I often hear teachers and parents talk about how math skills build on each other in a way that other subjects do not: you have to know how to add before you can subtract, you have to know how to multiply before you use exponents. This is certainly true to an extent, however, I’m wondering if we’re reinforcing these modes by our overly linear curriculum maps (*ahem*). In an inquiry based approach of mathematics, we often preach about “multiple solutions or solution paths” or “multiple entry points.” If we believe what we’re selling, doesn’t that fly in the face of a laddered approach to curriculum mapping? Are we just paying lip service to the whole “multiple solution paths” bit because we know the real way to solve the problem?
I was talking with Kelly Renier (@krenier), director at Viking New Tech, and we began discussing the concept of “power standards” or “enduring understandings” or “What are the Five Things you want your students to know when they leave your class?” then build out from there. However, we didn’t discuss building those Five (or whatever number) Things out into linearly progressing units, but rather concentric circles.
Tasks and/or concepts may go in some ring of each of these concentric circles.
Think of it as an outward moving spiral.
However, standalone, this still operates somewhat linearly: you start with the middle stuff (which is allegedly easier or essential) and progress outward, just like you would at the start of a unit, progressing to more complex concepts. But we make an entire curriculum of concentric circles and rotate from concentric circles cluster to concentric circle cluster every few days, or even in a week, potentially moving outward from the center of each set of concentric circles along the way.
- Moving Part 1: Constructing units as concentric circles
- Moving Part 2: Rotating through and revisiting topics
That said, I’m not sure you could do Moving Part 2 without doing Moving Part 1. We probably need a name for this type of Scope and Sequence / Curriculum Map: Circular Curriculum Mapping? Iterative Curriculum Mapping? Archimedean Spiral Curriculum Mapping?
This is getting a little mad-scientisty, I realize. Still, this may have a few potential benefits.
1) Students get to revisit a general topic every few weeks, rather than a one-and-done shot at learning a concept.
2) Students have time to “forget” algorithms and processes and when they see a scenario they have to fight their way through it accessing prior or inventing new knowledge, rather than relying on teacher led examples. Yes, I consider this a benefit.
3) Teachers may formatively assess more adeptly.
4) Students may see math as a more connected experience, rather than a bunch of arbitrary recipes to follow.
5) It probably better reflects the learning process, which happens in fits and starts, and frankly, cannot be counted upon to be contained within a specified time frame.
Personally, I find this framework compelling to a point. I think it better exemplifies recent research and advocacy toward math education. It certainly is messier than a linear approach to curriculum mapping. Your syllabus could potentially look elegant and beautiful or ugly and convoluted. Your administrator might back you, she might not. I’m guessing if you were forced to follow a district scope and sequence, or your math department wanted to be teaching the same things at the same time this would be a non-starter.
So this is just a sort of framework I’ve been playing around with, mostly in my head and I thought I’d throw it out there. I haven’t really developed anything useful. I’d be interested to hear your thoughts. How would you feel about a framework such as this? Do you think it adheres to best practices around mathematics instruction? Would this just work to create more confusion within students? Just how impossible would this be to develop in a public school? Maybe some math departments or curricula are already doing this or something like this? And if it does adhere to best practices and it isn’t implementable due to external constraints, then there may be additional implications for a teacher, school and district. For now, we’re just trying things on. And possibly tearing things apart and starting from scratch. Again.
emergent math 
