Signs of understanding
Ability to reduce everything to elementary school mathematics signifies true understanding.
Ability to solve textbook problems is another important sign.
Reconstruction ability
One must be able to reconstruct important knowledge acquired so far, relevant to the subject being studied, from scratch. This must be done with minimal external reference, and with the power of one’s memory and abilities. This is helpful in preparing the mind for studies and in solidifying knowledge. For thoroughness of topics covered, a reference sheet may be used.
Being able to reproduce complex proofs without notes is an important skill. Eg: Pradeep presenting many proofs of statistical learning theory, VCD, packing number etc.. PratIka presenting ICML papers.
Checking understanding
Challenges and exercises
A common Atma-vaÑchanaM is to fool oneself into thinking that one’s skills and knowledge in a subject is sufficient for research. Hence, challenges of others, exercises and self-testing is extremely important. Experience shows these.
Thence you recognize holes in your knowledge and skills. The more holes you find in your knowledge and skills, the better it is.
Often work out small details while reading.
ALWAYS attempt for a reasonable time to work out the solution strategy yourself before looking up the proof or the solution to the problem. This is very important training in problem-solving in that area. It is also important in the development of intuition.
Tackling, not solving
The point is to tackle, not necessarily solve. This is very important training in problem-solving in that area.
Review attempt
Even if you read the solution, consider what you *could* have done to solve the problem.
Questioning and inquiry
NEVER be lazy about asking questions. Ask yourself as many questions (especially about the object of study) as possible.
Challenges
Other people ask questions to satisfy their own curiosity, and to desmonstrate a flaw in your knowledge and skills. Both of these are good.
Exams, homeworks, exercises are others.
Talk to people. Clearly state your ideas, so that they may criticise them.
Thence, challenges should be encouraged and embraced.
Exam strategy
Read up homework solutions and results; write them in cheat-sheet. Solve many problems as warmup. Beware the calculation mistakes. Use separate sheets for separate questions. Don’t over hydrate yourself. Sleep well. Solve easy, short questions first; solve many questions parallelly.