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In this article, we consider the phenomenon of improvisation by small groups of middle years students while engaged in rich mathematical tasks in a classroom setting. Working from the premise that improvisation comprises a spectrum of behaviour, we propose that there is a range of improvisational behaviours that may be observed as the students work together. We discuss four levels along the spectrum – interpretation, embellishment, variation, pure improvisation – and offer vignettes from our research to illustrate each of their characteristics. We argue that considering improvisation as a spectral behaviour is a valuable way to view students’ mathematical problem solving as it highlights how students draw on their own experiences and understandings while problem-solving, and how they vary in how far from the “script” they may venture in their discussions. Our results also show the importance of giving students time and space to “stay with” a mathematical task.

A spectrum of possibilities: levels of improvisational behaviour in middle school mathematics