How we can learn math, part I

Do you think in words? Is it in your native tongue? I know when I have certain thoughts, it's as if I'm talking to myself in my mind, but that's not always the case. One of my old college German professors told me that I will have turned a corner in my study of a foreign language when I no longer need to mentally "translate" the language, but rather begin thinking in the new language itself. Back then I agreed with him, and I still see his point, but not ALL thoughts, at least mine, are in any particular language at all.

This is most evident when I am working with technical information such as a math equation or designing a piece of furniture. When engaged in this type of activity, which I would include the act of teaching, itself, words and language are entire absent from my thoughts. I am consumed with forms and shapes and esoteric images that help me to process the information in the dark corner of my mind. Apparently, I'm not alone in this capacity.

Einstein, himself, wrote:

Words and language, whether written or spoken, do not seem to play any part in my thought processes. The psychological entities that serve as building blocks for my thought are certain signs or images, more or less clear, that I can reproduce and recombine at will.

Jacques Handamard, French mathematician and author of the book, The Psychology of Invention in the Mathematical Field (1945), a ground-breaking publication in which he analyzes what mathematicians think of when they actually do mathematics, said this of his own methods:

I insist that words are totally absent from my mind when I really think . . . even after reading or hearing a question, every word disappears the very moment that I am beginning to think it over.

If this is a common trait shared by those with an rational, analytical mind, that is, by those who are "good" at mathematics, then it should be surprising why most great mathematical minds cannot adequately describe their internal methods for their mental calculations.

One person who actually CAN describe how he accomplishes his remarkable mental calculations is Daniel Tammet, a 29 year-old Autistic Savant. His descriptions of his thoughts are also devoid of words or language, but instead, are full of unconventional imagery, consisting of shapes, colors, emotions, and even tastes! Known as Synaesthesia, in his mind, he says, each number up to 10,000 has its own unique shape, color, texture and feel. He can intuitively "see" results of calculations as a synesthesic landscapes without using conscious mental effort, and he can "sense" whether a number is prime or not. He has described his visual image of 289 as particularly ugly, 333 as particularly attractive, and pi as beautiful (he has event painted it--see picture above.) 6 apparently has no distinct image. In earning the European record for recitation of the decimal digits in pi (22,514 digits in 5 hours), he merely told the "story" of the synesthesic landscape he was seeing.

Although his feats are extraordinary, and attributed to his savant "disorder," his methods do offer insight to how the brain works, and possibly reveal some methods that extends well beyond mnemonics that the average person can employ to increase his/her mathematical abilities.

It would appear that analytical thought is independent from linguistic thought. Most mathematicians, physicists, chemists, inventors, etc. who have discovered something great, have not done it in one sitting. In fact, as many describe it, it came after months or years of careful study and analysis, during which time the solution eluded them. The solution, itself, often came rather unexpectedly, through no deliberate, conscious thought process. This is how Kekule discovered the Benzene ring configuration and how Gould invented the Laser. These "flashes of brilliance" were nothing of the sort. Their mind was prepared for a solution, and it processed the seeds of study and diligence to produce the fruitful discovery.

Now I'm no Einstein, Handamard, Tammet, Kekule, or Gould, but I am a math educator interested in reaching more students. So I ask the question: do any of these scenarios offer any insight into why so many students struggle relentlessly with mathematics and how to remedy it? How can we educators communicate, not only about the results of mathematical thought, but about the thought processes itself if we don't think about it linguistically? How can we concretely communicate about the abstract? How can we teach beyond the symbols, but reach to the fundamental mental abstractions of mathematical objects and ideas themselves?

It requires us to "invite" our students into a "fake reality" where things are strange and unfamiliar, a realm of pure abstraction and imagination. We can do it! We as humans have developed the capacity to visit this world, in which there is no direct or simple link to our real world. We cannot touch thing with our hands. We cannot smell things with our noses. We cannot see with our eyes, nor hear with our ears. We can only escape within our own mental landscapes and explore the terrain. This is a strange land for many, but as with anything, with frequent visits, it becomes less fearful, more comfortable, and even enjoyable.