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Dedicated to Teachers

It Takes a School to Develop Numerical Fluency

NumericalFluencyBlog 9.10.18This post was adapted from Chapter 11 in Developing Numerical Fluency by Patsy Kanter and Steven Leinwand

Changing your instructional practice can be hard.

In math, moving from merely gathering answers to expecting explanations and justifications requires a mindset shift that takes time and support to emerge. Moving from the convenience of worksheets to the complexities of eliciting and celebrating the thinking that accompanies alternative approaches to finding solutions is often a difficult transition.

But we can’t just make these changes in the “good” teachers’ classrooms.

For change to stick, we need to create a school culture focused on the purposeful development of numerical fluency as an integral part of K–5 mathematics in every classroom and among all students regardless of labels, languages, or learning needs.

Download a Sample Chapter of Developing Numerical Fluency

Authentic, meaningful fluency cannot be developed in a single lesson or a single classroom or even over the course of a single year. Instead, supporting the growth from parts and wholes to sums and differences to products and quotients takes a schoolwide commitment that is infused into every classroom and every pull-out program where mathematics is taught. It takes a shared understanding of exactly what numerical fluency is and what common classroom routines and practices support its development. And it takes a shared sense that what happens in grade x directly impacts what can be done in grade x + 1.

There are a range of strategies that schools can and must adopt to develop this schoolwide culture and commitment. We advocate for serious consideration and implementation of the following collaborative structures to build this knowledge, culture, and commitment.


Making Better Use of Meetings

The heart of every coherent, equitable, and effective mathematics program is the consistency of daily high-quality instruction that emerges, in part, from professional interactions that occur during grade-level planning meetings. The philosophy of “you do your thing and I’ll do my thing” too often shortchanges students and severely complicates matters for the teachers who will face these students next year. Making better use of grade-level and whole-school faculty meetings can help.


Grade-Level Meetings

Teachers tell us that their grade-level planning meetings are essential sharing and problem-solving experiences that strengthen the skill and knowledge of all participants. Starting with a goal of developing numerical fluency appropriate for that grade level for every student and acknowledging that worksheets, “mad minute drills,” and mindless practice are inappropriate strategies for meeting that goal, grade-level meetings are where teachers use high-quality resources and collaboratively create lessons that incorporate the processes and pivotal understandings we discuss in Developing Numerical Fluency.

Consider the one-hour meeting of three first-grade teachers all wrestling with the challenge of developing fluency with addition and subtraction facts. They know that their students need a stronger sense of appropriate strategies and they know that their students need to support their answers with justifications that explain the various strategies they use. This is the “strategy chart” they develop to post and use in each of their classrooms:



Faculty and Cross-Grade Meetings

More and more teachers tell us that the focus of faculty meetings is shifting from administrivia and announcements (easily conveyed by e-mail) to teaching and learning throughout the school. As part of this transition, faculty meetings become powerful opportunities for presentation of model lessons, schoolwide discussions of the shifts in practice required to strengthen numerical fluency, video analysis, analysis of student work, and celebrations of success. The question for teachers, coaches, and administrators is simple: Why bemoan an absence of needed fluency when small, universally implemented shifts can make such a difference? And why not use faculty meetings to address these shifts and build expectations that they will be widely implemented?


Using Video

Because most of the practices that develop numerical fluency cannot be done with worksheets or via independent student work, video is often the most useful vehicle for capturing the discourse, engagement, and teaching moves found in great lessons. When we use video clips as part of grade-level meetings or faculty meetings, we limit clips to no more than ten minutes and almost never let a video run for more than three minutes without pausing for discussion. In our experience, the video is a stimulus for discussion and a chance to envision what is possible in a real classroom in our own school with one of our own teachers. We tend to guide this discussion with only three questions:

  • What was impressive in what you observed? Why was that impressive and why do you think it worked for students?
  • What questions do you have about the teaching and the learning in the lesson we observed? What might you do differently?
  • On the basis of the video and our discussion, what one or two shifts are you willing to commit to making in your own class to better serve your students?

Consider a third-grade teacher who uses a number talk model to do an estimation task near the beginning of every mathematics lesson. How hard is it to capture the eight minutes of instruction on video using a phone or tablet? How hard is it to view, rewind, and review section by section such a video during a faculty meeting? And how hard is it to use the video to stimulate discussion of what is going on, how it is effective, and what would prevent the entire faculty from adopting such a practice?

The key for this and other approaches to work is to start small. Use volunteers who are willing to take a risk. Ensure a safe and respectful space for discussion. Focus, in the end, on the one or two changes that your team is willing to commit to making.


Teacher Collaboration

Beyond these larger-scale meetings, teachers also need ways to collaborate directly.


Combining Classes

It is not at all uncommon to find very diverse comfort levels with teaching mathematics among grade-level colleagues. We have seen teachers combine their classes and co-teach the larger group for the purpose of modeling and encouraging strategies that need to be practiced in all classes. Teachers tell us that they marvel at what their own students are able to do when working with a broader pool of students and under the guidance of another teacher. Seeing this change in behavior and engagement is the best motivator to incorporate these instructional shifts back in one’s own classroom.


Collegial Visits

It astounds us both how rarely teachers get to observe each other—not because of a lack of time, but because collegial visits just aren’t part of most school cultures.

But when we hear how instructional practices were reinforced because someone saw them in a colleague’s class or when we are told how one collegial observation “completely changed how I’m introducing multiplication to my students,” we realize how valuable and underutilized this collaborative structure is. In all seriousness, we ask, how hard is it to schedule some collegial visits as part of the agenda for grade-level meetings? How hard is it to take a music period or gym period once a month to wander into a colleague’s class? Whether done informally or more formally to see a specific strategy or lesson, we find classroom visits invaluable. In our experience, once-a-month collegial visits, far from being burdens, are powerful and exciting learning opportunities that come at essentially no cost.

Consider how two fifth-grade teachers can agree to reschedule their math block to facilitate each teacher being able to observe a math lesson on a topic of common concern while their students are in music or art. Consider that power of the end-of-the-day discussion about “what you did and what I did” and what seemed to work and why, as well as what didn’t seem to work as well and why. Consider how something this simple breaks down silos of isolation that make growth and improvement so much more difficult.



In our experiences, little is as effective as real time coaching/co-teaching with individual teachers and their students. Effective coaching is not sitting in the back of the room taking notes, but rather active involvement in the lesson, focused on a specific instructional practice, accompanied by the all-important post-lesson debrief. This is not the place to describe effective coaching practices, but it is the place to remind us that shifting practice and mindsets is not easy and unlikely to be done in isolation. Consistent coaching is one more powerful way to gradually create a schoolwide commitment to the development of numerical fluency.


Garnering Administrative Support

If you are a teacher reading this, it is the perfect opportunity to print this and share it with your principal. (Or point them to the entire book.) If you are an administrator reading this, you know that many of these collaborative structures require your support and encouragement.

Some strategies for providing the leadership necessary to support these activities include:

  • Minimizing spending meeting time on administrivia that can be disseminated by e-mail and maximizing focus on teaching and learning.
  • Participating in grade-level meetings to provide ongoing support and encouragement for sharing around issues of teaching and learning.
  • Working with a few volunteers to video record mathematics lessons that incorporate the development of fluency, and using these videos for faculty meeting viewing and discussion.
  • Encouraging and supporting a schoolwide commitment to collegial observations that create a much stronger “we’re all in this together” culture. Just as our students learn from each other, as educators, we have our colleagues as the first line of support in doing more and better on behalf of our students.

We urge you to go forth and start small. We urge you to begin with what seems simple. We urge you to then build on these initial efforts. We urge you to monitor what works and what doesn’t. And we urge you to remember that when every child leaves your classroom confident in their numerical fluency—because of what you have done—you change the world for that student, for every student, and for our collective good. Why else would we as teachers do what we do?


Learn more about Developing Numerical Fluency at Heinemann.com

Download a Sample Chapter of Developing Numerical Fluency


Patsy Kanter is an author, teacher, and international math consultant. She worked as the Lower School Math Coordinator and Assistant Principal at Isidore Newman School in New Orleans, Louisiana, for 13 years. Patsy is the co-author of Every Day Counts: Calendar Math and a consulting author for Math in Focus.

Follow Patsy on Twitter @patsykanter



stevenleinwandSteve Leinwand is the author of the bestselling Heinemann title Accessible Mathematics: Ten Instructional Shifts That Raise Student Achievement.He is Principal Research Analyst at the American Institutes for Research in Washington, D.C., where he supports a range of mathematics education initiatives and research. Steve served as Mathematics Supervisor in the Connecticut Department of Education for twenty-two years and is a former president of the National Council of Supervisors of Mathematics.

Follow Steve on Twitter @steve_leinwand


Topics: Steve Leinwand, Developing Numerical Fluency, Patsy Kanter

Date Published: 09/10/18

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