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.

*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)

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

-- Henri

You're making want to write more about my structure, Henri! So glad I landed on your posts. I think I'm finding it easier to talk about it now! https://lazyocho.com/2018/11/06/i-have-trouble-talking-about-my-teaching/

ReplyDeleteI find that the spirals have become circles with the "next" level of the spiral not any deeper than the current one.

ReplyDeleteWhat are you referring to?

DeleteExample:

DeleteThe 1st time "solving 1st-degree equations" comes around, students work on one- and two-step equations.

The 2nd time "solving 1st-degree equations" spirals around, they work on two-step equations.

The 3rd time "solving 1st-degree equations" spirals around, they work on two-step equations.

The 4th time "solving 1st-degree equations" spirals around, they work on two-step equations.

Rather than:

The 1st time "solving 1st-degree equations" comes around, students work on one- and two-step equations.

The 2nd time "solving 1st-degree equations" spirals around, students work on equations with variables on both sides.

The 3rd time "solving 1st-degree equations" spirals around, students work on equations for which students need to distribute and/or add like terms first.

The 4th time "solving 1st-degree equations" spirals around, students work on equations involving all of the above with rational coefficients.

I see this within a grade level, and, more importantly between grade levels.

The 8th/9th grader is beginning to be proficient when solving two-step equations, maybe with variables on both sides.

The 12th grader is proficient when solving two-step equations, maybe with variables on both sides, but has had no opportunity to develop proficiency with needing to distribute and/or add like terms first, nor with rational coefficients.

Then when those students arrive at the two-year college I teach at, they end up being placed into our lowest level algebra class because by the end of that course we expect them to be able to solve all of those equations by the end of that semester. Also, the first half of the unit on solving first-degree equations, they don't pay attention because they believe they already know all of the material. Then toward the end of that unit they are surprised to find out the equations they are expected to solve are much more difficult than the ones they are familiar with.

I see this over, and over, and over, for almost every topic.

Another example:

In 3-5 grade students learn to add fractions. The State test specifications say that students whould be able to add two fractions with denominators amoung 1,2,3,4,5,6,8,9, 10, and 12. Students in grades 3-5 never see a fraction with a different denominator, and for the next seven years students don't get any practice adding fractions whose denominators may factor into 2-4 prime factors, including say 7, or 11, or 13. I see just-graduated students who have no idea what is necessary to add two fractions whose denominators are say 35 and 105. (And others who do know, but who also choose to use 35*105 as a common denominator.)

etc.

.

I can see that you're in a frustrating situation. I am guessing that what you're seeing is the result of tracking. Your students were placed in low-expectations classes in high school, where the goal was not understanding, but the memorization of specific "steps". So, to take the example of linear equations: it is in fact impossible to memorize steps for every possible linear equation, so the students are over and over made to work on what can be memorized. I taught high school math for 32 years, and NEVER taught "one-step, two-step" equations. Instead I delayed equation-solving until students had a solid grounding in basic ideas such as like terms and the distributive law. Once you have that foundation, you can solve any linear equation, and don't need to memorize steps.

ReplyDeleteOne possible way forward for our profession is for high school math departments to discuss the ideas in NCTM's Catalyzing Change, which aims to help rethink high school math, and end the tracking of students into dead-end classes where what you describe is all too common.

Of course, I may be wrong, and your students maybe were not in low-tracked classes. But they're still the victims of a low-expectations, memorization-based approach to learning math.