In a post on Devlin’s Angle, Dr. Keith Devlin differentiates between “instruction/training and teaching/learning.” As near as I can tell, instruction consists of demonstration, drill, and testing and teaching consists of interaction, collaboration, and deep understanding. This is not a dichotomy. One person’s instruction could be another person’s teaching and vice versa. It depends on the sophistication of the observer and the learner.

Apparently Dr. Devlin remembers his early educational experiences as trivial if he remembers them at all. He observed a pattern, learned it easily and went on with his young life. Of course. He was in incipient mathematician. He may have had good teachers, as he admits, but the entirety of the teaching experience didn’t take, just the math skills and knowledge. Teaching techniques (instructional techniques) need to be age appropriate. A seven year old child’s abstraction abilities are limited. Dr. Devlin notes later in his commentary that he only really learned calculus when he had to teach it. Again, of course. At that point he was well-advanced in mathematical sophistication, had seen calculus employed in other areas of math and presumably science and had a personal and professional need to really know it.

Dr. Devlin says he has seen really good teaching/learning. I assume he observed other instructors (I mean teachers) and saw attentive students who asked good questions and appeared to be following the teacher’s organized, incisive arguments and examples as they responded to their teacher’s Socratic questions. I have seen this myself and even had a few days in the classroom like that. The characteristics that I have just listed of a “good” class session are generally seen as characteristics of good teaching, but, I say, not necessarily of good learning. My students’ performance on their quiz the next day does not always correlate with the previous “good” teaching day. Good teaching, at least in a classroom setting, needs to be followed up with individual student work that helps them form their own model of the subject at hand. This could include routine (drill) type problems and also problems that probe the concept from other directions. Learning happens inside the brains of the students as they work their way through the material. Dr. Devlin references the video of a girl Cena learning or not learning place value of decimal numbers. The little girl answered correctly in a classroom setting using for all we know unconscious cues but in the interview later she could not reproduce her apparent understanding. One could say that Cena had not been asked the question enough times and in enough ways. Not enough drill. Also the physical model that the interviewer was using may have foreign to Cena.

Dr. Devlin will be teaching a online course next fall. My advice, as if he needs any, would be to form lectures on his own unique understanding of the material (his own models) (or why teach the course) and to design exercises and examples of varying intensity that cause his students to build their own mental models. V.I. Arnold, according to Maxim Kazarian and Ricardo Uribe-Vargas in the introduction to Dynamics, Statistics and Projective Geometry of Galois Fields by V.I. Arnold, believed that ‘examples teach more than a formal proof.” also, if Dr. Devlin wants to change individual student’s brains, he might consider discouraging collaboration. If not, he will get plenty of papers with similar answers but the students will likely have very different levels of understanding.

Speaking of good teaching, I still remember a class period conducted by George Polya at Stanford on generating functions. A concept, by the way, that I used with ease just a few weeks ago. The generating function idea seems to have taken at that moment some 45 years ago, but I am not sure. Over the years I have read papers and books where such functions were employed and of course my mathematical knowledge has increased somewhat since those days. So I am not sure when I really learned about the modeling power of generating functions.

Thanks to Nick Chura for pointing out Dr. Devlin’s stimulating post.

“examples teach more than a formal proof.”

Very true. Generally speaking, mathematicians first discover mathematical truths inductively, by so called playing around or exploring, and then later try to enter the information in a deductive scheme as the end product. Unfortunately we teach math backwards. We present the end product and expect students to find examples on their own. This is like staring at a masterful architectural building and having no clue as to how it was constructed.