![]() (See also “ There’s more to maths than grades and exams and methods“.) But the transition from the second to the third is equally important, and should not be forgotten. The transition from the first stage to the second is well known to be rather traumatic, with the dreaded “proof-type questions” being the bane of many a maths undergraduate. This stage usually occupies the late graduate years and beyond. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory.This stage usually occupies the later undergraduate and early graduate years. The emphasis is now primarily on theory and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. re-doing calculus by using epsilons and deltas all over the place). The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g.This stage generally lasts until the early undergraduate years. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. ![]()
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