In this post, I want to argue that while I agree with the fundamental underlying idea of a spiraled curriculum, there is such a thing as overdoing the spiral. I will end with specific recommendations for better spiraling.

### Impact on Learning

Too much spiraling can lead to atomized, shallow learning. If there is too much jumping around between topics in a given week, or in a given homework assignment, it is difficult to get into any of the topics in depth. Extreme spiraling makes more sense in a shallow curriculum that prioritizes remembering micro-techniques. In a program that prioritizes understanding, you need to dedicate a substantial amount of time to the most important topics. This means approaching them in multiple representations, using various learning tools, and applying them in different contexts. This cannot be done if one is constantly switching among multiple topics.In particular, in homework or class work, it is often useful to assign nonrandom sets of exercises, which are related, and build on each other. For example, “Find the distance from (

*p, q*) to (0, 0) where

*p*and

*q*are whole numbers between 0 and 10.” (This assignment is taken from my

*Geometry Labs.*) At first sight, this is unreasonable: there are 121 such points. But as students work on this and enter their answers on a grid, they start seeing that symmetry cuts that number way down. In fact, the distances for points that lie on the same line through the origin can easily be obtained as they are all multiples of the same number. (For example, on the 45° line, they’re all multiples of the square root of two.) Nonrandom sets of problems can deepen understanding, but they are not possible in an overly spiraled homework system.

### Impact on Teaching

The main problem with hyper-spiraling is the above-described impact on learning. But do not underestimate its impact on the teacher. For example, some spiraling advocates suggest homework schemes such as “half the exercises on today’s material, one quarter on last week, one quarter on basics.” Frankly, it is not fair to make such demands on already-overworked teachers. Complicated schemes along these lines take too much time and energy to implement, and must be re-invented every time one makes a change in textbook or sequencing. Those sorts of systems are likely to be abandoned after a while, except by teachers who do not value sleep.*Algebra: Themes, Tools, Concepts*we tried to compensate for that by offering an Index of Selected Topics and Tools. We also included notes in the margin of the Teachers’ Edition: “What this Lesson is About”. But even with all that, a hyper-spiraled approach makes extreme and unrealistic demands on teachers’ planning time. In fact, some hyper-spiraled curricula lack even those organizational features. Without them, a teacher needs to spend the whole summer working through the curriculum in order to be ready to teach it. This can be fun if the curriculum is well designed (e.g. the Exeter curriculum), but no one should feel guilty if they’re not up to that level of workaholism.

### Spiraling Made Easy and Effective

So, you ask, what do I suggest? In the decades following the publication of my overly-spiraled book, I developed an approach to spiraling that:- is unit-based, and allows for going in depth into each topic
- is easy to implement and does not make unrealistic demands on the teacher
- is transparent and does not hide what lessons are about (most of the time)

*extending exposure*. The ingredients of this teacher-friendly approach are:

- Lagging homework and assessments
- Separating related topics
- Teaching two units at any one time (just two!)

-- Henri