Six Strategies That Can Lead to More Equitable Online Mathematics Instruction

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.

    References

    Black, P. & Wiliam, D. (1998) Assessment and classroom learning. Assessment in Education: Principles, Policy & Practice, 5(1), 7-74.
    Boaler, J. & Staples, M. (2008). Creating mathematical futures through an equitable teaching approach: The case of railside school. Teachers College Record, 110(3), 608-645.
    Butler, R. (1988). Enhancing and undermining intrinsic motivation: The effects of task-involving and ego-involving evaluation on interest and performance. British Journal of Educational Psychology, 58(1), 1-14.
    Carpenter, T. P., Franke, M. L., & Levi, L. (2003). Thinking mathematically: Integrating arithmetic and algebra in elementary schools. Portsmouth, NH: Heinemann.
    Hattie, J. & Timperley, H. (2007). The power of feedback. Review of Educational Research, 77(1), p81-112.
    Hiebert, J. Thomas, P., Carpenter, E., Fennema, K. C., Fuson, D., Wearne, P., Human, H. M., & Alwyn, O. (1997). Making-sense: Teaching and learning mathematics with understanding. Portsmouth, NH: Heinemann.
    Hiebert, J., & Wearne, D. (1993). Instructional tasks, classroom discourse, and students’ learning in second-grade arithmetic. American Educational Research Journal, 30(2), 393–425. Retrieved from https://doi.org/10.2307/1163241
    Hiebert, J., Carpenter, T. P., Fennema, E., Fuson, K., Human, P., Murray, H., Olivier, A & Wearne, D. (1997). Making mathematics problematic: A rejoinder to Prawat and Smith. Educational Researcher, 26(2), 24-26.
    National Council of Teachers of Mathematics. (2014). Principles to actions: Ensuring mathematical success for all. Reston, VA: Author.
    Smith, M. S., Boston, M., Dillon, F., & Miller, S. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 9-12. Reston, VA: National Council of Teachers of Mathematics.
    Smith, M. S., Huinker, D. & Bill, V. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in K – Grade 5. Reston, VA: National Council of Teachers of Mathematics.
    Smith, M. S., Steele, M. D., & Raith, M. L. (2017). Taking Action: Implementing Effective Mathematics Teaching Practices in Grades 6-8. Reston, VA: National Council of Teachers of Mathematics.
    Stein, M. K., & Lane, S. (1996). Instructional tasks and the development of student capacity to think and reason: An analysis of the relationship between teaching and learning in a reform mathematics project. Educational Research and Evaluation, 2 (1), 50–80.
    Stein, M. K., Grover, B. W., & Henningsen, M. (1996). Building student capacity for mathematical thinking and reasoning: An analysis of mathematical tasks used in reform classrooms. American Educational Research Journal, 33(2), 455–488.
    Wiliam, D. (2005). Keeping learning on track: Formative assessment and the regulation of learning. In M. Coupland, J. Anderson, & T. Spencer (Eds.), Making mathematics vital: Proceedings of the twentieth biennial conference of the Australian Association of Mathematics Teachers (pp. 26–40). Adelaide, Australia: Australian Association of Mathematics Teachers.

    Accountable Talk and Learning Walk are registered trademarks of the University of Pittsburgh.

    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

    4 Go-To Learner-Centered Routines to Bolster Math Discussions, In-Person and Online

    By Laurie Speranzo and Kristin Klingensmith

    Institute for Learning

    Educators who believe that every student has meaningful ideas to contribute can use learner-centered routines to ensure that every student is actively engaged in a math discussion. These routines create space for students to reason about the mathematics of the task in ways that make sense to them using their knowledge from both inside and outside of the classroom. Using learner-centered routines gives teachers multiple opportunities for formative assessment within a single lesson from which they can make adjustments in real-time to better meet where students are in their understanding.

    To School Leaders & Coaches

    These routines support learning at all ages, so consider if and how you might use these routines during in-person and online professional learning to support deeper understanding of the ideas being explored and discussed with other educators.

    Based on two decades of experience with in-person teaching and learning and several years of engaging in online teaching and learning, here are four go-to learner-centered routines to use while facilitating meaningful mathematics discussions in-person and online.

    1. Turn and Talk
    2. Stop and Jot
    3. Step Back
    4. Quick Write

    When used consistently, these routines enhance Accountable Talk® discussions in mathematics, support engagement, and scaffold learning.

    1. Turn and Talk

    Why Use a Turn and Talk?

    Turn and Talks create space for students in groups of 2 to 3 to share their ideas related to the mathematics being explored and hear the ideas of others. Turn and Talks are an informal way for teachers to assess (little “a”) where students are in their thinking. Teachers can and should use the insights gained from a Turn and Talk to inform the direction of the mathematics discussion.

    Tips for Using Turn and Talks

    Turn and Talks are low-risk talk opportunities that can be useful throughout a discussion because students get a chance to “try out” and “rehearse” their thinking alongside others. Turn and Talks are a great go-to after 10 seconds or more of wait time when students need to talk through their thinking before being ready to share with the whole group. If you are familiar with Accountable Talk moves, Turn and Talks are particularly beneficial when used in combination with the moves: challenging, pressing for reasoning, or expanding reasoning.

     

    Adapting Turn and Talks for Online Discussions

    In person Turn and Talks are as easy as turning and talking with a neighbor, and allow the teacher to listen in to get a feel for student thinking. Turn and Talks are not as easy to replicate online but are still important for supporting student thinking and engagement.

    For online Turn and Talks try

    • moving students into breakout rooms with just 2-3 people for a very brief amount of time.
    • asking students to mute their mics and say their thinking aloud to themselves.
    • establishing Turn and Talk partners so that students can chat privately with a pre-assigned peer via the chat box.

    Though online Turn and Talks do not afford the opportunity to “overhear” the Turn and Talk of others, all of the adaptations offer students’ the opportunity to put their thinking into words and to rehearse how they want to express their understanding. Because students cannot be overheard, teachers may want to have a few students share out to the class what they said aloud to themselves or what was shared during their exchange with a peer. If planned in advance, a teacher may pair a Turn and Talk with a multi-select poll to see which ideas resonated with the students following their Turn and Talk.

    2. Stop and Jot

    Why Use a Stop and Jot?

    Stop and Jots provide each student time to collect and record their thinking as it relates to the mathematical idea or relationship currently being discussed. Stop and Jots are commonly used during critical points of discussions and usually involve students adding something new to their written work. This provides teachers the chance to casually check on student thinking as they walk around and glance over students’ shoulders. The information gathered from glancing at students’ Stop and Jots can be used to determine the pathway of the discussion.

    Tips for Using Stop and Jots

    Because Stop and Jots start out as private thinking time, students tend to feel more comfortable expressing their initial, and possibly incomplete or unfinished, thoughts. Stop and Jots can ask students to write about how two concepts are related, create another or connect two representations, record another/different equation, describe a situation, etc. When using a Stop and Jot, teachers may look for patterns in responses, unique responses, and responses that suggest over-generalizations, to determine what to “take up” next in the discussion.

     

    Adapting Stop and Jots for Online Discussions

    During online discussions, it isn’t as easy to sneak a peek at what students jot, so consideration has to be given to low-risk ways for students to share their jotting.

    For online Stop and Jots try

    • asking students to write their thoughts on paper and having those who are comfortable share orally or, in the case of visual models or equations, by holding the paper up to the camera.
    • inviting students to type or draw in a shared document.
    • having students use the chat box to share their thinking with the teacher only or with the whole group.
    • using PollEverywhere.com (or a similar platform) so that students can text their responses anonymously and have them collected on a single shared screen.

    When making a choice about how to have students share their ideas after a Stop and Jot, keep in mind that some students may feel more comfortable sharing what they wrote verbally than actually sharing their writing. Things like spelling, sentence construction, and use of formal math language are not and should not be a focus during Stop and Jots.

    3. Step Back

    Why Use a Step Back?

    Step Backs offer an opportunity for students to reflect verbally on key learnings from across the entire discussion and to share their conclusions or generalizations with the whole group. Step Backs offer teachers a glimpse into how students are taking stock of the ideas being discussed and their progress toward the mathematical learning goal. They also allow students to again hear the salient and mathematically critical ideas that surfaced during the discussion.

    Tips for Using Step Backs

    Step Backs are best used following big “ah ha” moments related to the learning goals for the lesson. These moments may occur in the middle or at the end of a discussion and sometimes both. Use Step Backs to create space for students to ponder and consider the Why of the big “ah ha” moment(s).

     

    Adapting Step Backs for Online Discussions

    Step Backs do not need to be adapted for online discussions, but they may be used more frequently to help students hold on to critical ideas as understanding is constructed.

    For online Step Backs try

    • pre-determining several places during the discussion to do a Step Back so that the discussion has multiple summary points that culminate by the end of the discussion.
    • having at least one or two students “say back in their own words” the teacher’s summary so that others can hear it more than one time and students have a chance to paraphrase the key ideas.
    • varying who is responsible, teacher or students, for summarizing the discussion at a given point.

    4. Quick Write

    Why Use a Quick Write?

    Quick Writes create space for each student to put their thoughts into writing, allowing them to take stock of their knowledge, reflect on their learning, and/or apply their insights, generalizations, and conclusions in a new way. Quick Writes can and should be used as formative assessment because they contain evidence of each student’s thinking and reasoning, especially after engaging in an Accountable Talk discussion.

    Tips for Using Quick Writes

    Quick Writes can be used at the beginning of a unit of study to learn about students’ prior knowledge. Quick Writes can also be used toward the end of or following a math discussion or summary. In this case, it is important to make sure that the Quick Write relates directly to the mathematical learning goal of the lesson.  Additionally, students and teachers can gain insight about how learning is progressing over time by using Quick Writes after each lesson in a series.

     

    Adapting Quick Writes for Online Discussions

    Quick Writes can be used when teaching online in much the same way they are used in-person. The adaption to Quick Writes is mostly about how students will submit them.

    For online Quick Writes try

    • having students write a response in the chat but wait until everyone is done writing before sending it so that no one’s thinking is compromised by reading others’ responses.
    • setting up a Padlet (or similar applet) for students to submit their responses and comment on the thoughts of others, using a combination of multi-media options: text, pictures, drawings, and video.
    • having students submit their response via an online form or through email.
    • using a shared document or slide deck.

    When making a decision about how to have students submit their Quick Writes, consideration should be given to what the submission option affords. Some options only allow written responses, while other options offer the ability for students to create images and upload pictures or video.

    The second article in this two-part series will be released December 1 and will explore how these four learner-centered routines can be used to foster student voice and agency and support students in developing positive mathematical identities as doers of mathematics.

     

    Tell Us About the Learner-Centered Routines You Use

    • Which of these routines do you already use? How is it going?
    • Which of the routines might you add to your toolbox? Why?
    • What other routine do you recommend? Why?

    Tell us here.

    Tagged with: Accountable Talk® Discussions, Equitable Instruction, Formative Assessment, High-Leverage Teaching Practices, Online Instruction, Principles of Learning

    Differentiation Across Tiers of Math Instruction

    Laurie Speranzo

    IFL mathematics fellow

    Derek Stoll

    Math teacher at ELMS, Syracuse City School District

    What does it mean to differentiate instruction to meet the needs of every learner in a mathematics classroom? Giving different students different tasks does not yield a common learning experience; therefore, the availability of holding rich mathematical conversations with the whole class is lacking. But we know that we need to meet students where they are and support students in their exploration of the math. Thus, there is the need to differentiate the questions that are asked of students as they engage in high-level tasks.

    Differentiation in Tier One Instruction

    In the Syracuse City School District, Derek Stoll is a math teacher at Expeditionary Learning Middle School (ELMS). Derek shares the power of implementing tasks’ differentiating questions. “Creating opportunities for students to have access into a complex task allows students to get invested in the mathematical learning inside a classroom, as well as pushed at a variety of levels. Planning questioning based on problem-solving strategies has been really helpful in providing targeted feedback to different mathematical learners.”

    Derek refers to planning questions based on strategies. As students are working individually or in small groups, the questions that assess and then advance student thinking differ based on the work they produce; some students/groups may get questions that ask them to try applying their strategy to a different set of numbers, some may get questions that ask them to create a model that matches their thinking, and some may get questions that ask for them to write about their reasoning. However, all student are asked questions that push their thinking forward toward the goal of the lesson.

    Derek does name, however, that differentiated questioning is not easy. “This work is very challenging to plan (especially when you are the only math teacher in your building) but working with other sixth grade math teachers and testing out tasks on other teachers in the building help build a greater awareness of what types of strategies [students may use to solve the task] and what questions students will need to be successful.”

    Differentiation in Tiers Two and Three Instruction

    There is an additional layer to the work around differentiation that is occurring at ELMS this year. Under the leadership of principal Kevin Burns, and with Derek coordinating data collection, ELMS has taken on differentiating additional learning opportunities for students who have unfinished learning in their prior knowledge.

    In an attempt to ensure that students are receiving exactly what they need, the school has created dedicated time and data usage for Tier Two and Tier Three services. Derek says,“This year we have organized our master schedule to have every sixth grade student receiving a math intervention at the same time every other day. With this unique opportunity, we have utilized the SOAR materials from the IFL to create a series of universal screeners. Using these targeted screeners, we created specific subgroups that are addressing gaps of unfinished learning in 6- to 10-week cycles of intervention using the SOAR materials.” Knowing that Tier Two and Tier Three instruction is meant to be flexible and concept-specific, Derek adds that the SOAR materials “have been really powerful tools to specifically provide some targeted instruction and flexibility for students to receive mathematical instruction at their level and be moved based on students’ problem-solving strategies, mathematical conceptual understandings, and efficiency to solve problems within contexts.”

    Like with planning for Tier One differentiation, there are challenges here as well. Derek names that teachers providing instruction need “consistent and thoughtful professional development on the domain they are addressing with a math specialist that can look at student work (using progress monitors also created with the SOAR materials) and align the SOAR intervention lessons based on what students need.” As Derek notes, progress monitoring is essential to ensure that the identified instructional supports are effective and that student understanding is advancing. Tiered instruction may need to be modified based on the data collected during progress monitoring.

    One measure that has been instrumental in the implementation of differentiated intervention time at ELMS has been team meetings. The meetings allow time to delve into looking at student work and analyzing students’ unfinished learning and next steps of individual students based on what they need. While an ambitious undertaking, the model that ELMS has adopted this year speaks to meeting the needs of the students at every level of instruction, truly defining differentiation in the math classroom.

    Tagged with: Formative Assessment, High-Leverage Teaching Practices, Math, Partner Spotlight