*This post was written by both Sam and his guest blogger, Bowman.*

At the Klingenstein (Klingon) Summer Institute, we (the illustrious Sam and Bowman) participated in planning a unit in the style of a Performance Based Assessment, which is very similar to method of planning advocated in Understanding by Design. Normal people just call it backwards planning. Whatever you name it, the core of this philosophy is that **Enduring Understandings** should be the focus of curriculum design, and not skills or learning activities. With a third (also illustrious) Calculus teacher, we put together a rough draft of a unit planned in this manner.The big idea of the unit was **the Relationship between Limits and Rates of Change**. We decided that by the end of the unit, we wanted students to understand that: *curves can be conceptualized as a joining together of almost linear pieces*; and *an infinitely fine approximation of a quantity is often needed to yield the exact value*.

This was a difficult process. We thought it might be helpful to share some thoughts about planning this way and share some of the ways that we approached backwards planning at KSI. Here is the completed, though still rough rough rough, product **[update: see the bottom of this post for another group’s work]**:

**What do you think an Enduring Understanding is?**

**SJS:** It’s funny. When I was first introduced to UBD at my school for curriculum mapping (barf, BARF, BARF), it was exactly at “enduring understanding” that everyone threw up their hands and gave up. Partly because we’d show it to the consultant, and she’d say “no” and then give us no guidance from there. Partly because it forced us to grapple with exactly what it is we wanted the kids to learn, and dig down to the core of what we really valued about what we taught.

At the Klingon Institute, our math leader (also illustrious) said something that really spoke to me.** He said “an Enduring Understanding is something you want your kids to remember 5, 10 years from now.”** It sounds lofty, even corny. But when I took a moment to really think about it, it struck me: I need to know what it is I truly care about, and this is it. But thinking in this way — what truly is at the core, mathematically, of what you’re teaching? — is terrifying and hard. Especially if you’re something who has always focused on units and skills. It is also exciting, because you get to come up with Big Ideas and use those as your lesson/unit/yearlong themes. But honestly, more terrifying.

The other half of what an Enduring Understanding is, is that “it has to be something general, but not vague.”

If a student asks you “why are we learning this?” and the best you can do is say “well, you’re learning that so you can learn calculus [or X]…” and then they get to calculus [or X] and you say “well, you’re learning this so you can do engineering and open doors…” I’ve been known to do this. BARF. How unsatisfying for a student. And if I’m not mistaken, *every time you say something like that, you yourself get a sick, guilty feeling.*

A really good enduring understanding should put a stop to this infinite regression, and those guilty pangs you feel. Because you know exactly what it is you want a student to take away — and you can tell ‘em, loud and proud. Okay, it may not always be sexy, but it is something fundamental they can latch onto now.

**BD:** I think that everything that Sam just said is perfect, and considering we learned about these at the same program, I’m not all that surprised that we have a similar enduring understanding about what enduring understandings are. The only piece I would add is that it helped us to start all Enduring Understandings with the sentence “I want students to understand that…”

**SJS:** I’m going to emphasize that “I want students to understand that…” is needed when creating your enduring understandings, but you cannot lazily make it into a skill. Something like “I want students to understand that when solving a radical equation, there may be extraneous roots” sucks. That’s too specific, and is a single skill. It should be something that applies more generally, like “I want students to understand that sometimes the best way to count in math is to not count.” (That might be for a unit on combinatorics.)

**BD:** One additional point I would like to make though is that, though the final product (your list of enduring understandings) is hugely helpful, I found the most helpful part of curriculum design like this was collaboratively going through the process of trying to figure it all out. Attempting to articulate ideas and sift through the wide world of math to find the meat forced me to think so deeply about my curriculum. To be honest, I’m not sure how useful a list of someone else’s enduring understandings would be to you. It’s like taking someone else’s lesson plans – unless you think about it and modify it to be your own, it’s hard to implement in your own classroom.

**SJS:** I also think the process was valuable, but I’d disagree with Bowman about not finding others’ enduring understandings useful. I have limited time, am sometimes (often) lazy, and I can get on board with someone else’s enduring understanding if I buy into it. Like, for example, Bowman is going to come up with a whole host of enduring understandings for calculus, and I’m going to steal them. Right, Bowman? Right? Why reinvent the wheel when Bowman will carve it for you?

**BD:** If you’re happy with slightly-to-horribly-misshapen wheels…

**How can Big Ideas and Enduring Understandings help you organize your curriculum?**

**BD:** It’s easy to get caught up in using the book’s sequencing of content, but thinking about big ideas and enduring understandings can help rearrange everything else to help promote those enduring understandings above everything else. For example, next year we will explore solids of a known cross section before solids of revolution, because the enduring understanding in solids is that if you stack up a bunch of infinitely thin cross-sectional areas, you can create a solid. Solids of revolution are really solids of known cross section too, just with circular cross sections – the revolution is just a way to construct the solid, not the main idea behind the integral. By talking about solids of known cross section first, it might be a good way to highlight the deeper idea without getting caught up in a multitude of smaller ones.

**SJS:** You might think textbooks give us Big Ideas — quadratics, conics, etc. But Big Ideas are not topical, but transcend topics. As for how they can help me organize my curriculum, I don’t know yet. I do think they are going to be the anchors of a class.

**BD:** A list of big ideas in mathematics that we generated with a group of 13 awesome math teachers at KSI: models, functions, dimension, relations, transformations, estimation, comparison, distributions, measurement, operation, conjecture, representations, rates of change, logic, proofs/reasoning, inference, mathematical objects, classification, systems and structure, definitions, inverses, algorithms, patterns, symmetry, equivalence, infinity/infinitesimal, and discrete vs. continuous.

**SJS:** We like this list, but it seemed too birds eye view for us. When we worked on it, we found it made more sense to just zoom in a wee bit. Our big idea again was **“The Relationship between Limits and Rates of Change”**. It’s not like there’s a right answer to how to do this. You have to do what’s useful to you.

**BD:** So what’s the different between a BIG IDEA and an ENDURING UNDERSTANDING then? The point of big ideas is to give you thematically lynch pins around which to organize your curriculum instead of the typical CH 1.4, CH 2.3 – i.e., what ties all these topics together? The same big idea can occur across many different math courses. Then the enduring understandings are the learning outcomes that you want to come out from exploring these big ideas (see above for a much better description).

**How do you assess Enduring Understandings in SBG?**

**SJS:** Right now I honestly have no idea. Right now I’m thinking of making SBG 70% of the grade, and Big Things (projects, enduring understanding assessments, problem solving) 30% of the grade. Or something like that.

**BD:** The big thing that I am going to add to my class next year is writing for informal assessment. Even if it doesn’t count for standards grades in SBG, I think that I might just keep a list of enduring understandings for my own purposes and informally assess the students as I go through the semester. Then, when larger assessments come around, I will explicitly focus review around enduring understandings. Since standards in my SBG-hybrid system only account for around 40% of the grade, I think I will keep my SBG standards to be skills and focus my summative assessments around larger ideas, though I will make sure to be explicit about this with my students. This of course is not perfect, but like Sam, this is something I’m wrestling with.

**How can you do this sort of work without having Noureddine (who was our curriculum group leader) giving you feedback?**

**SJS:** That’s why we have blogs — for feedback! But I suppose the some questions you can ask yourself (regarding if you have a good Enduring Understanding):

1) Is it general, without being vague?

2) Do multiple “skills” fall under the mathematical principle/idea your Understanding encompasses?

3) Of all the content related things, is this something you’d want a student to remember 5 or 10 years from now? Honestly? REALLY? Okay now, really?

**BD:** One of my goals next year is going to be to more effectively utilize the resources at my school, i.e. the other teachers in my department. The more people that collaborate on something or check out your work, the better chance it has of being something valuable. I know this is a general principle, but I would have the temptation to not go to other members of my department because they don’t already plan like this – I am definitely not going to fall into that trap next year.

**Does this sort of thinking re-orient (re-frame?) the way you look at teaching, or the meaning of what math class is?**

**SJS:** For me, it’s helped me see the value of looking for a bigger picture. It’s complicated, the question of “what do you teach?” Right now I teach skills, and I can do that pretty well. But skills for what? That’s the real question I’ve done a bang up job of dodging. So when I worked on this, it forced me to countenance that head on. What do I really want to give to my students, mathematically? [1] Example: It’s not completing the formula, say, but it’s the idea that you can transform a non-linear equation () into a linear equation to help you solve it (by reducing it into ). I suppose it makes you think more about the larger themes of a class, or something.

*[1] I’m not talking about habits of mind, or those sorts of things. What I’m talking about here is purely mathematical content.*

**BD:** When you have a group of unruly students who will be sitting in front of you for 45 minutes every day, it is easy to get caught up in the day to day of lesson planning. The first thing I always jump to is the learning activities. To give myself a bit of credit, I think I often had an idea, though subconscious, of the bigger picture, but by never spelling it out for myself, I could never really spell it out for my students either. It’s hard to take the time when you start planning to think about big ideas, but I found in just the one unit we planned together that once we had identified the enduring understanding, solid learning activities were so much easier to come up with. This hierarchy has helped me see that I can’t really implement learning in my classroom until I frame what learning really means in terms of big ideas and enduring understandings.

**SJS:** Here’s a gedankenexperiment. If you asked your kids at the end of your course what the big mathematical takeaways were, what answers would you get? If I asked my kids that question at the end of the year… well, it would be a crusty hodgepodge of things. They don’t know my mathematical goals for the course, and clearly that’s because I myself don’t know my goals. Not broadly, not meaningfully.

**What was the most frustrating part about curriculum design like this?**

**BD:** I am someone who is good at working within a framework and tweaking that, but this involves rethinking the whole conceptual framework of your class. Also, it was frustrating to realize that all of my SBG standards were skills, and that I didn’t ever explicitly identify the big ideas. Being self-critical without being self-deprecating (and not in the funny way) is tough for me, but that’s partly what this process is for.

**SJS:** So many things, so many things. The gads and gads of time it took. The supreme annoyance when we couldn’t come up with a good Enduring Understanding or Big Idea. Our inability to easily come up with good assessments to check exactly what it is we wanted the students to learn. But mostly, it was what Bowman noted: realizing that even though I’ve taught Algebra II and Calculus for four years, I don’t really have a sense of what it is I truly want students to get out of it.

**Are you going to change your teaching because of this?**

**SJS:** I want to say yes, but I don’t think it’s something I’m going to be able to do wholesale. I think I’m going to try to do only one unit using this sort of planning — but do it really well. (That’s what our illustrious Klingon curriculum leader suggested.) And build up from there, each year. This is all a little lofty for me, and it’s no magic bullet for student understanding.

**BD:** Even if I don’t formally plan units this coming year with this method, I am happy to have my thinking shifted to be more in terms of big ideas and enduring understandings. Like, after you spend forever looking for new shoes, all you can notice about other people is their shoes – I’m hoping that even if I teach the way I did last year, I will be able to pick out the big ideas in the process and focus on those. Then codifying and formalizing unit plans into grand designs like this will be much easier in the future.

**SJS:** Kudos, sir, kudos. We can use this year to brainstorm these enduring understandings, as we teach and ask ourselves, forlornly, “what the heck is the takeaway from the rational root theorem?”

**BD:** Good luck with that.

**Update: **Another Klingon group said we could share their unit planning (for Algebra I) which is below.

OK this is a ramble and possibly pointless. but the post intrigued me. … be forewarned. I really am intrigued about where this process goes with you two…. I’m now getting many students who are 5, 10, even (eek) 15 years out of high school.

It’s been a bit refreshing to see that in at least a couple of cases, they are remembered really important things that I cared about a lot. In most cases this happened in my AP Stats class.

* I’m designing a study for the NBA – here are all of the issues that will prevent me from doing the study I want to do. I need to brainstorm some practical work-arounds so that the data can be used to make some conclusions.

* My professor’s explanations for this genetics problem are definitely not making sense to me. These calculations make sense to me. He told me I’m wrong. I don’t think I am. Can we collaborate? (PS my students was right – the professor was mistaken in one area).

* Even when I was kicked out for heroin, I knew that I was smart and capable and could do good things. I’m finally doing that. I’ve been clean for four years. You never made me feel like I’d be a failure. I knew I could do it – you believed in me.

In each case, I was floored that these kids came to me and said what they did. I was not close with any of these kids, nor were they “favorites.” They were certainly nice, and I cared about them, but it was scary that whatever I did had that much impact. I worry about whether I had a negative impact on kids as deeply too.

I guess what I’m trying to say is that some of the powerful take-aways after 5 or 10 years may not be even “big concept” based, but even bigger picture: more “habits of mind” or “self-concept” based 5 or ten years later. I always have tried to design lessons and run class with “enduring concepts” in mind, but how does one measure that? This is not to say that curriculum shouldn’t be designed around enduring understandings – on the contrary: I try to do the same! But somehow, as I get older and more crotchety, I wonder how to measure what kids really leave with 5 or ten years down the line. I’m thinking again about how we know it’s working. I would love to see a long-term longitudinal study about the impact of such efforts by teachers like you in the professional lives of the students impacted. I think that if a study like this were ever done, the results would be powerful and meaningful.

OK – enjoyed the post – sorry to ramble! BT

ps Chloe just declared her love of Fr33dom. Check it out.

So, my math department (Karl + I) went through the process of mapping our entire curriculum via UBD over the past year and a half. Our approach was to use the UBD framework (Enduring Understandings -> Essential Questions -> Skills -> Topics & Content) at both the department and course unit level. For example, an EU for all math courses was “Formulating and solving math questions in the real world takes creativity, perseverance, and discipline.” much akin to some of the Common Core Mathematical Practices Standards. Then at the course unit level, we had EUs like “Differentiation and integration can be applied to other coordinate systems” for polar/parametric/vector calc. and “Pattern recognition allows us to count high numbers of things efficiently.” for counting/prob. We had only a couple hours of instruction about UBD, so apologies in advance for what we botched or misunderstood.

Some thoughts from my experience:

1. I enjoyed thinking about the biggest of big pictures of what math should be about.

2. I realized that many of the skills and much of the content I taught was only circuitously connected to the understandings I wanted to achieve.

3. As a result, I questioned whether I had chosen the “right” EUs,

4. Which made me laugh and realize how easy it is to fall in love with skills/content.

5. In retrospect, many of our course unit EUs were way too specific, and would never pass the 5-year test.

6. Often, it felt like UBD was better suited to other academic subjects than to math. I had a hard time thinking of Essential Questions as motivating skills (or perhaps, the understandings and questions I thought of did not lead directly to the skills I already taught and assumed to be important). Yes, the area problem motivates integral calculus, etc. but I struggle to connect most of the Precal / Alg. II dots.

7. We were simultaneously implementing SBG. Lots of head scratching and bit of math department identity crisis.

The biggest question on my mind: How do you assess understanding? Especially ENDURING understanding? Projects and group work with a communication emphasis are where I started, but those are baby steps. Maybe I should cold call students a few years down the road and ask them what they remember. I feel like the answer I’d get would probably be a nugget of content like “the derivative of sine is cosine.” Skills are so much easier to teach, remember, and assess (esp. with SBG)!

I think I believe that skills and understanding are equally important in math. UBD has skills well down the hierarchy, while SBG tends to place them first and foremost. Perhaps the opposite is true for capital-u Understanding. In my own classes, I think there’s plenty of room for me to push toward the bigger picture thinking and understanding that UBD attempts. Probably not going to use it as the overall structure though.

Looking forward to hearing how it goes for you two, and anyone else trying to reconcile these philosophies/systems.

Ok, true confessions: tl;dr. But this sounds like good stuff, I’m just skimming through a lot of backlog.

Wanted to add, though, that it’s worth getting (or borrowing) a copy of the original Understanding By Design book that kicked off the backwards design focus. I did after hearing about the ideas second hand, and struggling with applying it to math. What I found is that the original book had no prolem whatsoever using core skills as the EU for examples in math classes. So don’t be ashamed to say, look, being able to read and create trig graphs is the EU for this unit. It doesn’t need a more high-level justification. That said, if there are Bigger ideas to target above the level of skills, great. But that’s not the only “real” way to do backwards design.

The other thing I see as being less of a conflict is meshing with SBG. Your SBG topics are basically your EU’s, although maybe you’d want to elaborate on them to make it more clear. Or at least that’s one possibility.

I appreciated josh g.’s comments regarding meshing UbD with SBG, specifically that it’s okay for the enduring understandings to be fairly narrow content-oriented concepts. This is reassuring for those struggling to use UbD in math. For those interested in bigger picture kinds of understandings, I found the book More Good Questions: Great Ways to Differentiate Secondary Mathematics Instruction by Marian Small & Amy Lin helpful. Some examples of what they call Big Ideas are:

“Algebraic reasoning is a process for describing and analyzing generalized mathematical relationships and change using words and symbols.”

“Comparing mathematical patterns or relationships either algebraically or graphically helps us see that there are classes of relationships with common characteristics and helps us describe each member of the class.”

“Many equivalent representations can describe the same pattern or generalization. Each representation may give more insight into certain characteristics of the situation or generalization.”

Sam and Bowman, your UbD unit seems great, but it sure looks like a lot of time went into it. I can see how the collaborative effort would pay huge dividends for understanding the material and planning a very coherent instructional strategy. However, if I had time constraints, I think I’d choose to forgo all of that work in order to spend the effort making formative assessments or locating/developing hooks for the lessons or working on multimedia supports (ala Dan Meyer). I’m not saying the process is bad or anything, just that if I was limited on time I’d choose to put my effort elsewhere. And, since we’re using SBG, we really have done backward design anyway when we identified learning goals and created multilevel rubrics and created a sequence of assessments for the goals.

Finally, a quick comment regarding Performance Assessment. I’ve seen these go so wrong. And I’ve also seen a ton of work go into some big project design, but when it comes to actually implementing the project there isn’t enough class time or it’s a poor use of time or whatever. I like how an educator (I wish I remember who; it could have been Marzano I guess) described performance assessment: it’s when you observe the student performing the skill or concept. This approach gives so much freedom because no longer does it mean it’s a big project or real-world application. A teacher can observe a student performing math in an authentic setting just while they’re doing some normal seatwork task or while a group is discussing or working their way thru a challenging problem.

Thank you everyone for such helpful comments. This blogging thing rocks – every single one of this comments really helped me put this all in perspective and made the monumental effort that we put into thinking about this more reasonable/feasible for my curriculum.

FWIW, this website has the new Common Core State Standards redone using the UBD framework. Kind of cool.

http://qualityinstruction.org/

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