By Laurie Speranzo and Joe Dostilio
Having students put their work on chart paper can serve the class’s discussion of math. Gallery walks have their place. But effective charting is a meaningful routine to use as it “serves as a public record of the intellectual efforts of and meaningful contributions from the learning community that can be referenced later or in future discussions.” (Klingensmith, Speranzo, Dostilio, Bill, 2020).
In an previous article we unpacked four learnercentered routines that afford math students greater access to classroom discourse. We examined those same four routines through the lens of equitable instruction in a followup article. Now, it is time to take on the fifth learnercentered routine: charting!
Learnercentered routines work to



 spur discussion,
 increase student ownership of understanding, and
 create space for student voice


So, what is charting?
Charting IS 
Charting IS NOT 


But how do we routinely create useful, authentic charts that support students as a means of preparing to and engaging in Accountable Talk® discussions?
♦ Supporting Accountability to the Learning Community
The learning community is a group that actively listens and reacts to each other’s thinking. Often in a discussion that is accountable to the learning community students will be asked to agree or disagree and explain why. Students may be adding on to each other’s thinking.
By keeping track of the growing thinking publicly, students have a chance to not only hear, but see the progression. Students need to hear ideas multiple times, process them, and see them. Charting aids in being able to see the mathematics—in all its representations! It is very hard to follow the verbal description of a graph but having it displayed allows a deeper level of access.
Charts are Created from Student Thinking and Student Work
To that end, it is crucial that the students’ ideas be the starting point of the charts. There may be a strong hankering to design the perfect chart and laminate it for coming years! 🎵 “Let it go!” 🎵 We all attend to things in our lives where we have an investment; this is true of charts as well! If students contribute to the chart, they are more likely to refer to it throughout (and even after) the lesson or unit. Using a piece of studentgenerated work or idea to coconstruct a chart not only honors students as authors of mathematics, it promotes students engaging in being mathematicians, working together over time to figure things out and moving toward precision through their shared process of learning.
Charts are Publicly Visible and Don’t Disappear
Large interactive screens/touchsensitive boards in the classroom are amazing! However, the work disappears when the next image or task goes up on the screen. The same with student whiteboards that get erased regularly. Creating a class record of thinking or longterm tool requires a public display that can be referenced as students engage in small group and whole group discussions, and after. That means you may need to use chart paper sparingly, but strategically choose what you want to showcase for the unit of study.
NOTE: Using interactive white boards allows for collaboratively created and edited work to be sent to interactive notebooks. Students can refer to the cumulative version of the class thinking later when it has landed in their personal accessible space. It can also be brought back up on the interactive white board as needed. Technology can be impactfully used as long as it is available when the students want to/need to/ask to access it and not only when the teacher wants to show it.
♦ Supporting Accountability to Content Knowledge
Being accountable to the students’ content knowledge means that there needs to be a focus on helping the students develop conceptual understanding. Use of representations—physical, visual, contextual, verbal and symbolic—helps students construct knowledge and see the structure of the mathematics. Public charting of the representations that students are using or studying helps students attach verbal language to the models. And the representations get cleaner and more precise over time.
Charts Become More Precise as Students Add to Their Thinking Throughout a Lesson or Unit
In this classroom at the Environmental Charter School in Pittsburgh, PA, the teacher started with a studentgenerated definition of proportional relationships. They added to and refined the definition as they learned more throughout the unit. You can see the use of student sticky notes to indicate needed changes and teacherrecorded ideas that came out of the wholeclass discussions.
This teacher in Fabens Independent School District outside of El Paso, TX, asked students to record their thinking about the dimensions and volume of cones. Students started with what they knew and added to the chart over two days of exploration. When students were working on problems related to conic volume, the teacher regularly asked students where they could go to find helpful information and this chart became a visual goto for students as they explored additional 3D figures.
♦ Supporting Accountability to Rigorous Thinking
In math, there are ways to press on rigorous thinking so that students move forward in their conceptual understanding. Two ways we can do that are to:




 Connect mathematical representations and
 Press for mathematical reasoning.



Connecting mathematical representations means being able to study at least two – maybe there are two representations you know and feel comfortable with or maybe you are looking at one you have in your toolbox and one crafted by another person. Having public access to what is being connected is crucial.
Charts Capture Students’ Conceptual Understanding
Unpacking mathematical reasoning takes time as students figure out not just the “what” but the “why” of the math. In the scenario below the students are both




 connecting the symbolic tables and equations to the visual representation of the graph, and
 figuring out the meaning of the key features as they related to the solution of the systems of linear equations (point of intersection, slope and yintercept of each equation, solution determined via substitution):



As students worked on this problem and discussed different solution methods, student language was not mathematically precise from the beginning. As the teacher coconstructed the chart with students, the teacher added mathematical vocabulary as students made connections between the graph and the context of the situation
Student 1: The lines cross when it’s 7.
Teacher: When what is 7?
Student 2: Wait, they’re both 30 at 7.
Teacher: So, the point of intersection is (7, 30). What does that mean in the context of the problem?
The precise mathematical vocabulary, “point of intersection”, was added to the chart only after students discussed connections between representations and collectively made sense of what it means to be a “point of intersection” during the wholeclass discussion.
How does charting support equitable teaching and learning?
When only a few students, like the ones who raise their hands first and often, are the only ones who contribute to a discussion, the classroom is not equitable. Consider these questions.


 From whom are teachers hearing?
 Whose ideas are being charted?
 What assumptions are being made about students’ ability to contribute to the class knowledge?
 Why is it important to get multiple student perspectives?

By charting collectively, the entire student community can feel that their thinking is important, even if it is messy or imprecise or even off the mark at first. Consistently and publicly revisiting and adding to/revising a chart as a class provides space and opportunities for all voice and sends a clear message that learning is a collaborative endeavor that happens over time.
So, what might you try or do more of? Consider:


 How can you use a student’s or student group’s work to launch a classroom chart?
 How might you have students add to someone’s thinking to ensure all of the big ideas are made public?
 How can you use student work and student language to make for a stronger reference tool?

Thanks to

References
Klingensmith, K., Speranzo, L., Dostilio, J., & Bill, V. (2020). Accountable Talk Mathematics Discussions: Teacher’s Guide. Pittsburgh. PA: University of Pittsburgh.
® Accountable Talk is a registered trademark of the University of Pittsburgh.