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I recently saw an episode of 60 Minutes about Jacob Barnett, a 13 year old boy who is currently attending advanced physics classes at a local university and was portrayed by the show as being a child prodigy/genius. In the interview, he demonstrates how he's able to perform mathematical transformations in his head by visualizing the numbers as colored shapes. As he explains in the linked clip:

This is 2 * 27: This is 2x27

He visualizes multiplying 2 by 27 as a series of colored triangles.

Is this a known phenomenon, and if so, does it have a name? Do other individuals visualize mathematics like this? Or is this just a 12-year old kid... embellishing (to be polite) his thought process?

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As background context, there is a more general literature on the important role of visualisation in learning mathematics. A Google Scholar search for "visualisation in learning mathematics" yields many interesting articles including this and this. –  Jeromy Anglim Jan 27 '12 at 0:20
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I can "see" some numbers as colors.. though It looks like the correct term is synesthesia, I've never heard of it.. –  user1135588 Jan 27 '12 at 16:11
    
Arent they all generally just called Savants? –  abhiii5459 Apr 17 '12 at 7:27
    
@abhiii5459 not really. "Savant" would be a good description of Jacob Barnett, but I was wondering if other individuals visualize mathematics like he described. They don't have to be savants. Also, the term "savant" isn't specific to visualizing numbers. –  Josh Gitlin Apr 17 '12 at 11:41
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up vote 16 down vote accepted

Sounds like a form of Synesthesia, in particular it sounds like Number Form Synesthesia mixed with Grapheme-color Synesthesia:

A number form is a mental map of numbers, which automatically and involuntarily appears whenever someone who experiences number-forms thinks of numbers. Number forms were first documented and named by Francis Galton in "The Visions of Sane Persons". Later research has identified them as a type of synesthesia. In particular, it has been suggested that number-forms are a result of "cross-activation" between regions of the parietal lobe that are involved in numerical cognition and spatial cognition

It doesn't seem exactly like what he does but it could be a form of it. It's hard to imagine just how it would be to perceive the world as a synesthesic person does.

It sounds like he's learned to apply his visual-spatial skills to "arrange" the number forms to aid his calculations.

Synesthesia is covered in detail in the book Synesthesia: a union of the senses, though I'm not aware of any specific case studies on individuals with this exact set of skills.

A term to describe people with Synestesia is Synesthesic.

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Wow, that's fascinating. It's also really interesting that you state It sounds like he's learned to apply his visual-spatial skills to "arrange" the number forms to aid his calculations because this occurred to me after hearing him explain. I thought, Hmm, this sounds like how a GPU can perform certain tasks much better than a CPU! –  Josh Gitlin Jan 26 '12 at 21:29
    
Reading Wikipedia further, it appears that Richard Feynman had Synesthesia. He may not have all the same skills as Jacob, but given that he was a Nobel prize winner, he was clearly a very intelligent man! –  Josh Gitlin Jan 27 '12 at 20:27
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I have number Synesthesia, actually, and numbers do indeed appear as a specific pattern inside my head -- I have never not seen numbers this way, and have even drawn my number map for my own edification and to show people what I'm talking about when I try and explain my synaesthesia to them.

Numbers are not individual digits to me; they are one huge entity. I am the opposite of this young man though -- I couldn't calculate myself out of an open door! In fact, I have significant deficiencies when it comes to numbers, math, and equations. It may as well be a foreign language, seriously.

On the other hand, I am in the gifted range of intelligence (not being pompous; just putting my Synesthesia in context) and have an almost freakish ability to remember details, names, faces, and facts.

I also have some characteristics of perfect pitch and recall music in the correct key. Music and numbers are related, and perfect pitch is within the scope of Synesthesia (some argue it is a form of Synesthesia; others believe it's not).

So, yes, the term is Synesthesia, and there are, I believe, five basic kinds.

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I have observed in several people that synesthesia can be an obstacle to learning procedures on the objects involved. One reason is that the procedure does not blend harmoniously with the inner perception. For example, most way of multiplying involve decompositions and intermediates whose synesthetic perception may clash with the view of the starting numbers and the final result. You should not consider computation a foreign language forever but try to relearn small numbers add and multiply from the very beginning to make them match your own personal perceptions. –  ogerard May 7 '12 at 4:46
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