# ThankThankThank You Bowman

I’ve been stuck in Grand Jury Duty for the past week and half, and for the weeks before then I was trying to eek every last fun thing I could think to do in New York City before jury duty and school started. It’s been a pretty ridiculous few weeks, and these weeks have almost nothing to do with school. I needed it, but now I’m feeling the anxiety creeping over me like some evil, insidious ivy. I’m still in the see no evil, hear no evil state of mind, and I will be in it until after jury duty is over. I’ve resigned myself to that, even though it’s doing nothing to dissipate that growing anxiety.

In these days, we’ve been fortunate to have Bowman Dickson write some fabulous guest posts:

Bowman did a bang up job. Not only did every post have something immediately useful and concrete, but heck if Kiki ain’t a fantastically engaging writer too!

I wanted to thank Bowman for agreeing to guest blog.

And more importantly, Bowman has started again blogging on his own blog, which he’s devoting to math teaching now, at the appropriately named http://bowmandickson.com/. I don’t know what to say except he’s so much of the real thing. Not just with this blog, but with teaching. He’s jumped in and isn’t scared to take risks and try things out, but he does so with such depth of thought coupled with passion for student learning, that I can’t help but hate him for being so much more awesome, and love him for being so much more awesome. You would be exercising crusty and reprehensible judgment if you didn’t add his blog to your reading, like, now.

And he tweets too at @bowmanimal.

Thank you Bowman! Peacocks for lyfe!

# Math Taboo

I participated in a great twitter conversation the other day where we brainstormed a few strategies to help make our courses more accessible to English Language Learners (we used the hashtag #ELLmath, the approximate transcript is here if you are interested). It was a great start to what needs to be a running dialogue for me, as I teach almost 100% students for whom English is not their first language. If anyone has any ideas about #ELLmath, I would love to hear them in the comments. The conversation reminded me of a little idea I had last year, playing the game Math Taboo to help students expand “definitions” to actual understandings of concepts. Now, I’m sure other people do this, and a quick Google search leads me to believe it’s not all that novel, but while discussing #ELLmath, it struck me as a particularly good exercise for ELL students.

The idea of the real game is to get your partner to guess a word by describing without using any of the five taboo words, which are usually the first words that anyone would go to in a description. So the obvious math equivalent is to pick a term that you are throwing around in your class and get students to describe it without using their go-to math descriptors.

We played during beginning-of-the-year-review as a class, with the word to guess already known to everyone, and I gave students a chance to take a stab at verbalizing a definition without using the taboo words, one at a time until we got an acceptable description. However, this could easily be adapted to be a much more interactive activity (though its creation might take just a bit of time).

#### So why play this?

Whenever working one on one with students, I found myself trying to diagnose why they were not understanding a problem. I would ask them things like, “Well, what is a derivative anyway?” and they would often answer with something that I found acceptable, but perhaps could have been just something that they had figured out should be said as the “correct” answer. Even if they weren’t saying book definitions (which would actually be easier to deal with), many times they were using my informal definitions – words that they had internalized about the concept that might not actually display a deep understanding, but that I had been mistakenly accepting as evidence of learning. Definitions are important, but assuming that those are indicators of deep understanding is, of course, very problematic, no matter where those definitions come from.

So, this Taboo game serves a two-fold purpose: learning for the students (by forcing them to think deeply about a mathematical concept; by having them trade in math jargon for conceptual understanding; and by hearing classmates describe something in more accessible vernacular) and learning for me (by seeing how well students actually understand a concept; and by seeing what language students use to talk math in the hopes that my mathematical narrative can better reflect theirs in the future).

#### Alternative game: In how few words can you express this definition?

I have never tried this game I’m about to describe, but the idea is to start out with a long definition from a math textbook and see how few words you can use to express the same idea. Delving into the Twitter world this summer I have realized how wordy I am, and the process of editing my tweets down has made me realize how many words I use that are unnecessary. Twitter forces me to think about what is the core of my idea, which led me to think up this exercise. This could be done competitively (give groups 5 minutes to brainstorm), or you could do it countdown style, trying to lower the number of words by one each time. This could get students to really consider what is important about a mathematical concept and to get them to realize that the thing itself is more important the words you use to express it.

# Feedback on SBG

This is going to be a super short post, directed to you SBGers. One of the things we talked a lot about this summer at PCMI was feedback! [1]

Here’s my question about feedback and SBG.

Let’s say you give a skill test with 3 or 4 skills on it, and you want to give feedback. Let’s also say that you give a numerical score for each skill. Besides that score, what sort of other feedback do you give on the test?

I’m wondering how explicit you get with identifying the issue, and what the goal(s) of you feedback is/are. Do you…

…circle where the student first started going awry?
… write a sentence (or two! or three! or more!) using words to explain where the work went awry?
… focus on something other than where the error crept in?
… give the total solution, written out for ’em?
… give some broad thoughts, encouragement?
… do something else?

Basically, I am super curious what your feedback looks like — in addition to the numerical score. The reason I ask is this… we talked a lot about feedback, but we didn’t talk about what effective feedback looks like. And for me, I don’t tend to give too much feedback. I’ll do some circling, some encouragement, I might write a sentence or two if the error is easily identifiable, but then I expect the student to figure the rest out on their own, or by working with a colleague.

And at first, as we talked more and more about feedback at PCMI, I felt a little guilty for the paucity of my feedback. But the more I thought about it, the more I thought: I think what I do is actually good for them. I don’t want to do all the work for them — they have to be a little more proactive. But I also wonder what and how others do, to see if there are things to steal to improve!

I wish I had scans of actual assessments to show you and have you give suggestions about how you’d mark them up, but in lieu of that, I’d like to hear how you do it (or would do it).

[1] One thing that struck me was how feedback given with a grade tends to be ignored by students, but feedback given without a grade tends to be really considered.

# Our Experience with Understanding by Design

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…

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 ($x^2+6x-3=0$) into a linear equation to help you solve it (by reducing it into $x+3=\pm \sqrt{12}$). 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.