In most math curricula, students work on a single topic at a time. (When I taught elementary school, decades ago, I noticed that if we’re working on subtraction, it must be November! But the same applies at all grade levels.) The idea is that is that by really focusing on the topic, you are helping students really learn it, before you move on to the next unit. Unfortunately, that is not how retention happens. It is much more effective, when learning a new concept, to see it again a few weeks later, and again some time after that. Thus the concept of spiraling. Years ago, the Saxon books distributed homework on any one topic across the year, typically with one or two exercises per topic on any given day. Some more recent curricula do facilitate that sort of homework spiraling by including review homework in addition to homework on the current topic after each lesson. The algebra textbook I coauthored in the 1990’s is spiraled throughout: not just in the homework, but in the makeup of each chapter and many lessons. This idea was so important to us, that there is an image of a spiral at the start of each chapter! (If you have the book, check that out! Or just look at it online.)
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.
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)
I have written a fair amount about this, under the heading 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!)
Implementing these policies does not require more prep time, or more classroom time, and it creates a non-artificial, organic way to implement “constant forward motion, eternal review”. It helps all students with the benefits of spiraling, but without the possible disadvantages. You really should try it! Read an overview of this approach on my Web site: Reaching the Full Range