What Is Bansho?

Source: Facilitator’s Handbook: A Guide to Effective Instruction in Mathematics, Kindergarten to Grade 6 – Teaching and Learning Through Problem Solving, produced by The Literacy and Numeracy Secretariat Professional Learning Series

To make public the mathematical thinking students use to solve problems, teachers need a way of organizing the work so everyone can see the range of student thinking. Such organization lets students see their own thinking in the context of the similar thinking of others. Students are expected to follow and be able to describe all the work represented – not just their own. They listen to the explanation of other students and restate, in their own words, the strategies the other students used.

Mathematical ways of talking are modelled and practised – resulting in the creation of a safe math-talk community. All students have a chance to learn more about the math used in developing solutions and to clarify their understanding of the concepts and procedures.

Through the careful management of discourse, the mathematics is made explicit. Japanese educators call this teaching strategy bansho. We will call this process of organizing, displaying, annotating and discussing solutions bansho as well. Bansho engages the teacher in examining student work, organizing it and displaying it to make explicit the goals of the lesson task.

For example, if the goal is to show as many polygons as possible with an area of four square units, the teacher creates an arrangement of student work along the following lines:

• On the left of your display wall, post squares with areas of four square units and bases parallel to the bottom of the page.
• To the right of that, post rectangles with bases parallel to the bottom of the page. Solutions that show similar mathematical thinking are arranged in vertical rows that together look like a concrete bar graph.
• Next, show irregularly shaped figures composed of squares and rectangles.
• In the fourth column, post parallelograms with areas of four square units.
• Next, post triangles with areas of four square units.
• Then, on the right of the display, show other polygons with areas of four square units.

The bansho process uses a visual display of all student solutions, organized from least to most mathematically rich. This is a process of assessment forlearning and lets students and teachers see the full range of mathematical thinking used to solve the problem. Students have the opportunity to see and hear many approaches, and they are able to consider strategies that connect with the next step in their conceptual understanding of the mathematics.

Bansho is not about the assessment of learning, so there should be no attempt to classify solutions as Level 1, Level 2, Level 3, or Level 4.