To teach mathematics we need to be aware of the following: the interconnectedness of mathematical concepts; the role of proof in explaining mathematical phenomena and illuminating this interconnectedness; the necessity of doing rather than observing mathematics; the necessity of using visualisation to grasp concepts and theories that get further removed from experiential knowledge; and finally but not least, the power of self discovery from the reverse flow: examples ->identification of pattern or structure -> proposition or theorem -> proof.
Many of the slide presentations here were produced when I worked in mathematics outreach at the University of Sheffield.
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