I learned subtraction back in the heady days before global warming, when we trudged uphill both ways to school in three feet of fresh powder. It was a process, both the trudging and the learning, because walking through snow is only fun when you have nowhere to be and because borrowing the one never made much sense.
You may remember. You’re trying to take 99 from 183, so you line them up vertically:
Except nine doesn’t go into three, so you need to borrow from the 10’s column, and then nine can go into 13, but then nine doesn’t go into seven so it needs to go into 17 and then, well, whatever, doesn’t someone have a calculator watch or something? Eventually, you figure it out, but it’s a pain and conceptually complex.
Fast forward a number of years, and I found myself on the subway, watching a school-age child dutifully doing his homework, a subtraction problem set, during his commute home. But instead of using the column method, he had some sort of horizontal line on his paper. I was fascinated. I didn’t ask him anything because that would have been creepy and weird and no one needs that on the F train at 3:30 in the afternoon, but I did some investigating when I got home.
He was using a number line, a newfangled way to do subtraction. It’s all the rage, apparently. You could go down the Google vortex in search of details or you could just watch this video, narrated by an unintentionally condescending British dude:
That’s fun. The concept has its roots in the discoveries of John Wallis, an English mathematician from the 17th century who served as chief cryptographer for Parliament and the royal court, and who is given credit for inventing the sideways-eight infinity symbol. He also helped develop infinitesimal calculus and has an asteroid named after him. (Impressive, but can he insert a link into a blog post? Doubtful.)
At first I thought it was really cool, and then I realized basically all you’re doing is changing a subtraction problem to an addition problem.
Anyhow. Back in the days of the colonies, the number line helped the masses conceptualize the idea of negative numbers, giving visualization to the fact that there was something below zero, i.e. to the left. Today, it has been co-opted to help youngsters learn subtraction.
It’s not a perfect system. A group of researchers studying the indigenous Yupno people of the Finisterre Range in Papua New Guinea found that number lines were not as engrained into the human brain as previously thought. “Our study shows, for the first time, that the number-line concept is not a ‘universal intuition’ but a particular cultural tool that requires training and education to master,” said Rafael Nunez, director of the Embodied Cognition Lab and associate professor of cognitive science in the University of California-San Diego Division of Social Sciences. “Also, we document that precise number concepts can exist independently of linear or other metric-driven spatial representations.”
At first I thought it was really cool, and then I realized basically all you’re doing is changing a subtraction problem to an addition problem. Someone using the number line gets the correct answer—assuming he or she understands the concept behind column addition and can add up the numbers—but do they understand the theory behind the subtraction? To me, admittedly someone who has never been trained in educational practices, some combination of the line method and the column method would seem like the best practice. But it’s interesting to see how how things like this evolve.
Now, please don’t get me started on the “1 up, 1 down” method.