
How to Study Mathematics (2017) - kenny87
https://www.math.uh.edu/~dblecher/pf2.html
======
e0m
By far the best explainer of mathematics I've seen anywhere is 3Blue1Brown:
[https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw](https://www.youtube.com/channel/UCYO_jab_esuFRV4b17AJtAw)

These videos do an incredible job of illustrating how to intuitively arrive at
an answer by composing many of the parts you need to build a proof for more
complex topics.

~~~
nothis
It always saddens me how mathematicians seem to look down on “intuition”.
Maybe higher math as a full-time job is just hard work and stubborn precision
but for me, an intuitive, visual look is probably getting me closer to
understanding than any cold-hard-facts book does. Not to mention how much
easier it is to appreciate the beauty of it.

~~~
dvt
> It always saddens me how mathematicians seem to look down on “intuition”.

They do this for good reason. I had to take, as part of my philosophy
concentration, a bunch of mathematical logic classes. First Order and Second
Order logic, as you might imagine, are pretty simple. The rules _make sense_
in a very intuitive way. I'd get 100 on exams just because I'm good at
programming. But when I had to study metalogic, model theory, Henkin Proofs,
and Godel's Theorems, that intuition quite literally flies out the window. I
guess my point is that most _interesting_ stuff is rarely intuitive.

~~~
goldenkey
That is ludicrous. The intuition is just harder to find, harder to grasp. But
it is still there because these theorems are logical consequences of the base
axioms. And the base axioms are logical consequences of our desire for
sensible assumptions.

~~~
dvt
This is provably untrue and all I need to do is think about how weird stuff
gets when increasing dimensionality of "household" shapes like circles and
squares. It literally goes against every intuitive fiber of my body.

If you think that coming up with theorems about, e.g. infinitary algebra, is
intuitive, then I guess you're just a lot smarter than I am. Infinities always
throw a wrench into our intuitions. Something being a "logical consequence" of
something else is an incredibly simplistic way of looking at things, not to
mention that I'm not exactly sure what that has to do with intuition anyway.

~~~
sdenton4
It /becomes/ intuitive with time, thought, and exposure, as you pick up more
tools for understanding the problem space.

That 'non-intuitive' aspect of high-dimensional spheres having very small
volume relative to a hypercube becomes a point that builds intuition for how
those spheres behave, once you're familiar with it. And then you can start
throwing out things that seem intuitively wrong, and following intuition
towards new ideas.

~~~
dvt
You just described the _opposite_ of intuition.

~~~
tashi
At this point a mathematician might step in and say that you all need to
define your terms. You're basing your arguments on two subtly different
definitions of the word "intuition."

~~~
dvt
The differences aren't subtle. I'm using the run-of-the-mill definition you
might find in a dictionary[1], what other definition could you possibly be
using?

[1] [https://www.merriam-
webster.com/dictionary/intuition](https://www.merriam-
webster.com/dictionary/intuition)

~~~
emmelaich
From that I would choose 2.c, with the emphasis on _evident_

> _the power or faculty of attaining to direct knowledge or cognition without
> evident rational thought and inference_

There is such a thing as developing your intuition.

I think your meanings are not qualitatively different.

------
dboreham
Something I would add :

Find out where the mathematics you're learning comes from : who first
developed it? what problems where they trying to solve? why were they trying
to solve those problems? what problems does it solve for us today?

In my mathematical education I noticed that I had a more pleasant time and
felt more motivated to learn the material when the teacher gave us this kind
of background story. Since most teachers don't do so today, the student
typically needs to get on the Internet to do their own research.

~~~
mbag
This happened almost through all my formal education. The knowledge was just
presented, as something that was obvious. There was very little discussion of
motivation, or what problems was person/people trying to solve. I understand
that there is so much knowledge to cover during high school and college, but
IMHO, this causes students to develop mindset, that you either see solution
instantly, or you are just not smart enough to crack the problem.

~~~
soVeryTired
I think Bourbaki are partly to blame for this state of affairs. Vladimir
Arnold had a lot to say on the subject: he was a big fan of keeping the
intuition and motivation for development in the subject clear.

At times he went a bit overboard, but he makes some valid points in this
lecture: [https://www.uni-
muenster.de/Physik.TP/~munsteg/arnold.html](https://www.uni-
muenster.de/Physik.TP/~munsteg/arnold.html)

~~~
goldenkey
I don't know if I can agree with that. I bought every book Serge Lang
produced. He was a member of Bourbaki. His Linear Algebra book is far better
at providing intuition and motivations than the rest. Yet it is also rigorous.
Lang is #1 for Mathematics books in my world.

~~~
soVeryTired
Fair - I don't doubt that there are some great educators in the Bourbaki
school. But as Arnold mentions in his lecture I do think mathematics lost
something when it embraced formalism so fully.

For me, the definition "A group is a set of transformations on an object such
that..." is _so much_ more enlightening than "A group is a set G together with
a binary operation * such that..."

Only yesterday I was trying to learn about differential forms. Most of the
notes I found online introduce the wedge product in a deeply unhelpful way, by
listing some axioms that it satisfies and deducing results from the axioms. It
takes hours of work to understand why those specific axioms were chosen. For
me - and maybe I'm wrong here - that's Bourbaki's formalist approach in a
nutshell.

It was only when I found Terry Tao's notes [0] and Dan Piponi's notes [1] that
I could actually see the use of differential forms. It's an unfortunate state
of affairs for the discipline that in order to learn about X, you have to
google "X intuition", since it's not given to you as a matter of course.

[0] [https://terrytao.wordpress.com/2007/12/25/pcm-article-
differ...](https://terrytao.wordpress.com/2007/12/25/pcm-article-differential-
forms/) [1]
[https://github.com/dpiponi/forms/raw/master/forms.pdf](https://github.com/dpiponi/forms/raw/master/forms.pdf)

~~~
goldenkey
I totally agree. A lot of times you have to search for intuition rather than
it being provided. Rote memorization is useless because mathematical research
requires the intuition to give your mind the right direction to go in.

------
cle
Memorization is so underrated. Memorizing the fundamentals and having them
available for instant recall is _hugely_ valuable, especially when trying to
grok a new concept.

I generally buck the standard advice and memorize first, before trying to
understand. Understanding is much easier for me if I can easily hold
everything in my working memory.

~~~
joe_the_user
I wouldn't personally find advice like this useful for me. I suspect it varies
with "personal style". I grasped mathematical concept easily from an early age
and I never made an effort to memorize things.

If anything, I saw those coming from less-advanced math founder on advanced
math (arithmetic to algebra, algebra to calculus, calculus to advanced
subjects) because they attempted to deal with the subject based on
memorization rather than grasping the basic point.

Generally, I did wind-up committing a lot of content to memory but it was and
is much easier, even "effortless", when I knew the reason for each part of the
content.

Of course, language is slippery and we might be talking about exactly the same
thing. So I wouldn't give blanket advise - each person has to find their own
learning style (or styles, if it varies by discipline).

~~~
gowld
Rote Memorization is awful. Memorizing by practicing so much that the details
become "muscle memory" is the foundation of progress in learning.

A basketball player has memorized how to shoot a free throw, not by cramming
by rote, but by practicing.

~~~
jacobolus
The best basketball players start by spending years and years just playing the
game all the time, as recreation.

The equivalent for mathematics is solving problems that you find personally
relevant and interesting and which are at the limits of your current
abilities. NOT just doing piles of rote calculation. Often solving a hard
problem requires a big pile of intermediate calculations, but those a tool in
service to some meaningful purpose, not an end in themselves.

Forcing students to work through thousands and thousands of trivial
calculation exercises is horrible pedagogy (and a very inefficient use of
time).

It’s like doing nothing but shooting drills and getting quizzed on the NBA
rule book for years before ever trying to play a game.

~~~
mamon
>> The best basketball players start by spending years and years just playing
the game all the time, as recreation.

No, they start by joining a club where they get professional training, from a
very young age. Just playing the game all the time won't make you any better
at it. There are millions of children that spent significant part of their
childhood playing basketball for fun, and never went past amateur level.

This is just a conclusion from Ericsson famous paper[1]: you need DELIBERATE
practice, not just recreational playing.

Playing for fun will just develop bad habits, which you then will have to
unlearn, so recreational playing might actually be an obstacle to learning.

[1]
[http://www.nytimes.com/images/blogs/freakonomics/pdf/Deliber...](http://www.nytimes.com/images/blogs/freakonomics/pdf/DeliberatePractice\(PsychologicalReview\).pdf)

~~~
jacobolus
As you say, the key to skill improvement is playing deliberately and getting
meaningful feedback in a tight loop, ideally at the edge of your abilities.

But there are many aspects to any complicated skill, including tiny subskills,
combination of various subskills, and at a higher-level meta skills like
knowing when to change strategies, etc.

The problem with only sport games as a mechanism for feedback is that (a)
there’s not necessarily a good way to correct and try again right away if you
want to practice one particular thing, (b) situations that arise are not
homogeneous enough to compare feedback across trials, (c) there are periods of
waiting in between direct practice, (etc.?).

As a result, people who are training to be basketball players spend only like
half their time playing realistic games, and the rest of their time on a
variety of drills and workouts including general aerobic and strength
training, practice running sideways and jumping, shooting / passing /
dribbling drills, practice setting screens / rebounding / driving / inbounding
the ball, larger-scale coordination drills, and so on. These let the players
focus their conscious attention at one level at a time, and repeat a situation
with analytical feedback in between.

But those who improve fastest are the ones who figure out what they need to
work on (or have a mentor/coach to figure out what they need to work on) and
then get lots of feedback about that particular area, in an intentional way.
This can even be done within the context of a realistic full game by choosing
what to direct attention/focus to.

But you’re missing my basic point, which is that all the skills in basketball
are situated within a context. It’s obvious how they fit into playing a game.
Students _start_ with the high-level context (by watching / playing full
games), and then break it down into small pieces they can work on separately
or jointly, and then frequently practice putting those pieces fully back
together in the context of the game. I doubt you’ll find anyone seriously
training to play basketball who doesn’t play in a semi-realistic game-like
situation at least once per week.

In classroom education, often the high-level context is completely missing,
and students end up memorizing / drilling on a pile of atomized facts and
trivial very low level skills without ever practicing combining those in
larger settings, without ever getting a very fast or effective feedback loop
going, and without ever having any motivation for going through the practice.

In the case of mathematics education, someone being taught 1-on-1 by a skilled
tutor can move I’d say about 4–10 times faster than a student only watching
lectures, doing completely independent book reading / trivial homework
exercises / quizzes, and then getting back graded homework/quizzes a week
afterward. That student can be assigned dramatically harder problems, can be
helped along by limited (but well placed) hints, and can be given direct
feedback/corrections at every level (from basic arithmetic computations
through higher-level strategy to meta skills like strategy
selection/evaluation). This is something that can’t be done in basketball
because basketball is a group activity; you can’t watch a single player go
through the game, and pause/backtrack everyone to the beginning of a situation
whenever that player makes a mistake or suboptimal play.

------
gtani
Some books on proof, theorems/axioms, set theory,
epsilon/delta/continuity/limits/differentiability,
natural/rationals/reals/countability etc before heading into your first proof
based LA or analysis sequence:

\- Kevin Houston "How to Think Like a Mathematician"

\- Keith Devlin "Intro Mathematical Thinking"

\- "How to Study as a Mathematics Major" Lara Alcock (for some reason,
she/Oxford Press has 2 books with seemingly identical content under Math Major
and Math Degree titles)

~~~
dvfjsdhgfv
My favorite is Burn Math Class: And Reinvent Mathematics for Yourself

~~~
goldenkey
Seems well liked but whats is up with the cover? Most people who are into
mathematics aren't zealous rebels bent on burning the world. I feel like the
marketing director for the book screwed up big time. I won't judge the book by
its cover though.

[http://amzn.to/2CrgFDI](http://amzn.to/2CrgFDI)

~~~
dvfjsdhgfv
I actually don't remember the cover. The title is provocative but adequate.
You could rephrase it as "Could you reinvent modern mathematics if it
disappeared?" \- but then probably less people would buy it.

------
BadMathBook3
I bailed on high school math, thinking I'm math dumb.

In my late 20s I decided to try again, but jumped straight into calculus. And
at first regretted that decision. However, I got lucky by stumbling upon this
book:

[https://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/...](https://www.amazon.com/Calculus-Made-Easy-Silvanus-
Thompson/dp/0312185480)

It "reads" like a book, with the ideas given context. I had an "ok" connection
with Algebra, and the book explained the rest well enough for me.

In school, the textbooks were loaded with symbols, but not enough description
-- I guess they relied on bored teachers making minimum wage to do that part.
I went to a school with poor academic showings (but connections to state
superintendent of ed got them a grant for football facilities).

Coincidentally, this book goes well with the technique described here:

[http://www.pathsensitive.com/2018/01/the-benjamin-
franklin-m...](http://www.pathsensitive.com/2018/01/the-benjamin-franklin-
method-of-reading.html)

------
bjourne
What was the name of that 30-something woman who had never studied mathematics
before, but then finished a degree in quantum physics in two years? Her name
has been posted on HN at least two times before. I guess what she did is
exactly how you should study mathematics if you want to become got at it.

~~~
sebg
Think you're talking about Susan Flower and two of her articles:

If Susan Can Learn Physics, So Can You -
[https://www.susanjfowler.com/blog/2016/8/26/from-the-
fledgli...](https://www.susanjfowler.com/blog/2016/8/26/from-the-fledgling-
physicist-archives-if-susan-can-learn-physics-so-can-you)

and

So You Want To Learn Physics... -
[https://www.susanjfowler.com/blog/2016/8/13/so-you-want-
to-l...](https://www.susanjfowler.com/blog/2016/8/13/so-you-want-to-learn-
physics)

~~~
badpun
Sadly (given her immense passion for physics), she ended up in software (doing
SRE).

~~~
systemBuilder
Hey I am in SRE. At one point last year there were 7 ex professors sitting
within 15 feet of my desk including me (and I did a PhD in theoretical
computer science at the same school as this math professor whose article we
are reading). I don't think it is an underrated career.

------
alexbecker
I left math after college for software engineering, but reading _A
Mathematicians Lament_ recently re-kindled my love for it. It is tragic how
intuition, technique and mental models are left out of modern mathematics
education and writing.

It occurred to me that while I learned in college how to show, using Galois
theory, that quintic equations are not generally solvable by radicals, I had
no idea how Galois theory really relates to the process of finding roots. So I
went back and derived the quadratic formula, the cubic formula, and sketched
the quartic formula to see how the process used the ideas formalized by Galois
theory and where it breaks down. I've tried to write the result up in a
motivated and understandable way, instead of like a math textbook:
[https://alexcbecker.net/mathematics.html#the-quadratic-
equat...](https://alexcbecker.net/mathematics.html#the-quadratic-equation-and-
beyond)

------
graycat
I studied a lot of math, pure and applied, taught it, applied it, published
research in it, etc. so developed some ideas relevant to the OP.

For

> To the mathematician this material, together with examples showing why the
> definitions chosen are the correct ones and how the theorems can be put to
> practical use, is the essence of mathematics.

Is good, but more is needed.

(1) Plan to go over the material more than once. The early passes are just to
get a general idea what is going on.

In such passes, for the proofs, they are usually the near the end of what to
study and not the first.

(2) When get to the proofs, for each proof and each of the hypotheses (
_givens_ , assumptions), try to see where the proof uses the hypothesis.

Next, try to see what are the more important earlier theorems used in the
proof. So, sure, in this way might begin to see some of how one result leads
to or depends on another and have something of a _web_ , acyclic directed
graph, of results.

And try to see what are the core, clever ideas used in the proof.

(3) For still more if you have time, and likely you will not, can use the P.
Halmos advice, roughly,

"Consider changes in the hypotheses and conclusions that make the theorem
false or still true."

(4) But, note that to solve exercises or apply or extend the theory, need some
ideas. So, where do such ideas come from? In my experience, heavily the ideas
come from intuitive views of the subject.

So, my best suggestion is to try to develop some intuitive ideas about the
material. Definitely be willing to draw pictures, maybe on paper, maybe only
in your head.

In the end, a solution or proof does not depend on intuitive ideas, but
finding a solutions or proof can make use of a lot in intuitive ideas.

For research, most of the above applies, but IMHO there are more techniques
needed.

------
adamnemecek
The Franklin method from yesterday still applies.

[https://news.ycombinator.com/item?id=16390046](https://news.ycombinator.com/item?id=16390046)

Also a lot of books these days have wolfram mathematica code and it works
surprisingly well even for some more abstract parts of math, To get a good
intuition.

~~~
aje403
The greatest book review of all time, sort of similar approach:
[https://www.amazon.com/review/R23MC2PCAJYHCB](https://www.amazon.com/review/R23MC2PCAJYHCB)

Put the book down when you get to the example or proof of a theorem, keep all
the theorems and examples before it in your head, and see if you can reason it
out yourself. Can't promise that your head won't hurt afterwards

~~~
BadMathBook3
Those prices seem comical. Rudin passed nearly 8 years ago. Who is this
benefiting? America is off its rocker.

Google the title + PDF

------
sampo
This is more "How to study pure mathematics", when the aim is to understand
how the theory is build, so that you learn to contruibute to the theory by
discovering and rigorously proving your own theorems.

I don't think I've ever seen a "How to study applied mathematics". How do
applied mathematicians, physicicst and engineers (who apply mathematics to
real world problems) study mathematics, when they use it as a tool? How much
or little emphasis do they give to proofs and theorems?

~~~
dboreham
All mathematics gets applied sooner or later, no?

~~~
aaachilless
Maybe, but “applied mathematics” is conventionally more similar to “how to
apply mathematics” than it is to “mathematics that is applied”. Studying how
to apply mathematics is qualitatively different in many ways than studying
pure mathematics.

~~~
dboreham
Ah. I genuinely did not know that. In Scotland we did not have Applied
Mathematics (as a school exam subject) so I guess I just assumed it was about
physics problems (there was an A-level with that name in England). It never
came up again in my Mathematics career..

------
miobrien
Awesome.

A related thread from two days ago for those who missed it:

[https://news.ycombinator.com/item?id=16372454](https://news.ycombinator.com/item?id=16372454)

------
skybrian
One thing they don't talk about is how you decide what to study. Before
spending a lot of time on a particular topic in math, you have to decide
whether it's worthwhile studying it at all.

It's apparently just assumed that you're taking a course so the decision is
made for you.

~~~
sincerely
I'm willing to bet that a course in post-secondary school is the context for
the vast vast majority of people studying mathematics.

------
commandlinefan
One thing I figured out on my own that I wished I had realized sooner (which
I'm trying - so far unsuccessfully - to impress on my 14-year-old son) is
that, when reading math books, they follow a similar pattern. They describe a
concept, show an example problem fully worked, and then discuss the
ramifications of that concept, followed by another concept, followed by a
fully worked problem, etc. I started making it a habit to try to work the
example fully-worked problem by myself, based on the description that preceded
it, before reading through the author's work. I was amazed how much better I
was able to understand what he presented, and how much better I did working
the exercises in the chapter afterwards.

------
backprojection
Ha, Dr. Blecher's webpage on HN! That's great.

I did my PhD in math, in large part because I enjoyed his class on advanced
linear algebra so much, and later on real and functional analysis.

------
dsacco
This is good. I think two principal components should be emphasized when
studying mathematics.

1\. Proper preparation. There are textbooks at even the graduate level which
have no formal prerequisites and which are largely self-contained. Technically
speaking, someone with no prior background but a strong mathematical maturity
could tackle these, but it might take them an inordinate amount of time to
really grasp the material. For example, if you understand things like
mathematical induction and proof by contradiction, you can learn analysis
before you've been exposed to calculus, or category theory without abstract
algebra. But it's far from ideal because you'll probably need to go over the
same material several times and struggle with it.

Furthermore, even the same subject within "advanced" mathematics can have
wildly different depths of coverage depending on the author. Pinter's _A Book
of Abstract Algebra_ is probably approachable for anyone currently reading
this comment, or even high school students. Dummit-Foote is a step beyond
that, and appropriate for undergraduates who are already immersed in a math
degree. But Lang or MacLane-Birkhoff would be significantly more challenging
without first building up to them.

Sometimes this is not just a question of depth, but also of pedagogical style.
You can get a lot of satisfaction by learning analysis from Rudin for the
first time, but it's really a rough go of it if you're not prepared for the
terse definition->theorem->proof->remark->definition->theorem->proof->remark
style of writing. On the other hand, Tao's _Analysis I_ and _Analysis II_ are
much more approachable (similarly, some writers, like Halmos or Munkres, are
praised for their exposition in introducing otherwise complex material).

Ideally someone looking to study a subject should introspect about whether or
not they are prepared for that subject overall. Once they've confirmed they
are, they should read the first 10 pages of five or so well-recommended
textbooks on the subject _at their level_ , then choose to stick with the one
that has the most approachable exposition style for them.

2\. Proper study. When studying any given textbook (or videos, lectures, etc)
it's really important to understand that mathematics is an _active_
discipline. You cannot learn it by reading it. The process that has worked for
me is the following: first, read through a chapter without taking any notes.
Do so quickly, but not quite so quickly as skimming. When you come across
things you don't know, compartmentalize them a bit and keep moving forward to
the end of the chapter. The idea is to let the chapter's new material
percolate a little before you begin actively tackling it.

Next, start over at the beginning of the chapter and write down every single
definition and theorem as you read. Before reading the author's proof of any
given theorem, try to prove it yourself for at least 10 minutes. Then compare
your work to the author's, and copy their proof meticulously in order to learn
the method. Continue on to the end of the chapter.

Finally, there will probably be anywhere between 5 - 20 exercises at the end
of the chapter. Solve a meaningful fraction of these exercises, and don't look
up the solutions to any of them until you've struggled with them for a good
half hour or so (each). When you do look up the solutions, make sure you check
multiple proofs for the same exercise so you can understand how the chapter's
material can be applied in different, flexible ways.

Mathematics has always exemplified a central belief of mine, which is that
humans learn under conditions of _optimal struggle._ Even though it feels like
being mired in hopeless complexity while you're struggling to complete a
particularly difficult problem, you're _actively learning_ the subject by
doing it. But it's a question of efficiency. You want to aim for a subject
_and_ a presentation of that subject which is difficult enough to be _just_
out of your current capabilities, but not so difficult that you can't follow
its exposition.

~~~
FourSigma
This comment should be way higher up.

------
lowtec
Notation and syntax are areas that I have struggled with in higher level
mathematics. Can anyone recommend a guide or resources useful to understanding
things like converting set theory, sequence and series style problems into
equations and solving them?

~~~
tkzzbneig
Number theory's as good a place to start as any. You'll learn proof techniques
and see a lot of notation in contexts that require varying amounts of
background knowledge. It's feasible that an undergrad intro class would be up
your alley if you're a software dev.

[https://ocw.mit.edu/courses/mathematics/18-781-theory-of-
num...](https://ocw.mit.edu/courses/mathematics/18-781-theory-of-numbers-
spring-2012/lecture-notes/)

------
Philipp__
I always strugle with memorizing part. I am never able to reproduce word for
word what I’ve learned or read, but I can put it into context and show my
understanding pretty well. Here, with math theory, there is no alternative. I
failed exams so many times only because of that vocal reproduction of learned
theory that required word for word knowledge. I am having math exam on
integral theory in 2 days and here I am sitting and lookin at screen. :)

~~~
jolmg
> with math theory, there is no alternative.

You can do the exact same thing you mentioned with math. Theorems aren't
arbitrary; there's a way they're derived. If you learn how the theorem comes
to be, even if you can't remember exactly, you can often remember enough to
come up with the theorem again.

I think the way you learn is the ideal. Knowing the context behind the facts
you learn gives them worth, and the redundancy of being able to derive the
fact from the context also makes it better integrated with the rest of your
knowledge and easier to remember in the long term.

~~~
Philipp__
I totally agree with you, but I feel like with math theory there is always
good chunk of knowledge that needs to be somehow inserted in your memory _as
is_. That’s the most irritating part for me personally. If i find a way to
overcome it or make it more accessible, I think I will enjoy math theory much
much more. :)

~~~
Smaug123
I'm afraid it's just practice. Lots and lots of solving problems until the
definitions, methods, and techniques are burned into your mind.

Once you have the background knowledge, everything is a lot simpler. Learn
whether each theorem has an easy proof or a hard proof. If it has an easy
proof, you can forget it (deriving it when you need to; an example is the
contraction mapping theorem [1]). If it has a hard proof, you can forget it
and look it up when you need to.

[1]: [https://www.patrickstevens.co.uk/how-to-discover-the-
contrac...](https://www.patrickstevens.co.uk/how-to-discover-the-contraction-
mapping-theorem/)

------
bluetwo
I wonder if anyone here has a background in instructional design and what they
think of this tutorial.

~~~
hollander
Do you need a background in instructional design to have an opinion on this?
;-)

~~~
bluetwo
No. Not what I'm saying. I'm asking for an option from someone with particular
experience. That doesn't invalidate the options of others, snowflake.

~~~
AnimalMuppet
Drop the "snowflake". Gratuitous personal insults don't belong on HN. You have
a valid point without it. Leave it at the valid point, without the personal
dig.

~~~
bluetwo
Anyone who claims to be offended by an honest question on behalf of some
unnamed party is a snowflake, in my opinion.

~~~
AnimalMuppet
Feel free to hold that opinion. But saying it is counterproductive. It leads
at least some of your readers to dismiss you as a jerk, rather than hearing
your message.

[Edit: Oh, yeah. It's also against HN site guidelines. Saying such things
repeatedly can get you banned.]

~~~
bluetwo
When someone misinterprets a rather clear communication and claims to be
offended by the result, I'm going to point that out. If that offends you,
that's your problem.

Or, look at it as a life-hack: _Don 't waste your time protecting the feelings
of a non-existant group of people._

------
Myrmornis
It should emphasize more that you have to read very slowly, and often re-read
the same section.

------
Ldorigo
This text should be required reading for ANY university-level math course. It
made me go from "ugh, not math again" to "yay, I'm going to learn something
new!" in a matter of days.

------
applecrazy
This advice also applies to more advance mathematics at the high school level,
such as AP calculus BC or even linear algebra.

Source: have followed this technique indirectly for the past few months

------
kenny87
One of the best explanations of the subject that I've seen.

------
nailer
If you can't read the page:

    
    
        window.document.body.style['max-width'] = '650px'

------
Hasz
I cannot stress the importance of working problems yourself enough. There is
literately no substitute.

------
perseusprime11
Mathematics for the million is a great book if you are looking to reignite
your passion for math.

------
iron0013
Thanks for this, everyone should take the time to review this material

------
HaoZeke
So basically.. Mug up everything and hope you have exactly the same thought
process as the course instructor.

This is a disgrace.

In any case. Definitions vary from book to book so mugging them up ad verbatim
would hardly help.

------
HackinGibsons
Love it

------
rublev
I've always wanted to learn mathematics but have no idea where to start, I
have no formal education in anything.

I tried the Ivan Savov books and they seem great but I have no way to judge
the quality of any resources.

Where do I start as a complete beginner? I want books not online resources
like Khan etc.

I would eventually like to get into Geometry but I have no foundation
whatsoever, just basic algebra.

~~~
mkl
Measurement, by Paul Lockhart, may be helpful to start with. It carefully and
clearly derives mathematical principles from basic ideas, in a very
approachable way.

Here's how it describes its content:

Part One: Size and Shape

In which we begin our investigation of abstract geometrical figures.
Symmetrical tiling and angle measurement. Scaling and proportion. Length,
area, and volume. The method of exhaustion and its consequences. Polygons and
trigonometry. Conic sections and projective geometry. Mechanical curves.

Part Two: Time and Space

Containing some thoughts on mathematical motion. Coordinate systems and
dimension. Motion as a numerical relationship. Vector representation and
mechanical relativity. The measurement of velocity. The differential calculus
and its myriad uses. Some final words of encouragement to the reader.

~~~
rublev
I have absolutely no mathematical background.
[https://www.reddit.com/r/mathbooks/comments/1ehupq/totally_d...](https://www.reddit.com/r/mathbooks/comments/1ehupq/totally_different_kind_of_math_book_measurement/)

Here people are saying you need a solid pure math background. I skimmed
through it and it gets complex very quickly. Around page 58 with the formulas
I get lost.

Thanks though, I think I'll be ready for this after Gelfand and Kiselev. Can't
wait.

