RE: Studying Mathematics Thread
March 8, 2018 at 1:54 am
(This post was last modified: March 8, 2018 at 2:03 am by Kernel Sohcahtoa.)
IMO, when studying math, it is extremely important to not get discouraged when encountering a challenging concept, proof, definition, theorem, etc. A couple of days ago, I encountered a proof dealing with the power series and the concept of the radius of convergence. When I got to sentence 3, I was unsure how the author drew a particular conclusion, and as a result, I got stuck and ultimately decided to take a break from it. However, today I was thinking more clearly and realized that if I took the negation of a particular definition (in this case, the definition of a bounded sequence), then the sentence made much more sense. Thus, difficult concepts are a part of mathematics: sometimes I'm fortunate enough to be able to wrap my head around them; however, there are times when it is necessary to let some difficult concepts go and make a note of them, so that you don't waste time staring into space when you could be learning new/important material that comes more easily to you (sometimes covering new material may provide more insight about the challenging concept). Hence, you can always come back to the difficult stuff later or ask someone who knows more about math for help, especially if the concepts are fundamental to successfully understanding the topic one is studying; there's no shame in asking for help, as it is part of the learning process.
Also, IMO, when studying higher level mathematics, it is crucial that one has a solid understanding of mathematical proofs. IMO, when taking a course in mathematical proofs for the first time, it is important that students do the following: give themselves ample time to digest the various proof-writing concepts, especially the basic proof-writing techniques; be willing to practice and work various proof exercises, so that they will gain first-hand experience navigating the proof-writing process (this often involves at least a draft of one's thought processes which is eventually harmonized into a coherent final proof); learn how to read condensed proofs via spotting key words that give away basic proof-writing techniques and being able to fill in the missing details via considering how one would get from A to B via theorems, definitions, previous knowledge, etc.; when writing final/condensed proofs, be sure to keep any clarifying notes (especially the analysis of proof, which IMO, is where the proof is actually explained and understood) that explain the fine details of a proof, because these notes will save you the time of having to figure things out again, especially if you haven't reviewed the proof for some time (days, weeks, months) and are somewhat rusty with the concepts, definitions, theorems, etc., that are involved in the proof. IMO, reading the condensed proof of another author makes one appreciate the value of including an analysis of proof with the final condensed proof.
Also, IMO, when studying higher level mathematics, it is crucial that one has a solid understanding of mathematical proofs. IMO, when taking a course in mathematical proofs for the first time, it is important that students do the following: give themselves ample time to digest the various proof-writing concepts, especially the basic proof-writing techniques; be willing to practice and work various proof exercises, so that they will gain first-hand experience navigating the proof-writing process (this often involves at least a draft of one's thought processes which is eventually harmonized into a coherent final proof); learn how to read condensed proofs via spotting key words that give away basic proof-writing techniques and being able to fill in the missing details via considering how one would get from A to B via theorems, definitions, previous knowledge, etc.; when writing final/condensed proofs, be sure to keep any clarifying notes (especially the analysis of proof, which IMO, is where the proof is actually explained and understood) that explain the fine details of a proof, because these notes will save you the time of having to figure things out again, especially if you haven't reviewed the proof for some time (days, weeks, months) and are somewhat rusty with the concepts, definitions, theorems, etc., that are involved in the proof. IMO, reading the condensed proof of another author makes one appreciate the value of including an analysis of proof with the final condensed proof.