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.  

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.

Why Don't Psychologists Like Facts?

Recognizing and embracing fact is necessary to understanding reality, but the complex system of  thought and activity that depends on factual evidence (The Scientific Method) is labeled "cognitive distortion" by psychologists. It is no mystery as to why science education in the U.S. is failing miserably.