Mental mathematics with mathematical objects other than numbers: The case of operation on functions

• Students generated three sorts of strategies: algebraic/parametric, graphical/geometric, and numerical/graphical.
• For these strategies, students generated specific cues from the tasks to solve the problems.
• Students were fluent in navigating between algebraic, numerical and graphical aspects of functions.
• Data analysis illustrates the potential of mental mathematics for studying functions.
• Strategies conceived through problem-posing allows to be attentive to the creative and adaptive nature of students’ strategies.

This article reports on a study, part of a larger research program, focused on issues of mental mathematics with mathematical objects other than numbers. The study is about operations on functions in a Cartesian graph environment with two groups of 30 high school students (grade-11). Grounded in aspects of the enactivist theory of cognition, the research aims at characterizing students’ emerging mathematical activity by analyzing the strategies they put forth in this mental mathematics environment. It illustrates how students pose their own problems when solving tasks and thus made emerge tailored strategies for solving the very problems that they posed. The data analysis highlights three specific approaches that students engaged with/in for solving tasks: algebraic/parametric, graphical/geometric, and numerical/graphical. This characterization offers understandings of how students have engaged in and succeeded in solving the various tasks, leading to a discussion of the generation of strategies for solving these tasks. Triggered by the nature of students’ engagements, the article closes with future research avenues and issues to investigate in mental mathematics.