This past summer, I was invited to participate in a Math Assessment Workshop for AERO (American Education Reaches Out), sponsored by the U.S. State department. For anyone who doesn't know what AERO is (I didn't really), it's basically Common Core for international schools; the goal is to create "a framework for curriculum consistency across grades K-12 and for stability of curriculum in overseas schools, which typically have a high rate of teacher turnover." The math standards are essentially the same as Common Core (in fact, I think they're the precursor to CC, but I'm not solid on my history there). The workshop was led by Erma Anderson (@ermaander), an impressive individual with a wealth of knowledge who I'm glad to have met and been able to work with.

We were a small group of 8 teachers from schools around the world, and from all different age groups (2 K-5, 3 MS, 3 HS). The rest of the group had participated in workshops before for the MSIS (Math Specialist in International Schools) program run by AERO; I was sort of an outsider who slipped in because my wife is doing MSIS, but now I want more!

Now, I thought I was pretty up-to-date on SBA in the math community (see the previous post for my history), but this workshop turned me on to a new framework I'd never used, read about, or seen before: the four "Claims for the Mathematics Summative Assessment" from the Smarter Balanced Assessment Consortium. The purposes of the workshop were to 

1. Try to write, or adapt, some "good" problems to assess these aims
2. Try to write a general rubric, which turned into more of a student profile, based on these claims.

We started the week writing a few assessment problems in grade-level groups, then critiquing them all together. I found this process challenging in a good way - frustrated by things I hadn't thought about, questioning my assumptions, occasionally questioning why we even teach this stuff anyway, etc. It made us really examine our goals in assessing our students, and recognize some inadequacies in many of the assessments we currently use. 

After getting our feet wet with this process, we turned to creating a student profile based on the four claims. We started out calling this a "rubric", which led to a lot of confusion about the purpose of the document. Once we changed our focus to creating a profile, we started coming together towards a final product. This process took two days, but we felt pretty good about having created a document that we all felt comfortable applying K-12

​After this, we got back to writing and critiquing assessments. The high school group borrowed heavily from Illustrative Mathematics problems, and the following are three problems we felt pretty good about. 

If anyone reading this is interested in checking out any of the K-8 problems, contact me.

Overall, it was energizing to be part of this group of math teachers who were focused and interested in what they were doing. I hope we can keep in touch through #AEROmaththinktank on Twitter.
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Processing...
Because of my unfamiliarity with the claims, I needed to do some independent study and research to help get my head around this way of approaching assessment. Here are two big ideas I'm going to try to use in assessment this year (but this will take some time, and I've got a lot of newness to deal with this year).

1. The four claims are the boss:
These are the things we are always assessing in math assessments, regardless of the specific learning target or subject matter. They should be considered whenever we are designing assessments. Part of Erma's instruction included a link between the four claims and the standards for mathematical practice (SMP); I'm much more familiar and comfortable with the SMP, so I found this helpful. The mess below is me visualizing this (and playing with MS OneNote on a new touchscreen computer).

2. Clusters are learning targets
I spent my first five years trying to write learning targets based on specific standards. Anyone who's done this knows how muddled it can get. Erma blew my mind with the idea of using the Common Core clusters as larger learning targets for assessment's sake. This is fitting in with the idea of a SBA "skills list", which I haven't used before, and the following is the beginnings of my attempt to do this for my geometry class this year. I'm using New Visions' curriculum as a jumping off point here.

Ok, I'm stopping there because the first day of work at my new school is tomorrow! Just had to get some of this down and out of my head before getting into work.

Where I'm coming from
I got my teaching license in 2003. None of my coursework for teaching, including my student teaching, had anything to do with standards. 
Then I got a job outside of the traditional school setting, out of the loop as far as current best-practices and educational reform were concerned. I kept this job for 8 years.
In 2012, I got back into classroom teaching by landing a job at brand new international school in China. The leadership of the school was pretty progressive, and decided from the beginning to go with the most current research-based practices.
I still remember the staff meeting where we were introduced to Standards-Based (SB) Grading, Assessment, and Reporting (I'm going to use SBG, SBA, SBR, and try to explain why I differentiate later). Few of the staff (some with much more experience than I) knew about or had much experience with SB anything, and it was kind of a shock to most of us. Oh yeah, BTW, this meeting happened AFTER two or so weeks of instruction, AFTER some of us had already distributed syllabi, grading scales, etc. I had no idea what anyone was talking about - the only grading I knew was percentages and ABCs. I cried in the bathroom that day...
Pretty quickly, watching others struggle with this, I came to the understanding that my lack of experience was an advantage. I didn't have to deal with, or unlearn,  years of assessing any other way, I just had to get my head around doing it this way. I was also the only secondary math teacher in the school, so I had an incredible, and often intimidating, level of freedom in developing my curriculum and classroom practices. So I bought in, did the work, and started learning. 
Needless to say, I learned more through the experience of teaching than I ever had in any class about teaching. I learned more through personal research, struggling with frustrations, searching for my own answers, than I ever have from professional development. After five years with this school, I feel like I have a relatively good grasp on the idea of SB, although I'm still struggling with the practice and implementation.

Where I am now
As far as I can tell, so is everyone else. I ask educators and administrators about their implementations whenever I can, and I read a lot of blogs and articles on the subject. Over the last five years, only one educator I've spoken with said that his school had "completely figured out" SBG; further conversation revealed that what he really meant was that his school had aligned a 1-4 grading scale with a percentage grading scale in a way that the majority of teachers, students, parents, and other stakeholders accepted.
Everyone else tells the truth; it's a journey, a learning process that no one seems to have nailed down completely yet. There are great ideas out there, but there doesn't seem to be anyone (other than that guy) who's willing to say they've got all the answers and they know exactly how it should be implemented.
I like to separate SB, especially when it involves grades, into three areas that help me think about my own practice. These distinctions are mine, from my experience, and may be different from others'. (they may also be wrong! :)

• Assessment (SBA): This is the spirit of the whole thing. You start from a standard, make it into a learning target, and design instructional activities (formative assessments) and summative assessments by which students can show that they've met those targets. Teachers assess the degree with which students have shown proficiency.
• Grading (SBG): This is the method by which teachers assign a category, or grade, to represent a student's level of proficiency. 1-3, 1-4, 1-5 scales are common, rather than letter grades, and each level comes with a descriptor (my last school used Emerging, Approaching, Proficient, Exemplary) and they usually reference a rubric which tries to explain what each level means. SBG also makes little or no use of formative assessments (homework, quizzes, classwork) in calculating the final grade, focusing instead on a student's performance on summative assessments. This is where the big controversy with students, parents, and teachers who are used to traditional grading scales and levels. It also seems like every school is coming up with their own (different) way to address this discomfort.
• Reporting (SBR): This is the report card, the way student progress is communicated to students, parents, other schools, etc. My last school had us report on content domains from the Common Core as well as and overall "grade" and a narrative paragraph (yeah, report cards took a long time).

​Where I'm going
Earlier this summer, I participated in an assessment workshop for AERO which gave me a whole new way to think about the standards, and I really want to write a post about it later. The shift that's rolling around in my head involves using clusters (not specific standards) to come up with targets, and couching the targets in the four claims from the Smarter Balanced Assessment Consortium.

In a few days, I'm heading out to start a new job in Pakistan! I don't know everything about how things work there, but I don't think they're using SBR yet (kind of a relief to me). They use percentage scales and letter grades for reports, but I've been reading a lot on how other teachers are doing SBG within their own classrooms, even if it's not a school-wide practice, and even if they eventually have to show a letter grade. Overall I don't think I can do assessment any other way, so SBA will be a part of what I do no matter where I teach. 

If anyone made it this far, thanks for reading. I hope to keep posting on this journey, reading about what others are doing, and refining my practice.


People who've helped me think about this (not an exhaustive list, INPO):
Follow the links for some great posts on SBG
Michael Matera
Dan Meyer
Dane Ehlert
Jonathan Claydon
Nora Oswald

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After 5 years of living and teaching in Dongguan, China, we're making the move to Lahore, Pakistan this summer (after some down-time in Florida)! Never mind the geography - I'm also considering a whole lot of changes in my teaching.

• IB School --> AP School
• Integrated Math --> Traditional pathway (Alg1, Geometry, Alg2)
• Using notebooks? Maybe? Jonathan Claydon  (@rawrdimus) has just about got me convinced. And I might be stealing his whole Precal course... Seriously, check out http://infinitesums.com/ for some great resources.
• Did an assessment workshop for AERO last week that totally blew my mind, so there will definitely be some assessment changes. Met a whole bunch of great educators in the #AEROMathThinkTank (@catyrmath, @adeacetis, @davejwaters1, @SonalPatel4, @ptang123, @DanielLadbrook) led by Erma Anderson (@ermaander).

I hope to get to the last two points in a little more detail in the blog this summer, but right now the sun is shining and the pool is calling (or the beach if we feel adventurous).

I learned about proofs in my high school geometry class the way most people probably did: two columns; statements and reasons; lots of arcane symbology and abbreviations; notebooks filled with reasons; and a lot of this from the textbook:

I don’t want to get into the reasons I found (still find) this boring and overly formal; let's just say it’s been one of my least favorite subjects to teach. Honestly, my response to this in the past has been to gloss over proof in favor of applications and activities or problems that make use of all those theorems. So I haven’t ever really taught it well.
This year as I approached congruence (8th grade Integrated 1) and similarity (9th grade Integrated 2) with dread, I took some time to look at the standards and think about the progression I was going to use, the reason we teach this in the first place, the tools I like to teach with, and the results I was looking for. I slowly began to piece it all together in a way that made sense to me (and kind of makes me feel silly for not realizing this earlier).
Integrated 1:
Construction → Rigid Motion Transformations → Congruence Proofs
I spent more time than usual on construction this year, but less time than usual on hammering the prescribed constructions (copying, bisecting, perpendicular and parallel, etc.). Instead, I treated it more as an avenue towards exploration and creativity with this construction project (thanks Pam Rissmann for the idea).

​This was a good way for students to get used to the tools (both compass/straight edge and GeoGebra). I didn’t really formally assess the classic constructions; I figure they’ll need to be able to do most of those things to get through the transformations sequence.
So I set up transformations as a series of construction tasks, using paper constructions to bring students to an understanding of how imprecise they are, then moving to GeoGebra. We start off with the whole toolbar, and eventually whittle it down to basic tools to show we know what we’re doing (inspiration from euclidthegame.com).
The big change in mindset for me came with really focusing on the definition of congruence as the result of rigid motion transformations. So I decided that that’s how we’re doing proofs this year. Instead of two column proofs, the students are moving triangles around, and learning a lot of GeoGebra tricks in the process

Integrated 2:
Review rigid motions → Add dilations → Similarity proofs
In reviewing the previous year's work on transformations, we made a big deal about the link to congruence and similarity, and defined these terms solely using transformations. They walked though a couple of these with some assistance, and today they started on their partner projects, working on visual proofs of the angle bisector theorem, and narrating them in a screencast.

Am I missing anything?
Seriously, I'm trying to think of a time after my high school Geometry class where I needed two-column proofs, and I can't come up with anything. If there's something I'm overlooking that my students are really going to miss out on, please let me know!

My students and colleagues all know how much I love Desmos, and how much I rely on it in my classroom. I've been playing around with doing an art project using Desmos for awhile, trying to find the right time, the right content connection, the right group of students, etc. I've thrown the idea out to different groups of students as an alternative or enrichment in a couple of cases, but never really prepared for it or made it an intentional goal. The way I presented and assessed it over the last couple of weeks with my 9th grade Integrated 2 class is the closest I feel like I've come (yet) to doing it right. I also feel like a lot of other teachers must do something like this, but I couldn't find anything that really fit what I wanted to do (although Jon Orr's Beautiful Functions was kind of an inspiration).
The content
I've got a growing love affair with "functions" as a specific area of focus in my instruction, and a growing understanding of how they fit into all the other areas, and how they fit into the grades 8-10 currriculum I've been working on for the past 4 years. Grade 9 is the first time we really dig into function transformations, and we're also at a point where we've had some experience dealing with functions that are a little more "interesting" than linear and exponential (mainly quadratics, but we also dabble in radicals and cubics). 
Tech and Prep
Working up to this assignment included the following desmos activities.
​1. Transformations Review by Suzanne Von Oy
This worked as a nice intro, with answers students could check. Also, the "library" of functions used matched up nicely with where we were.

2. Match My Function by Jon Orr
​This was nice because it asked for the functions in function notation, something I was shooting for in the project. It also let students play around with unfamiliar function types, and drove home the idea that transformations are universal, and apply to all functions, even ones we don't know yet.

3. Make a Face! by Me!
I'm still in the process of finding the right balance between spending my time making my own materials and spending my time searching for existing materials. This was fun to make, and focused in on some of the main aims of the project. I think it got the kids ready pretty well. Building it also gave me a better idea of the time and effort it would take for the students.

The assignment
Part of the "success" of this project, I think, was that I put some thought into the requirements, and into how I would assess it. Not perfect (I'd like to find a way to make it a little more "mathy"), but at least it's clear. Here's the Google Doc of the assignment and rubric.

Providing a model
I remembered to do this this time! It always helps, especially with the language I was looking for at the end.
"Link: https://www.desmos.com/calculator/8y5nzp27wy 
This is a recreation of the mask of one of my favorite comic book characters, Grendel. I used an image from this link for a guide: https://www.redbubble.com/people/bighairmonkey/works/3126293-grendel. For the nose, I used two transformations of the quadratic parent function, one of which was a vertical reflection so that the parabola opened down. I shaded between them using the top function as a restriction for the shading on the bottom function. For the eyes, I used exponential, quadratic, and absolute value functions, which I dilated, reflected, and translated, then used inequalities to shade. I found some of the shapes in the mask difficult to reproduce using functions, because they curve back in on themselves. This would make them fail the vertical line test, so they couldn’t be functions. In order to reproduce these shapes correctly using functions, I would have had to write many different small functions. Instead, I decided to simplify the shapes to make them easier to draw with fewer functions."

The Results
I gave the students two 90-minute class periods to work on this so that I could make sure they got off to a good start, and then two weeks outside of class to put in whatever amount of effort they wanted. I'm really happy with what they came up with. (The one on the top left's supposed to be me, BTW :)

The Follow-Up
To take care of a 40-minute Friday class, students showed off their work to the class, practicing their academic language. Fun way to finish off the week.

Every once in awhile (sometimes in a LONG while), a lesson just clicks. It's some combination of the right planning, the right group of students, the right materials, etc. that just works. For myself, I'd like to start keeping track of those lessons so that I can build on them and refine, polish, or just re-use them in the future. For others, I hope I can provide some materials and ideas to help build a great lesson in other classrooms. I'm going to try to keep posting these as the "Lessons I Like" (LIL) series on this blog.

A few weeks ago, I had the opportunity to take over a colleague's classroom for a day to introduce his students to GeoGebra. This was my first time fully teaching another teacher's class, and there are some parts of that dynamic that I would do differently next time, but the lesson went well.
Setting the scene

• 18 7th grade students, 1-1 laptops, partners
• No previous experience with GeoGebra
• Geometry- Basic rigid-motion transformations

Managing tech introduction
Introducing a new piece of technology can be super frustrating. I've learned from experience that it's best, at least for the instructor, if -at least for the preliminaries like account creating, passwords, etc.- I can keep all of the students on the same page. Otherwise, they can sort of get all over the place and the room becomes really difficult to manage. 
To curb this frustration and keep everyone together, I use Red/Green Cards (RGC) (two sheets of construction paper, one red, one green, sandwiched together and laminated, then cut in quarters).
​I've been playing around with these for awhile, but this is the first lesson where I was really intentional about using them as a management tool, and they worked pretty well. Here's the idea:

• Students sit in pairs, each with a RGC on their desk/table

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• Students may not touch their own cards. Instead, they need to prove to  their partner that they've completed the task, and the partner flips the card to green

​Why this works

• Gets partners interacting with each other
• Gets students used to the idea of confirming their work with peers before going to the teacher
• Gives instructors a clear visual cue as to the progress of the entire class

Introducing GeoGebra
I decided to have the teacher set up a GeoGebra group to keep track of student progress. If you use GeoGebra in class, but haven't tried their groups, I strongly suggest you check it out.
Here are my slides for the introduction:

Some things to note

1. Google Chrome is the recommended browser for GeoGebra
2. If your school uses Google Apps for Education, signing in to Chrome and using Google accounts whenever possible makes password management a lot easier.
3. I learned the hard way that usernames matter when you're trying to assess and give feedback and using many different apps and accounts. Alphabetical order keeps everything lined up.
4. Acknowledge the content, but start with something fun! This is the hook. You just introduced your students to a powerful tool for learning, and there's no better way to get exploring than by starting with something fun, maybe even a little silly.

Which brings me to...
The content
I was lucky enough to stumble across this bundle of transformations exercises on tes.com by Mark Horley. It includes a paper worksheet and a bunch of GeoGebra files, which I've uploaded to GeoGebra (they can be found here).
I really like this blend of paper and tech, and the directions are very clear. These students would have benefited from a little more of an introduction, but they seemed to get into the activity pretty quickly. For a 40 minute lesson, this was a great introduction to one of my favorite math tools. Special thanks to Elizabeth (@eab69), Lilian, and Marty for supporting.

​I had a really rough class last week with my 8th graders. They'd been using a Chinese word pretty regularly in class that I suspected was a swear word. So I decided to call them out on it. When I heard a student using the word, I'd ask them what they said. And what did they do? As you can probably imagine, they totally lied to me. Then their classmates backed them up. 
I left that class pretty furious and went directly to the Mandarin teacher to confirm my suspicions (it's like the worst word you can say in Mandarin). Then I started stewing... Losing sleep... Plotting my revenge...
Because of the schedule, it turned out that I didn't see this class again for a week, which was probably a good thing. I asked a colleague for five minutes with them at the end of her class the next day, and I let them know that I knew what they were saying, and I was really disappointed with them. I tried to make sure they understood that it wasn't the profanity that really bugged me (I use profanity, although I don't throw it around in public), it was the dishonesty. Also, that this was an indicator of a bigger problem with this class: the lack of academic language in our daily discourse. 
Silence... Blank stares... Some guilty looks... Some grins...
So then we all had a week to think about it. I spent a lot of mental energy on it, although I doubt they did. I got an email from one of the students involved apologizing for the profanity, but didn't hear anything else.
The next time I had them in class, I started out by making sure they understood the problems:

1. Dishonesty- I can't teach this class the way I want to if I can't trust you.
2. Academic Language- The language we use in class directly affects our ability to respond to directions and show our understanding using the language of the classroom, regardless of our home language.
3. Responsibility- All of us need to take responsibility for the language we use in this class. It's not always enough for YOU to do the right thing; sometimes you need to take the time to encourage those around you to do the right thing. It's your learning, so take charge!

And then I went on strike. "I can't teach you until we figure this out. I've tried everything I know how to do, and it's not working. You fix it. I'll be at my desk when you've got a solution."
I sat at my desk in the corner and left them to have a conversation. Slow start... leaders emerge... They ended up having a pretty decent conversation for 8th grade students. 

• Punishment! - But punishment doesn't work.
• Only English in class! - But sometimes you need things translated that you don't understand
• Why should I care? - I care!

They ended up passing around a whiteboard and each writing some personal agreements on it about how they would contribute to a solution​. Also came up with some norms/rules addressing when/why mother tongue is needed (I was actually impressed with this: it's pretty close to what I would have had them do if this was teacher-led). 

After about 45 minutes, I stepped in and gave a little direction to solidify and wrap things up, and then led them through some roleplaying scenarios (1 plays teacher and 2 play students).

• Teacher is teaching and you have no idea what he's talking about. What do you do?
  • Raise my hand, ask to have it translated, partner translates.
• Partner is speaking about math in non-academic language
  • Ask partner to repeat in AL, help her if she needs it.
• Partner is speaking in English, about math, but using incorrect terminology
  • Correct partner by paraphrasing using AL
• ​Partner is speaking in any language about something not related to math class
  • "We're in math class. Let's stick to the material."

Then I actually got to finish up class by teaching a little math!

The school trip is coming up, so again, we have a long period of time until our next class. I feel like they did some good work today, and I hope it sticks. At least it got them talking and thinking about this.
Take-aways, Lessons Learned:

• I need to be more intentional about classroom management
• I need to remember to involve students in formulating classroom norms
• I need to do more of this in the beginning of the year (while responding to a problem can be a powerful way to learn and grow, it might be better not to have the problem in the first place)
• I need to remember to give myself and my students space to reflect when things go poorly