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.  

Neurotypicals are the true Black and White thinkers

The non-scientific term ‘black and white thinking’ pops up like a rash all over on the internet, in popular pseudoscience websites about autism and Asperger people. It also appears in articles written by psychologists, and has many synonyms: concrete, literal, all-or-nothing, dichotomous, polarized, and primitive thinking.  

I was shocked when I first encountered the claim that Asberger individuals are limited to black and white thinking; I personally use several thinking processes, including visualization, in order to navigate and understand the world. Each type of processing has its function; one or more processes may be appropriate or useful in any particular situation. More about that later.

Let’s start with some uses of 'black and white thinking' in articles, since writers rarely include a definition. Psychologists conveniently leave out 99.99% of what life and the universe are about: FACTS, a prejudice that effectively erases what an Asperger is. 

From the book – Asperger for Dummies

Concrete “black-and-white” thinking. Many people with Asperger Syndrome don’t understand unspoken rules, due to difficulties with interpreting figures of speech and nonverbal communication. They tend to rely solely on the words themselves. Difficulty with perceiving nonverbal communication causes significant challenges during social interaction. The same characteristic holds true for people with more classic autism. However, this trait tends to be more startling in those with Asperger Syndrome, due to their often superior verbal skills. Despite the fact that many persons suffering with Asperger Syndrome have high IQs, due to challenges in understanding the unspoken rules of employment, they’re greatly challenged in being successful at a typical job.

From this description we must conclude that having “superior verbal skills” leads to black and white thinking and that rules of employment ought to remain undefined and hidden – which is ridiculous; this common practice furthers discrimination, unequal pay and anxiety in employees. Overall the paragraph leaves the impression that neurotypicals are too lazy to acquire verbal competence and that reading body language is the most important qualification for employment, not talent, skills and hard work. As an Asperger individual who has worked in the business sector, “rules of employment” is code for kissing the boss’s ass and spending many hours of each day pretending to agree with those who have power in the hierarchy.

     If “mind / body reading” is so important, why do businesses and individuals rely on contracts? If one must be in the physical presence of another human in order to "read" facial expressions and body language, how did the telephone ever get off the ground? Books? Email? Written letters? Faxes?      

 

Asperger’s Syndrome and Nonverbal Learning Disorder from Orion Academy: California, a school for children with Neurocognitive disabilities. “…are very concrete thinkers, viewing everything dualistically—black and white, all or nothing, good or bad; keep in mind that everything you say to them will be taken literally.”

This statement is breathtaking in its black and white characterization of Asperger people. The use of “everything” and “everything you say” is indicative of black and white thinking. The writer sounds angry; the real point of the mini-tirade is that children are expected to be obedient, but that Asperger children are egalitarian, operate on facts, and have a terrific instinct for lies. 

Black-and-White Thinking
Another associated trait (with Asperger’s) is 'black-and-white thinking.' There are several definitions of this term: The fallacy that anything must be true of either all or none of any group (an instance of the Fallacy of False Dilemma).

1. 1.A tendency to evaluate anything as all good or all bad.
2. 2. A mode of thought concerned with definite, observable facts (e.g., "The sky is blue," "The Eiffel Tower is in Paris") as opposed to vague judgments ("Chewing with your mouth open is inappropriate")

3. A mode of thought that utilizes a scientific theory of the Universe. 

4. Mathematical thinking.

This list contains a huge clue as to why social people mistakenly claim that Asperger individuals exhibit black and white thinking. We operate on FACT, whereas social individuals abhor facts. Many people who are considered to be normal, such as religious fundamentalists, deny that facts exist. Facts are inconvenient because they get in the way of social coercion, manipulation of individuals and groups, and the incessant negotiation of status with the social hierarchy. Lies, not facts are the basis of social interaction.

Asperger's individuals show higher than normal Fluid Intelligence

Superior fluid intelligence (abstract reasoning ability) in children with Asperger’s disorder

• Mika Hayashi^a, Motoichiro Kato^a, Kazue Igarashi^b, Haruo Kashima^a

Abstract

Asperger’s disorder is one of autistic spectrum disorders; sharing clinical features with autism, but without developmental delay in language acquisition. There have been some studies of intellectual functioning in autism so far, but very few in Asperger’s disorder. In the present study, we investigated abstract reasoning ability, whose form of intelligence has been labeled fluid intelligence in the theory of Cattell [Cattell, R. B. (1963). Theory of fluid and crystallized intelligence: A critical experiment. Journal of Educational Psychology, 54, 1–22.], in children with Asperger’s disorder. A test of fluid intelligence, the Raven’s Standard Progressive Matrices Test, was administered to 17 children with Asperger’s disorder and 17 age-, gender-, and FIQ-matched normal children. The results showed that children with Asperger’s disorder outperformed on the test of fluid reasoning than typically developing children. We suggest that individuals with Asperger’s disorder have higher fluid reasoning ability than normal individuals, highlighting superior fluid intelligence.

Asperger's: An Unsought Individuality

Being diagnosed as Asberger's confers a strange unsought individuality.

We exist at the edge of the human domain, not looking inward to an anthropocentric land of incessant squabbles and breathtaking cruelty, but deeply, into the complex manifestation of matter and energy that is our earth. We also look outward to the inhuman universe, whatever it may be. We see what social humans refuse to see. Humans are not the summation of evolution, but one flicker in an emerging image.