Saturday 18 January 2020

Subtraction, Borrowing, Carrying, and other Naughty Words, A Brief History

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 in Germany 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. 

These notes from Jeff Miller's website on the First Use of Mathematical Terms explain the different attitudes about the "borrow" in subtraction, "BORROW is found in English in 1594 in Blundevil, Exerc.: "Take 6 out of nothing, which will not bee, wherefore you must borrow 60" (OED2)."

In October 1947, "Provision for Individual Differences in High School Mathematics Courses" by William Lee in The Mathematics Teacher has: "The Social Mathematics course stresses understanding of arithmetic: 'carrying' in addition, 'regrouping' (not 'borrowing') in subtraction, 'indenting' in multiplication are analyzed and understood rather than remaining mere rote operations to be performed blindly."

The + and - symbols first appeared in print in Mercantile Arithmetic or Behende und hüpsche Rechenung auff allen Kauffmanschafft, by Johannes Widmann (born c. 1460), published in Leipzig in 1489. However, they referred not to addition or subtraction or to positive or negative numbers, but to surpluses and deficits in business problems (Cajori vol. 1, page 128).*(Jeff Miller, Earliest Known Use of Math symbols)

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].  Robert Recorde used the term in The Ground of Arts (1543) and Denniss and Smith (Robert Recorde, the life and Times of a Tudor Mathematician) describe the term as "a term that had already gained some currency".   The word, "borrow" may not have been commonly 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 and if you include a digital image, I will sacrifice a student in your honor)

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. "
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

And I have just found a copy of a 1773 Encyclopedia Brittanica that has "borrowing" used in the process of subtraction

Many teachers are equally uneasy about the term "carry" used in addition, subtraction, and multiplication.  Jeff Miller's Earliest Use of Math Words offers "according to Smith (vol. 2, page 93), the 'popularity of the word 'carry' in English is largely due to Hodder (3d ed., 1664)'."
I'm not sure what words were used prior to Hodder's book, but the motions on early Greek sand trays, the Chinese Abacus, and even early mechanical calculating instruments like Pascal's "Pascaline" calculator using clock-like mechanisms to transfer the "carry".  I imagine they had a word for the mathematical idea that made this happen.  (anyone?) 

The gelosie, or lattice, method of multiplication that many teachers prefer to the more traditional approach, includes this carry in the way the lines are counted.  This method is essentially the method by which Napier's Bones, (from 1617) operated as shown below to show the multiplication of 425 by any single digit. 

To find  3 x 425, drop down to the white 3 and read across, combining numbers in the same diagonal.  the first digit is 1.  Next we take the 2 and add to the 0 to get 2, so our first two digits are 12.  Now the 6+1 gives us 7 for the next digit, and then we are left with only the five, for 1275.  If you want to multiply by a multidigit number, we would perform the same process for each of the values, and offset the tens, hundreds, etc as the traditional method of multiplication. 

I received this comment on the blog, "I have never confessed, I think, to checking, yet I have several older geometry and algebra books, and one arithmetic book.

New Arithmetic (Boston, 1889, 15th edition) uses only the word "subtraction," points out that storekeepers count up (adding) rather than actually subtracting, and gives no hints to performing the operation, only a series of exercises.


So I later wrote a short post on this "anti-subtraction" approach that I believe most good number folks use frequently. (Thank you Jonathan!)

I'm sure most middle school teachers have encountered mistakes like the one in the picture above due to students using an algorithm to replace understanding subtraction.
Many years ago I became alarmed when I saw it happening frequently in my Alg II classes on a test question that appeared in a testing program used at the school... it wasn't in my syllabus, but still... these are bright kids. I had always been aware of a few students who were dependent on their calculators for very trivial questions, but in following up on the particular subtraction issue, I became aware of how many of my students had almost no skills in what I call "mental math".

Now in my history I had been a regional director for the math council, and had talked with elementary teachers at conferences and in school visits about many areas of math outside my personal teaching experience, and on a few occasions when it was requested, I laid out a suggested topic map for units on teaching subtraction by the method commonly called adding-on using an idea I called number-roads. (not original, but I don't know from whom I claimed it)

In essence the idea was to develop facility with tens compliments and hundreds compliments as part of the pre-development and then use the "roads" to allow something similar to skip-counting, but with variable skipping, to get the differences between numbers. In my high-school classes I dispensed with "number roads" and simply encouraged the use of "linking numbers" to pave the way for counting on.

A student given 307 - 128 would be encouraged to set up the problem with the numbers spaced apart more than is typical, and then put a few easy linking numbers in between. These would depend on the student, and would change as they got better at mental math, but might include 130, 150, 200, 300. They could then make little gap-markers between each pair (2 for the gap between 128 and 130... twenty more for 130 to 150...etc) and simply add the gaps...

When I met resistance by students who felt they were fully functional with conventional algorithms, I would make them solve a couple of problems with time or distance involving bases other than ten. If they handled these, I would assume they understood and used subtraction well enough to do well with whatever approach they used.

I was amazed at how many of the students became fond of challenging themselves with doing the problems mentally after awhile. We would do two or three problems in the review at the beginning of each class and they often became quite skilled. When we had guests and they wanted to show off I would pick a problem which particularly favored counting on, something like 1004 - 397, but it really became unnecessary as they quickly progressed.
Their interest gave me a beautiful lever when we got to working with polynomials. I would show that (10x+5)^2 allowed you to mentally calculate the square of numbers ending in five... and 38x42 could quickly be viewed as (40-2)(40+2)... We would continue to the end of the year with a few moments of mental drill each day, and the kids who returned for pre-calc and calculus seemed to retain the skills for the most part.

I don't teach Alg II anymore, and still mention mental math once in awhile in pre-calc, but I've never gone back to the regular teaching of mental math skills.

I wonder if counting-on is taught in the elementary grades at all, or how subtraction is approached.
I did find a UK educational site where the term "counting on" was used to describe an approach very much like what I have taught...and they even have an interactive Excel spreadsheet to teach the method.

I got some nice additions in the comments for this post, and want to share them.  Most of what I know, I learned from the comments section of my, and other, blogs.

"I do something similar, early, with many groups. I like to convince them that there is an easier way to do something, or that I can reteach them to do something easier, or faster. Their ability? check. My skill in teaching? check.

Plus it helps, profoundly, with number sense.

I call mine "no-carry subtraction" and simply add to the subtrahend and dimuend. 307 - 128 (add two) 309 - 130 (add 70) 379 - 200 (pass the problem to your less-skilled younger brother or sister - you've made it so easy, even they can do it)

I do recognize that in my head, subtraction is distance, and 713 - 284 is 16+13+400 (in that order, don't know why) which comes closer to your roadmap.

Jonathan"  I suspect this is the same Jonathon as before, are you still out there Johnathon?

Another wrote "Anonymous Anonymous said...

You have to allow negative numbers as well as positive for really efficient mental math. For example, for 307-128, my first intermediate is 308, so I get -1+180, rather than having to hold half a dozen numbers in my head.

But I've never been good at memory work, so perhaps my methods are not optimal for others."
My rule one, understanding stuff beats memorizing stuff every time.
I wrote back that, "I think the touch points adjust depending on the skill... with a little experience they would see 307-128 as 72 + 7 + 100... seeing complements of 10 and 100 (and 1000) comes pretty quickly and numbers are grouped to make the remaining addition easy."
This space for your comments..........

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