Problems with Visual vs. Abstract Thinking: Calculus

Examples from my college days preserve vividly the difference between those who think visually and those who are adept at abstract thinking. Although visual 'talent' as it was sometimes referred to, was a big bonus for studying geology, from geomorphology (landforms), to mineralogy (the crystal classification system) to structural geology (deconstructing relationships in space and time), in order to earn a degree, geology students were required to pass three semesters of Calculus and Calculus-based physics. I am not an abstract thinker; once mathematics leaves 'Concrete World' my visual mind simply goes blank - literally.  

No problemo! My visual brain could spin the wooden models that (no longer) are used to learn crystal forms, while other students were on the verge of emotional breakdown when tasked with this test. 

First panic, then strategy: I signed up for Calculus based physics before taking a Calculus course. I had to 'see' what Calculus looked like in the real world. It worked; I passed physics by sacrificing a few points on the mathematics while storing equations as whole images, and giving them names like, "the double violin with a thingy under the roof" and attached the image to a process for solving the equation. I had no idea what the equation described mathematically, but I could usually solve it. Calculus texts aid this approach since each chapter is organized by groups of similar equations.

 

I had no problem with the math that describes geologic processes like stream flow, because I could easily see the realization in nature. So, physics itself was understandable as it applies to the familiar world of motion, energy and behavior of objects, but getting through Calculus as a foreign language was like decoding signals from a very advanced and very alien civilization. 

Graphs! A visual window into the abstract language of mathematics.

From Paul's online math notes: http://tutorial.math.lamar.edu
"Now, let’s take a look at just how we could possibly get two tangents lines at a point. This was definitely not possible back in Calculus I where we first ran across tangent lines. A quick graph of the parametric curve will explain what is going on here. So, the parametric curve crosses itself!  That explains how there can be more than one tangent line.  There is one tangent line for each instance that the curve goes through the point."

 

For me graphic images, which can be quite a distraction due to their beauty, do help, but in order to 'do math' I had to approach maths as foreign languages. The analogy is fitting: one can learn the forms and grammar of another language, but never become fluent, nor speak it like a native.  

Visual Thinking: Neurotypicals Have No Clue

Neurotypicals have no clue as to how visual thinkers think. This is easily demonstrated. Visual thinking is achieved WITHOUT WORDS. That's why it's VISUAL thinking. Visual thinking therefore takes place without concepts. It's  concrete. Many animals are visual thinkers. To read a description, please go to Post 4/10/14 - Childhood: A Female Asperger Mystery 

1. Arranging concepts graphically is not

visual thinking.

              

2. Arranging concepts by drawing squiggly lines instead of straight lines is not visual thinking.              

  3. Drawing words instead of using type fonts is not visual thinking.

 Visual thinking starts with the

one-to-one correspondence of memory with visual information. 



Visual perception and visual memory are very old in evolution, and are considered primitive by people who believe that verbal language is superior. However, images are packed full of information about the environment that word language simply cannot duplicate.
Illustration from Don Hoffman, Visual Intelligence 

My experience of 'the world' is predominantly visual. My information processing is unconscious. I didn't have to learn to think visually, it's normal for me. Despite scoring high on verbal aptitude tests, I must consciously translate my visual thoughts into written language; it's easy if I'm talking about objects and processes and not concepts, which can have little meaning. 

Translation from visual to verbal description is exhausting, like having your dominant hand tied behind your back and being told to write a journal entry each day for the rest of your life, and that you are only allowed a vocabulary of  10 words. Visual vocabulary is HUGE and common verbal language is so limited and distant from concrete-visual reality, that communication, is at times, nearly impossible and it's best to walk away.  

Neurotypical language does not describe physical reality, it expresses ideas about reality, most of which are incorrect. 
While necessary for functioning in modern social environments, verbal language has become so nonspecific in everyday use that it no longer connects with anything real; it's devoid of content and meaning. What passes for communication is the repetition of empty phrases and the recitation of supernatural scripts. In visual thinking, the image is the content and meaning.

The social demand to use and respond to verbal-written language has diminished visual processing in modern humans; the process is there, but has atrophied; brain resources are redirected or subsumed by words. Visual thinking was likely negatively selected during human domestication. Evolution has a rule: What you don't use, you loose. Research does confirm that typical humans are very poor at visual discernment and memory. The remnant population for whom thinking is predominantly visual is very small. I think the world had better hang on to us before we disappear.

From Don Hoffman, author Visual Intelligence, UC Irvine 2004.

"This seminar is a highly illustrated and accessible introduction to visual intelligence, informed by the latest breakthroughs in vision research. Perhaps the most surprising insight that has emerged from vision research is this: Vision is not merely a matter of passive perception, it is an intelligent process of active construction. What you see is, invariably, what your visual intelligence constructs. Just as scientists intelligently construct useful theories based on experimental evidence, so your visual system intelligently constructs useful visual worlds based on images at the eyes. The main difference is that the constructions of scientists are done consciously, but those of your visual intelligence are done, for the most part, unconsciously."