CONTENTS Introduction Mathematics Learning, Maths & AI Penrose, Maths & AI Mathematics Understander Contextual Memory System Heuristic Applier/Learner Conclusion

This presentation is in eight parts. The introduction will give a brief history of this research and explain the methodology and motivation. Second, we look at mathematics, and in particular, the nature of mathematical experience. Third, we consider the work of AI in getting machines to do mathematics. Fourth, we look at Penrose's critique of AI, and in particular his view that a program cannot understand mathematics. Fifth, we give an introduction to the Mathematics Understander, MU, and how it in some sense does understand mathematics. Sixth, we look at the underlying Cognitive Architecure of MU, the Contextual Memory System, or CMS. Seventh,we consider Morgan's work on the Heuristic Applier/lLearner, HAL, which models the learning from worked examples.In the conclusion some limitations and possible applications will be mentioned.