Reflections on Learning—Part I: Routines for Reasoning and Generalizing the Four Essential Instructional Strategies

“The Four Essential Instructional Strategies keep focus on the mathematical thinking while providing access to learners.”

It’s hard to believe it’s June 8th.  The last day of school for students is June 21st, which is crashing in! As I look back over the year, I want to be sure to capture some of the learning that’s been important so that it doesn’t get lost when thinking about and planning for next year. Here is the first in what I hope will be a series of posts.  

Routines for Reasoning: Fostering the Mathematical Practices in ALL Students (#fosteringMPs) by Grace Kelemanik (@GraceKelemanik), Amy Lucenta (@AmyLucenta)and Susan Creigthton is a book that has pushed my own thinking and taken my work to a deeper level. It amplifies the Standards for Mathematical Practice (SMPs) which can get neglected in pursuit of content. I contend, as do the authors, that you aren’t fully teaching the Common Core if you aren’t engaging students in the SMPs. This book is about teaching math by thoughtfully and deeply engaging students in the SMPs and doing so in ways that support each/ALL learners and give them tools to become independent in using them. Routines for Reasoning takes the SMPs to a much deeper level giving ways to empower students to “Think Like Mathematicians”.  From the “Avenues of Thinking” to the Four Essential Instructional Strategies, to the actual routines in the book, I will be unpacking my own learning of this book for a long time.  

The authors of Routines for Reasoning make the case for three Avenues of Thinking: quantities and relationships (SMP 2), structure (SMP 7), and repetition (SMP 8). These three avenues support SMP 1: making sense of problems and persevering in solving them. If students and teachers aren’t doing that then the others aren’t happening. The remaining SMPs (3, 4, 5 and 6) support each of these avenues. For more information you can watch this Ignite. Herein begins the brilliance of their work. 

Each of the routines in this book are organized to support student learning giving students independent think time, partner discussion, whole group discussion and independent reflection. Additionally, each routine incorporates Four Essential Instructional Strategies, which keep focus on the mathematical thinking while providing access to learners. They are:  

  • Ask-Yourself Questions  
  • Annotations  
  • Sentence Frames and Sentence Starters 
  • The Four Rs: repeat, rephrase, reword and record 

They seek to support learners in gaining access to the math and in giving them tools that can be used to become more independent learners over time. If we want students talking productively about their mathematical ideas then we need to teach them how and what to share, orient them to one another’s thinking and to the mathematics.  

I think these Four Essential Instructional Strategies are foundational. In fact, I believe they should be a foundation in all professional learning related to math pedagogy. With that in mind, I have been intentional about calling them out and using them our work with Observing for Evidence of Learning (OEL) Labs (a structure similar to Learning Labs/Math Labs work the University of Washington and others are doing, or the studio lab idea that Teachers Development Group offers) as well as other forms of professional learning.  

Ask-Yourself Questions  

In each of the routines in this book students are introduced to “AskYourself Questions” so that students think mathematically and can internalize these questions to begin to ask themselves whenever they are doing math. Teachers use them to model how these questions can guide our mathematical thinking. Some examples of Ask-Yourself Questions are: 

Think Quantities 

  • What can I count?  
  • What can I measure? 

Think Structure 

  • Does this problem remind me of another I’ve solved?  
  • How can I chunk this express/visual/number?   

Think Repetition  

  • Is there a process I keep repeating?  
  • Do I keep repeating the same set of calculations?   

When working alongside teachers and students we worked together to create a poster, so to speak, of the above 6 questions. Teachers referred their students to them when they got stuck, and excitedly shared that students successfully used these prompts to get themselves unstuck! (Side note: we had to take the posters down during our state testing. That’s fodder for another blog post.) What I pay attention to when sharing Ask-Yourself Questions with students and teachers is that they become a problem-solving tool that is much more mathematically powerful than using “key words” a practice I am committed to eradicating.  


“Annotations can help you draw students’ attention to important connections or relationships in a problem” (p. 33). Use of color, specifically, can help students make connections between different representations and mathematical ideas.  Annotations can become a tool for mathematical thinking (SMP 5), allows students to be more precise (SMP 6) and can be used to support mathematical arguments (SMP 3).  

Their use of annotations makes me think more critically about how I record student thinking in a math routine like number talks or number strings. I’ve always seen color as powerful and that our recording matters, but this year I’m finetuning this attention to ways our recording/annotations help support students seeing connections between students’ thinking and the structure of their thinking.  For example, intentionally showing two students’ different ways of adding on shown on a number line and stacking them in order to make it easier for students to see the similarities and differences in their thinking.  

Sentence Frames and Sentence Starters 

The sentence frames and starters used in Routines for Reasoning are intended to support students staying focused on the kinds of mathematical thinking that you want students engaging in and support those who struggle with the language to use and support them in ways of collecting their ideas. They are also used to focus discussions between partners and with the whole group.  

This year I’ve focused more on the use of sentence frames and starters to focus conversation in classrooms.  I mentioned earlier that students need to learn how and what to share. One way we’ve been doing this is giving more specific prompts and posting them so that students can refer to them. For example, when wanting students to see the relationship between problems in a number string, we asked them specifically, “How can I use the previous problem to help me solve …?” By verbally and visually projecting this prompt, it helped focus students’ attention to the connections between problems and their underlying structure and their discussions were richer.  

The Four Rs: repeat, rephrase, reword and record 

The Four Rs are important in helping students stay focused in and make sense of the various discussions that occur throughout a routine. “…it falls upon you to facilitate discussions with your students in each of the routines, and this strategy provides a structure for helping you do so” (p. 36) It goes on to highlight some teacher question and decision points as well as actions to take (see below). The Four Rs are easily generalizable across discussions. That said, this strategy is the one that requires the most in-the-moment decision making (p. 37). This strategy serves as a way to amplify the mathematics and the mathematicians in discussions. It also serves as an anchor, in terms of the recording of student ideas and their use as an artifact, or anchor of their understanding.  

Teacher Questions and Decision Points: 

Did all my students hear the idea?  

  • If not, call on a student to repeat what was said.   

Did all my students understand the idea?  

  • If not, call on one or two students to rephrase what was said in their own words  

Is now the time to press for precision of language?  

  • If so, then reword saying the idea again substituting mathematical or academic language that you want students to develop.   

What do I need to capture in the language and thinking for reference by my students?  

  • Record by publicly capturing the salient ideas using language you want students to use. 

Of all the strategies this is probably the most complex to incorporate for me because I can only plan so much for it, and in the moment, I might be attending to 17 other things. I think this strategy has such potential for elevating discussions. It feels a lot like pushing JELL-O up a hill: I can attend to maybe 2 of the questions above but not all 4, at least so it feels natural, which is why I want to make it a focus of my own work next year.  

Lots of things to think about! I’m curious what you are thinking about as you reflect on your year? What this post causes you to think about?  

Stay tuned for my next post. Reflections on Learning—Part II: Routines for Reasoning and Capturing Quantities …  

Week 2 #MTBoSBlogsplosion: Building Our Mathematical Identities

“We teach who we are.”

My favorite part of my job is being in classrooms and working with teachers and students supporting mathematics teaching and learning. I’m inspired every day by students and teachers. One of my goals is to change how students, teachers, and parents experience mathematics, in essence, change the culture of how we see, think about, and engage around this subject.  I want mathematics to pique curiosity and make sense, and for our student- and teacher-mathematicians to reason and think critically and deeply. And, I want us all to see the mathematics around us; it’s everywhere! We all know friends, family, co-workers or acquaintances who don’t identify themselves as mathematicians. For some, mathematics was lorded over them, and they may have a phobia of mathematics due to what Jo Boaler (@joboaler) refers to as “math trauma” they suffered along the way. Knowing no other ways, we often teach how we were taught.

I encourage teachers in taking risks and I share bite-sized ways to “hack” math class. I know the term “hack” might offend some and I don’t mean to do that, or to underestimate the immense expertise involved in teaching. Teaching is complicated work. Most teachers don’t want to change everything but will try something. Hence, the use of “hack.” I leverage a few instructional activities, predictable routines (IAs) that have a high yield in terms of student learning and opening up mathematics. Giving teachers choice in where they’ll begin gives them control and ownership.

In some classes, those hacks start by changing one aspect of a lesson by using IAs such as Quick Images, Number Talks, Counting Collections, Mathematizing Read-Alouds, Numberless Word Problems, Noticing and Wondering, etc. to nurture number sense and problem solving, engages students in the Standards for Mathematical Practice, and build a community of mathematical thinkers and problem solvers. In my own practice, I started with Number Talks. What I learned was invaluable: how to listen to my students and to see and understand math how they saw and understood it. This changed everything, and I want other teachers to experience this. In other classes the hack might start with flipping the problem solving; engaging students in problem solving (using the strategies of Noticing and Wondering or Numberless Word Problems) first rather than after they “know the skills.”

Together we work through the emotional challenges of change and not knowing where that change might lead. While hopeful for what these changes may bring about, there are also concerns. Concerns worth addressing: “How will I respond?” or “What will I say, or ask, next?” Additionally, we tackle the content and pedagogical impacts of opening up the mathematics and shifting the ownership of the knowledge to the students. For example, when planning with teachers around addition and subtraction, I wonder aloud with them if it is always (or ever) necessary to subtract? Are there times they might want to add up/on instead? As teachers try these “hacks” the students respond: they want more! “This is math?” they say when engaging in a Number Talk! Yes, in fact, that is math! “When can we do that again?”

In The Heart of a Teacher: Identity and Integrity in Teaching the author, Parker Palmer (@parkerjpalmer), starts off with, “We teach who we are.” If this is true, then I want teachers to identify themselves as mathematicians. Only then can we hope to build our students’ identities as mathematicians allowing them to see the beauty and joy of mathematics. Coming alongside, and investigating mathematics and students’ thinking around mathematics, we are learning together and changing our identities. Sharing our successes and challenges with our peers continues to shift our mindsets.

I wonder what has worked for you in supporting others in building their identities as mathematicians, or in building your own? I’d love to hear from you.


Week 1 #MTBoSBlogsplosion: One of my favorite sites–WODB

“That one doesn’t work. There are only two that have flat surfaces that are circles.”

This is my first attempt at blogging. I have been thinking about it for a long time and realized I was spending too much time pondering the title of my blog, the topic for my first post, etc. All things that don’t really matter. Thanks to some advice on exploring the MTBoS,  and some encouragement from colleagues, I decided to stop pondering, dive in, and continue to build the site as I go. I joined Twitter in April 2015 just before arriving in Boston, MA for the Annual NCTM Conference. I’m still learning about Twitter but have found it, thanks to #MTBoS, to be welcoming and a vital source of professional growth. I so appreciate #MTBoS for all the support and encouragement in just getting started blogging. They have a great challenge to blog each week in January and even give you prompts to blog about. So, here goes …

I’m interested in creating more opportunities to engage students (and educators) in sense-making, reasoning, critical thinking and problem solving while also engaging their sense of curiosity and wonder about mathematics. With that in mind, I chose WODB as one of my favorite websites. (There are a LOT of great ones from which to choose!)

Every August, I have the opportunity to work with all K – 5 teachers new to my district. Typically, I include  a couple of what I think of as high-leverage instructional strategies. Things like: Quick Images, Number Talks/Mental Math, Noticing and Wondering. This past August, I decided to take a bit more time talking about how teachers can mathematize their classrooms. A colleague pointed out that we spend a lot of time at elementary, typically, creating a literacy-rich environment but not very much time making it a numeracy-rich one. So, I mathematized information about me and turned it into a poster as a way to show teachers how they might share about themselves with their students in a way that highlights mathematics. (Thanks, Pinterest.) I also shared children’s books about mathematics and mathematicians, and estimation 180 and WODB both sites that are highly engaging and get kids and adults talking about mathematics. We discussed ways to build our (students and adults) identities as mathematicians through these various instructional activities.


Fast forward to December. One of my colleagues, Hillary Chandler (@hillylilly) decided to try WODB with her first graders. She chose the image above. Hillary allowed students to digitally ink their ideas about which image didn’t belong. One student started by saying that, “the basketball didn’t belong because it is the only sphere”.

Then crickets. She waited. And waited. Then, Hillary prompted, “What if I said the brownie in the top right didn’t belong. Why might I think that one doesn’t belong?” After some thinking, a student said, “It’s the only one that has vertices.” Students then began making different claims:

  • “I pick the brownie because it doesn’t have two flat surfaces that are circles.” Others pointed out, “That one doesn’t work. There are only two that have flat surfaces that are circles.”
  • “The cookie doesn’t belong because it’s the only one that’s pink.”
  • “I think the cake doesn’t belong because it’s the only one that has writing on it.”

A bit more discussion ensued with ideas being tossed out and then retracted because they “didn’t work.” Students would choose an attribute that only worked for two of the images. One student said that the image he’d picked didn’t belong because it didn’t have a circle as two flat surfaces. Another student said, “That one doesn’t work. There are only two that have flat surfaces that are circles.” Loving this discussion and the students’ use of vocabulary.


What impressed me is their comfort level with sharing ideas with one another and that their peers could push back on thinking, and their discussion of attributes and definitions. Christopher Danielson (@trianglemancsd), in his blog, Talking Math with Your Kids, shares the importance of and ways we can talk math, or mathematically, with kids in order to instill a love of mathematics, reasoning and making and supporting conjectures.

I don’t know about you but my math classes growing up NEVER included anything quite so fun and interesting, which would have nurtured my logical reasoning.

So, I wonder, how do I support more teachers in growing this type of discussion and reasoning in their practice? How do I support Hillary and her students in their growth? I’m curious what you find has helped to engage students in meaningful mathematical discourse and building young mathematicians. I’d love to hear your ideas.

Math on!