Learn by Doing

Comments on the importance of exercises, from the Preface to THE RISING SEA: Foundations of Algebraic Geometry by Ravi Vakil:

Finally, if you attempt to read this without working through a significant number of exercises (see §0.0.1), I will come to your house and pummel you with [Gr-EGA] until you beg for mercy. It is important to not just have a vague sense of what is true, but to be able to actually get your hands dirty. To quote Mark Kisin: "You can wave your hands all you want, but it still won't make you fly."

and:

0.0.1. The importance of exercises. This book has a lot of exercises. I have found that unless I have some problems I can think through, ideas don't get fixed in my mind. Some exercises are trivial – some experts find this offensive, but I find this desirable. A very few necessary ones may be hard, but the reader should have been given the background to deal with them – they are not just an excuse to push hard material out of the text. The exercises are interspersed with the exposition, not left to the end. Most have been extensively field-tested. The point of view here is one I explored with Kedlaya and Poonen in [KPV], a book that was ostensibly about problems, but secretly a case for how one should learn and do and think about mathematics. Most people learn by doing, rather than just passively reading. Judiciously chosen problems can be the best way of guiding the learner toward enlightenment.

(cf Gateways to Mathematics (2004-05-20), Expect Nothing (2012-02-20), Enso (2012-02-29), Potentially Useful Ideas (2018-06-26), Small Eyes Notation (2020-02-22), Think Cleanly (2022-03-01), ...) - ^z - 2022-03-16