The person who truly taught me to teach mathematics in a deeper and more productive way is W. Gary Martin.
He showed me, by example, how to stop talking so much about my own understanding, and listen carefully to student understanding.
In particular, he demonstrated for me how, when a student is presenting their own problem solution to the class, to pay careful and concentrated attention to whether other students were understanding.
Over the years, as a teacher educator in the UK and the United States, I have helped student teachers, as well as practicing teachers, to focus less on their own explanations and focus more on what their students understand. Less talk and more listening.
This point was raised again in the April 6, 2011 GenerationYes blog: part 4 of 4 of a series on Khan Academy.
The blog post cites Gary Stager as saying that “anytime you go to ‘help’ a learner, pause and think about whether you are taking away an opportunity for them to learn it themselves.”
Gary Stager summarizes this as “Less us, more them.”
Many years ago at a National Council of Teachers of Mathematics conference a well known mathematics educator, now a distinguished Professor and Dean of a major College of Education, showed a video of a student and herself. In the video the student showed how to find one-half and then one-fourth of a cookie.
The student paused when asked to find one-third of the cookie. The well known math educator explained to the student that it might be easier to replace the cookie by a candy bar and proceeded to approximately mark a drawing of a candy bar into thirds.
The student who to that point had been quite talkative, now remained silent.
I was so struck by the inappropriateness of this “showing” that I dubbed this type of instructional interference with a student’s thought processes as “cognitive theft”.
The math educator’s intervention did not solve the original problem, did not even show how to find one-third of a candy bar, and took away – stole – from the student the chance of continuing to think about the problem.
Well meaning, I suppose, but who wants a well-meaning brain surgeon?
Equally, who wants a well meaning teacher stealing from students the opportunity to think?
The GenerationYES blog discusses perceived failings in the Khan Academy’s instructional videos from many perspectives.
These videos are not directly stealing the opportunity for students to think, but they are emphasizing mathematics as a bunch of techniques that one learns by being shown by one’s teacher.
This seems, to me, to promote a passive consumer attitude to learning mathematics: a show me how to do it” attitude.
Whereas learning mathematics usefully, productively, flexibly and deeply requires active participation in solving problems.
The “show it to me” attitude is traveling dangerously close to cognitive theft.
In the long-run it is indeed cognitive theft: a student is stealing from themselves the opportunity to learn deeply, and, as it turns out, more joyfully.