The reality of unconscious mental processing leading sometimes to the Aha! moment is increasingly recognized by the scientific community. Research attention is shifting to conducting experiments with an aim to understand more about this effect. A study just published in Thinking & Reasoning (2017) is called “Insightful solutions are correct more often than analytic solutions”. The authors (Salvi, Kounios, Beeman, Bowden, and Bricolo) conducted a variety of puzzles comparing insightful solutions to analytically determined answers. They were able to show that those answers coming into consciousness from unconscious processing were more likely to be correct than answers constructed through conscious reasoning. A timed deadline was imposed on each puzzle which created anxiety among test subjects. A majority of answers tossed out just before the imposed deadline appeared to be either a guess or analytically derived and often wrong. The authors conclude that encouraging insightful solutions to learning and problem solving requires a relaxed atmosphere where the factor of time is not present. The need for incubation time is mentioned in my book on page 56 where an argument is made that formal education is too often run by the clock thus imposing a deadline on thought. As this research report indicates, deadlines short circuit the use of intuitive processes hence sometimes leading to wrong solutions.
How often are we accustomed to saying that a book was better than the movie? Srinivasa Ramanujan, a young Indian clerk is effectively profiled in a 2016 film called, “The Man Who Knew Infinity” based on a book of the same name written by Robert Kanigel. The book is indeed better than the movie, the reason being that in 373 pages of text Kanigel has room and time to plunge into a really complex problem. The puzzle of intuition, examined in my own book using Michael Polanyi’s tacit theory of knowledge as a guide, collapses into an enigmatic cognitive black hole when applied to the task of explaining Ramanujan’s mental magic. The finest mathematicians of our century admit defeat. Ramanujan’s capability is, to a degree, inexplicable. The movie skillfully illustrates the problem but fails to give a viewer the full perspective needed to understand the significance. We watch Ramanujan, played magnificently by Dev Patel, making an effort to earn a living for himself, a new, young wife, and his mother. Unfortunately, details of his early schooling are missing from the film. Throughout the movie, we watch Ramanujan rattling off seemingly complicated formulas. However, where did they come from? His mentor Hardy (played by Jeremy Lyons) not only wants to know but reasonably insists they be accompanied by formal, logical proofs. We are not informed that Ramanujan was provided access to several key books while a pre-college student. The first was a text on trigonometry that by age 13 Ramanujan had fully absorbed. The other was an improbable tutor summary of all the mathematics needed to pass a formidable Trinity College exam at the time. A first Ramanujan mystery presents itself. Why was he so singularly enthralled with mathematics to the degree of obsessively studying the 5,000 formulas in the exam preparation book? Scholars have subsequently poured over this book trying to find within its pages a pathway to his eventual brilliance. Aside from its expectation that a reader should prove all the assertions displayed therein, analysis of the books contents fail to display how Ramanujan might have extracted divine inspiration from its pages. Perhaps we may infer that Ramanujan received, in effect, an undergraduate degree in mathematics through his self-studies and some unidentified math courses taken at local colleges (before flunking out over a refusal to bother with any courses other than math). Today we apply the term “grit” to describe his unyielding passion.
The movie essentially begins with Ramanujan filling his notebooks with conjectures and theorems of a highly original nature. These gained the attention and admiration of Indian mathematicians leading to his introduction to Professor Hardy at Trinity College in England. We watch his arrival in England and subsequent relationship with Hardy. The film makes much of the drama surrounding Ramanujan leaving his wife at home in India but this story is probably more Hollywood than reality.
The primary mystery of Ramanujan is the source of his inspiration. Kanigel in his biography of Ramanujan describes a warning by the great mathematician, Jacobi, that excessive control over the form and pace of learning will constrain the future capacity of the learner to be creative. Ramanujan escaped that formal control by exploring mathematics on his own filling notebooks without reference to the usual problems listed at the end of textbook chapters. This characteristic of Ramanujan’s early process of learning offers a hint toward our understanding of his creativity. However, it is not enough.
The movie then gives us a sketchy and truncated version of how Ramanujan ended up at Trinity College. Most of the film shows us the problems he faced there and his interesting intellectual and semi-friendship with Hardy who skillfully managed his transition from a largely self-taught genius to a gifted mathematician capable of explaining and proving his assertions. During the sparring matches between Ramanujan and Hardy (the highlight of the film), Hardy asks in exasperation just where the magical-seeming assertions came from. Ramanujan mysteriously attributes his wisdom to whispers from his family goddess or dreams planted in his head by the same divine source. The ever practical and atheistic Hardy simply cannot understand such mystical claims. I continue to search within the tacit theory of knowledge developed by Michael Polanyi and the related new neurological research on avalanches of neural networks lurking underneath our conscious scenes for an answer. Meanwhile, I highly recommend the film and then more eagerly recommend the book.
Tacit knowledge is all about generating the “AHa” effect which embeds understanding into a more intuitive form of cognition than analytically encoded symbolic notation. I am always happy to see someone working hard to accomplish that as Kalid Azad does in his mathematics teaching blog. He complains (correctly) that most explanations for difficult ideas are offered in a logic based format (seems the best way by experts) rather than through a holistic approach. I recall as a former professor of physics the standard approach to teaching a topic in physics was to begin with an equation and show how that equation is derived. Students typically find the derivation quite baffling and may desperately memorize the steps. An alternative is to offer some exposure to examples of how the physical system behaves. Throw in a few problems, and then later work through the derivation when the learner has some grasp of what the topic is all about. Watch how Azad explains topics in mathematics to see an example of how this is done. He points out how a particular explanation may not work for you when a different one does so you might need to cast around looking for just the right kind of explanation to suit your own prior understandings. Kalid has several excellent books on Amazon to help anyone learning math develop an intuitive sense of what it all really means. Check out the calculus book here. His books make an excellent present for grandkids taking STEM in high school and college. The students themselves are not likely to realize they need his books but a parent and grandparent could help them get ahead by making sure the kids have these books to get them ahead in their education.