Students will internalize questioning only if they find it useful. The next section explores how to plan a discussion in which students’ presentations and questioning build the new mathematics of the lesson. When students experience insights and build new mathematical concepts through discussion of classmates’ work, they will be hooked on the power of questioning.
Perhaps the most difficult aspect of a problem-solving lesson is neriage (pronounced nary-ah-gay) – a Japanese term for “kneading” or “polishing” students’ ideas through discussion. As shown in the figure below, neriage occurs after students come up with solutions to the problem. The point of a problem-solving lesson is not simply to solve the problem; it is to develop important new mathematical ideas through neriage that helps students build a bridge from their current solution methods to these mathematical ideas.
Neriage provides the bridge from students’ current understanding to the new concept or procedure you want them to understand by the end of the lesson. (This understanding may be fragile at first, to be strengthened through future use.) Questions to help you plan the neriage are:
- What knowledge will students bring to the problem?
- What new understanding will they build during the lesson? (What actions or words will show their new understanding?)
- What experiences and insights will help students progress from their initial knowledge to the new understanding? For example, what do you expect students to learn from the task itself and from comparing different solution approaches?
- What specific features of different solution approaches will be important for students to notice? What questions will help students notice them?
Summary and Reflection
Following the neriage discussion, be sure to save time (at least 5 minutes) for the class to generate a lesson Summary, followed by time for students to write individually in their math journals about what they learned. The Summary can be initiated with a question like “What did we learn as a class today?” followed by sharing ideas. As you plan the neriage, you should have developed a clear statement of the summary you hope will emerge from the lesson, but it is powerful to use students’ own language to create the Summary on the board. If students do not summarize their learning in the way you hoped, this is powerful (if disappointing!) feedback for you.
By copying the Summary from the board into their math journals, students may be able to solidify their thinking; they also create a record of their learning from each lesson that can provide a valuable future reference. After students have written the Summary in their notes, they can immediately go on to write their Reflections, which highlight their own personal learning from the lesson. The Journals tab provides further information on nurturing student Reflections and using them to introduce each TTP lesson.