HN Books @HNBooksMonth

I highly recommend the courses on brilliant.org. They are not axiomatic, but they heavily focus on building up deep insight rather than mechanical problem solving -- https://brilliant.org.

That said, for a rigorous proof-based approach to high school math, you may enjoy "Basic Mathematics" by Lang: https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/03879...

The comment you replied to received loads of responses, and many of the answers ignored the actual request, and plugged their favorite math books instead. In case you missed it, there was at least one suggestion that gave a real answer to the question: 'Basic Mathematics' by Serge Lang[1]. It's a streamlined but somewhat dry textbook that starts with the properties of arithmetic and ends with precalculus. You might try reading through the table of contents to get a better idea of whether or not it will solve your problem!

[1]https://www.amazon.com/Basic-Mathematics-Serge-Lang/dp/03879...

I found the article rambling and mostly pointless...

However, if you have a weak background in math and want to get up to speed before going into calculus and beyond, I have 2 suggestions.

1) Lial's Basic College Math[1] is adequate and will get you up to speed. 2) Serge Lang's "Basic Mathematics" is great and will cover all you need to go into a rigorous theory based college math class.

[1] http://www.amazon.com/s?ie=UTF8&field-keywords=lials%20basic... The editions basically the same... pick the cheapest

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

⬐ blazespin

The article wasn't pointless at all. He's saying that a conceptual understanding is important, but so are things like flashcards. I know growing up my ability to quickly do simple math and recognizing patterns quickly always helped me speed along quickly.

⬐ goodJobWalrus

she. the authors name is barbara oakly.

⬐ ClashTheBunny

The problem is that unless you are also able to approach things like a constructivist, you will end up with a bunch of disjoint facts. I think that most people just don't understand math well enough to understand this key. The more math I know, the more I wish I hadn't memorized anything, but derived more things. There are many cases these days that I will have to go back and derive an equation for the first time because I was never taught the equation.

Her department is "Industrial & Systems Engineering", she doesn't need to know much more math than how to do arithmetic and a Chai squared. Not that I know more math than she does, it's just not a true part of the mathematical disciplines. She and I probably both just use the tools handed to us and don't understand the beauty or the ugliness. One of the big problems with math today is that nobody understands that arithmetic isn't math. It would be like a baseball player being called a woodworker because she used a wooden bat.

⬐ ivan_ah

Since we are on the topic of math textbooks, I will suggest the No bullshit guide to math and physics which is a math textbook written specifically for adult learners. See http://minireference.com/ for more info.

<discl>I'm the author</discl>

⬐ throwaway6822

I've been looking for something just like this.

Do you plan to release an ePub or mobi-format book? I'd prefer to read it on my ereader, and PDFs don't reflow on smaller screens.

⬐ aluhut

Just get Calibre and convert your pdf to epub.

⬐ ivan_ah

I've been trying to get the .epub working for a long time, but its not easy to convert all the equations and make them look nice. Recently I found some very good new tools[1], so hopefully I'll add .epub/.mobi to the eBook bundle soon.

Do you know of any math books that are available as .epub? I'd like to see how they implement equations... PM me if you would like to be a beta tester.

[1] https://github.com/softcover/softcover

⬐ clarry

It's an interesting looking book, and I'm somewhat inclined to buy it. Bookmarked!

I agree that you don't need to read thousands of pages to learn calculus. However, I don't want to stop at calculus. Basically, what I'd really love to have is a "mother of all maths textbook" -- a thick and heavy tome that compacts information from all the other thick and expensive books (which I'm never going to read) and different fields of mathematics. With enough detail that you can actually learn from it -- so it shouldn't be just for review and looking up formulas you couldn't memorize. I'd like to call it a reference book I can forever keep in my bookshelf (under my bed) and always look in it if I'm unsure about something...

If someone has book suggestions, I'm all eyes.

⬐ dobbsbob

The Princeton Companion to Mathematics is exactly what you are looking for http://press.princeton.edu/titles/8350.html

⬐ maxerickson

I hope you get a well informed answer. There are famous (so to speak) examples of such things:

http://en.wikipedia.org/wiki/%C3%89l%C3%A9ments_de_math%C3%A...

⬐ ivan_ah

I've looked at a bunch of these math compendiums while researching what to include in my book, and this one seemed the best so far: http://www.amazon.com/Mathematics-Content-Methods-Meaning-Do... The writing isn't very hand-holdy, but it covers a lot of important topics, and without too much fluff.

For a more "math for general culture" I'd recommend this one: http://www.amazon.ca/Mathematics-1001-Absolutely-Everything-... which covers a lot of fundamental topics in an intuitive manner.

I have both books on the shelf, but not finished reading through all of them so I can't give my full endorsement, but from what I've seen so far, they're good stuff.

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

That thread recommends many very few good books, but probably mostly books too hard at first for the participant who has posted this new thread.

I'll recommend a couple of books from that thread:

http://www.springer.com/physics/book/978-0-306-45036-5

http://www.amazon.com/Mathematics-Short-Introduction-Timothy...

I agree with the recommendation of An Introduction to Mathematical Reasoning in this thread.

Another participant has already recommended my favorite for background reading, Concepts of Modern Mathematics by Ian Stewart.

http://www.amazon.com/Concepts-Modern-Mathematics-Ian-Stewar...

Get that right away.

Sawyer's A Mathematician's Delight is surely also good (I've read other books by Sawyer).

http://www.amazon.co.uk/gp/product/0486462404/

Read those for background as you get my favorite overviews of mathematics: Basic Mathematics by Serge Lang and Numbers and Geometry by Joseph Stillwell.

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

(Basic Mathematics is mostly high school level math, with a minimum of fuss and bother, and good exercises.)

http://www.amazon.com/Numbers-Geometry-John-Stillwell/dp/038...

(Numbers and Geometry is mostly undergraduate level math, with very good explanations and excellent exercises.)

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.