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,"
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.
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.
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?
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.
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.
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:
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.
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.)
[...] 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.
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.
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.
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?