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?