An Investigation of Subtraction Algorithms from the 18th and 19th Centuries - The Great Algorithm Debate

The equal additions algorithm, decomposition algorithm, complement algorithm, and Austrian algorithm for subtraction are represented in printed arithmetic books between 1700 and 1900. Was one algorithm preferred over the other algorithms during this time period? Table 1 shows our sample of printed arithmetic books examined and lists the algorithms that were utilized in them. If a book had more than one algorithm in it, it was listed more than once and has an asterisk next to its name.  

Table 1. Summary of Algorithms Used in Arithmetic Books from 1700 to 1900

Thomas Weston (1729) mentioned both the equal additions algorithm and the decomposition algorithm in his text without indicating a preference for either algorithm. Equal additions and decomposition are present in Joseph Ray’s (1856/1877) texts. Ray (1856) included both algorithms, presenting the equal additions algorithm first and the decomposition algorithm second. Later, Ray (1877), despite introducing the decomposition algorithm first, gave preference to the equal additions algorithm (see Figure 12). Figure 12 shows Ray’s explanation of how to compute \(805-637\) using his second method, equal additions.

Figure 11. Subtraction using equal additions, with preference given to this method “especially when the upper number contains ciphers [zeros],” on p. 35 of Ray’s new practical arithmetic, by Joseph Ray, published in Cincinnati, Ohio, in 1877. (This image has been reproduced from a Google Book with free access.)

Ray may have provided two explanations but he definitely advocated for equal additions by stating the equal additions algorithm was “less liable to error.” Ray was not alone in this opinion (see, e.g., Osburn, 1928). There were numerous studies in the early 1900s comparing algorithms and many believed the equal additions algorithm to be the “preferred” method. While many claimed that the equal additions algorithm was “less liable to error,” not everyone agreed that this was the case. Chauncey Lee (1797) called the equal additions algorithm “circuitous” and stated that the decomposition algorithm was a “more simple, natural and easy mode” (p. xi). J. Brookes (1776) preferred the complement method. J. T. Johnson (1938, p. 27) wrote:

[T]he equal additions method was not found in any German text examined, the decomposition method being chiefly used in that country. On the other hand, the decomposition method was not found in any French book examined, published later than 1820, the equal additions method being the prevalent method in France. The complementary and equal additions were the outstanding methods in Italy and England. The Austrian method was found in very recent English texts but did not appear in any Italian books examined.

It is interesting to note that all four algorithms – equal additions, complement, decomposition, and Austrian – are represented in printed books probably used in the U.S., as shown in Table 1.

Despite the presence of all four algorithms in printed books with authors as advocates for each, only two algorithms are present in cyphering books. When Nerida Ellerton and Ken Clements (2012) analyzed their private collection of 280 cyphering books, they found 51 of the cyphering books specifically stated a rule for “subtraction.” Of those 51 cases, 33 described “equal additions” and 18 the “complementary method.” The fact that not a single cyphering book in Ellerton’s and Clements’ personal collection of 280 cyphering books, dated between 1701 and 1860, utilized the decomposition algorithm indicates that decomposition was perhaps not an implemented subtraction algorithm in North America during this time. If it can be assumed that cyphering books displayed what students actually studied at the time, the fact that not a single cyphering book contains the decomposition algorithm prompts two very important questions. First, why was the decomposition algorithm not widely used from the 1700s to the early 1900s in cyphering books despite prominent authors, such as Warren Colburn, advocating the algorithm? And, second, given that the decomposition algorithm is prevalently used and advocated in modern classrooms, when did this transition occur?   

In the early 1900s there was not a “standard” subtraction algorithm as there is today. In fact, in his Handbook to Smith’s Arithmetic, David Eugene Smith recommended no specific algorithms. According to Smith (1905), “for a book to insist upon one of these would be to confuse the children in a school where another method is working satisfactorily” (p. 56). Despite Smith not recommending a “standard” method, there was a “great debate” in the early 1900s upon what should be the chosen algorithm (Johnson, 1938; Osburn, 1927; Ross & Pratt-Cotter, 2000). Many studies were devoted to determining which algorithm caused the least error and in the early 1900s the majority view was that the equal additions algorithm was superior (Johnson, 1938; Osburn, 1927). In fact, W. J. Osburn asserted at the end of his study (1927, p. 246):

The superiority of equal-additions (carrying) over decomposition (borrowing) is ‘as certain as taxes’ and ‘almost as certain as death’.

With the equal additions algorithm prominent in cyphering books and with studies in the early 1900s supporting the superiority of the equal additions algorithm, it is quite surprising that this algorithm is hardly present today in schools in the United States.

The ideas of John Heinrich Pestalozzi, a Swiss-German educator, were a possible influence for the selection of the decomposition algorithm. During the period from 1820 to 1880 Pestalozzi’s ideas became well known and around this same time the algorithms debate began (Ellerton & Clements, 2012). Osburn (1927) argued that one of the reasons for the increasing popularity of decomposition was directly tied to Pestalozzi (Osburn, 1927, p. 242):

It was the Pestalozzian movement for object lesson which brought in the decomposition method, the reason being that decomposition can be illustrated with bundles of splints. 

Both Pestalozzi and Colburn urged other teachers to adopt an inductive approach, where the rules or algorithms did not precede the learning, an approach the current cyphering books and printed books did not support (Ellerton & Clements, 2012). The influences of normal-school reformers, the introduction of written examinations into schools for children, and in particular the implementation of the first “standardized” examinations by Horace Mann are all possible reasons not only for the changes in terminology used, but also for the need to use a “standardized” algorithm. The definitive reasons for switching from the equal additions algorithm to the decomposition algorithm are not clear and deserve further study.

^‡Although Robert Record, or Recorde, first published his Grounde of Artes in London in 1542, the 1658 copy was examined. During this time, additions and changes to books often were included as additional pages at the end of the book. The editing process for Record’s Grounde of Artes seems to have worked this way, keeping the original intact and just adding a new section to the back. Record died in 1558; therefore, 17th century publications of his arithmetic book were most likely revised by others. Images from the 1543 edition of the Grounde of Artes can be seen here in Convergence.