# The Problem of Pedagogy in Advanced Mathematics DevFeed: [The Problem of Pedagogy in Advanced Mathematics]() Original publisher: [Read original article]() Published: 2026-05-11T00:00:00Z Content type: opinion Language: en Sources: [Susam Pal]() Topics: [Mathematics](), [Math and Logic]() Tags: [advanced](), [mathematics](), [opinion](), [students](), [theory]() ## AI overview The article argues that pedagogy remains a serious problem in advanced mathematics. It focuses on graduate-level textbooks whose proofs are often presented as high-level outlines, leaving students and even professional mathematicians to reconstruct omitted intermediate steps. It advocates explanations that are correct, complete, and accessible to reasonably motivated students. ## Source excerpt It is a commonly held opinion that educational institutions could do more to improve the pedagogy of mathematics. This is especially applicable to primary and secondary schools, where students are first exposed to mathematics as a formal subject, along with other 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 of effort to understand and justify each line. Sometimes a 10-line argument in a textbook might expand into a 10-page proof if the student really wants to convince themselves that the argument works. I am not a mathematician, but out of personal interest, I have worked with professional mathematicians in the past to help refine notes that explain certain intermediate steps in textbooks (for example, Galois Theory by Stewart, in a specific case). I was surprised to find that it was not just me who found the intermediate steps of certain proofs obscure. Even professional mathematicians who had studied the subject for much of their lives found them obscure. It took us two days of working together to untangle a complicated argument and present it in a way that satisfied three properties: correctness, completeness and accessibility to a reasonably motivated student. There is a reason why jokes like 'proof by obv