One day, students in my eighth-grade math class were working in pairs. Two boys, Nainoa and Keoki, called me over to resolve their dispute over differing answers. After reviewing their work, I confirmed that Nainoa was correct. Nainoa shouted, “Ha, told you so!” Keoki dropped his head to his desk, whispering, “Whatever.” I scolded Nainoa, but it was too late—the damage was done, and Keoki shut down for the rest of the class period.
Later that day, I reflected on the incident. The lack of humility and detrimental effects of Nainoa’s words lingered. Keoki, shamed for one incorrect response, was now less likely to take mathematical risks. The person who used the most cognitive energy? Me, the teacher, since I was the one who did the work to determine which student was correct.
The Value of Productive Discussion in Math
Putting that poor result aside, the scenario of two students discussing different answers is exactly what we want in math classrooms. Common Core Standard of Math Practice 3 is: “construct viable arguments and critique the reasoning of others.” Such discussions create the perfect opportunity to challenge students like Nainoa to extend their understanding beyond getting the correct answer or doing the task.
Nothing is more challenging for a student who has the correct answer than to explain why someone else is incorrect in their reasoning because alternative methods, especially those that generate incorrect responses, are hard for the student with the correct answer to make sense of.
What happens when we’re able to make sense of such alternative understandings, viewpoints, processes, or methods? We become more intelligent human beings, with more holistic conceptual understandings of content. In addition, we utilize and develop the essential 21st-century communicative skill of empathy and are better suited to thrive in our increasingly diverse society.
So what went wrong in the scenario with Nainoa and Keoki? I had not set up an effective routine to optimize their interaction. In order for an academic discussion routine to work effectively, I found that I needed these three elements:
- heterogeneous grouping of students to ensure greater opportunities for discourse around diversity of understandings,
- student understanding of the value of heterogeneous grouping for their development, to motivate them to engage in productive discussions, and
- a protocol for discussion that ensures all students are supported in these discussions.
I was lucky enough to have Nainoa and Keoki arrive at different answers, but to ensure that productive discussion happened on a regular basis, I needed to start intentionally using heterogeneous grouping. To help students see the value of collaboration in heterogeneous groupings, I came up with the following “Tiers of Understanding,” which I put up on a poster for my students to reference:
Tier 1: Do—complete the task.
Tier 2: Explain the process used to complete the task.
Tier 3: Empathetically explain the thought processes used by another student to complete the task.
Doing a task is the lowest level of understanding—explaining is higher, and the highest is explaining empathetically. Through heterogeneous grouping, the student struggling in their ability to do the task is supported by the higher performing student, who seeks opportunities to empathetically explain. This creates differentiation through inclusivity—personalized value for the mixed levels of learners in working together.
If I had known to teach Nainoa and Keoki the value of empathetic explanation, they would have looked at their different answers as a learning opportunity.
Teaching Students to Empathetically Explain
I now teach students a three-part protocol for discussing divergent answers when they’re working together.
1. Listen to the other person.
2. Try to see how they could be correct—maybe you’re both correct. Math is not fixed, as there are many avenues to arrive at one solution, and solutions can appear in many equivalent forms.
3. If you believe the other person is incorrect, explain how you are correct, and/or how they are incorrect. It is the responsibility of the person with the correct answer to rectify the misunderstanding.
The most valuable part of the protocol is the last part—that it’s the responsibility of the student with the correct answer to help their partner correct their misunderstanding. It was not Keoki’s fault he had the wrong answer, and since Nainoa had the correct answer, he could have tried to assist Keoki through empathetic explanation. Even in a case where the student with the incorrect answer initially thinks they are correct—which was the case for Keoki—the three steps above will lead them to realize where they went wrong.
Sentence stems such as “You were thinking... because...” or “I can see how you tried to...” are excellent scaffolds to help students start crafting empathetic responses.
It’s now been six years since, in my second year of teaching, I started intentionally grouping students heterogeneously and generated the 3 Tiers of Understanding and the protocol for empathetic explanation to guide students in their collaborative work. These steps have helped me create a more engaging, student-centered classroom by requiring that more of the cognitive work of coming to terms with math concepts be done by the students.
After I had been using this process for a while, I saw my students gravitate toward collaborating with each other. They recognized the value that empathetic discourse had for both students in heterogeneous grouping. Most significantly, this created a class culture grounded in kākou, Hawaiian for “we’re all in this together,” which is what the culture of classrooms should be.