HN Books @HNBooksMonth

I guess any introduction to real analysis should cover that.

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

Unfortunately, I don't know any online references out of the top of my mind.

What would be a good book series (preferably a classic one that's stood the test of time) on math (Algebra, various Calculus topics, Statistics, etc)?

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.

The publishers will tell you that it is because it is a lot of work to produce new editions every couple of years to keep the textbook up to date.

I'm pretty sure they are lying, at least as far as that explaining the high prices. Here is why I believe this:

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

http://www.amazon.com/gp/product/0471000078

These are Apostol's excellent two volume undergraduate calculus textbook. Volume I is $231.25, and volume II is $189.49.

They were around $22 each when I bought my copies in 1977. I'm sure that NONE of the subsequent price increase is due to revisions and updates, because there have been no revisions or updates. Volume I is still the second edition (from 1967) and volume II is also still the second edition (from 1969). (There has been no need to revise or update them).

Based on inflation, these books should be about $80-90 per volume now (if we assume that $22 in 1977 was a reasonable price).

⬐ WalterBright

Yeah, I bought my copies of I and II used in 1977 for about $10 each. I was shocked at how high the Amazon prices for them have gone.

⬐ nova

> Based on inflation, these books should be about $80-90 per volume now (if we assume that $22 in 1977 was a reasonable price).

And based on the fact that books and other "intellectual" goods may be expensive to create, but only as a one-time investment, and all the years that have passed they should be dirt cheap by now.

In fact, they should be in the public domain, if we had something resembling a fair copyright law.

⬐ textminer

Maddening how so many classic textbooks have new versions every few years that add a few nominal features ("Calculus in the Real World!" "Online Tutorials That're Worse Than What Someone Else Made on YouTube!") with the real goal of shuffling around the exercises so a student can't do his or her homework without having the correct edition. Completely squashes resale markets.

⬐ FuzzyDunlop

Apparently nothing is sacred when it comes to trying to make a quick buck.

I've Silvanus sitting in my shelf, but am yet to look into it yet.

A couple of recommendations (not specific to just Calculus):

- What is Mathematics? (Courant http://www.amazon.com/Mathematics-Elementary-Approach-Ideas-...)

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

- Mathematics from the Birth of Numbers (http://www.amazon.com/Mathematics-Birth-Numbers-Jan-Gullberg...) This book was written by a Swedish surgeon without any background in Mathematics. He started working on this when his son started attending university. A recommended read.

- The Calculus Lifesaver (Adrian Banner). This book is supposed to be a guide for students to crack their exams. But I found the book surprisingly informative. http://press.princeton.edu/titles/8351.html

- Godel Escher Bach. I've read only the first couple of chapters. My interest in mathematics was rekindled to a great degree by Godel and the Incompleteness Theorem. (http://en.wikipedia.org/wiki/Kurt_G%C3%B6del#The_Incompleten...)

- http://us.metamath.org/. The concept alone makes me happy! Metamath is a collection of machine verifiable proofs. It uses ZFG to use prove complicated proofs by breaking it down to the most basic axioms. The fundamental idea is substitution - take a complicated proof, substitute it with valid expressions from a lower level and keep at it. It introduced me to ZFG and after wondering why 'Sets' were being taught repeatedly over the course of years when the only useful thing I found was Venn diagrams and calculating intersection and union counts, I finally understood that Set theory underpins Mathematical logic and vaguely how.

- The Philosophy of Mathematics. From the wiki: studies the philosophical assumptions, foundations, and implications of mathematics. It helped me understand how Mathematics is a science of abstractions. It finally validated the science as something that could be interesting and creative. http://plato.stanford.edu/entries/philosophy-mathematics/

I think the Philosophy of Mathematics should be taught during undergraduate courses that has Maths. It helps the students understand the nature of mathematics (at least the debates about it), which is usually pretty fuzzy for everyone.

⬐ rsanchez1

If you've only read the first few chapters of Godel Escher Bach, you should really set a goal to continue reading. The book is filled with so much good information presented in a digestible format. Topics are slowly revealed throughout the book until you just get it. It's a great experience.

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?

⬐ invalidOrTaken

Having just gone through the entire statistics playlist of Khan Academy (about 10 hours of video) in about a week, I think I can offer some data here. I ran a running instance of clojure next to KA, and I wouldn't go on to the next video until I'd replicated what Sal had done in Clojure. This seemed to be extremely helpful, especially because there are no practice problems for the stats videos.

I think a quick-and-easy way to get feedback is essential. For some lessons there are practice problems, but for others, a student who wanted to maximize learning/minute spent watching video would be wise to at least open up Excel or something.

⬐ squidsoup

This is an excellent idea and an opportunity to kill two birds with one stone (improving my mathematics and learning clojure have been on the cards for a while now). I'm a professional programmer, but thanks to the over-specialisation of the British education system, only did two years of mathematics in high school. Increasingly in my work I find myself struggling with relatively basic statistics concepts and think it's about time to try to educate myself.

⬐ ChuckMcM

I'm sorry but it read a bit much like 'hey if you can visualize it you can learn it' but that is a cruel joke to someone who can't visualize anything. Sort of like laughing at someone for not being able to see the number in a color blindness test.

Some people just don't visualize. Not even a little bit. And I'm not sure its "just because they never learned to." Myself, I've always seen the 'picture' in my head and even dream in full technicolor (like this means anything) but my wife of 20+ years just can't. She is definitely smart, graduated with a CS degree from USC and is a much better planner than I will ever be, but those questions where you see a flat piece of paper with a bunch of dotted lines on it and you need to guess the shape it will be if they were all folded, just can't see it.

When I was growing up I used to think they only put those kinds of questions on tests so that everyone could get a few answers right, they were just that easy for me.

So Newport's thesis that if you can visualize it you can gain 'insight' is no doubt true for some people, but it certainly isn't a panacea for teaching complex subjects.

If you've ever seen the online math courses that Stanford did [1] under the EPGY program, it has some excellent tools that seem to work well for a variety of learning styles. Worth a look, and just down right priceless if you're home schooling your kids.

[1] http://epgy.stanford.edu/courses/math/

⬐ Radix

I think you make an important point, but I'm not sure the post takes any particular focus on visual representation other than a graph is generally an easier way to intuit what a derivative is. His repeated use of the word concept suggests insight for him requires a more general abstraction.

As an aside, why do or did people claim there is visual learning aside from spatial learning? I don't experience visual and spatial imagination as different things. (With reasoning about time always assumed.)

⬐ tokenadult

why do or did people claim there is visual learning aside from spatial learning?

I'm pretty sure that those are distinct neurological processes, as revealed by the differing individual deficits that patients can have after suffering strokes. But I don't have the medical references at hand, and you have certainly seen many sources that combine writing about both, as I have.

⬐ pfedor

[...] other than a graph is generally an easier way to intuit what a derivative is.

Even that is a matter of personal preference. I honestly believe it's easier to get the concept of a derivative by linking it to instantenous velocity.

⬐ psykotic

There are many different ways of thinking about mathematical concepts like derivatives. The more you know, the more deeply you know them, the better.

Here's a random example: Marsden and Weinstein define derivatives in their out-of-print textbook Calculus Unlimited without limits. The tangent to a graph at the point x is the boundary between two line pencils, one of lines entering the epigraph at x, the other of lines leaving. There's no limit-taking of chords. It's a simple and neat definition that connects with classical notions of tangency.

In his essay On Proof and Progress in Mathematics, Thurston lists a dozen other definitions or conceptions of derivatives in his personal arsenal, some very sophisticated. But even those among his definitions that are elementary and have roughly the same scope there is a difference in their psychological affordances, and that can make all the difference.

⬐ ellyagg

Although, as with anything I believe, new information could change my mind, I don't currently think "learning styles" exist in any meaningful sense. Some information or concepts are better learned through, e.g., visual aids, but that's the nature of the information not the learner.

This video by Harvard-educated cognitive psychologist and professor Daniel Willingham is relevant:

http://www.youtube.com/watch?v=sIv9rz2NTUk&feature=playe...

⬐ None

None

To this day I still pull out Apostol's Calculus textbooks. If you're looking to learn calculus really well, or just brush up, these are the ones. http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...

⬐ kenjackson

IMO the theorems in Apostol are tricky reading for most first learning Calculus.. Although probably for the HN crowd are appropriate. But I can't argue with the content or preparation it provides.

I personally prefer Stewart as I think it is appropriate for a wider range of ability, while still being excellent.

⬐ krambs

Ha, yeah. Apostol definitely was a bit painful at first. But it felt like deriving calculus helped me learn better and remember it longer than more rote methods.

⬐ Darmani

Apostol is quite dry and unmotivated. A grad student friend of mine recently wrote a textbook to replace it, and has been testing it on the freshman honors calculus course here.

⬐ gaurav_v

I would strongly suggest Jerry Shurman's Calculus and Multivariable Calculus texts, which are available for free on his website: people.reed.edu/~jerry/

⬐ shadowpwner

This is pretty cool. Are there answers for the exercises?

⬐ gaurav_v

He posts answers to the exercises on his website as the course progresses each year. This year there are no answers because he's not teaching the course; he's on sabbatical.

Thanks for the recommendation. $154 new from Amazon and it was written in 1966. Killer reviews though. Thanks again. http://www.amazon.com/Calculus-Vol-One-Variable-Introduction...

volume 1 of the best series out there...

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

⬐ None

None

I think it would be enlightening if you could provide the textbooks you buy.

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.