A short post today with a question for our readers.
A number of years ago there was a popular piece by Alison Blank titled Math is not linear, which gave a number of ideas about the order in which we teach mathematics. A curriculum writer has to grapple with the fact that, although math is not linear, time is. Hermione Granger’s time turner does not actually exist. Tuesday comes after Monday, and Tuesday’s lesson comes after Monday’s lesson and, in the end, a teacher has to decide what to teach on each day; that is, they have to decide on a linear order in which to teach mathematics. The gist of “Math is not linear” is that that order need not be a dry march through a logical hierarchy of topics. You can, as Blank says, go on tangents, foreshadow topics to come, connect back to previous topics, and give students problems that create a need for a new topic. These are all great ideas.
Our question is: what other ideas do people have to make sure that the sequence of lessons in a course makes sense to students and makes sense mathematically? Do you recommend any books or articles that might help answer these questions? We have some ideas and will be writing some posts about them, but want to hear from the community as well. Please feel free to share your thoughts in the comments, or on Twitter with @IllustrateMath, #timeislinear.
I think an important factor is not just looking at what skills and knowledge are necessary to learn topic “A”, but also has the student had enough time to master those skills and knowledge. In other words, before we can teach students how to add fractions with unlike denominators, we need to be sure they understand how to construct equivalent fractions with different denominators.
However, it is unlikely we can teach the equivalent fractions on Monday, and then teach adding fractions with unlike denominators on Tuesday. An important piece is not just which topics need to come earlier, but also consider how much time students need to master the topic before building on it.
In addition to paying attention to the sequencing of topics and lessons across a program, it’s important to map out arcs for central mathematical practices. These arcs do play out in the sequencing of topics, but they are more concerned with the uses of abstraction, continuity, formalization, encapsulation, and other styles of work. For example, the arc of functions might proceed from a recipe for a repeated set of calculations to a mathematical object that can be graphed, composed with other function, inverted, and so on. Here, the arc is about encapsulation. Another related example (made explicit in CCSSM): the arc of polynomial algebra starts with expressions as devices to model various situations and proceeds to the abstraction that polynomials are elements in an algebraic system that behaves a lot like the ordinary integers and that can be used not only to describe situations but also to create tools for keeping track of calculations (as with the binomial theorem).
Much of this design is the province of the writers, but I think it’s important to let the kids in on how these arcs are progressing across a program.
Al