It is a commonly held opinion that educational institutions could do more to improve the pedagogy of mathematics. This is especially true in school, when students are first exposed to new subjects. Poor exposition can turn students away from mathematics for a lifetime. Only the highly motivated ones continue to engage with the subject. This is very unfortunate because mathematics is a beautiful subject and it is filled with wonder. It also teaches rigour in reasoning, clarity of thought and the discipline of constructing arguments from first principles to obtain intricate and often beautiful results. What is perhaps less known is that pedagogy is a problem even for graduate-level mathematics students and professional mathematicians. The proofs in many graduate-level mathematics textbooks are, in my humble opinion, not really proofs at all. They are closer to high-level outlines of proofs. The authors simply do not show their work. The student then has to put in an extraordinary amount…
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