Found in 12 comments on Hacker News

ColinWright · 2016-06-07 · Original thread

As always, it depends ...

If you just want to learn the maths relevant to your specific interests then you can simply pick a page on wikipedia, build a tree of topics you need to cover, then start to knock them off one by one, building a web of knowledge as you go. Then ask questions on http://math.stackexchange.com/ making sure you take the advice about how to ask questions the smart way[0][1].

If, on the other hand, you want to get into studying maths generally and build your maths study skills, then I would recommend starting with a really good maths text book and work through it, doing all the exercises, reviewing earlier material, and taking it seriously. Two options are Spivak[2][3] which claims to be about calculus, but is really about analysis, or "Sets and Groups" by Green[4]. The latter is great to create the underlying basic knowledge you need for cryptography, but more, it teaches you how to do maths properly.

You could also just pick something you think is interesting on Khan Academy[5] and go for it.

But having said all that, it's tough to get back into maths, and you need to make sure you really understand your motivation. Most people don't want to write a book, they want to *have written* a book. Most people don't want to study maths, they want to *have studied* maths. If you're not serious, you won't succeed, especially with no one to track your progress, answer questions, and generally encourage, coax, support, and inspire you, it will be tough.

How well motivated are you?

[0] http://www.catb.org/esr/faqs/smart-questions.html

[1] https://www.mikeash.com/getting_answers.html

[2] http://computo.fismat.umich.mx/~fhernandez/Cursos/Calculo201...

[3] http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...

[4] https://www.amazon.co.uk/Sets-Groups-Course-Algebra-Second/d...

playing_colours · 2015-05-08 · Original thread

You can try "Data Science from Scratch" [0] to get some taste. It uses Python to teach essentials of data science, and ML altorithms. The code quality is very good, and there is an introduction to Statistics, Maths and Python to start.

Then you can continue with improving your maths (Linear Algebra [1], Calculus [2], [3]) and moving on with Statistical Learning [4] [5]. I am personally going now through this plan.

[0] http://shop.oreilly.com/product/0636920033400.do

[1] http://www.amazon.com/Linear-Algebra-Right-Undergraduate-Mat...

[2] http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...

[3] http://www.amazon.com/Calculus-Manifolds-Approach-Classical-...

ludicast · 2015-03-12 · Original thread

I agree. As far as calculus goes, I am more enamored with books like Spivak's (http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...) that take a proof-centric approach to teach calculus from first principles.

Incidentally, for those who want to learn linear algebra for CS in a mooc setting there are 3 classes running at this very moment:

https://www.edx.org/course/linear-algebra-foundations-fronti... (from UT Austin)

https://www.edx.org/course/applications-linear-algebra-part-... (from Davidson)

http://coursera.org/course/matrix (from Brown)

The first 2 use matlab (and come with a free subscription to it for 6 months or so), the last python. One interesting part of the UT Austin class is that it teaches you an induction-tinged method for dealing with matrices that let you auto-generate code for manipulating them: http://edx-org-utaustinx.s3.amazonaws.com/UT501x/Spark/index... .

And of course there are Strang's lectures too, but those are sufficiently linked to elsewhere.

tokenadult · 2013-09-22 · Original thread

I'd like to edit this some more during the edit window for this comment. To start, the books by Israel M. Gelfand, originally written for correspondence study.

http://gcpm.rutgers.edu/books.html

http://www.amazon.com/s?ie=UTF8&field-author=Israel%20M.%20G...

An acclaimed calculus book is *Calculus* by Michael Spivak.

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...

Also very good is the two-volume set by Tom Apostol.

http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...

http://www.amazon.com/Calculus-Vol-Multi-Variable-Applicatio...

Those are all lovely, interesting books. A good bridge to mathematics beyond those is *Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach* by Hubbard and Hubbard.

http://matrixeditions.com/UnifiedApproach4th.html

A very good book series on more advanced mathematics is the Princeton University Press series by Elias Stein.

http://www.amazon.com/Fourier-Analysis-Introduction-Princeto...

Is this the kind of thing you are looking for? Maybe I can think of some more titles, and especially series, while I am still able to edit this comment.

tokenadult · 2011-03-23 · Original thread

Cal Newport, the author of the submitted blog post, draws comments both here on HN and on his own blog pointing out that deep understanding of a subject doesn't necessarily equate to VISUAL thinking about a subject. There is a big literature on "learning styles" and some attempts by some schoolteachers to categorize children by what their preferred learning styles are. When I have taken learning style questionnaires, and when I have asked my wife (a piano performance major and private music teacher) about this, the answer on learning styles is "all of the above." I personally think, based on my observations of successful learners of a variety of subjects, that learning styles are themselves learnable, and a learner with a deep knowledge of a particular subject will know multiple representations of that subject. My wife has had many piano performance courses, and also music theory and ear training courses, and has learned visual representations of music both in the form of standard musical notation and in the form of "music mapping,"

http://www.amazon.com/Mapping-Music-Learning-Teachers-Studen...

which she has found very helpful.

As for mathematics, the subject I teach now, I have always cherished visual representations of mathematical concepts, for example those found in W. W. Sawyer's book Vision in Elementary Mathematics

http://www.amazon.com/Vision-Elementary-Mathematics-W-Sawyer...

http://www.marco-learningsystems.com/pages/sawyer/Vision_in_...

But other mathematicians who taught higher mathematics, for example Serge Lang, recommended memorizing some patterns of multiplying polynomials by oral recitation, just like reciting a poem.

http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...

The acclaimed books on Calculus by Michael Spivak

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...

and Tom Apostol

http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...

are acclaimed in large part because they use both well-chosen diagrams and meticulously rewritten words to deepen a student's acquaintance with calculus, related elementary calculus concepts to the more advanced concepts of real analysis.

Chinese-language textbooks about elementary mathematics for advanced learners, of which I have many at home, take care to introduce multiple representations of all mathematical concepts. The brilliant book Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States by Liping Ma

http://www.amazon.com/Knowing-Teaching-Elementary-Mathematic...

demonstrates with cogent examples just what a "profound understanding of fundamental mathematics" means, and how few American teachers have that understanding.

http://www.aft.org/pdfs/americaneducator/fall1999/amed1.pdf

http://www.ams.org/notices/199908/rev-howe.pdf

Elementary school teachers having a poor grasp of mathematics and thus not helping their pupils prepare for more advanced study of mathematics continues to be an ongoing problem in the United States.

http://www.ams.org/notices/200502/fea-kenschaft.pdf

In light of recent HN threads about Khan Academy,

http://news.ycombinator.com/item?id=2348476

http://news.ycombinator.com/item?id=2350430

I wonder what Khan Academy users who also have read the submitted blog post by Cal Newport think about how well students using Khan Academy as a learning tool can follow Newport's advice to gain insight into a subject. Is Khan Academy enough, or does it need to be supplemented with something else?

rsheridan6 · 2011-02-25 · Original thread

Most books have problems and most have solutions guides for about half of the problems. I'm surprised nobody's mentioned Spivack yet:
http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...

katovatzschyn · 2010-11-02 · Original thread

```
Anybody have insight into how to actualize these
nuggets into some semblance of a self-learning course?
```

Buy Calculus by Micheal Spivak. Solve at least one problem every day. Make it ritual and a daily requirement. Watch MIT lectures for corresponding chapter you are on.To learn this, don't trouble over the path and reason at present. Buy the book and start. Right now.

http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098...

Buy it. To learn this- buy it and start. Right now.

jimmyjim · 2010-07-21 · Original thread

Spivak's Calculus, http://www.amazon.com/Calculus-4th-Michael-Spivak/dp/0914098... is a good bet. But it's very theory-involved.

tokenadult · 2009-06-14 · Original thread

"A textbook author in Toronto made enough money from his calculus textbook to afford a $20 million house."

I know that calculus textbook.

http://www.amazon.com/Calculus-Early-Transcendentals-James-S...

It's workmanlike, but certainly not inspiring like Spivak's textbook.

http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098918/

tokenadult · 2009-05-07 · Original thread

Interpreting that as a request to name the textbooks I find useful, I'll do that here.

Elementary mathematics:

Primary Mathematics

http://www.singaporemath.com/Primary_Math_s/21.htm

and

Miquon Math

http://www.keypress.com/x6252.xml

Secondary mathematics:

The Gelfand Correspondence Program series

http://www.amazon.com/Algebra-I-M-Gelfand/dp/0817636773

http://www.amazon.com/Functions-Graphs-Dover-Books-Mathemati...

http://www.amazon.com/Method-Coordinates-I-M-Gelfand/dp/0817...

http://www.amazon.com/Trigonometry-I-M-Gelfand/dp/0817639144...

http://www.amazon.com/Sequences-Combinations-Limits-Library-...

and

Basic Mathematics by Serge Lang

http://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/038796...

and

The Art of Problem Solving expanded series

http://www.artofproblemsolving.com/Books/AoPS_B_Texts_FAQ.ph...

When a student has those materials well in hand, it is time to work on AMC and Olympiad style problem solving,

http://www.artofproblemsolving.com/Books/AoPS_B_CP_AMC.php

http://www.artofproblemsolving.com/Books/AoPS_B_CP_Olympiad....

and also the best calculus textbooks, such as those by Spivak or Apostol.

http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098918/

http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...

http://www.amazon.com/Calculus-Vol-Multi-Variable-Algebra-Ap...

Elementary reading:

By far the best initial reading text is

Let's Read: A Linguistic Approach

http://www.amazon.com/Lets-Linguistic-Approach-Leonard-Bloom...

but there are many other good reading series, including

Primary Phonics

http://www.epsbooks.com/dynamic/catalog/series.asp?seriesonl...

and

Teach Your Child to Read in Ten Minutes a Day

(I devote more time than that to reading instruction, typically, because I use multiple materials)

http://www.amazon.com/Teach-Your-Child-Read-Minutes/dp/14120...

and quite a few others. There is more junk than good stuff among elementary reading materials, alas.

mechanical_fish · 2008-09-09 · Original thread

A snapshot of my bookshelf's "math" section, which really hasn't changed much since I was in high school and hadn't taken calculus:

W.W. Sawyer, *What is Calculus About?* and *Mathematician's Delight*

Courant and Robbins, *What is Mathematics?*

Hogben, *Mathematics for the Million*

Steinhaus, *Mathematical Snapshots*

Ivars Peterson, *The Mathematical Tourist*

Davis and Hersh, *The Mathematical Experience*

Polya, *How to Solve It*

Huff, *How to Lie With Statistics*

McGervey, *Probabilities in Everyday Life*

Raymond Smullyan: *The Lady or the Tiger*, *Alice in Puzzle-Land*, others

Anything by Martin Gardner. I happen to have picked up *Mathematical Magic Show* and *Mathematical Circus*, but I'm sure there are many other collections.

I also recommend cryptography stuff. David Kahn's *The Codebreakers* is not really a math book, but it is awesome and it stars mathematicians, as does Simon Singh's *The Code Book*. You could read Schneier's *Applied Cryptography*.

This is HN, so I would be remiss if I didn't point out that you can learn a lot of fun *and* useful math by reading SICP, Knuth, or any good algorithms book.

If anybody out there knows a good, spirited statistics book addressed to someone who knows calculus, tell me. I keep planning to go through *Fundamentals of Applied Probability Theory* but I never get around to it; see "Related Resources" here:

http://ocw.mit.edu/OcwWeb/Electrical-Engineering-and-Compute...

Having said all of that: I have a Ph.D. in physics/EE, so I've *got* to tell you, if you haven't tried calculus you haven't lived. ;) I'm not sure how to go about learning calculus in a *fun* way for a *mathematician* -- I took fairly standard first- and second-year college courses in calculus and physics and learned it that way. The folks on Amazon seem kind of enthusiastic about Spivak:

http://www.amazon.com/Calculus-Michael-Spivak/dp/0914098918/...

Many consider the book's presentation of the topic utterly beautiful, bordering even on the spiritual.