Some years ago, in 1996 to be precise, I read this article:
Davis, G. E. (1996) What is the difference between remembering someone post a letter and remembering the square root of 2? It was in the Proceedings of the 20th Conference for the International Group for the Psychology of Math Education.
I’ve been thinking about it ever since.
You can find the whole article here.
I mention it here because that article made me, and still makes me, think about the importance of students remembering math, to quote Davis, as more than “a rule-driven, bloodless, passionless, activity, situated nowhere in time or space.”
He writes, “Why, however, should a young student’s mind actively engage with mathematics when they don’t yet know what it is about? It is not, after all, and every day subject that most people talk about as they do the weather, their health, or the activities of their friends and neighbors. But perhaps it could be…”
In this article, Davis writes about the importance of of developing mathematical imagery that make abstract concepts something more “real”, and creating memories that situate ideas in time and place, with a sense of meaning. The idea that “abstract knowledge, once obtained retains within it elements of the original context” seems likely a reasonable idea but one that I think we do not pay enough attention to.
Davis references a 1951 book by E.J. Furlong, A Study in Memory, stating that to “recollect, reminisce, and retrospect” – with imagery – is essential to mathematical thinking.
It’s this development of a “rich set of images” about mathematics that interests me. These images are more than pictures we might draw on paper, but the actual contexts in which we have experienced mathematical constructs. The students who have participated in the Random Walk lesson have a certain “lived experience” and imagery associated with coins flips and random behavior because of the experiment. I have found that they refer to their actions in the lesson years later when they talk about randomness.
I don’t claim that makes this lesson the “best” lesson for teaching this topic. But I do argue for the importance of viewing any math concepts as part of a story – a story in which each student is a character, not just a reader. The Random Walk lesson has been particularly effective in engaging students as protagonists of a personal story of math, one that goes beyond “I remember I learned about that math topic in 7th grade, but that’s about all I remember.”
There are certainly lots of ways to engage students, but I think those that promote students’ development of the “episodic memories” of which Davis writes have a particularly important effect on students’ learning of abstractions. Math can be something we talk about in the same way we talk about much of the rest of our lives.