Mathematical discourse can clarify student thinking, address misconceptions, highlight student work completion, showcase multiple approaches, and build high level connections across mathematical content and standards. Through the final three of five practices for orchestrating mathematical discussions (Smith & Stein, 2019), selecting, sequencing and connecting, we have a powerful approach to address misconceptions we notice during formative assessment.
This approach is ideal when we diagnose a misconception and elicit strong evidence of student thinking that suggests why the misconception occurs. Working backwards, the goal is for students to make connections that clarify this misconception; so, we also must select and sequence student work in a way that leads students to focus on that connection through discussion. To reiterate, only apply this method when you’re confident of the reason(s) why the misconception exists and you can use existing student work in-the-moment to lead students to discover these reasons.
| 1. Determine the type of student work you’d need to help illustrate and clarify the misconception |
a. What would the best example of the misconception be? b. What ways of approaching the task would help clarify it? |
| 2. Select two or more student work samples that can diagnose a misconception and reconstruct conceptual understanding |
a. At least one of the work samples must demonstrate the misconception in question b. The other samples could be a variety of methods and approaches that lead to the correct solution, an unfinished solution, a unique yet correct solution, or any combination of these. |
| 3. Sequence the work samples to tell a story |
a. For example: i. Start with the misconception and discuss how the student’s method demonstrates mathematical reasoning yet where that reasoning breaks down ii. Continue with a method that shows correct reasoning where the former example broke down. Discuss the implications of these differences on the solution. iii. Continue with another method that, while different, demonstrates understanding of the concepts. iv. Compare this different method to the original work sample to see the misconception through a new lens. v. Connect all samples to name accurate conceptual frameworks that students should transfer to future tasks. |
The example above is one of countless ways to diagnose misconceptions and re-construct conceptual frameworks through discussion of student work samples. When implementing this method in-the-moment, allow yourself to take indirect pathways to Connect especially if that allows you to show multiple methods for solving the task.
Students will often make connections when you’re sequencing student work. Allow them to do so and adjust the questions you ask while continuing to sequence what you’ve selected. These opportunities can be great to justify their reasoning and summarize what they’re learning.