Using students’ misconceptions and errors as building blocks

Learners make mistakes for many reasons, including insufficient time or care. But errors also arise from consistent, alternative interpretations of mathematical ideas that represent the learner’s attempts to create meaning. Rather than dismiss such ideas as “wrong thinking”, effective teachers view them as a natural and often necessary stage in a learner’s conceptual development. For example, young children often transfer the belief that dividing something always makes it smaller to their initial attempts to understand decimal fractions. Effective teachers take such misconceptions and use them as building blocks for developing deeper understandings. 

There are many ways in which teachers can provide opportunities for students to learn from their errors. One is to organize discussion that focuses student attention on difficulties that have surfaced. Another is to ask students to share their interpretations or solution strategies so that they can compare and re-evaluate their thinking. Yet another is to pose questions that create tensions that need to be resolved. For example, confronted with the division misconception just referred to, a teacher could ask students to investigate the difference between 10 :– 2, 2 :– 10, and 10 :– 0.2 using diagrams, pictures, or number stories.

By providing appropriate challenge, effective teachers signal their high but realistic expectations. This means building on students’ existing thinking and, more often than not, modifying tasks to provide alternative pathways to understanding. For low-achieving students, teachers find ways to reduce the complexity of tasks without falling back on repetition and busywork and without compromising the mathematical integrity of the activity. Modifications include using prompts, reducing the number of steps or variables, simplifying how results are to be represented, reducing the amount of written recording, and using extra thinking tools. Similarly, by putting obstacles in the way of solutions, removing some information, requiring the use of particular representations, or asking for generalizations, teachers can increase the challenge for academically

advanced students.