Brave, Safe Spaces

Teachers are the guardians of spaces that allow students to breathe and be curious and explore the world and be who they are without suffocation. Students deserve one place where they can rumble with vulnerability and their hearts can exhale.

– Brene Brown

Yesterday, I watched one of my amazing coach colleagues deliver a demo lesson in a 6th grade classroom that I did a demo in earlier this year. I promised her: you will love teaching in this classroom. In watching the demo of this ELA strategy (more on that on a different day), I once again noticed everything I loved about this group.

  • Everyone participates.
  • They cheer each other on.
  • They laugh with and enjoy each other.

So, how do you get to this place? Where 6th grade is not a chaotic madhouse at the end of the school year, but a calm, safe spot where students know exactly what is expected of them as a member of the community?

Of course I asked the teacher! She told me she did her entire Master’s thesis on building classroom community. In her student teaching experience, she was surprised in a second grade classroom where blending started immediately day 1. Shouldn’t a community be built first? So that’s what she does.

She explicitly tells them, “I won’t let you fail.”

During the demo, the coach communicated the same safe space to the students. She built rapport with the students through a calm, casual voice. She appreciated them out loud, telling them “I am so impressed by how you walked into the room.” She connected with them where they were at, introducing the lesson (about analyzing text evidence for meaning – no easy feat) with Tupac, Taylor Swift, and Kevin Hart. She taught by walking around the room, getting close to each of the groups and making eye contact. In group discussions, she circled around the room, listening in to the students, asking questions, pushing on thinking.

Two students yesterday especially caught my eye. I know they’ve struggled in the past. They were especially vulnerable, both standing up to analyze a quote on the board. One started to sit back down, but with the class cheering him on, he pulled out an answer that was a diamond in the rough. Where it could have slid into a shaming event – the teacher might have corrected the grammar, the stuttering, the “close but no cigar” – instead the teacher praised the diamond inside, which was the insight into the quote that was personal and insightful. The class “clam clapped” (tapping fingers together like a clam but without the crazy noise to annoy the class next door). The second student started to share his response and then said, “No, I just can’t say it right.” The coach said, “I totally get it. Tell me some of the words in your head right now.” The student started and stopped. The coach smiled and waited as he regathered his thoughts. Soon, he brought out another wonderful insight into the quote they were studying. Clam clapping ensued.

There is a common thread among all the teachers I work with who have classrooms similar to these. It is an empathetic, relaxed demeanor that communicates, “I’ve got you. I’ve got this.” These teachers know what they want the students to accomplish, and are convinced every student can accomplish it. They are well aware that their students are bringing in lots of baggage in many different forms. They know this because they have taken the time to get to know every student in their classroom. They are also well aware that not every student will perform in the same way. When they talk about their students, they don’t focus on the baggage. They focus on the growth. They don’t categorize the students as “low” or “high”, they categorize them all as learners. These teachers talk about successes on the part of the students, and when their students fall short of success, these teachers look for ways to grow their own practice instead of blaming the student. Their quest for knowledge is ongoing – students are always bringing to us new challenges to overcome and the world of education is ever evolving.

Teaching is a difficult job, and none of what I’ve described above is easy. It is tempting to slip into the blame and shame routine as the pressures of administration, testing, and parents mount. In this week of Teacher Appreciation, I am grateful to all who work tirelessly to prioritize and create these brave, safe spaces for our students to grow not only as learners, but as courageous and vulnerable human beings.

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.

On My Soapbox About Elementary Math

I’m still digesting and processing all the great learning I had at NCTM San Diego 2019! It is truly awesome to think of all the knowledge and brainpower I was surrounded by in San Diego.

With that said, there is still something that doesn’t sit well with me, from a few years ago, that was reiterated at this conference. Steve Leinwand told a large group of coaches in a workshop I attended a couple of years ago that a mediocre K-5 education can be overcome, but 6th grade is where math really takes off. Linda Gojak echoed that in her presentation, but added that she thinks 5th grade is really that important turning point. In talking with coaches at different elementary school districts, I’m learning that various school districts de-emphasize the importance of math at K-5.

It just doesn’t sit well with me.

At the same time you’ll hear that algebra is the “gateway to college access” (WestEd, 2015), and that students who fail Algebra 1 the first time are unlikely to pass it the second time. Performing just slightly above the proficient line on the CST’s in 6th grade offered only a 50/50 chance of passing Algebra. I’m looking for similar data from SBAC assessments but haven’t seen an updated version yet. WestEd details a summer intervention between 6th and 7th grade as a support, acknowledging the limitations of such a short time frame.

But I still don’t understand! If a proficient (albeit barely) performance on our state standardized assessments only gives you a 50/50 chance at success in algebra, then why aren’t we concerned about the mathematics education of K-6? Isn’t this a huge equity issue? Who gets access to algebra, and thus college?

They don’t start learning mathematics in 6th grade. It starts in kindergarten. We can’t *assume* it started before that. Of course, in the ideal world the babies were counting everything in sight with their parents, comparing more and less, starting to subitize small quantities, etc. Their early numeracy trajectories could have started well before kindergarten. But for others, it didn’t. The opportunity gap has begun before they even stepped into their brightly colored kinder room.

I can’t accept that the choice of college is already limited upon walking through the kinder door. We have to do better than that. However, our elementary teachers are charged with the critical goal of teaching children how to read – no easy task by any stretch of the imagination. It’s hard work, stressful, and compounded by a society that doesn’t value elementary education – as demonstrated by the “teacher pay penalty” (EPI, 2018).

Is this why we give K-5 the “pass” on mediocrity in math? We just can’t burden elementary anymore?

I can’t accept that either. In the meantime, our students suffer and the opportunity gap widens. We know what is working from other countries – here is a very partial list (NCEE, n.d.):

  • Longer school days
  • Preschool
  • Early childhood focus on literacy AND numeracy
  • Ongoing teacher development and collaboration

It’s not a secret. In order to compete globally, we have to provide a stellar education from Pre-K through 12th grade. Even with the limited amount of time in American school days (this really has to be changed but it is a subject for a different day), we can’t expect even a preschool teacher to spend the entirety of every day on literacy. Early numeracy, conceptual understanding, basic fluency, and positive math interactions are critical to their secondary education. We cannot wait to focus on 6th grade – at that time they have had 6 prior years of math “mediocrity” that has potentially established math anxiety, math hatred, or math avoidance. Undoing 6 years of learning is too tall a task for a summer intervention.

Instead of accepting mediocrity, let’s invite success. The opportunity gap has already started before they walked through that kinder door. We must change that math narrative into one where the algebra gate is readily open to all, due to their stellar K-5 preparation.

They Just Don’t Know Their Facts

They Just Don’t Know Their Facts…

I’m pretty sure I’d be rich if I had a dollar for every time a teacher has said that to me. To be fair, I’m pretty sure I’d be rich if I had a dollar every time I said it in my career. But, when you know better, you do better, so here is what I now know.

My career started as a student teacher in 1998. That was the first time I heard the complaint about not knowing facts. The answer was always – they need to practice their flash cards at home. If only they practiced. If only their parents made them practice. If only they weren’t lazy. If only they cared about school. If only they paid attention.

“If only” and “just” may be the enemy words of progress and innovation. It takes complete responsibility off of the person saying them. I started to realize this later on in my career, and began listening carefully to those words. These are the words of oversimplification, of self justification, and of saving face in front of a situation that seems out of hand.

“Yet” is the new buzzword. So, “They just don’t know their facts” turns into “They just don’t know their facts yet.” It’s an improvement for sure – a growth mindset that might lead to some new learning around how students “know facts.” It still boils math down to “just” – a word that continues to scare me. Are these students really just one flashcard away from knowing their facts? Or is there something else that we haven’t considered?

My new recent learning is sadly not “new” at all. I hope that the more people that share learning, the more rapidly it spreads, so that we can stop blaming the students and start finding the solutions.

Fact Fluency Through Games, Not Timed Tests

First, if you haven’t read Jo Boaler’s Fluency Without Fear, then stop right here and go read it. If she doesn’t convince you to stop giving timed tests, then I’m not sure what will. However, for those beginning the journey away from timed tests, the easiest entry point is to practice with games. Her “How Close to 100?” game is actually a lot of fun, even as an adult playing with students. Plus, when you get kids playing games, they now have an independent activity that frees you as a teacher to confer with students, hear their math explanations, and probe thinking.

Games engage students to build fluency while simultaneously building number sense.

In my career, games became my initial step away from timed tests. If they weren’t going to memorize them from timed tests (anxiety, lack of feedback, practicing wrong answers…) then wouldn’t they memorize them from play?

Sure. For many students, the games did the job of practice and developing fluency. For others, they needed a different kind of help. I relied on multiplication charts to help them out when we were learning a different concept that needed the fact fluency to be there so that their cognitive energy was freed up for the other thinking. Why teach what a computer can do inevitably faster than a human can?

Yeah, but…

“Not everyone has a grid in front of them to draw a picture, and they need to be able to do problems like 342 x 74 with relative speed.”

“They are still failing their math tests.”

“They won’t have a multiplication chart on the SBAC.”

“I don’t have time for these games when I need to get through the textbook.”

Hmm. 🤔

So these are questions I had asked myself at one point, and then asked others, and now have others ask me.

The Math Wars in the 1980’s and 1990’s created some serious debate over the role of technology in math. I think the Common Core does a good job bridging this debate – students must be able to understand number relationships. If we focus on that development of number sense, then we truly can rely on technology for the speed piece. If we neglect the sense making, then no technology is going to support our work.

So let’s focus on the statement, “They are still failing their math tests.” I have been spending time giving the IKAN and GLOSS assessments from the Georgia Numeracy Project, which has based their work on NZMaths from New Zealand. For background info and directions for assessment, check out Graham Fletcher’s page here. The IKAN written assessment gives numeracy stage scores in 4 domains – Number Sequence & Ordering, Fractions, Place Value, and Basic Facts. Which one do you think was the lowest? Which was the highest?

Out of every class I’ve scored (at this point I’ve scored 12 classes of about 28 kids each), every single class had the highest score in Basic Facts, and the lowest score in Place Value. Now I realize that this is in no way scientific research, but the trend is fascinating. Maybe we are missing the point when it comes to “they don’t know their facts”?


What is going on behind these facts and this place value idea? Here is the document I’m currently digesting:

Now I’m putting my fourth grade teacher hat on, and I’m thinking, but this seems so basic? The thing is, do we really know that our students have had enough experiences with number to move them through this trajectory? If my students don’t know “know their facts”, then can they visually recognize something like this:

Can they tell me what one more is (without recounting)? One less? Yes? What about this? How far away is it from 10? From 0? Can they tell me two numbers that make up this number? Do they see the 3 and 2 or the 4 and 1? Yes? Then what about this?

And why do things like this need to be searched for and even bought from TPT, when they should be free to all? Under my fourth grade teacher hat, which up until a few years ago didn’t have this knowledge, all of this seems like too much. Just have a go with the flashcards and don’t be lazy. But I know better, and now I do better. The thing is, in the classroom, I affect 28 students for one year. How do we spread this knowledge, so that all teachers are armed with the understanding, as well as armed with the materials and skill to dig deep into children’s mathematical understanding?

The Basics of Fact Fluency

Procedural fluency is skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

Common Core State Standards Initiative, 2010

Flexibility • Efficiency • Appropriate strategy use • Accuracy

There’s the definition . Does a timed test measure any of these? How do you know? Before you can say, “They just don’t know their facts,” we have to carefully reflect on this definition. How did we prepare them for flexibility in choosing strategies? How did we prepare them for accurate calculation? What did we do to help them understand efficiency? And how have we spent our time to help them see appropriate strategy use vs. ineffective strategy use?

In conclusion, dig deeper.

We have decided upon some screeners (including the GLOSS along with the IKAN, as well as some ideas from Build Math Minds) to get at some answers to these questions. Yes, many of these are one on one, but when in doubt, ask the students! The interview process can take some time, but it is enlightening to find out how students are thinking – not to mention on the job professional development for the teacher. If we can understand where a breakdown in understanding has happened, we can intervene more precisely. I’m also interested in doing more work with math Running Records, which involves more interviewing to understand what strategies students are using to add, subtract, multiply, or divide. If our end goal is for them to be fluent but flexible with numbers, then it would help us to know where their current level of understanding is.

In the meantime, my fervent hope is that teachers take the time to do Number Talks (or Fraction Talks) every day. I’m not sure there is a more powerful way to help students explore number relationships. For our teachers, we are working on a repository of resources so that Number Talks are not something they have to spend lots of time searching for – the resources will be at their fingertips. Games will also be provided so that teachers can easily pick from games that will not only engage but help students build number sense.

For me, I’m working on coming up with a more straightforward answer to “They just don’t know their facts.” This post is a start, but nowhere near a final version! I’ll be seeking out more understanding with my colleagues, with Twitter, with the research, and most important with the students we serve. Would appreciate your comments below!

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.