Paradigm, No. 16 (May, 1995)
Essex CM1 4TN
It is commonly supposed that in the old days children knew their tables and that they got to know them by memorising statements (once two is two, two twos are four, etc.) arranged in columns (the two times table, the three times table, etc.), often involving chanting in unison with the rest of the class. If this were the case, it would be expected that textbooks would have provided support to this process by, as it were, printing the lyrics. A historical examination of textbooks, however, shows this to be far from the truth.
In this article two particular features of the presentation of multiplication tables (or, more accurately, the multiplication table, as it was always referred to) in textbooks are considered namely, the range of numbers to be included in the table and the organisation of the results on the page.
The texts in question are all in English and cover the period from around 1300 (when what is thought to be the earliest arithmetical manuscript in English was written) to 1900, after which separate textbooks tended to be written for the Primary and Secondary schools, the latter not usually including tables at all and the former often giving results in partial form at different stages. (However, there was, and still is, considerable variation.) In all some 45 texts are considered. Dates in the body of this text refer to year of original publication.
One of the most serious weaknesses of the so-called traditional tables is that they ignore the powerful commutative principle (e.g. 3 x 4 = 4 x 3) and thus require children to learn nearly twice as many results as they need. It is therefore startling, perhaps, to find this principle recognised in the very earliest text The Crafte of Nombrynge (c. 1300):
This is the multiplication table in its most pared-down form. It is triangular because it uses the commutative principle. It starts at 2 x 2 and goes up only to 9 x 9 since this is all we need in our base-ten system of numeration, though we also need to know how to multiply by 10! The table is therefore reduced to the essential 36 facts, as opposed to the 132 facts traditionally learned.
Sacrobosco (John of Holywood) in his manuscript The Art of Nombryng (1488) extended the table to 10 x 10 and included all the reversals in the square:
but Robert Record in The Ground of Artes (1543), effectively the first arithmetic textbook in English, reverted to the triangular form and the basic 36 facts:
Wingate (1630) and Cocker (1678) both used the number square form given by Sacrobosco, but only going as far as 9 x 9. Ward (1707) presented the table in the form of an array of arithmetical statements:
which was interesting for several reasons. He used the x sign; rare in arithmetical textbooks until the end of the 19th century. He started at 3 x 3 -- even the two times table was too obvious to bother with! He omitted reversals and explained why.
There followed a period in which there was something approaching agreement regarding the presentation of the multiplication table. Some 20 authors from Hill (1711) to OGorman (c 1840) all extended the table to 12 x 12 and, with a few exceptions, used as their form of organisation a number square (e.g. Hutton) or a tapered array e.g. Fisher.
Hill, although using a number square, merely added 12 to the square used by Wingate and Cocker.
and thus neatly formed a bridge between the early authors and this middle period. The other exceptions in this period are also interesting; Dilworth, like Ward, omitted the two times table and gave:
Vyse used a number triangle going up to 12.
and Guy and Joyce wrote out the full array from 1 x 2 up to 12 x 12. (Joyce also offered the 12 by 12 number square as an alternative.)
If the 130 years up to 1840 were, in general, notable for their consistency, the period from Foster (1842) to Mackay was remarkable for its variety. Of eighteen authors examined:
One went up to 9 x 9; 9 to 12 x 12; one to 12 x 13; one to 12 x 20; 3 to 20 x 20; one offered both a 9 x 9 square and a 12 x 12 square; one offered both a 9 x 9 square and a 9 x 9 array; one gave a 9 x 9 square embedded in a 12 x 12 square embedded in a 20 x 20 square!
Nine authors used number squares of one size or another, eight used arrays of various kinds and one used both forms.
Over the entire period (1300-1899) number triangles or squares were nearly twice as popular as arrays, and taking the table up to 12 x 12 was twice as popular as all the other ranges put together. Only nine of the 45 texts gave the traditional tables.
It is interesting to speculate what these authors would have chosen to do had they been writing today. There is little doubt that, responsive to the market-place as they were, they would have cut out the obsolete 12 times and the 11 times that was a bridge to it. They might even have gone all the way back -- could it be that the author of The Crafte of Nombrynge had it right first time?
1. Robert Steele, The Earliest Arithmetics in English (OUP, 1922).
2. Edmund Wingate, Arithmetique made easie (1630).
3. Edward Cocker, Cockers Arithmetick (1678).
4. John Ward, The Young Mathematicians Guide (1707).
5. John Hill, Arithmetick (1711).
6. Daniel OGorman, Intuitive Arithmetic (c. 1840).
7. Charles Hutton, Complete . . . Arithmetic (1764).
8. George Fisher, Arithmetick (1719).
9. Thomas Dilworth, The Schoolmasters Assistant (1743).
10. Charles Vyse, The Tutors Guide (1775).
11. Joseph Guy, Guys Tutors Assistant (1823).
12. J. Joyce, A System of Practical Arithmetic (1807).
13. W. Foster, Elements of Arithmetic (1842).
14. J. S. Mackay, Arithmetic, Theoretical and Practical (1899).
Reference list of texts
The first date given is the date of the first edition. The remainder of the reference refers to the edition used by the present writer. The entry in square brackets is the printer of the first edition, if different from the edition used.
1. 1307, Anon, The Craft of Nombrynge (Reprint in Steele, Earliest Arithmetics in English).
2. 1488, Sacrobosco, The Art of Nombryng (Reprint in Steele).
3. 1543, Robert Record, The Ground of Arts (Edward Dod, 1654).
4. 1630, Edmund Wingate, Arithmetique Made Easy (M. Flesher for P. Stephens & C. Meredith, 1630); as Mr. Wingates Arithmetic, 15th edn. (J. Phillips et al. 1726).
5. 1634, Thomas Masterson, Mastersons Arithmetic (G. Miller, 1634).
6. 1678, Edward Cocker, Cockers Arithmetic, 54th edn. (R. Ware et al. 1753) [T. Passinger and T. Lacy].
7. 1707, John Ward, The Young Mathematicians Guide, 11th edn. (C. Hitch et al. 1762) [John Taylor, 1707].
8. 1711, John Hill, Arithmetick, 8th edn. (R. Ware et al. 1750) (D. Midwinter, 1711].
9. 1715, William Webster, Arithmetick in Epitome, 2nd edn. (C. King et al. 1716).
10. 1719, George Fisher, Arithmetick, 7th edn. (Charles Hitch, 1748) Fisher, [H. Tracy 1719].
11. 1743, Thomas Dilworth, The Schoolmasters Assistant, 19th edn. (Richard and Henry Causton, 1834) [H. Kent, 1743].
12. 1751, Francis Walkingame, The Tutors Assistant, 72nd edn. (Longman, et al., 1834).
13. 1754, Daniel Fenning, The British Youths Instructor, 13th edn. (J. Johnson et al., 1806).
14. 1764, Charles Hutton, A Complete Treatise of Practical Arithmetic, 11th edn. (C. G. & J. Robinson and R. Baldwin, 1801) [J. Wilkie, 1764].
15. 1766, John Mair, Arithmetic, 5th edn. (Bell & Bradfute, 1794).
16. 1775, Charles Vyse, The Tutors Guide, 13th edn. (Wilkie & Robinson et al., 1807).
17. 1778, John Bonnycastle, The Scholars Guide to Arithmetic, 5th edn. (J. Johnson, 1788).
18. 1788, Nicolas Pike, A New and Complete System of Arithmetic Newbury-Port (John Mycall, 1788).
19. 1794, Joseph Saul, The Tutors and Scholars Assistant, 4th edn. (J. Hertley and C. Lew, 1807).
20. 1797, John, Practical Arithmetic, new edn. (Brett Smith, 1840).
21. c. 1800, Thomas Hodson, The Accomplished Tutor, 2nd edn. (H. D. Symonds et al., 1802).
22. 1807, J. Joyce, A System of Practical Arithmetic, new edn. (Longman etc. 1835) [J. Johnson, 1807].
23. 1823, Joseph Guy, Guys Tutors Assistant, 28th edn. (Cradock & Co., et al. 1860) [Baldwin & Cradock, 1823].
24. 1830, Augustus de Morgan, The Elements of Arithmetic, 3rd edn. (John Taylor, 1835)
25. 1832, John Hind, The Principles and Practice of Arithmetic, 4th edn. (Deighton etc., 1842).
26. 1833, J. Crossley and W. Martin, The Intellectual Calculator, 85th edn. (Hamilton, Adams & Co., et al., 1858?).
27. 1833, J. W. Colenso, Arithmetic, new edn. (Longmans Green & Co., 1874).
28. c. 1840, Daniel OGorman, Intuitive Arithmetic, 6th edn. (Macliver and Bradley, 1853) [for the Author, c. 1840].
29. 1842, W. Foster, Elements of Arithmetic (Simpkin and Marshall and W. Woodward, 1842) [Simpkin and Marshall, 1842].
30. 1843. W. H. Crank, A Theoretical and Practical System of Arithmetic (John W. Parker, 1843).
31. 1845, Benjamin Greenleaf, Introduction to National Arithmetic, new edn. (Boston: Roberts, Davis, etc., 1861).
32. 1854, Robert Wallace, The Elements of Arithmetic (John Cassell, 1854).
33. 1854, J. R. Young, Arithmetic [Orrs Circle of Science series] (W.S. Orr & Co., 1854).
34. 1854, Barnard Smith, Arithmetic for Schools, rev. edn. (Macmillan, 1896).
35. 1855, J. Cornwell and J. G. Fitch, The Science of Arithmetic, 11th edn. (Simpkin, Marshall & Co., 1867).
36. 1899, Mackay, J. S., Arithmetic Theoretical and Practical (Chambers, 1899).