Why a Math Writing Rubric?

I Googled “rubric” just to see what kind of images came up, and up popped Rubik’s Cube. This is actually a perfect symbol of the way I feel about any rubric placed in front of me. It’s almost there, but not quite right. No amount of tweaking ever gets it to where I want it to be. And if I somehow get to that sweet rubric spot, someone else will look at it, and see the cube above. The tweaking begins again.

So what is the point of a rubric, and in our case, a math writing rubric? As we debated and wordsmithed and debated some more, we knew that we wanted it to be :

  • a tool for feedback to students AND
  • a tool to inform instruction

If we reduce a piece of writing down to just a grade, then only the grade is recorded in the student brain, not the corresponding learning. We considered a single point rubric, but also thought about whether teachers wanted to write more feedback or highlight feedback on the rubric itself. To save teacher time, the single point rubric has been tabled, but may be revisited at a later time. We considered the feedback from teachers in our last cohort and used their thinking to improve the rubric.

We borrowed language from the SBAC. When stuck on a rubric for a piece of work, I like to use the score description words. Does this feel like a Full & Complete answer, or at least a Reasonable one? Maybe it’s Partial, and not quite Reasonable. Now that I have a feeling on where it falls, I can look at that column to find bullet points or adjust up and down accordingly. We purposely left score “points” off. While we might have labeled this 4, 3, 2, or 1, we chose instead to avoid the idea of giving it a “score”. I know this is a tough one for teachers, who have requirements at varying levels to have documented grades in the gradebook.

Coherence between the lessons and the evaluation were important to us. We used language from the writing lessons we have been developing for our own use as the criteria for evaluation. What we saw from scoring tons of papers using our old rubric is that we would have decent explanations from some students – who had serious math misconceptions. It is lovely to read their writing to understand better their thinking process and misconceptions, so in terms of explanation and justification, they would have scored much higher. But in a holistic rubric, where one of the requirements was correctness, we ended up with a whole slew of the same low score. In using criteria, we hoped to discriminate between correctness and justification in order to better target the needs of the student or group.

We also wanted to link the criteria to the Standards for Mathematical Practice (SMPs) so that we could build awareness and understanding of them. Writing is not the only way to evaluate the SMP’s, but it certainly lends itself to being one way to see how a student is using the practices.

The Rubric (version 2.0)

Curious about version 1.0? Click here.

I have to admit, to put so much work into something that looks so simple is really quite humbling. Not to mention, even though this is not my work alone, I’m definitely feeling vulnerable sharing it out. With that said, I’m also constantly inspired by Graham Fletcher’s quote, “All of us are smarter than one of us.” So, I put it out here for feedback and sharing.

How Would I Use It?

First, our lesson series basically has students come up with these on their own. In introducing this rubric, I’d show how it connects to what they have identified as valuable in mathematical argumentation. Once students have had that grand conversation, then we’d go over the wording in this rubric, and trace how it changes from stage to stage. Academic language such as “accurate”, “element”, and “logic” would need to be addressed, along with any others that may be confusing to students.

We’d evaluate preselected anchor papers together, which is part of our lesson series. Discussion among partners, groups, and whole class would help calibrate our understanding of each level and criteria.

After some lessons on how to write mathematical argument (a topic for another day), we’d use one row for partners to give feedback to each other. As teachers, I’d use a preselected row to give feedback on. As proficiency grows, I might use the rubric to score a class and look at where the scores fall. Do I have a lot of Full & Complete for direct response, but only partial for justification? Might be a need for a whole class reteach. Do I have a small group that have Partial for direct response? Might pull a small group and work on identifying the question or statement to defend, and how to make sure I’ve directly responded to it.

I always put myself back in my classroom shoes, where I know my next question is “How do I grade it?” Now that I know more, I’d tell myself, “Don’t. Give feedback instead.” Eventually though, I want to report progress. So, once they’ve had plenty of feedback, I might assign a task as an assessment – when I know they understand the expectations. In my grade level PLC I’d want to determine what constitutes an “A” across the classrooms. I might collect tasks along the way as a portfolio, and look for growth. I also might play around with the concept of A/B/C/D and decide which descriptor warrants the A. Quite honestly I’d err on the positive side as long as I saw a progression in their learning. It’s not the grade that matters. It’s the learning.

Our Next Steps

For us, our next steps are to test this out with some class sets that we’ve saved from our writing series. We want to calibrate and see if there is anything we are missing or inconsistencies that make the rubric confusing. We will update our series of lessons to include this version 2.0 of our rubric. And this summer, we present again. We’ll offer extended coaching in the fall, learn more with some new teachers and share out with our prior early adopters. During that time, we’ll measure again for growth with this new tool. We predict we will see more targeted data to help us better identify needs for our lessons and/or task selection.

Leave thoughts in the comments! I’d love to hear your tweaks to our rubric, or ideas for using in the classroom.

Math Writing at Empower 19

ASCD Empower 19
Chicago, CA
March 16-18, 2019

Just got back from Chicago, and have so many thoughts rolling around in my head!! Blog ideas about navigating the conference, Ron Clark, Doris Kearns Goodwin (this is my new role model – her storytelling, her presence, her speaking ability and intelligence are all qualities I aspire to), presenting do’s and don’ts, and writing ideas are all swirling around in my head. So, in trying to stay focused, I’m starting with my big passion – communicating mathematically.

There were few workshop options in the math writing/communicating world, and the first one I went to was titled Elevating ELL Discourse in the Mathematics Classroom, presented by Sherry Ayala and Sylvia Olmos. Although the title didn’t reference writing, the summary did. Much of this workshop turned out to be similar to work we have done in our district around the 5 Practices. Lately as I’ve attended workshops, I’ve noticed quite a bit of similar learning, which leads me to look for the “nuggets”, or the new piece of learning that I can attach to prior knowledge. In this case, my nugget was how we teach and practice math vocabulary.

Students partner up, and explain their thinking around a problem, such as 324 – 165. As a teacher, you’ve considered what vocabulary they may use around this problem, such as subtract, add, ones, tens, hundreds, regroup. You provide the students with a small table, so that the listening partner can tally the number of times the partner used the vocabulary words.


From there, the listening partner can paraphrase back to the speaker what s/he said, and the speaker can self assess. Why didn’t I use regroup (or ones/tens/hundreds/etc). Did I need to? How could I explain differently so that I could use a different vocabulary word? I’m super curious to try this out in a classroom to see how students respond. The vocab words chosen are all based on what you have taught and what you expect them to use – being aware that the flexible thinkers you have molded will likely be able to find ways to explain that would change the vocab you are looking for.

The other workshop I went to was titled Strategies for Teaching Effective Mathematical Communication, from the book by the same name, presented by Laney Sammons and Donna Boucher. This one intrigued me because it was directly on mathematical writing, and I’m searching for different research based opinions and ideas to round out the work we have been doing so far. Again I saw many similar ideas to what we have already been doing with discourse (oral communication ideas, using CRA in math progressions). I noticed that this workshop also brought up the idea of developing math vocabulary, including the use of the Frayer model (along with a foldable that I’ve referenced in many of my own trainings).

Another nugget from this workshop was the inspiration to redo our criteria on our rubric. Their suggestion, based on work from the Ontario Ministry of Education, was to examine:

  • Precision regarding details, strategies, observations, and calculations
  • Explanations of assumptions and generalizations made
  • Clarity in logical organization
  • Cohesive arguments presented
  • Elaborations that explain and justify mathematical ideas and strategies
  • Appropriate use of mathematical terminology

I think that this list makes for a nice set of criteria beyond what we have used up to this point, and the work now will be to blend it with our current rubric:

So what is next for my math writing partner in crime and myself? We are looking at streamlining our presentation so that the focus is not the lessons that are prepared for classes, but instead an approach more linked to disciplinary literacy. An approach in which the vocabulary is introduced, clearly understood and practiced. One where there are multiple criteria for success and we can narrow lessons down to one or two of those criteria each time to develop, strengthen, and support their communication in writing. Plus, we need to go over the book, Strategies for Teaching Effective Mathematical Communication, to look for more ways to refine our thoughts and our rubrics.

The last nugget, that I should probably save for my navigating the conference post, is: Just because you didn’t attend the session doesn’t mean you can’t spy on the notes! So the Connecting Math and Literacy session by Alex Kajitani was at the same time as another session. In his presentation, he had this wonderful quote from Marilyn Burns. Besides wondering why I hadn’t seen it before, I was struck by the clarity it brought me.

Writing in math class isn’t meant to produce a product suitable for publication, but rather to provide a way for students to reflect on their own learning and to explore, extend, and cement their ideas about the mathematics they study.

Marilyn Burns
Educational Leadership, 2004

Spoiler alert – that one will be in our next presentation. Thanks to all who developed and presented at the conference! You challenged us to develop and refine our own thinking, and I am grateful for it.

2019 Goals: Writing

In this last of my 2019 goals series, I’m focusing on writing. Hence the blog! I figure if I’m going to help teachers and students write, then I need to be in the process myself. There is actually a record of me saying I hate teaching writing on Facebook, back in 2009 when posting answers to 25 questions about yourself was trending. Shortly thereafter, I came to discover that I actually loved teaching writing. It was through a professional development series that was actually based in technology that I grew to love the writing process. It’s funny how things can work out that way. Take something I love (technology) and then embed something I’m not interested in (writing) and a spark ignites. I’d like to think that same thing happens for our students as well.

So how did I go from hating writing to loving writing? The process was based in helping my students navigate their brand new laptops (our grade level had been given a laptop cart to share). While they were writing letters to pen pals, I started to realize that I couldn’t try to correct everything possible in their writing. It would sound unnatural – not like a 4th grader. I also learned that there would be many, many questions in this process, and I couldn’t spend 10 minutes with each child. Writing conferences became more like writing drive through’s, where I would know of one or two items in the student’s writing that s/he was working on, and I would focus my brief attention there. I never focused on spelling during these writing stages – I’d file that information away for a lesson to be done shortly thereafter. Plus, I made sure they were aware of how to use spell check! The whole world of writing opened up to me – I realized the power of a mini lesson and the power of meeting students where they were at. Now, Google docs (through the use of Google Classroom) really makes the writing process a fun one for teachers and students. Click here for a short overview of what I love about Google Docs.

I’m super excited to take a group of teachers to see Carol Jago at the end of January, as she presents on opinion writing. I love that she is not only talking about the process, but also how to provide feedback and manage the paper load that teaching writing creates.

One of my biggest interests is in the area of mathematical writing, and will likely be the focus of many future blog posts. Currently, I’m working with many teachers on a set of lessons based out of Think It, Show It Mathematics.

This was suggested to me after attending a workshop from a professor at UC Davis, where she had shared lessons they had created from this book. So, we have worked with them and tweaked them for the needs of our teachers. At this point, we’ve had rave reviews on our PD. Teachers who have really focused on mathematical writing are finding that by having a rubric by which to evaluate the work, along with language for the writing itself, has helped their students’ explanations soar. Here are a couple of examples of fourth graders’ recent math writing. Note the use of transition words and a clear statement of the answer.

Clear statement of the answer. What might be your next steps with this student?
What understandings does this student have?

I just happened to grab a couple of snapshots as I was in the room for another purpose – to observe their collaboration as partners! But, I was so struck by the fact that after collaborating with partners, there weren’t any questions of “How do I explain this?” It was fascinating to not only observe their conversations, but then see them explain their thought process so clearly on paper.

I’m looking forward to doing even more investigation into mathematical writing with my teachers, through a few different lenses. First, there’s work from UConn including the original recommendations from the Elementary Mathematical Writing Task Force. Linked there, but I think also important to link here is the article, “Why Should Students Write in Math Class?” from Educational Leadership (2017). Last, is a piece that I found on Twitter a week ago called 17 Prompts For Writing in Mathematics. See a glimpse below:

So that’s it – my three big goals this year are to investigate Intervention, Listening & Speaking, and Writing. Keep in mind though – all of these investigations will also come through the lens of coaching and support, so I certainly intend to blog about coaching as well. What are your goals this year?