Scaffolding students' thinking in mathematical investigations

Mathematical investigations are loosely-defined, engaging problem-solving tasks that allow students to ask their own questions, explore their own interests and set their own goals. The value of investigations for students lies in their complexity. Scaffolding plays an important role in supporting students' high-level engagement by encouraging divergent and creative thinking. Scaffolding is "a process that enables a child or novice to solve a problem, carry out a task or achieve a goal which would be beyond his (or her) unassisted efforts." Scaffolding provides the opportunity for students to develop their independence, sense-making and self-confidence whilst working mathematically. However it is incorrect to assume that all conversations between teachers and students amount to scaffolding because not all result in high-level thinking and reasoning in students. Understanding what is ineffective and why, is one way to improve teachers' pedagogical practice. This article describes some of the issues that teachers might encounter when scaffolding students' thinking during mathematical investigations. It describes four episodes in which a teacher's interactions with students failed to support their mathematical thinking and explores the reasons why the scaffolding was ineffective. As a background to these episodes, the authors first provide an overview of the mathematical investigation. The article concludes with some recommendations for scaffolding during investigations.