic by Paul Klapper (1934), he gives an example both with and without the markings, and calls the form without the markings the "recommended form --- no 'crutches' should be permitted." The very use of the word crutch seems to confirm Professor Raimi's assertion that the marks were viewed as a weakness to be avoided or overcome.

However in the article Klapper states that, "This method is the favorite of many teachers who hold that it is very simple because it can be demonstrated objectively with dimes and cents and that it can be habituated quickly. Others are opposed to it because it requires a second set of number facts -- the subtraction combinations." The evidence seems to suggest that the use of a the borrow markings were common in America well before the publication of Brownell, but it may not have been common in textbooks because, as stated by both Professor Raimi and the Klapper book, it was viewed as a weakness.

I have also found another early use of supplemental marking of a problem. This example, using the equal additions method, comes from a 1873 copy of Charles S Venable's A Practical Arithmetic. Here is a copy of the paragraph from page 25

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