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How to Study Higher Math and
Do Proofs
Higher math relies heavily on
definitions, proofs, and notation, so we will concentrate
on those areas. (If you are studying geometry please
visit the
mathforum
website) The first thing you can do
is find a good
book; ask your professor,
other students, or check my Book
List for
a reference. A good book can
make the difference between learning the material and
progressing through the book, and getting stuck on poorly
explained examples and overly difficult
exercises.
M.I.T.(Massachusetts Institute of
Technology) has an open courseware
website
were all the syllabi, readings, exams,
assignments, and projects of all their courses are posted
online, free for anyone to access. You can use their
website to find good book references and to help
plan what
you should study. Before we begin, it
would be great to review techniques on how to study any
topic in general; criticalthinking.org
has great outlines on how to
study.
If I had to condense this entire page
into one sentence it would be; if you want to learn
how to do proofs you must emulate
the reasoning processes of those who write the
proofs. The more you emulate the more your mind
shapes itself to think like those who write proofs, until
you can write proofs on your own.
Proofs are more involved; you should
first learn some of the basic logical proof structures,
direct, contradiction, and converse. From experience, I’ve learned that the
following materials teach you how to write proofs
correctly, but don’t teach you how to
actually derive or
deduce proofs in a
specific area of mathematics (please think about this it
is a subtle point). There
are many good books on how to do proofs, here are some of
best:
Proofs and Fundamentals: A First Course in
Abstract Mathematics by Ethan D.
Bloch
How to Read and Do Proofs: An Introduction to
Mathematical Thought Processes by Daniel
Solow
Here is a decent proofs
book available for download:
Bridge to Abstract Mathematics - Mathematical
Proof and Structures, 1st Ed. - Morash, Ronald
P.
Download Directly
Here are two very basic
introduction to proofs (the books above are
preferable)
Intro
1 Intro2
Here is a website with a
good introduction to proofs.
How to Write Proofs
Write out every example, definition,
and problem in the book mentioned above. You can use
the proofs book concurrently with your classes
Mathematics builds upon itself;
nearly everything you learn depends upon the mastery
of prior subjects. Ideally, you should start applying
the techniques from the beginning of the book/semester.
If you are in the middle of your book/semester the best
thing to do would be to go back to last section you fully
understood, then apply the techniques below.
Sometimes an entire subject may be
too advanced for your level of mastery, if so you may
need to go back to the last subject you fully understood
and start from there. You will eventually master the
original subject, but you need to understand prior
subjects first.
Once you get your textbook, read each
section carefully, afterwards go back to the beginning of
the section and memorize all the definitions, rules, and
notation. Then, for each section, write what you
memorized on a blank piece of paper one by one from
memory without looking in the book, they must be exactly
the same as the ones in the book if not, write them down
again until they are "perfect". You want to learn the
definitions precisely not in generalities. Apply
each definition to an example, apply each rule to an
example, and apply each notation to an example and so on.
You should have a ready example for everything you
memorized.
Writing everything down is
important because when we are first learning a new
mathematical concept it is difficult to see, in our
mind’s eye, what exactly is going on with the
notation. In addition, it is difficult to keep track of
all the little details.
In higher math notation plays an even
bigger role than it does in basic math. When proving a
theorem there is a big gap between the plan and the
execution. When looking at a theorem or simply testing
the theorem for several cases, it is sometimes obvious to
us that it is either true or false. We usually have an
idea on how to prove it, but an idea is not enough, the
name of the game is execution. The only way we can prove
it is manipulating the notation and dealing with its
little rules, this is the reason why proofs are so hard.
Our execution of the idea is constrained by the rules of
the notation; this is a good thing for our idea may be
wrong. The notation is what keeps us honest and guides us
towards truth. Sometimes we may need to create new ideas
and new notation to prove a conjecture, if this is the
case, it may not be provable with our current knowledge,
but then again you can never know until the conjecture is
proved.
We are going to take a different
approach to learning higher math due to the difficultly,
at first we are simply going to memorize and write an
explanation for everything.
After you have read the section, go
back and look at the examples and proofs, write down
every single example and proof and work out
every little detail. Look at each example and
proof and find out how it works, what definition they
used, what examples they used, what concepts they used,
what methods they used, what strategies, and what tools
they used to get the answer. Write an explanation for
every step, you want to relate it to other steps and
then to the proof as a whole. After this, memorize them
and memorize what you learned. Then write down every
example and proof one by one, from memory, without
looking in the book, they must be exactly the same, if
not write them down until they are perfect. Pay
attention to all details no matter how small I cannot
emphasize this enough. If you are looking for some good examples of proofs you can go to the website below:
Proof Wiki
The worst attitude would be to
view each step in a proof as a line that follows strictly
from the previous line. There are many logically
possible steps that could follow from any line; you
should look for an explanation why this particular
step was taken. In order to do this you must
understand how this step relates not only to the previous
or following steps but how it relates to the entire
proof, how does it relate to the result, how does this
step advance the line of reasoning. You want to
understand the reasoning process of the person who wrote
the proof. Of course as a novice, this may be
difficult, but if you practice this when the proofs are
easy, it will be easier to do this when the proofs are
more advanced.
Since our memories aren’t
perfect we will usually forget a little step in our
example or proof, since the step is so little you will
usually be able to fill in the step using your own
reasoning, at first even the little steps will be
difficult, so simply look in the book analyze the little
step and complete the example or proof. Write down the
example or proof again from memory until it is perfect
without looking in the book.
Memorization is important because
it trains your brain to pay attention to little details,
which are very important in higher math. If you feel that
you still do not understand the proof(s) at an
“intuitive” level, review the proof(s)
everyday until your understanding becomes
“intuitive”.
After a while, you will begin to
notice that many proofs use the same strategy over and
over again, you will start relying less on your memory
and more on your reason, try to imitate the other proofs
you memorized, if a certain method, tool, strategy, or
trick worked for a similar proof try using it
again. Afterwards go over
the examples and proofs in your mind and in full detail
without writing anything. Take some time everyday to
review or rederive critical proofs, this way you will
remember important proofs and build your
“intuitive” understanding of the subject. The
more you practice rederiving proofs to theorems in the
way the book presents it, the more your mind will begin
to shape itself to prove other theorems.
You can also deliver yourself a
3-minute formal lecture on the most important proofs, or
you can write a 15-minute essay about the proofs. Review
the day’s work in the evening, the week’s
work on Friday, and the entire material you have covered
once a month.
If you come across something you
don’t understand, analyze your difficulty so you
can state specifically what you don’t understand.
Afterwards go to your professor to help you understand
difficult concepts.
Higher Math also relies heavily on
special examples, you will know which examples are
special by noting how many times they are used in the
book; memorize and understand these examples down to the
smallest detail. Most of the time theorems are applied to
these special examples. Special examples are used as
counterexamples to conjectures, and theorems (if you drop
one or many of the conditions in a theorem).
Some textbooks have very little
examples or explanations; if this is the case then you
should find a better book.
Amazon allows the public to rank books; I suggest you
search on
Amazon for the highest
ranked book in your subject
and at the appropriate
level.
You can also get a
Schaum’s Outline
and work through their examples, or
take a look at
Springer – Verlag’s
problem books in
mathematics. You can also
visit
Mir Publishers
they also have many good books in
advanced mathematics. Otherwise, get yourself another
textbook with more examples and better explanations.
Please check my Book
List.
Work through the easy textbook and use it to understand
your hard textbook. You can usually always find another
textbook in your subject even if it is Univalent
Functions in Teichmuller spaces, or a course in
Homological Algebra. Go to your college library, public library
or bestbookbuys.com
and search for your
subject.
After you have followed the procedure
for a couple of sections you may feel confident enough to
rederive the solution to an example or the proof
to a theorem without looking the in the book. Keep
pushing yourself to re-derive the proofs, if you
can’t re-derive the proofs simply memorize them and
follow the procedures above. Review the most important
concepts and proofs everyday; this will help build your
intuitive understanding of the subject.
Rederiving or attempting to
rederive the proofs in the book allows you to fully
understand how you could have derived those proofs
from your understanding of the concepts. This is the end
goal.
If you reach a point where you can
no longer progress then you may want to ask for help on
the internet please check my links page.
Once you can do this reliably
(meaning your proofs look exactly the same or similar to
the ones in the book) and in a reasonable amount of time,
you may simply work out the details of every example and
proof without memorization.
If you follow the procedure throughout
the entire semester, you will begin to notice that the
proofs on the homework assignments and tests will start
getting easier.
Checking your Proofs; The
Power of Negativity
Once you think you have a proof,
checking it is almost as important as deriving it.
Generally, your proof will contain many little parts and
any one of those parts may be wrong. What you should do,
is take you mind off of the proof(s) for about an hour or
so, watch some TV, take a walk, or do something unrelated
to your proof(s). Afterwards come back to your proof(s)
and try to find something wrong with each proof.
Take a negative attitude and believe that each one of
them is wrong, try your best to find something wrong with
each proof no matter how small. If you do find something
wrong correct it. Sometimes if what you find wrong is
critical for the proof then you may have to modify the
entire proof and start over. Otherwise, if after a while
you cannot find anything wrong then most likely your
proof is correct.
We would accomplish many
more things if we did not think of them as
impossible. - C.
Malesherbes
A mathematician is a device
for turning coffee into thereoms. - Paul
Erdos
Imagination is as vital to
any advance in science as learning and precision are
essential for starting points. Let me warn you to beware
of two opposite errors: of letting your imagination soar
unballasted by facts, but on the other hand, of shackling
it so solidly that it loses all incentive to rise.
-
Percival Lowell
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