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Thursday / August 16

5 Shifts in a Math Teacher’s Role

Last week, I had the opportunity to work with a group of STEM educators. One question I posed to this group was, “What do you feel a teacher’s role should be during collaborative student work?” Participants were to share their ideas by inputting up to three words, using an online tool from The results (below) formed a word cloud, which displayed the most popular roles indicated by the largest font.


I’ve used similar tools to summarize responses to questions focused on student engagement with many other audiences. But this time, the results struck me in a way that I had never experienced before. Guide? Facilitator? I wondered, during collaborative student work–‘guide’ or ‘facilitate’ what? I was thankful for the candid responses and yet, simultaneously, challenged with the results. (Note – To add some context, I should reveal that I entered the teaching profession in an era where teachers were taught to be ‘a guide on the side, and not a sage on the stage.’)

However, to guide means to “show or indicate the way to someone,” and the definition of facilitator is one “that makes an action or process easy or easier.” Put these together and you might have a recipe for low expectations, where learners are permitted to avoid engaging in deep levels of mathematical reasoning and problem solving.

Not Our Role During Collaborative Problem Solving

I regrouped my thoughts, and then stated that although these results are often consistent with what I observe in some math classrooms, I believe we need to rethink our roles related to engaging all learners. Here’s what I mean.

Some teachers (during the planning phase) establish mathematics goals to focus learning (NCTM, 2014), and then communicate learning outcomes with students early and throughout key times in a lesson. They form collaborative student groups with assigned roles, launch into sharing an activity involving a rich mathematical task, negotiate materials needed, communicate expectations related to how student work will be evaluated, share the time allotted, etc. And then, very often, three things occur with respect to the teacher’s actions.

First (and sometimes almost immediately), the teacher begins visiting student teams and posing a list of pre-determined questions, many designed to help get a sense of what a student knows and is able to do. What happens next has become a predictable pattern of behavior during many of my classroom observations. Students very patiently pause the work they have been assigned, and together, awkwardly attempt to answer the teacher’s premature questions. I imagine “thought clouds” popping up above their heads, revealing some of their initial, in-the-moment reactions…


When satisfied (or frustrated) with the responses, the teacher moves onto the next group of students and repeats this process. The prior group of students eventually recover, regroup, and go back to working on the assigned task. Meanwhile the teacher visits all groups repeating the same process.

The second and third things that I see, all too often, are the teacher monitoring on-task social behaviors (versus mathematical behaviors), and then keeping track of (and announcing) the remaining time left for students to complete their work. Re: the latter, I have yet to visit a classroom without a clock hung prominently where all can see it, and the behaviors I’m referring to are not necessarily those associated with directly supporting student engagement with the Standards for Mathematical Practice (National Governors Association Center for Best Practices, Council of Chief State School Officers, 2010). Now, don’t get me wrong… there’s a time and place for posing purposeful questions–both those intended to assess student understanding, and those that can help advance student thinking (Smith & Stein, 2018). There’s also an appropriate time and place to teach students how to engage in and self-monitor appropriate social actions and behaviors. All of this connects back to learning intentions.

A Good Time for ‘Guiding’ and ‘Facilitating’

If you’re still reading up to this point… then I should acknowledge that there certainly are times for both ‘guiding’ and ‘facilitating’ during a mathematics lesson. In Everything You Need for Mathematics Coaching (McGatha, Bay-Williams, Kobett, & Wray, 2018), the authors describe eight Shifts in Classroom Practice, with direct connections to the Effective Mathematics Teaching Practices (NCTM, 2014). These shifts serve as a guide to building mathematics proficiency for all students through increased student and teacher engagement.

As a self-assessment, consider where you feel your own practice lies with regard to five of the eight Shifts, each central for engaging all learners.

Source: Retrieved from Tool 4.1 (Connecting Shifts to Engaging Students Self-Assessment) of the companion website for Everything You Need for mathematics Coaching: Tools, Plans, and A Process That Works: Grades K-12 by Maggie B. McGatha and Jennifer M. Bay-Williams with Beth McCord Kobett and Jonathan A. Wray. Thousand Oaks, CA: Corwin.

The “Engaging” Shifts

How do these Shifts connect with your role in providing opportunities for all students to engage with meaningful mathematics? Let’s take a look.

In Shift 2: From routine tasks toward reasoning tasks, the teacher’s role includes implementing high cognitive demand mathematical tasks that allow students to reason and problem solve using multiple representations, strategies or pathways. It also gives students uninterrupted opportunities to make sense and persevere in exploring and reasoning through tasks that allow them to “… draw on their funds of knowledge (i.e., the resources that students bring to the classroom, including their home, cultural, and language experiences)” (NCTM, 2018, p. 32).

Shift 4: From show-and-tell toward share-and-compare lends itself to providing forums for (and yes) facilitating meaningful mathematical discourse. The key is that the teacher’s role is to strategically invite student participation in ways that help to facilitate student opportunities for “… making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem-solving” (NCTM, 2014, p. 10). This Shift is about giving students time and space to to share their ideas with each other. It’s about teaching and holding them accountable for actively listening, honoring, and even critiquing each other’s ideas, as a way to build and deepen their shared understandings.

Helping students reflect on what they know and are able to do, while also prompting students who have temporarily ‘hit a wall’ by posing questions to advance their thinking (Smith & Stein, 2018) is what Shift 5: From questions that seek expected answers toward questions that illuminate and deepen student understanding is all about. Making this thinking both public (NCTM, 2018) and visible is where opportunities for student engagement and deep levels of learning can take place.

Shift 7: From mathematics-made-easy toward mathematics-takes-time remains a cultural struggle for many in the U.S., as a prominent belief held by some, is that the purpose of doing mathematics is just the opposite. However, when a teacher communicates caring and confidence in students, while holding high expectations and providing opportunities and time to struggle with tasks, the capacity for a student to enhance their mathematics identity and sense of agency increases (NCTM, 2018).

Formative assessment in action takes place when teachers emphasize Shift 8: From looking at correct answers toward looking for students’ thinking through connections between their planning and in-the-moment use of practical classroom-based assessment techniques such as the Formative 5: observations, interviews, Show Me, hinge questions, and exit tasks (Fennell, Kobett, & Wray, 2017). A teacher’s role during collaborative student work is to closely monitor, capture, and record observations of their thinking; intervening only when more information is needed, or to prompt students with questions to advance their thinking (Smith & Stein, 2018); and to avoid making high-cognitive demand problems easier or sharing our own solution pathway(s).

A Math Teacher’s Role

‘Guiding’ and ‘facilitating’ can be critical aspects of engaging all mathematics learners. Sometimes though, as teachers, we fail to give students the uninterrupted time they need to engage in collaborative problem solving, mathematical reasoning, and productive struggle. All students must be encouraged to lead discussions of their strategies, making their “thinking public” (NCTM, 2018, p. 34), with the entire classroom community valuing potential mistakes and errors, and using students’ mathematical ideas as teachable moments and springboards for later success. We have an important job to do in helping all students engage in deep mathematical reasoning during collaborative work, and sometimes this means that our role is to simply stay out of their way!


Fennell, F., Kobett, B. M., & Wray, J. A. (2017). The formative 5: Everyday assessment techniques for every math classroom. Thousand Oaks, CA: Corwin.

McGatha, M. B., & Bay-Williams, J. M. (2018). Everything you need to know for mathematics coaching: Tools, plans, and a process that works for any instructional leader, grades k-12. Thousand Oaks, CA: Corwin.

National Council of Teachers of Mathematics. (2018). Catalyzing change in high school mathematics: Initiating critical conversations. Reston, VA: Author.

National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.

National Governors Association Center for Best Practices, Council of Chief State School Officers. (2010). Common core state standards – mathematics. Washington D.C.: Author.

Smith, M. S. & Stein, M. K. (2018). 5 Practices for orchestrating productive mathematics discussions, 2nd edition. Thousand Oaks, CA: Corwin.

Written by

Jon Wray is the Coordinator of Secondary Mathematics in the Howard County Public School System in Maryland. He is also the Project Manager of the Elementary Mathematics Specialists and Teacher Leaders Project. Jon recently served as an elected member of the National Council of Teachers of Mathematics (NCTM) Board of Directors. He is a former elementary classroom teacher, mathematics resource teacher, and past president of both the Association of Maryland Mathematics Teacher Educators (AMMTE) and the Maryland Council of Teachers of Mathematics (MCTM). Mr. Wray is a recipient of the MCTM Outstanding Teacher Mentor Award and was selected as his district’s Outstanding Technology Leader in Education by the Maryland Society for Educational Technology (MSET).


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