
Tips for Success in Undergraduate Math Courses (2002) - isolier
https://web.stanford.edu/class/math53/jasp.html
======
griffinmichl
I mostly aced my math degree. The keys for me were:

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.

~~~
tnecniv
I agree with this, except for 2. I need to write things to internalize them,
and I get a lot more out of lectures I take notes in because it keeps me
focused.

~~~
baby
> I need to write things to internalize them.

Have you tried air writing? Or just "doodling" some notes when in a lecture?
It worked for me.

~~~
stdbrouw
This. By not worrying about whether your notes are actually complete /
sensible, you get the cognitive advantages of writing stuff down without the
distraction. You can still make a proper summary after class if need be.

------
WalterBright
Haha, this is very similar to the list I came up with when I was flunking at
Caltech:

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.

~~~
pmalynin
I'll got a bit further if I may, but its mostly a tip for the instructors:

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...](https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140904.pdf)

Lecture 3: More Numbers and Rationals
[https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090...](https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140905.pdf)

Lecture 4: Irrational Numbers
[https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/2014090...](https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140908.pdf)

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...](https://www.math.ualberta.ca/~xinweiyu/117-118.14-15/20140910.pdf)

And so on; By rebuilding your knowledge in this deliberate way it becomes
easier to work with more difficult topics later on.

~~~
appleiigs
I went to the UofA 20 years ago. The math profs at that time were so bad I got
turned off math forever and went into business school for finance. I have many
stories, but the worst was when one prof quit half way through the semester
because he demanded silence in class but didn't get it. Then the math dean
said we could learn the rest of the course by reading the textbook with no
lectures and the entire grade would be from the final exam. Seems like they
have improved since then.

------
analog31
My math grades had some ups and downs, and the key variable was doing the
problems / proofs over and over. Likewise for physics courses, which involved
a lot of the same kind of stuff.

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.

------
yeowMeng
(Other useful strategies not mentioned)

* 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

~~~
vacri
HN really needs 'bulletpoint' markup...

~~~
jonsen
HN can do

    
    
      * bullet
      * points

~~~
grzm
True. As you've noted elsewhere, this is intended for code. It doesn't work
very well for text. You have to break the lines yourself rather than relying
on the browser. If you break wrong, those with small viewports end up with
nasty side scrolling.

~~~
jonsen
_Videos_

Watch 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)

 _..._

~~~
grzm
That would be one way to do it.

------
hyperhopper
For my first classes this was true. Now I'm getting courses that fewer
students take, and it goes like this

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.

~~~
xapata
Strange. That was the case for my first couple years, in classes with hundreds
of students, where the university hadn't yet shed its "look to your left, look
to your right" mentality. Later classes had fewer students and engaging
professors, since they'd weeded out many of the worse students.

~~~
dilemma
Universities sabotage undergraduate education by conducting lectures in
foreign languages. Interesting.

~~~
VLM
Its more a side effect. Everyone knows everyone cheats on the TOEFL yet it
would be incredibly awkward in academia to discuss that issue, so its kinda
glossed over to avoid massive interpersonal drama. But, yeah, any prof or TA
you have who doesn't know English is a known cheater, violation of honor code,
etc. It does make you wonder if their CV or transcript or reference letters or
published papers are as fabricated as their obviously fake TOEFL scores.

~~~
pkd
Toefl is a surprisingly easy test to pass even with a strong non-native
accent. I know people with mediocre to bad command on English who achieved
scores of over 100 on the online test. There is no reason to cheat the system.

~~~
xapata
That may be, but several of my classmates in grad school told me that they
regularly wrote their papers in Chinese and sent them to a translating service
for English. These were students who were conversational in English. It makes
you wonder how much the translator changed the content.

~~~
pkd
I am not acquainted with the Chinese system, but here in India, you have to
personally go to the designated test centers and take the test. At the center
there is some Airport level frisking before you get to see your computer and
you need to have your passport with you to prove your identity (if I remember
correctly). You are also not allowed to wear jackets/overcoats/caps while
taking the test. Really very little chance for gaming this system.

~~~
xapata
That's for standardized tests. I was referring to homework assignments in the
US.

------
unimpressive
For the memorization aspect, I would say use spaced repetition software:

[https://en.wikipedia.org/wiki/Anki_(software)](https://en.wikipedia.org/wiki/Anki_\(software\))

Source: I actually did this and it improved my grade 55% in Calc I.

~~~
arrmn
I've used Anki for some of my courses but I don't know how I should use it for
math courses. Can you maybe share your calc Anki cards?

~~~
unimpressive
My calc anki cards have copyrighted material in them so not off hand no.

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.)

------
isolier
There is some pretty obviously advice in here; basically take notes and do
your homework for the sake of comprehension and not merely completion. But,
"keep a list of hard problems" is pretty good advice that, I think, most
people do not adhere to.

------
chrismealy
My breakthrough in math came when I started studying with other people. Best
way to learn something is to explain it to somebody else. And you're a lot
less likely to get stuck on something for half an hour if you have other
people to help you. By studying with people I aced calc just by going to class
and keeping up on homework.

~~~
saberlynx
Depends on who you are. I can't listen to other people talk about math without
getting confused myself. Other people think about math differently and often
throws my own thinking into a loop. I need to study alone and ask questions if
truly stuck.

------
mfsch
The issue I have with lists on how to be successful in a class is that they
rarely provide much guidance on how to prioritize the work. Of course students
are successful in a class when they read & understand the material before the
lecture, follow the lecture attentively, redo all example problems, solve &
understand all assignments and maybe read a few textbooks on the subject. The
difficult part, at least for me, is how to decide what learning activity will
help your understanding in the most any given moment and how to split your
time between the different classes and other activities.

------
gizmo686
I'm a math major.

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.

~~~
kmill
As a math PhD student who is supported through teaching undergraduates, I have
to spend a lot of time convincing my students that they have misunderstood
what math is and they do in fact have to memorize the definitions of the
words. It used to be in high school that many of them could ride right through
a course relying only on their native intelligence to derive or deduce
everything, but this tends to bite them when they refuse to believe that each
word has a specific definition in the subject --- and many times they are
defined the way they are due to a few hundred years of effort by many smart
people trying to figure out the best way of expressing some mathematical
phenomena. For instance, "continuous" doesn't mean "can be drawn with a
pencil" \--- while that is a fine intuition, if you need to prove something is
continuous from first principles you had better remember that it means the
function equals its limit at each point. Another is "linearly independent,"
though the usual error students make is wishful thinking that you only need to
check that no vector is a scale multiple of any of the others.

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!

~~~
saberlynx
As a math PhD student I concur. I got through most of my undergrad memorizing
nothing. It was easy back then. The definitions were simple and intuitive.
what's a group? I hold an intuitive picture in my head which can be translated
into words if I want. What's the Taylor series of cosine? If you want to do
serious work in applied math you'd better have that on the back of your hand.
You can derive it every time of course. You'll just take WAY longer. What is
are the homology groups of a group with integer coefficients? Ha. Good luck
doing anything with it if you haven't memorized it. Not memorizing might work
in undergrad. I've reluctantly come to the conclusion that it doesn't work
anymore. In fact not memorizing anything is biting me in the ass. It seriously
hampers my workflow to have to look up the definition of Calc iii stuff. If
you don't have some things mmeorized you won't even understand lectures. They
don't take the time to let you get used to a definition in grad school. They
give it to you and just RUN with it. They expect that out of you. They don't
remind you. They expect that they can just give you the definition and start
using it to prove theorems right away. Most of the time they don't give you
the intuition for it either. No examples. You need to get used to the language
of math and do it fast.

~~~
pbhjpbhj
>What is are the homology groups of a group with integer coefficients? Ha.
Good luck doing anything with it if you haven't memorized it. //

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.

------
hluska
The most useful thing that I did in any of my undergraduate courses was to
make a point to go to all of the office hours offered. No matter whether the
office hours were with the professor or a TA, I made a point of going, asking
any questions that I had, and showing them the methodology that I was applying
to a project/assignment/etc...

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!!

------
thisisforyou
I'm curious how many of these are also on the list of 'tips for success as an
aspiring mathematician'. I just finished reading Benoit Mandlebrot's memoir
and he repeatedly talks about how poor academic standing is in predicting a
person's ability as a researcher. As someone who is not a mathematician I'm
genuinely curious.

------
douche
Find a section of the math course that is being taught by a professor that is
actually fluent. Math is rather infamous, particularly in introductory
undergraduate courses, for having professors, brilliant though they may be,
who speak broken, incomprehensible English.

------
saberlynx
I'm still trying and most of the time failing to apply all of these things as
a grad student.

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quickben
What worked for me: 1\. Get a solutions companion to your calculus book. 2\.
Solve everything twice.

------
rawnlq
Terry Tao (Field Medalist) has a lot of great writing on this subject:
[https://terrytao.wordpress.com/career-
advice/](https://terrytao.wordpress.com/career-advice/)

------
viach
Imho, you only need motivation. And where to get it, that's the question.

------
pdm55
I love GeoGebra, [https://www.geogebra.org/](https://www.geogebra.org/). What
a (free) resource to explore Math for oneself!

