On page 81 in my book, there is a brief discussion of “feedback loops”. This teaching/learning strategy is so obvious and falls so easily out of any understanding of Polanyi’s tacit theory of knowledge that I am amazed to see it pop up in a recent research study as though a major discovery is being reported. Investigators at Florida State University claim that the “Mathematics Formative Assessment System” (MFAS) applied to kindergarten and first-grade students demonstrated learning gains of six weeks to two months worth of extra instruction. The technique calls for teachers to assess what the students understand, what they do not understand and then offer the right additional learning experiences to fix the missing or error-laden material in the students’ mind. Students are asked to explain their reasoning thus exposing to the teacher any misconceptions or procedural failures on the part of the students as they work on problems.
The real news here is uncovering the practical classroom management techniques needed to manage the formative assessment tasks. Since the MFAS process is different for each student, there is a challenge for teachers to deal one-on-one with each learner and develop the right instructional intervention to solve the cognitive problems experienced by each child. Perhaps a search for additional information on this could be found using the name, Mark LaVenia who was listed as the “methodologist” on the Florida State Team.
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.
A recent blog written by Ian Bogogt offers a few gentle concerns about the new trend for offering “Massive Open Online Courses”. Note the added commentary from readers as well. I see a typical criticism expressed that seems to forget the “Adaptive Learning” approach that can be configured into online education. Without the adaptive part, the material offered in a MOOC does seem much like the same old story where content is shoved at a student to be absorbed sometime other than in a classroom (it’s called homework). Read the text and then come back the next day for class discussion. If the “homework” is in the form of a video it might, for a while, offer content in a fashion sufficiently novel to attract the interest of a student but after another “while” isn’t this just the same thing as studying the content in a text? I guess it is unless you add the adaptive learning part. As the student struggles with the online homework there is instant feedback that comes from the online computer delivering the goodies to be absorbed. Quick questions judge in real time how well the student is grasping the content and equally quick decisions are made by the computer regarding what the student might be having trouble understanding. Now the program branches off to offer special added help to get the student back on track. This is a key feature to the Polanyi theory of tacit knowledge where insight comes after the artful give and take between student and tutor. If a MOOC instructional design fails to offer adaptive learning as a core part of the instruction, then of course all you have is a rather elaborate system for delivering content without feedback; might as well just read a textbook.
I have enjoyed surfing around websites such as Newton’s and Area9 where highly sophisticated computers are solving Benjamin Blooms “two sigma” problem. Back in the mid-nineteen eighties he led efforts to compare regular classroom instruction (lecture and summative testing) to mastery techniques and other specific learning modalities. He concluded that the master/apprentice relationship (completely personalized tutoring) was measurably superior to other educational practices. The problem then became one of expense. Schools cannot afford to hire teachers for each student. He asked for studies combining classroom instruction with combinations of other techniques to sneak up on the superiority of one-on-one instruction. The new adaptive, computerized learning systems are moving dramatically in that direction.
The theory of tacit knowledge fits right in and offers theoretical support to these new learning systems. Check out some of the key features of tacit learning. One is the need for feedback to the student. After identifying some specific error in the conceptual mental makeup of the learner (deduced automatically by the program), the computer can assign the right kind of instructional experience to get the student passed their error in thinking. Tacit knowledge warns us that misconceptions are a severe barrier to forming the desired “tacit integration” and adaptive learning systems are well suited to overcoming those kinds of barriers. The self-paced nature of adaptive learning meets another requirement for formation of tacit knowledge which is the need for incubation time. The mind does not automatically form tacit integration’s on the schedule of a school system but when self-pacing is factored in that incubation time is allowed. I will be exploring how these adaptive learning programs aid in the formation of tacit knowledge further.