I knew the day of reckoning was coming. As my son’s math homework became more complex and we had to add columns and rows to the multiplication tables we used for finding mnemonic patterns, long division loomed on the horizon. I’m not arithmophobic. I aced second-year Probability and Statistics, got through Calculus 201 (albeit barely), enjoyed Analytical Chemistry, and researched early modern account books and reckoners for part of my dissertation. Even in my career as an English/DH professor, I happily populate spreadsheets, devise complex formulas for norming and analyzing grades, and crunch through encoding whenever I get the chance. But I missed the unit on manual long division in grade 5, and never did master that dark art. Not until yesterday, that is.
I skipped an entire year of the BC elementary school curriculum. My school’s solution to a bored kid in Grade 5 was to move her abruptly to Grade 6 after Christmas. Whether or not the academic benefits outweighed the dire social consequences remains an open question. But one thing is certain: I was hopelessly confused by this process that all my new peers seemed to have mastered, ignorant of its purpose, and too shy to ask for help. Somehow I compensated—first by working out the answer laboriously in my head and then by resorting to a calculator as soon as we were allowed to bring one to school—and got all the way to my late 40s without needing the skill. But now my 10-year-old son is halfway through the very bit of the curriculum I skipped.
When he asked for help with his math homework on Tuesday, I scanned the page and saw the symbol I’d been dreading: that alarming combination of closing parenthesis with a long bar. (Apparently, ⟌ has no name. No wonder I’ve been filled with nameless dread for nearly forty years! It’s some consolation that Unicode thought to include it. Look up U+27CC if you ever need to insert the glyph into a document.)
Geeky asides aside, I had to confront my fear of long division (a phobia that, like U+27CC, lacks a name). My first move, though, was avoidance: “Hey, honey, how about if I cook supper tonight and you help with homework?” I thought my son would be happy with this arrangement, but it transpired that he prefers my partner’s cooking. So I had to confess my true motives. With my phobia outed and my culinary skills in question, humility was really the only option.
I listened in on the lesson and heard my own confusion and resistance in my son’s voice. “What is this arrow for? Why can’t I just do it my head? The answer is obvious! What’s the point of all these lines and remainders?” The homework coach patiently explained that the learning outcome was mastery of the process, not production of the right answer. We have to learn the process on simple problems so that we can scale it up to less tractable problems, he said. How many times have I said similar things as a professor? So there was no excuse for not mastering the process of manual long division by remainders, despite the ubiquity of devices that calculate a million times faster than I ever could.
The next night we moved the white board into the kitchen and I became the student. Within three minutes, I was wielding the dry-erase marker myself, reckoning quotients from random numerators and denominators. The cleverness of the method—essentially an algorithm that breaks down long division into a series of shorter divisions—is deeply satisfying.
My first thought after performing two or three divisions in rapid succession was “This is genius! Who invented this technique?” And then of course I had to ask the question in a more formal way in the library today. English mathematician Henry Briggs (1561-1631; see ODNB or Wikipedia) usually gets the credit for teaching the long division algorithm in this particular way.
In 1597, Briggs was appointed the first professor of Geometry at Gresham College in London, endowed by Sir Thomas Gresham who built the Royal Exchange in London. In 1616, Briggs wrote the preface to the English translation of John Napier’s Mirifici Logarithmorum Canonis Descriptio (1614). The printer of A description of the admirable table o[f] logarithmes with a declaration of the most plentiful, easy, and speedy use thereof in both kindes of trigonometrie, as also in all mathematicall calculations (STC 18351) was Nicholas Okes, best known to Shakespeareans as the printer of the Pied Bull Quarto of King Lear but best known to MoEML and much admired by me as the printer of most of the mayoral pageant books. Long division has been hovering, unseen, on the periphery of my research life for a long time.
Inventor of logarithms, Napier recognized the impediment that complex calculation presents to mathematical investigation. “There is nothing,” he wrote, “that is so troublesome to Matheticall practise, nor that doth more molest and hinder Calculators [people performing calculations], then the Multiplications, Divisions, square, and cubical Extractions of great numbers” (STC 18351; Sig. A5r).
Briggs seems to have been a good teacher. At “Gresham house,” he “publickly taught the meaning & use of this [Napier’s] booke.” Given that not everyone could attend his classes, he aimed in his preface to “give some taste of the excellent use” of the book. He wanted to make clear that the techniques described in the book had an application. This particular book wasn’t ultimately about long division, which was merely a technique for performing the calculations necessary to produce logarithmic tables, but the message is pedagogically valuable.
When my son needs help with long division again, I will try to historicize the method and explain that it is simply a way of breaking down and rendering on paper something he understands quite well in the abstract. There are other algorithms that predate Brigg’s long division, and perhaps my son would find one of them more appealing. I will try to explain that the technique is not an end in itself, even though his math textbook presents a culturally and historically specific method as a universal law. And I will happily draw arrows and divide with him.