An Administrator’s Six Spheres of Influence in Mathematics Teaching and Learning: Part 3

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To help you be effective instructional leaders in your schools or districts, I want to delve further into what I refer to as your Six Spheres of Influence in mathematics teaching and learning.

Your Six Spheres of Influence:

  • The lens through which you observe teachers
  • Your expectations regarding the posting of lesson objectives
  • Your requirements related to lesson structures
  • The ways you hold teachers accountable to provide feedback to students
  • The structures you have in place for intervention
  • How you support teacher planning

In Part 1 of my blog series I discussed observations, and in my second post I shared ideas regarding lesson objectives. Here I will share my views on the third sphere, and I will make sense of the remaining spheres leading up to the 2018 Model Schools Conference. My session on June 25 will provide a window into schools and classrooms where these spheres are operating in ways that increase student achievement in mathematics.

Your Requirements Related to Lesson Structure

There seems to be a trend to expect the use of a lesson structure often referred to as “I do, we do, you do” during daily mathematics instruction. It is my position that this structure is being overused in the math classroom. Its over-application is similar to the way the requirement to post the lesson objective is over-applied because the expectation is for its use in every lesson, every day. The good news is that there has been some push back recently. However, in an effort to hold on to a structure that many see as being no longer appropriate to support all aspects of teaching for rigor, people are responding by saying, “you can enter gradual release at any phase.” How do you gradually release something by beginning when it is already released?

The entire idea of gradual release of responsibility is to begin with the teacher (who’s in control of the sense making) modeling a problem or idea. P.D. Pearson and M.C. Gallagher originally coined the term gradual release of responsibility in a 1983 article on reading comprehension. The teacher then moves to a more facilitated role by supporting students to engage in the task along with the teacher, in essence replaying what the teacher has shared. Finally, the teacher relinquishes control so that students can demonstrate their understanding.

Just as with posting lesson objectives at the start of the lesson—the sphere discussed in my last post—I do believe there is a time and place for gradual release. This structure is absolutely appropriate for teaching procedures in mathematics. If the goal of the lesson is to teach long division or polynomial division, I strongly encourage teachers to first model the process, then provide guided instruction as students apply the procedure with the teacher, and finally, allow space for students to practice the algorithm independently, thus supporting what is described as “I do, you do, we do.” The implementation of gradual release, without modification, is appropriate for procedural lessons. What about lessons that are more conceptual in nature? This is where gradual release needs to be replaced, not revised. You should use your sphere of influence to replace gradual release with Layers of Facilitation when the lesson is conceptually based, my co-authors and I argue in Making Sense of Mathematics for Teaching Grades 35, published in 2016.

Layers of Facilitation

  • The teacher facilitates the whole class to engage in meaningful tasks through questioning.
  • The teacher facilitates small groups to extend the learning initiated in the whole-group setting.
  • The teacher facilitates individuals to provide evidence of their understanding of the learning goal.

The Layers of Facilitation describe a lesson structure that is more student-centered than gradual release of responsibility. The structure aims to privilege classroom discourse while maintaining a focus on the learning goal for the lesson. The teacher implements a task through the use of whole-class discussion. The teacher supports students to engage in the task through questioning rather than modeling how to solve the task for them. Full-class discussion around a problem is followed by students all working on the next problem (or set of problems) in concurrent small groups with the teacher pushing into groups to provide support through questioning and to collect evidence of student understanding. Finally, individual accountability is supported as students work on problems on their own to provide evidence of where they are relative to the learning goal for the lesson. It is during this time that the teacher can focus on intervention, the topic of your fifth sphere of influence.

The key here, again, is that the students are doing the sense making, and the teacher is supporting them to meet the learning goal through the chosen task and the questions used to support the implementation of that task. Consistent with the theme in each of my posts, the key to increasing student achievement is through the use of flexible instructional structures that are guided by the learning goal. A critical role of administrators is to use your spheres of influence to guide teachers to support students’ mathematical sense making by critically analyzing instructional structures.

Check back soon for the next installment of the Six Spheres of Influence to learn how the ways in which you hold teachers accountable to provide feedback to students might influence student engagement during mathematics instruction.

The views expressed in this article are those of the author and do not necessarily represent those of HMH.


Join more than 5,000 educators in 100+ sessions at the 31st Annual Model Schools Conference in Orlando, Florida, from June 25–28, 2023, where you can learn more about school improvement plans.

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