Mathematics Department Stanford UniversitySurvival Tips for Math 51H1. Overall approach to the class:Most days in the Math 51H class you will be presented with new material,usually at a pace which makes it difficult to fully appreciate the subtleties andsignificance of the various definitions and results. For this reason it is essentialthat you take accurate notes of what has been covered, so that you can makea detailed review later (as discussed in point 2 below). In an honors class it isusually not the case that the lecturer simply “follows the text,” so the takingof accurate notes for later review becomes doubly important.Try to keep in mind at all times that the various results and concepts shouldall fit together as an integrated whole which ultimately makes sense and whichseems down-to-earth and logical.If your level of assimilation of the material reaches a point where this is thecase, then you will find that very little “memorization” and “rote learning” areinvolved; such a state of mathematical maturity can only be attained throughconsistent effort and constant review of the material. Helping you to achievesuch a state is one of the main aims of Math 51H. Try not to be discouragedtoo quickly if you do not feel you are achieving this immediately—many stu-dents with outstanding mathematical potential have trouble in the first stages.This may be partly explained by the fact that many have an experience in highschool where rote learning and routine application of formulae are almost ex-clusively what is emphasized; in many cases students, even those with excellentmathematical potential, have never developed the skill of being able to sit downand think about a problem which initially seems to be quite new to them andis not simply a rehash of a “problem” which you have seen a number of timesbefore in just a slightly different guise.Keep in mind also that in Math 51H all results will be proved—virtuallynothing is to be taken on faith except for the basic axioms of set theory andthe real numbers. So you should never be in a situation of using results whichyou do not know how to prove rigorously.2. Reviewing Lecture Material:After each lecture (the same day or early the next morning) you should askyourself at least the following questions: What were the main 4 or 5 pointscovered in today’s lecture? Can I at least state those results? To what extentdo I already understand these results? For example suppose you look at thedefinition of “subspace of Rn” (as on p.5 of the text) and “span” as on p.6 of thetext. The natural tendency is simply to read those definitions and to say “yesI certainly understand that.” But you have to accept that “understanding”applies at various levels; for instance can you easily and directly apply thedefinition of subspace and span to prove that if you are given vectors v1, . . . , vk∈Rnthen the span of v1, . . . , vkis a subspace of Rn. If you can’t do that more orless immediately then it is fair to say that you certainly do not “understand”the concepts of “subspace” and “span,” at least not at a level where thoseconcepts become useful. The same principle applies to just about every topicwe cover in 51H—you really have to develop the ability to seriously ask yourselfthe question “do I understand that?”1As part of any review of basic material (either one or several lectures’ worthof material) you should examine special or “extreme” cases of definitions andtheorems, as well as standard examples. Also try to see if there are unusualor non-obvious cases of the result which have important consequences or whichtie it to results or concepts covered previously. Remember, a primary overallaim is to be able to view the totality of the material as an integrated whole,with many parts which fit together in a clear and elegant way, so pay particularattention to how the different parts of the course fit together. Try to constructsome sort of chart which graphically represents how the different results, andtheir proofs, are connected, and keep this under constant review. You’ll besurprised how useful, and at the same time complex, this can be.3. Homework:The weekly homework is of paramount importance. It is the main way inwhich you come to terms with the material presented in the the lectures, andstart to really assimilate it, “understand” it in at least the sense discussed inpoint 2 above, and apply it to problems which are at least one level removedfrom direct “rote” application. You’ll notice that only 15% of the total score forthe course is allocated for homework; that is done not because the homeworkis of lesser importance than the tests (on the contrary it is the most impor-tant learning tool provided to you), but rather to encourage you to tackle thehomework with more concern about using it to build your understanding andless worry about getting a good score. Clearly if you get too much help, giveup and ask for help before you have even seriously thought about what couldbe involved in the solution of the problem at hand, then you might get a goodscore on the homework but have gained very little in real understanding of thematerial. Of course if you are completely stuck on one of the problems andeven after thinking seriously about it you are unable to get a start, then youneed to ask for help. But even in those circumstances you should not make themistake of asking the TA “how do I do this problem?” Rather you should askfor a hint as to what is involved in getting the problem started, and then goon from there. If you later find you are still stuck then of course you can againask for help; but do keep in mind that real mathematical ability is generallyonly developed by struggling with the various problems set in the course, notby imitating someone else’s solution to those problems.Each week you should review the homework (and all test problems) afterthey have been graded and returned to you. With the aid of the solution set,actively work out where your problems were and what is still not clear. Keep anongoing list of things that are causing difficulties, and review it regularly. Tryto analyze this list to see if you can identify the basic difficulties from which allother problems/confusion arise. This practice (keeping track of your difficultiesand analyzing them as thoroughly as possible) is a very important part of themathematical learning process.4. Reviewing for Tests and
View Full Document