Found in 6 comments on Hacker News
psyklic · 2019-01-06 · Original thread
I highly recommend the courses on They are not axiomatic, but they heavily focus on building up deep insight rather than mechanical problem solving --

That said, for a rigorous proof-based approach to high school math, you may enjoy "Basic Mathematics" by Lang:

earthicus · 2018-07-28 · Original thread
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!


oldbuzzard · 2014-10-02 · Original thread
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] The editions basically the same... pick the cheapest


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,"

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

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.

The acclaimed books on Calculus by Michael Spivak

and Tom Apostol

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

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

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.

In light of recent HN threads about Khan Academy,

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?

tokenadult · 2009-08-11 · Original thread
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:

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.

Get that right away.

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

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

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

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

tokenadult · 2009-05-07 · Original thread
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


Miquon Math

Secondary mathematics:

The Gelfand Correspondence Program series


Basic Mathematics by Serge Lang


The Art of Problem Solving expanded series

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

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

Elementary reading:

By far the best initial reading text is

Let's Read: A Linguistic Approach

but there are many other good reading series, including

Primary Phonics


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)

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

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