While trying to get my lovably dorky son to put his math down and go to sleep, I came across this mistake in his work:
He is seeing this equation symmetrically — good! — but in this case he’s using mirror symmetry. From this point of view, the right side of the equation needs to be read from right to left. Isn’t this nice? That’s how equations should work.
It goes to show: a certain amount of mathematical knowledge is the ability to see things in a particular way. This is most evident in areas of math that involve shapes and stuff, but it’s just as true in arithmetic and algebra.
Jeffrey K. Bye recently shared a paper with me that explores this. The title is “Mastering algebra retrains the visual system to perceive hierarchical structure in equations” and, yeah, it’s about that. “Mathematical practice is undeniably perceptual,” they write. Some people might be inclined to say that none of this matters to the mathematician. But the algebraic notation, only a few hundred years old in most cases, is often sensitive to mathematical meaning:
While low-precedence operations like addition require a full symbol (p + q), multiplication requires only an abbreviated symbol (p•q) or no symbol at all (pq). If the visual system were sensitive to such regularities, then the hierarchical structure of algebra could be read off directly from an expression’s layout. And, indeed, people are sensitive to these visuospatial norms.
Part of learning algebra is learning to see certain kind of letters and numbers as especially significant. This is a cognitive thing, but it’s also a perceptual one. A trained eye looks at these expressions differently — they have structure! We see expressions as composed of “chunks” of mathematics:
The claim of the paper is that learning algebra literally retrains your eye to see strings of symbols as objects. Their premise is that you can tell the difference between a trained and an untrained eye by how much attention is necessary to answer a question. The researchers would show participants an expression like w x a + c x f, and then ask a question about it. For example, sometimes they would color two of the letters and asking if the colors were the same of different. The goal wasn’t to catch some people answering incorrectly. The idea was that the amount of time they spend answering each question could be significant.
Their results suggested that when the mathematically significant variables are involved, people familiar with that structure answer the question with less required attention. Or, in their more jargony words, “participants who had mastered the hierarchical structure of algebra exhibited object-based attention for algebraic sub-expressions.” Right, that.
They raise another interesting point, which is that teachers and others sometimes just hate when kids aren’t thinking about things when they solve problems. The mistake that my son made at the start of this post was not the result of thinking — it was just his gut “intuition,” really a perceptual error. But when people get really good at math, they don’t think about things all the time. First they think, then it becomes “automatic.” And one of these paths towards automaticity is the retraining of the visual system:
The current results suggest that relying on visual processing might be a boon, not a barrier, to mathematical reasoning. This might come as a surprise. Confronted with evidence of students’ reliance on misleading, superficial visual strategies in algebra, some have argued that mathematical training should avoid and even suppress perceptual strategies (e.g., Kirshner, 1989; Kirshner & Awtry, 2004). For example, when asked to solve 4 + 4/2 + 2, some students might be led to answer “2,” incorrectly, because of the superficially tempting, perceptually strong 4 + 4 and 2 + 2 groups. Indeed, we sometimes found evidence for perceptual grouping around addition, rather than multiplication, particularly among participants who had yet to master the hierarchical syntax of algebra. But the fact that novices use perceptual strategies to arrive at incorrect answers does not imply that experts abandon such strategies entirely. [My emphasis – MP] Instead, experts may refine those perceptual strategies so that they become reliable, robust, and rapid routes to correct solutions (Goldstone et al., 2010; cf., Hutchins, 1995, and Rumelhart et al., 1986).
Do you have to explicitly teach students to see the structure of expressions and equations? Or does it just happen on its own when kids learn a lot of algebra via the usual concepts, procedures, skills? I don’t know. I don’t think this paper knows either.
After my son made this mistake I first said, please just go to sleep. We’ll talk about it in the morning. But when he insisted, I said, this is not how we read equations. We read both sides of the equation like we read English, left to right. Oh, he said, and he started trying to erase. Yosef, I said, please go to sleep. And then I walked out.
There’s something powerful and just a little bit scary about this kind of learning. We learn what we learn, and at some point along the way it forever changes how what we see when we look out at the world. It structures it, and then we just can’t avoid it. We’re teaching ourselves to see objects where before there were none. We can do that with algebraic expressions, OK that seems fine, but we must be doing this all the time. You know how first impressions always seem to be right? We’ve done that to ourselves, man. What we see is often not so different from how we think, something good to remember.