For the last two years, I’ve adopted a problem-based, discussed-based approach for algebra 2. The whole curriculum is interleaved, meaning that big ideas are parsed and revisited over long periods of time (weeks or months) to improve retention. At any given time, students are learning small parts of a few different units. This allows for extended exposure to the topics that my kids learn. This is not how a curriculum is commonly viewed because, with this model, there are no traditional units. By “traditional” I mean thematic (e.g. Unit 6: Logarithmic Functions). Instead, these thematic units are broken down and served piecemeal to students over long stretches — mainly through problems. The sequencing of this model is indiscrete and quite messy.
As evidenced by that bewildering opening paragraph, I find all this terribly hard to communicate with others. I made my best effort to describe it here. It is often referred to as spiraling. I think Henri Picciotto does a good job of articulating it.
Thematic units have the advantage of being simpler…and easier too, I think. They are a slow-moving mass of closely-related topics that stays for a little while and then leaves when the next one comes along. Everything in them is directly linked and, therefore, these units make it easier for students to draw connections between mathematical concepts. At the same time, they encourage the isolation of facts and skills. Because related ideas are all lumped together, these units offer an easier pathway to a deep understanding in a short amount of time. Or at least the illusion of deep understanding.
These units make everything easier for the teacher, too. Thematic units and their associated lessons are far easier to plan and execute. The whole process linear; the focus of each lesson is based on the previous. There’s no untidy looping in and out of concepts, no systematic revisiting of big ideas over time. The concepts march in a clean, single-file line.
This is my guess as to why textbooks and traditional forms of curriculum have adopted thematic units. Seen in this way, they make the most sense both for the student and teacher.
But easier doesn’t make it better, right? When learning is hard, when it places a higher cognitive demand on the learner, isn’t it more meaningful? By helping students learn something small, then forget it, and then recall it after a reasonable amount of time — and iterating this process again and again over the course of the school year — can’t we help ideas cement? By not blocking out content, and instead spacing out practice and frequently assessing on the same topics at greater depth, do we help students better retain it? There’s research that says yes. Make it Stick by Paul Brown really helped me understand this.
There’s no denying the challenge that this creates for teachers. Tracing how concepts mature over the course of weeks or months is not easy. Adjustments to the sequencing can be tricky, too, because concepts are so tightly intertwined. I’ve been personally building the lessons and sequencing for two years and its still not right. Granted, I only work on it during the school year — and pretty much on the fly. Nonetheless, at least compared to traditional units, I’ve found it far more demanding and unusual to plan. And I haven’t even mentioned the loneliness — I have met no teachers during this time who are doing similar work with their curriculum. Who knows, maybe one day I’ll be disciplined enough to sit down and formally document my sequencing for other teachers to understand — and to start a conversation — but, I’m not worried about that.
I begin another school year in two weeks. I have realized that coupled with the fact my students retaining more information than ever, the messiness of interleaving has awakened and excited me these last few years. (I could do without the loneliness, though.) After years of marching in a single-file line, interleaving has made my curriculum work more of a dance. It’s interesting and lively. It moves. It sways. Fortunately, I work under an assistant principal and principal that given me the autonomy to do this. They decided to accept the consequences of the risks that I inherently took on when I decided to throw my units out the window. They trusted me even though none of us fully knew what I was doing. I think that I was at the right place at the right time because I’m not sure many other schools or departments would be on board with such a break from the norm of traditional units. My students and I have learned so much.