Tips for Success in Undergraduate Math Courses (2002)
web.stanford.edu
web.stanford.edu
1. Read and get a basic understanding of the material before lecture.
2. Attend every lecture and take notes very few if any notes. Most of what you need is in the book and math is about understanding. You cannot grok and write at the same time, and it's better to try to grok while an expert is explaining it to you.
3. Do a shitload of problems / proofs, depending on the class. Be honest with yourself when you don't fully understand something and stick with it until you do.
Math is different from other subjects, and you need to treat it that way.
Have you tried air writing? Or just "doodling" some notes when in a lecture? It worked for me.
2. I have seen a lot of very good students in my math bachelor, and none of them took notes. Furthermore, I've seen students taking a lot of notes, even some writing theirs in LaTeX and sharing it to the rest of the class to later flunk the year.
1. I wish I would have done that. But depending on your school, this might be just impossible.
I think the key problem for most people is speed: if you need to spend so much effort just on the note taking that your brain can't process/understand what you're writing, then obviously taking notes is a bad idea. I don't think this is accounted for in the aforementioned studies.
I remember seeing a study that showed that people who take notes on paper do better vs people who take notes on a laptop. People who didn't take notes(or take very few) weren't mentioned.
A good review of the literature (from 1989) is found here:
My highschool calc teacher gave us the same advice and I carried it through uni.
I folliwed this same advice and went from being a B- to A avg student in just two quarters. If you can, review the material the night before so you can sleep on it and process it unconsciously.
In my case, another essential key was to attend both the professor's and the TA's office hours. Most of the time it was to discuss the material outside of the scope of the class. This ensured that I truly understood the subject matter and wouldn't forget it after the final.
However, all of us who did well did 3.
The reason for the discrepancy was that the lectures were quite different. Math and physics lectures tended to be focused on problems, and how to do them. The professor would walk through example problems, explaining as they went. I diligently wrote all of that down, and would refer to it when I did my homework. I found writing down the examples as we went a good way to make sure I was actually understanding each step.
I never took notes in computer science classes because the lectures were more conceptual - beyond the first semester, the professor was never walking us through how to code. They were explaining new concepts, not showing us techniques I would have to reproduce. I didn't see much reason in taking notes about concepts, I would rather think about those concepts during the lecture. I found taking notes a distraction. The only exception here was some algorithms lectures, where the professor walked us through how to prove particular things.
Basically, if I was being shown techniques, I wrote down the techniques. If I was being told new concepts, I just listened.
I'd add a few more things to your list. A lot of students hit a wall when they run into calculus. Partly this is the same old problem of mathematics being taught poorly in general, and there's also an annoying element of calculus classes being difficult on purpose (being as it is a topic which can involve an arbitrary amount of memorization and busywork) for stupid reasons. But additionally calculus relies heavily in having a firm grasp of advanced algebra, trig, and pre-calc, all of which equally suffer from the general problems of our educational system. Reviewing those subjects, probing for weaknesses in understanding, and shoring up any weaknesses can go a long way to making your life easier as you tackle more advanced subjects.
Also, as a general rule and a corollary to your third point: don't treat college as a race to get a valuable piece of paper, one that requires merely toil and busywork to acquire. Treat it as an opportunity to acquire new skills, understanding, and knowledge. The piece of paper might improve your earnings incrementally but the knowledge will not only change the way you look at the world it will unlock a great deal of potential in terms of what you can do (in your career and elsewhere).
P.S. since we're stating bona fides, here's mine: I got my math degree mostly by accident. It was something I was good at and a reasonable default choice, and I hardly had time to think about it before I graduated at 20. I don't "use" my math degree in my career typically but it's definitely made me a better developer and critical thinker.
Is there something like that in math? I'm thinking of getting from A to B in a proof or solution by introducing things like definitions, theorems, common algebraic manipulations, and so forth. If you can't just see those things in your head, and write them down, then you won't have a reserve of mental bandwidth to think about the higher level structure of the problem that you're working on, and you'll reach a level of complexity and abstraction where the cumulative effect of small mistakes prevents you from ever getting to B.
Is it poorly taught? That's certainly a possible problem. One thing I noticed when I taught math, was that the kids were never given a higher level explanation of what they were doing. Memorization and busywork are necessary but not sufficient. My students did not understand what "show your work" means. What it means is that math is a mixture of theory and performance art, like music. Math has a weird social role too... by the time they are in high school, most kids know that they are learning math with no expectation that they will ever use it. Parents treat it as some sort of obedience training.
The big difference between math and music is that the latter is a real-time applied form of the former, with "real-time" being the key. You can waste as much time as you want thinking over things and research solutions to the given math problem; the solution you arrive at after a week is as valid as someone else's solution written down in minutes.
I even had one class where the final was essentially a recitation of proofs we did all semester in lecture (notes, no book) in which you had 90 minutes to do 6 proofs. Things like Green's Theorem, proof of irrationality of pi (Niven's proof), proof of the Chain Rule, F.T. of Arithmetic, a bunch of other stuff.
Probably mostly in how bad are the materials used. Have you ever tried learning a completely new mathematical "thing" using just books, exercises, on-line courses and so on, but without having an "expert explain it to you"?
It's doable, but - being an auto-didact in many other areas too, so I can compare - it's so much harder than most other things you'd like to learn that it's not funny. I don't know the reason, but I doubt it's related to how "math is different" or "math is hard" because it's not: if you have enough intelligence to play with legos you can also do math.
1. attend all the lectures
2. attend all the recitation sessions
3. do 100% of the homework, and on time
4. make sure to understand every homework question and how to solve it.
5. write legible notes
This was good enough to get a baseline B. To get an A, you had to put in much more work.
I know this stuff seems obvious. But it took me a while to figure out I couldn't just coast and wing it like in high school, and many other students had the same experience.
Start from the basics. This makes such a tremendous difference I think, because we have these pre-conceived ideas about math, numbers, calculus, etc. (due to usually shitty high school education) that must be shattered and rebuilt from the ground up.
Here are the example notes that my undergrad (in fact, first college math class) professor used:
Lecture 2: Numbers https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090...
Lecture 3: More Numbers and Rationals https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090...
Lecture 4: Irrational Numbers https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090...
As you see there is a very clear and natural progression from fundamental assumptions of mathematics to common day objects, with justifications.
From this you introduce the notions of infinity, again with historical developments (math didn't just come about in the last 50 years!)
Lecture 5: e https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014091...
And so on; By rebuilding your knowledge in this deliberate way it becomes easier to work with more difficult topics later on.
This.. So much this for me. It took a long time for me to realise that I couldn't just wing it, as I did for school, and once I got that I went from failing to near top of the year.
I think it's important to get the "mechanical" work, e.g., step by step derivations, so it can be done quickly and accurately. For me, this is also resulted in committing the definitions, axioms, theorems, etc., to memory. During exams, being able to speed through this work gave me the spare time I needed to sit back and think about each problem, especially the one "surprise" problem in the set.
Oddly enough, I kind of treated the humanities courses in a similar way. I signed up for courses that were known to be graded based on mostly written work -- essays, papers, etc. Because all of these courses involved the same "mechanical" work, e.g., writing a paragraph supporting a concept, I also got very quick at it, which compounded itself in terms of getting assignments done quickly and writing fairly lengthy, coherent answers, in blue-book exams.
Note: It also helps a lot to actually enjoy the stuff. Attending the lectures helped me later in life, as I observed the teaching styles -- good and bad -- of my teachers. That experience has helped me with presentations and other kinds of interactions in my regular job.
* Watch video's on Khan Academy/youtube * Use Wolfram Alpha to confirm/deny yr solutions * Get a tutor who can explictly show you how they approach solving any arbitrary problem you bring them * Accumulate a list of math tricks that you can use and how to use them (example: how to long divide polynomials) * Print out old exams and pretend to take them ~3 days before final * Rewrite your lecture with each example problem on it's own page(s) * Create an index on your lecture notes so that you can quickly identify what type of problems and topics were covered in each lecture. * Identify yr strenghts - we can't all be good at everything - knowing yr strenghts will give you a foundation of confidence to build upon * Queue cards for memorizing all those trig identities * Expect to spend a lot of time - this is not always possible
* bullet
* pointsWatch video's on Khan Academy/youtube
Wolfram Alpha
Use Wolfram Alpha to confirm/deny yr solutions
Tutor
Get a tutor who can explictly show you how they approach solving any arbitrary problem you bring them
Tricks
Accumulate a list of math tricks that you can use and how to use them (example: how to long divide polynomials)
...
1. Skip lectures because the professors barely speak english and can't teach
2. Barely rush through the homework before its due, it doesn't matter if you understand it as it has no relation to the rest of the class
3. Find last semester's exam and study that for a morning, as the questions are pretty much the same.
Its a shame that the higher level courses that should be more interesting and useful end up being the most painful and disregarded.
https://en.wikipedia.org/wiki/Anki_(software)
Source: I actually did this and it improved my grade 55% in Calc I.
And since he comes up regularly on HN: Cal Newports Stealth Studying[1] is really efficient in an Anki deck
[1] http://calnewport.com/blog/2007/10/03/the-art-of-stealth-stu...
But generally what I would do is use them to memorize anything I needed to know while I was solving a problem but couldn't remember.
The important thing is to know how to write mathematical expressions in LaTeX syntax, because this lets you make non-crap cards. You might do something like:
\frac{d}{dx} \cos x = ?
On the front and then
- \sin x
On the back.
The important point is that between the front and the back you actively recall the information you're trying to remember. Undergrad mathematics can be thought of as a context free grammar with a set of production rules, memorizing those production rules is a major pain in the ass and this helps.
(Maybe other people don't have trouble with this, I nearly failed every math class in high school and am decidedly not a math prodigy. Your mileage may vary.)
Avoid cards where it takes a long time to state the back side of the card. Even if you know the answers, you'll get bogged down and it'll take too long to review. Answers should be short and easy.
Better to create 5 cards where you replace 5 parts of a formula with blanks and say "fill in the blanks", than one card that requires remembering 5 things.
This list reads like my strategy to do well in a math class without learning the material (literally; this method got me through numerical methods and advanced calculus, both classes that I had no interest in and took only to satisfy the degree requirements).
My thoughts on the particular points:
>Keep a list of THINGS TO MEMORIZE.
If this list is anything but empty you should feel bad. "Memorizing" something in math is a way to act (and test) like you understand it without understanding it. The one exception is in computationally heavy classes, where you memorize the completed solution of common forms. However, if your goal is to understand the material, you should be able to derive everything you are memorizing without thinking. The memorization is only to save you time on the exam by skipping the computation. Having said that, as I mentioned above, if you do not understand something but still want a good grade; memorize it. The professor will never see the difference.
>Watch for example problems.
Because these will be the problems on the test.
>Know how to do every homework problem assigned!
Yes. Also, know how the book/teacher wants you to do the problem. It is often times hard to write a problem that can only be solved one way, so it is sometimes possible to avoid using a method you don't like or understand.
> Start the homework at least a few days before it is due.
Yes. Also, if possible, consider homework due at the last office hours before it is actually due. Otherwise, you can't go to office hours for help.
>Keep a running list of HARD PROBLEMS.
Good advise if you are going the memorization route; otherwise, this is just memorizing the solution to a particular form of questions. Also, in my experience, if you are going for the understanding route, this list just does not stay relevant long enough to justify keeping it.
>When you get your homework back: Look over the things you got wrong.
Always good advice. Having said that, if you got a problem wrong (for reasons other than computational/algebraic mistakes), the bigger issue is that you thought you got it right. This means that you not only did not know how to solve the problem, but that you have misunderstood some concept that you need to learn.
>Find a quiet place, set a timer for the amount of time you'll have in the exam, and take the practice test. Don't look at the practice test before you do this.
Good test prep advice in general.
>If a problem is hard, skip it and come back later.
I triage problems much more aggressively. If a problem looks time consuming I skip it. If a problem looks like it involves thinking, I do enough work to verify that it actually involves thinking then skip it.
>Do a quick check of each problem to be sure your solution is reasonable. E.g. if the problem asks for a distance, is your solution positive?
Do this check after you finished the test. If you got something wrong, but didn't finish the test, then knowing you got it wrong doesn't help; you still didn't have a chance to correct it. Also, you are more likely to notice an incorrect answer after spending time away from the problem.
Having said that, sometimes you answer "feels" wrong as you are solving it. If this happens and you see where you went wrong, correct it. Otherwise, complete the incorrect solution (if feasible), and mark it. This gives you the chance for partial credit; and sometimes your feeling is just wrong, and the answer is weird.
>Write SOMETHING on every problem. The grader really wants to be able to give you some partial credit.
If you have time. If you really have no idea how to approach a problem, then your time would be better spent doing better on the rest of the exam instead of producing a plausible looking solution. Mark these problems, and, if you have time at the end, come back and look at them again.
Having said that, this is still good advice. If you think you know how to solve part of the problem; do it. If the part you know how to do requires you having computed something that you do not know how to compute, then clearly write "let a = thing I can't compute". Graders don't have time to look closely at your answer; make it easy for them to give you partial credit.
If you are answering a proof based question, and cannot figure out how to prove a particular fact that you need for your proof, consider writing "it is clear that". You will be amazed how often this works.
>When you've tried everything, go back to the problems worth the most points first.
Triage. Go back to the problems that you think you got wrong and can improve first.
>Given time, double check your algebra carefully!
If possible, verify your answers using a different method. For example, if you are asked to find the integral, verify your answer by taking its derivative. You are less likely to make the same mistake, and for many problems it is easier to verify an answer is correct then to find the answer
>After the Exam
Write down what you can remember about the problems you could not solve (including ones where you put down something that might be correct, but were not sure about). Solve these problems (using book/notes/TAs/etc if needed).
When you get the exam back, compare it to the list of problems you knew you didn't get. You don't care about these problems at this point; you already knew you missed them and worked through them. You learn nothing new by the grader telling you that you missed these.
The problems that you did not know you got wrong are where you should focus your attention. If you just forgot about them, then work through them like you did the ones you already knew about. If it was a computational/algebraic error, don't worry about it (but do do more practice if these errors cost you a significant amount of points). Pay special attention to problems that you thought you got right but didn't. These highlight the areas where you have a misunderstanding of the material.
Final remarks:
HARD PROBLEMS list:
As you might have noticed, I don't like this. What I do like is a concepts list. When you go to study, read through the list and make sure you understand all the concepts. It should be small enough that there is no point in creating a separate list for hard concepts; and what you consider to be an easy/hard concept will change as you get more practice with some things, and never see other things between the first month of class and the final.
Studying for the test:
As you probably noticed, my opinion is that many of these points are techniques to study for the test. This is a valid thing to do in school (after all, you are graded on the test, not understanding), but be aware when this is what you are doing. If you plan on taking another class that builds on this one, then this method will come back to bite you then.
In class notes:
Do not make them. The only reason you should be writing during a math lecture is to use the paper as a scratchpad to think about what is being said. Anything else is a distraction from the lesson. All the material should be in the book. If it isn't, you can ask the professor for a copy of his notes. If you want your own notes (which can be a good idea), write them after class. If you do take notes, keep them in a separate notebook. Interspersing them with problems and scratchpads just makes it more difficult to use them.
LaTex:
If you are taking math as a gen-ed requirement, ignore this. If you are going into a math heavy field, learn LaTex as early as possible. Write you notes in LaTex. Do your proof based homework in LaTex. This will pay off down the road, when you A) know LaTex and B) have a digital record of you previous math classes. Plus, turning in proofs written in LaTex make them feel far more correct, so you might get graded easier.
Study groups:
Form a study group of people with similar skills as you. If you try studying with people far better than you, then you will end up just having them give you the answer; in which case you are better off talking to a TA (who is probably better at explaining things). In a study group, you want to be part of finding the answer. Along the same note, if part of your group just gets the answer (and you are in the part that does not), start by talking with the other people who do not know the answer. That way you can figure it out together, instead of just being told what the answer is.
To be fair to the writer, it'd be a much longer and more difficult article if it was called "Tips for understanding concepts in math." I don't even know how you could approach that.
Also, what the heck is wrong with memorizing definitions? You can't prove something using nothing. Eventually, after doing enough proofs employing the definition (or axiom or theorem or whatever else you memorized), you won't have to even think about it anymore - similar to how you or I could find the first derivative of a polynomial without thinking about it or know the definition of an even number.
For my own studies, there are many many definitions and theorems I just have to memorize. It is true that I have familiarity with how the theory is all proved, but in the end I have to remember what all the main theorems say. For instance, there is no way you can figure out what "Alexander duality" is or the axioms of a "group" are by their names alone.
Re computational errors: a lot of times they tend to be hidden conceptual errors, since if you had understood the concepts better you would have detected the error!
Re hard problems list: I disagree that keeping such a list amounts to memorizing certain kinds of solutions. I try to keep one mentally, which I use to try to figure out what about the problems seem hard. The hope is that the hardness of the problems dissolves, leaving me with fewer problems and more insight. Even better is when the hard problems suggest other hard problems.
Re "it is clear that": I recommend you find a grading job on campus. The usual reasons this "works" is that the grader is overworked and forgets it is only clear to them, or the student didn't say any nonsense elsewhere so they've earned the benefit of the doubt. But don't delude yourself outside the exam. (The worst are students who make their work intentionally messy to try to throw off the grader. Believe me, the mess is transparent.)
I used to have a distaste for memorization, believing in an ideal of "real math" which was some fountain of pure understanding we might come to know, but over the years I realized that to even begin approximating that ideal you need fluency in the various languages of mathematics --- and to learn a language you need to memorize its vocabulary at the least!
I feel you're missing the point the parent was making. The point as I see it is not to try not to know the answer without deriving it but to not set out to simply memorise the answer.
Of course through use you'll commit certain proofs, formulae, corollaries or what have you to memory; but that's not what people generally mean by "memorising". When studying [undergrad honours, UK] I didn't set out to memorise things and often relied on working out formulae/results/theorems from first principles or from some other theorem. [Perhaps this is a physicist thing, I did joint honours in Phys/Maths]
Remembering and memorising are different approaches that lead to a similar result; however IME remembering when you didn't set out to is often much better. For me I could cram and memorise some results, but then that subject matter never stuck around long afterwards.
What you meant to say is memorization is necessary but not sufficient. If you really didn't memorize, you couldn't answer "What is a derivative?" It would be as unrecognizable as "What is a Jabberwocky?"
The mind is great at synthesizing material and discovering patterns, but you have to feed it the material first.
Von Neumann said: "In mathematics you don't understand things. You just get used to them."
While I can't guarantee that this resulted in better grades, I know for a fact that the professors who I built relationships with during my undergrad have been incredibly helpful during my career!!