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.

Well, this is indeed what we have been arguing and training people to do in Understanding by Design (as you imply by your reference to ‘enduring understandings’ from our work). I think you make a compelling case in math, especially. I have for decades been arguing against the so-called linear view of math education since i think it confuses process with product along the leins that Dewey argued a century ago. You can find numerous posts on my part about this at grantwiggins.wordpress.com

Hi Grant, as a fan of UbD from afar, yet not as familiar with the nitty gritty, do you have a sample of, say, a scope and sequence that adheres to the principles of UbD?

I’ll forward you one shortly. Meanwhile, this blog entry is suggestive:

http://grantwiggins.wordpress.com/2013/04/10/my-100th-post-so-why-not-bash-algebra/

Thanks for the post. I am redesigning my 8th grade map this August and needed a fresh perspective.

I have always struggled to create a map that reflects the need for spiraling and fluency. You have got me thinking that its the linear design of the map that makes this process so difficult. I am hoping this concentric approach improves the design and usefulness of my map. I am tired of creating maps that I never use in my daily practice. Would you like me to send the map your way when I’m done?

The concentric circles really resonate with me! I think you may be right about the linear curriculum maps reinforcing a we’re-done-with-that-topic mentality. In fact, the more I think about the idea of concentric circle curriculum maps (C3M?), the better I like it!

Isn’t this what “spiraling” is about? At its worst, spiraling is Everyday math, touching a bunch of subjects but not settling in on one. But spiraling as is used in CPM (a curriculum I’m not crazy about, overall) is pretty effective.

So when I teach linear equations, for example, I cover the basics using a modeling approach, then we move onto modeling inequalities (and I don’t know why Prof Wiggins says inequalities aren’t useful–in the real world, inequalities are HUGE in programming, much more “real-life” than linear equations). As they work on inequalities, I slowly introduce increasingly complex models, a process I continue as I move into systems. Then as I move into modeling quadratics, I bring back systems again.

In geometry, I might introduce coordinate geometry basics once, but then bring them back when working on transformations–reflecting a triangle over the line y = x+3, for example, or testing a rotation to be sure it’s perpendicular. Then bring it up again with diagonals and parallelograms.

In general, when assigning problems to work on a new concept, I include problems that require practice on a previous concept.

My tests never focus on any one topic; I always randomly choose from any concepts we’ve covered, although at least 40% of the test will be new material.

Again, I thought this was called “spiraling”, but perhaps I’m misunderstanding.

Incidentally, I totally disagree with Prof Wiggins on Algebra. It’s not that I think Algebra is essential to the real world, but the idea that anyone who can’t manage algebra can manage any of the concepts he mentions, much less non-Euclidean geometry, is a non-starter.

I suppose there is some overlap, particularly regarding the “rotation of concepts.” Although, I’ve honestly never seen much of a concerted effort to spiral at the middle or secondary level. Do you have an example of a scope and sequence or curriculum map that adheres to something like this?

Also, my impression of spiraling is that it still takes place in linear “units.” Concepts – such as statistics – are “covered” for a couple weeks, then moved on from, then returned to once, perhaps twice for the rest of the school year. I would argue that a concept should be “rotated through” much more frequently. But again, only hypothesizing here.

CPM is probably the best example. I’ve written much about my annoyance with CPM, so I’m not suggesting it as any sort of exemplar, but it’s where I got the idea of spiraling in assessments:

http://www.cpm.org/pdfs/information/sampleChapters/CCG%20Ch%203%20TV.pdf

Search for “Assessments” and go down to Individual Tests, where it says “It is strongly recommended that more than half of each test be made up of material from

previous chapters”. Also check the Review and Preview sections. Now that I think of it, most textbooks have the same thing–a review section of previous random topics at the end of each section. But CPM’s are more integrated.

“Also, my impression of spiraling is that it still takes place in linear “units.” ”

Definitely not how CPM does it, and definitely not how I do it. Every single test I give includes random questions on previous material. I expect kids to remember how to work certain problems–or, during the test, I’ll remind them (if they are weaker students). There’s some interesting research that testing on topics helps students learn the material; I definitely see some “aha” moments as the kids recognize oh, yeah, I’ve seen this before.

Sometimes I warn the students of earlier material by giving them some additional practice on it the week before, integrated with whatever topic we’re working on then. Most of the times, I don’t.

I’m pretty sure Geoff is looking for something a little more connecting than just practicing and testing previous concepts at a later date. But I’m just conjecturing here.

Interesting! It does scare me a little because I am comfortable with the linear approach, but I can see definite benefits from this change. I may try this out with a few lessons to see if this is something for my clasroom

I hear you, Andy. I’d love to hear how it goes with your experimentation – you’re already ahead of most of us by considering a relatively unique approach.

I think that good teachers with strong content knowledge most likely already teach this way whether or not the curriculum is actually structured this way. That’s the most important thing…strong content knowledge!!!

Probably. It’d be great to see some models. Show your work, everyone.

I like the concentric-circles idea. But I might replace the outer circle “amazing if you knew it” with “related thing of the student’s choice.”

Thank you for giving an example of your thinking with activities. It’s frequently hard to imagine what someone is considering if only a description of the curriculum is given, without specifics.

When I think of spiraling and textbooks, it’s Saxon (not CMP) that comes to mind. I’m contemplating something similar with the concept of pre-teaching ( planting-ideas.blogspot.com ) in my latest blog post.

Scott Hills

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