By Joe Dostilio and Laurie Speranzo

Institute for Learning

Every student has the right to be engaged in meaningful learning that draws on their unique abilities and backgrounds for making sense of mathematical ideas and relationships. Educators who believe this have looked for ways to engage all students in equitable mathematics instruction that advances each student’s conceptual understanding as they navigate this new world of virtual instruction.

These are three teaching practices that work in combination and can be used in virtual spaces for keeping every student engaged, talking, and supported in online learning.

  • Keeping Learning Focused and Advancing
  • Engage Students in Talk/Hear Student Voice
  • Engage In Formative Assessment

 For each of these practices, there are strategies that support equitable mathematics instruction when teaching virtually.

NCTM (2014) research shows that “Student learning is greatest in classrooms where the tasks consistently encourage high-level student thinking and reasoning and least in classrooms where the tasks are routinely procedural in nature” (Boaler and Staples 2008; Hiebert and Wearne 1993; Stein and Lane 1996) and “Not all tasks provide the same opportunities for student thinking and learning” (Hiebert et al. 1997; Stein et al. 2009).
Even when instructional time moves online, students need time to consistently engage in high-level tasks on their own and with their classmates in order to keep learning focused on sense-making.

Strategy #1:  Use Tasks that Require High-Level Mathematical Thinking and Reasoning

High-level tasks

  • have multiple entry points;
  • include or can be solved using a wide range of representations and tools; and
  • require problem-solving.

Focus tasks, a subset of high-level tasks, are a type of cognitively demanding task that requires thinking and reasoning that leads to sense-making, but they do not require the same level of investigation or problem-solving. Focus tasks explicitly press student thinking about a key mathematical concept or relationships by leveraging one of the following methods. Try using a focus task that

  • provides a model for students to interpret and make sense of the mathematics.
  • shows an accurate and an inaccurate solution, then asks students to analyze and name which is accurate and which is not, and explain why.
  • shares a way of solving that is not a traditional algorithm. Ask students to discuss the way of solving and apply it to a new situation or set of numbers.

Use focus tasks to help provide a steady diet of high-level tasks that leads to greater and deeper learning, especially in a time of potentially decreased instructional time due to the pandemic.

Investigating and discussing high-level tasks to make sense of mathematics supports students in seeing themselves as doers of mathematics.

Strategy #2: Provide “On Your Own” Time for Students to Begin Thinking and Reasoning about the Mathematics in the Task

Students need time to generate their own thinking and solutions to a task. Consistently giving students independent work time
  • honors each student’s thinking and develops their identify as a doer of mathematics;
  • provides formative assessment opportunities for teachers; and
  • makes it possible for students’ ideas and solutions to be discussed and compared in small breakout groups and in whole-class discussions.

When time is a concern, consider sending the task to student in advance with a pre-recorded set-up that supports students in understanding the context of the task (without unpacking or giving away the math concept for them). Ask students to engage in “On Your Own” time and come to the live class meeting time prepared to share and discuss their solutions.

It is critical to provide time so that students can process the task, formulate their own mathematical thoughts and solution path, and put their thinking on paper before engaging in discussions. Providing this time makes it more likely that every student—regardless of native language, identification, background, or grade point average—will share their ideas with others, instead of just hearing from those students who work faster and/or louder.

Carpenter, Franke, and Levi (2003) talk about the importance of student voice. “Students who learn to articulate and justify their own mathematical ideas, reason through their own and others’ mathematical explanations, and provide a rationale for their answers develop a deep understanding that is critical to their future success in mathematics and related fields.”

When engaging students in virtual classrooms, look for ways to keep students generating, talking about, and making connections between their ideas and solution paths. Making time for and providing ways for students to use their voice and agency will build positive math identities.

Strategy #3: Incorporate Manipulatives into Lessons

Representations help to clarify the reasoning of specific students, while also leaving a visible trace of the strategy, which allows other students to enter into and follow the mathematical thinking of their classmates.
NCTM (2017) Taking Action

Manipulatives allow students to show their thinking and provide a reference when explaining their reasoning. Consider these options for having students use and discussion manipulatives in virtual classrooms.

Using student manipulative representations allows students to discuss their own understanding and to comment on and explain the thinking of others. When using virtual manipulatives, sending out the link in advance is a good idea, as students (and adults alike) need to “play” with the manipulatives before using them for a task.

Strategy #4: Create Space for Students to Share Their Thinking

Victoria Bill, longtime Senior Math Fellow at the IFL, always says, “The person who talks the most, learns the most!” Use ways to provide time and provide space to ensure each and every student has opportunities to talk and for their voice to be heard.

Check out 4 Go-To Learner-Centered Routines to Bolster Math Discussions, In-Person and Online to read about some routines that prepare students to engage in discussions about deep mathematics. Consistently using these and other routines that get students to share their mathematical thinking, provides similar benefits as those listed above for “On Your Own” work time, honors student thinking, develops their identify as doers of mathematics, provides formative assessment opportunities, and makes it possible for students to generate solutions to share and compare with other students.

Wiliam noted that “the important point is that we must acknowledge that what students learn is not necessarily what the teacher intended, and it is essential that teachers explore students’ thinking before assuming that students have ‘understood’ something. In this sense, assessment is the bridge between teaching and learning” (2005).

As mathematical ideas and solutions are shared, translate some formative assessment best practices to the virtual world. When students say or write about what they think, these strategies honor that thinking and students develop a sense of worth of their ideas—that their ideas are worth sharing and will be discussed in the process of learning mathematics.

Strategy #5: Ask for Agreement/Disagreement . . . Then, Ask Why?

Asking every student to weigh in on whether they agree or disagree is a simple means of formative assessment that teachers may use throughout a lesson. In virtual classrooms, identify ways to continue to use agree/disagree to keep students engaged and provide on-going formative assessment throughout the lesson.

  • Ask students to use a Reaction or an Emoji.
  • Poll students using a simple “Agree” or “Disagree”.
  • Have each student send a message through the chat box; ask that the message be sent only to you when you want to make sure students are not swayed by others’ responses.

Then, prompt students to say more about why they agree or disagree! Without the justification of their agree/disagree stance, there is a missed opportunity for students to share their mathematical reasoning. If teachers are not in the habit of asking for students to back up their thinking, students may agree or disagree out of habit and not out of math content knowledge.

Strategy #6: Provide Effective Feedback

Learners engage better when feedback is focused on their work, identifies what they have shown in their work in regards to the mathematical learning goal, and provides an actionable way for moving forward from their work. Feedback that assigns a grade or score that can often send a message that the learning is complete and has been assessed (Butler 1988, Black & Wiliam 1998, Hattie & Timperley 2007).

When providing written or verbal feedback, call attention to what work the student(s) have done and pose actionable next steps that press the student(s) toward deeper meaning-making.

Characteristics of Feedback and Guidance
Feedback Guidance
  • Highlights mathematical ideas or strategies and lets the student know the benefit or usefulness of the idea or strategy.
  • Acknowledges student’s actual work.
  • Is focused on student’s work, not the student.
  • Conveys to students that they have an audience for their work, “I noticed…” “When looking at your work…”
  • Highlights components of the program that are valued (e.g., connections between representations, communication of mathematical reasoning).
  • Extends the student’s work from where the student is in his or her work to the next stage of the work.
  • Encourages the student to do the work, serves as a call to action, “Give it a try…”
  • Presents students with a challenge or a counter to their method.
  • Prompts the learner to consider the use of alternative representations, strategies, or processes.

Check out these two examples of feedback and guidance given to two different groups by a teacher who utilized Google Slides during small group work a focus task.


Provide time for the student to take action/respond and then respond/resubmit. As time with our students is often reduced in hybrid and remote learning, this process of providing effective feedback and having students take action and respond/resubmit can create a better virtual experience for students and strengthen student-teacher relationships.

Regardless of where students are in their mathematical thinking—not yet fully expressing their reasoning, exhibiting faulty or overgeneralized thinking, or even having the correct answer with sound justification— they all deserve feedback and guidance. Every student should be pressed to move forward from where they are to deeper mathematical understanding. This makes the instruction both equitable and differentiated!

Tell Us About the Strategies You Use When Implementing High-Leverage Teaching Practices in Virtual Classrooms

  • Which of these strategies that support equitable mathematics instruction do you already use in virtual classrooms? How is it going?
  • What other strategies that support equitable mathematics instruction do you recommend for virtual classrooms? Why?

Share your story here.


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    Tagged with: Academic Rigor, Agency and Voice, Equitable Instruction, Formative Assessment, High-Level Tasks / Curriculum, High-Leverage Teaching Practices, Math, Multi-Lingual Learner Instruction, Online Instruction, Principles of Learning