In my last post, I argued that, as teachers and math education leaders in a school or district, we need to free ourselves from the sequencing preordained by the textbook, and instead pay attention to what actually works with our students. In this post, I will present some general guidelines for sequencing topics, and some specific suggestions. All these ideas are based on 3+ decades in the high school math classroom, with somewhat heterogeneous classes.
Is the topic ageappropriate? Mysteriously, not much attention is paid to this. For example, tradition requires completing the square and the quadratic formula as topics for Algebra 1, In my experience, it is much easier to teach this in Algebra 2, to students who have a little more maturity. This in turn frees up precious time in Algebra 1 to reinforce the sort of basics that we find so frustrating when they are missing later on.
Can the topic fit in a single unit? Some topics are important, but difficult for some students. It’s a good idea to spread those out over more than one unit, and sometimes more than one course. One example of that is linear, quadratic, and exponential functions, which can be approached in different ways at different levels, from Algebra 1 to Precalculus. Another example is trigonometry, which can be distributed among Geometry, Algebra 2, and Precalculus.
When in the school year? Difficult and important topics should not be introduced in May! By then both students and teachers are tired, and there is little chance of success. Teach those topics as early as possible. There may also be some traditionally late topics that can be useful early on. For example, an exploration of inscribed angles has only very few prerequisites, and can be used to practice angle basics in an interesting context. I did this very early in my Geometry class. See my Geometry Labs (free download.)
Spiraling? There is much to be said for coming back to alreadyseen topics later in the school year. However spiraling can be overdone, and result in an atomized curriculum consisting of chunks that are so small that students don’t get enough depth on each topic. One easytoimplement policy, which provides the advantages of both spiraling and depth is to separate related topics. I have written about that in past blog posts (here and here).
Review? It is widely believed that one should start the year, and then each unit and even each lesson with review of relevant past material. Of course, I understand why that is a standard practice, but I believe it is counterproductive. Over time, it tells students “you don’t need to remember anything — I’ll make sure to remind you.” It is also profoundly boring for the students who don’t need the review, and takes away from the excitement of starting something new. It is far better to start with a solid anchor activity, and use homework, subsequent class work, and if need be outofclass support structures to do the review. It is especially catastrophic to start Algebra 1 with a review of arithmetic. The students who need it won’t get it, and those who don’t will be disappointed.
Anchor activity? If you can, start a unit with an interesting problem or activity (the anchor). It should be motivating and memorable, and it need not be easy. Examples of anchor activities are Geoboard Squares for the Pythagorean theorem (Lab 8.5 in Geometry Labs,) Rolling Dice for exponential functions, or SuperScientific Notation for logarithms. A good anchor brings together key content with good practices, and generates curiosity and engagement. It is something you can refer to later on, to remind students of the basics of the unit.
Start with definitions? No! Most students find it difficult to understand a definition for something they have no experience with. It is more effective to start with activities leading to concepts, and introduce vocabulary and notation when your students have some sense of what you’re talking about. For example, see the SuperScientific Notation lesson mentioned above. Another example: the tangent ratio can be introduced with the help of slope, without having to mention trigonometry or the calculator, instead using the 10centimeter circle. Once students can use the concept to solve problems, you can name it and reveal that there is a key on the calculator for it.
Concrete or abstract? Math is all about abstraction, but understanding is usually rooted in the concrete, so it is usually a good idea to start there. This can mean many things:

Discrete first, continuous later. Numerical examples first, generalization later. For example, work on the geoboard (both the standard 11 by 11 geoboard, and the circle geoboard) is strictly with specific examples based on the available pegs. But it lays the groundwork for a generalization using variables which would otherwise be impenetrable to many students.

Natural numbers to real numbers  almost any new idea is more accessible if you start with whole number examples — as in .

Kinesthetics (link) and manipulatives (link) do not accomplish miracles, but they can improve classroom discourse and provide meaningful and memorable reference points. In particular, algebra manipulatives can provide both access and depth to an essentially abstract subject, by way of a visual / geometric interpretation.

Tables and graphs can help provide a concrete foundation to the study of functions. This is sometimes described as modeling: you start with a concrete situation, use tables and graphs to think about it, and generalize with equations. This is the approach I use a lot in Algebra: Themes, Tools, Concepts and in my Algebra 2 materials.
From easy to hard? Well, that is certainly implied in the previous segment. However, I will now challenge that assumption. (What can I say, sequencing curriculum doesn’t lend itself to simple choices.) In my view, it is a good idea to start with somewhat challenging material, then ease up, and keep alternating between hard and easy. Starting too easy can give the wrong impression, that the unit will not require work. In fact, most of the above guidelines are best implemented as a back and forth motion: for example, after introducing vocabulary and notation, one needs to reintroduce the concepts. Likewise for most of these guidelines.
This is all fine, but how does one deal with externally mandated sequencing? Alas, I have no experience with this, as most of my career was at a small private school, and moreover I chaired my department (with plenty of input from my colleagues.) I can only suggest discussing these ideas with colleagues and supervisors! Also, most of the suggestions in this post address sequencing within a unit, and thus may be implemented anywhere if there is any wiggle room at all in the mandated sequence.
 Henri
PS: I’m offering two summer workshops (one on algebra, grades 711, and one with Rachel Chou on Algebra 2 / Precalculus. The workshops will include many of the bits of curriculum I linked to in this post. More info.
Notes:

Much of this post is based on one section of my article with the cheerful title Nothing Works.

Some of it is from a previous post on sequencing (Mapping Out a Course), where I propose a step by step process for doing just that.
Your thoughts about sequencing certainly ring true for me, all the way through. Perhaps I’m not crazy.
ReplyDeleteAnother aspect of the progression of a lesson that I try to keep in mind is strategic placement of tasks or exercises that the students will find encouraging. This relates to your observations about alternating between easy and hard. Every now and again, I want to include problems or task that will feel to the students like a celebration of what they have learned. For example, if a question was asked earlier in the lesson that the students knew they couldn’t yet solve, but then they realized in the later exercise that they now could solve it, then this is a celebration. I’m aiming for genuine implicit positive reinforcement. It is too easy for students to make genuine progress but not notice. Noticing progress is an important aspect of metacognition.
Scott Farrand
Hi Scott! Welcome to my blog!
DeleteYou make a great point! Those kinds of moments help students see they can grow mathematically, and are worth a thousand speeches about growth mindset!