# Enabling Teachers to be Real Teachers: Necessary Levels of Awareness and Structure of Attention

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## Abstract

Awareness is a complex concept comprising both conscious and unconscious powers and sensitivities which enable people to act freshly and creatively in the moment. In the case of mathematics teachers, and teachers of those becoming mathematics teachers, it is possible to lead students mechanically through a sequence of ritualised tasks by means of trained and habitualised reactions. But the result is that even though student attention is indeed directed, their behaviour trained, and their awareness educated to some extent, the students have not been taught in the fullest sense of that word. I argue that to be a real teacher involves the refinement and development of a complex of awarenesses on three levels, and that this is manifested in alterations to the structure of attention.

The problematic nature of what students are attending to when a teacher is teaching them, led to the conjecture that each technical term in mathematics and in mathematics education signals a shift in the structure of attention of people using that term, and that a corresponding shift is required for students to appreciate that term. Investigations of attention led to the development of Gattegno's very general but rather subtle notion of awareness into a three-layer structure which applies both to mathematics and to teaching, and so demonstrates why becoming a teacher is such a complex matter.

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