When we attend performances by musicians it is an aesthetic experience. We are living in the moment. If we are lucky enough to attend a great performance, we can be swept away by the beauty and passion of the event. Art, music, great scholarship are indeed high points of society.
Yo-Yo Ma plays "The Swan"
Herein lies the complexity of teaching and some of the pitfalls of the profession. We can mistake beauty for effectiveness, and this is very easy to do. Just as Sirens luring sailors with their enchanting songs, beauty, lucidly, and inspiration can lure a teacher to lose sight of the point of school and effective teaching.
I make my point by analogy. Going to see musicians like Yo-Yo Ma perform are an inspirational experiences. It is a necessary but not sufficient condition for studying to become a musician. If you want to be a musician, you have to play. Play lots. Play often, and reflect.
Similarly learning mathematics requires one to see the subject as meaningful. But one also has to do the hard work to build the mind. Listening to a lucid, inspirational math talk can be enlightening. BUT lectures do not build capacity to do mathematics with nearly all listeners, especially novices. Mathematicians already have the capacity to think mathematically -- we can learn what we need from a lecture much of the time. For students who are not yet advanced in mathematical thinking, they cannot take away the same lessons from the very same lecture.
If you want your novice cellist to learn to play, you can't just show a vid of Yo-Yo Ma, and say "There you go. Now go home and practice hard." It's not that simple. Telling isn't teaching. Likewise really understanding calculus, deciphering nested quantified statements in theoretical math courses, and learning to build differential equation to model a physical situation requires more than just following a recipe, processed by the professional mathematician (i.e. the equivalent of Yo-Yo Ma).
Students need a supportive environment, well-matched problems, time to be stuck, and opportunities to figure things out for themselves. It is this long, arduous, and rewarding process that unlocks potential. Good musicians know this. They don't just listen to someone else play. They also practice with intent with teachers, with collaborators, and with new music to keep their minds and hearts fresh. They experiment. They learn new skills, they interpret pieces in their own way. They do.
To put it simply, lectures and IBL methods should be used for different purposes. Lectures can be used to inspire. IBL methods should be used to build students' abilities and capacities to do.
The myth that "Teaching is Art" is one that lets us rationalize "aesthetics = effectiveness." I am not criticizing aesthetics, by the way. Beautiful ideas are why we are here. But there is a difference between showing students something beautiful and helping students become young mathematicians. Horses for courses, as they say across the pond. Or use the right tool for the job over here in the U.S.
What's the right mix of lecture and IBL? We don't know exactly. Data suggest a small percentage of the time should be lecture, and that as instructors talk more, students report less learning gains. Many of the most effective and experienced IBL instructors use IBL daily, and intersperse mini lectures as needed or at opportune times. Examples are (a) when the students have completed a body of work, (b) to showcase for students how certain ideas or techniques can be further used, (c) to summarize big ideas, (d) enculturation, (e) exposing students to things that there is not sufficient time for, (f) summarizing student strategies and proof techniques from the week,...
"Science is 1% inspiration and 99% perspiration." -- Albert Einstein
