Learning from history for teaching in the future

Reflecting on the learning process of topologist Stephen Smale, certain properties and characteristics of that process are selected and analysed: the question and anticipation of future mathematics, the generalising potential thereof, and the shift from after-image or `model of ...' to pre-image or `model for ...'. These features are explored for the benefit of the design of learning-teaching sequences in mathematics education, and are illustrated in a variety of examples.