At the moment, what is happening is the systematic undermining of the student’s intuition. If teaching is reduced to mere data transmission, if there is no sharing of excitement and wonder, if teachers themselves are passive conduits of information and not creators of new ideas in the topics they teach, what hope is there for their students?
The result is that the student becomes a passive participant in the creative act. Students are making statements to fit a preexisting proof-pattern, not because they mean them. They are being trained to ape arguments, not to intend them. So not only do they have no idea what their teacher is saying, they have no idea what they themselves are saying.
In place of meaningful problems, which might lead to a synthesis of diverse ideas, to uncharted territories of discussion and debate, and to a feeling of thematic unity and harmony in science and logic, we have instead joyless and redundant exercises, specific to the technique under discussion, and quite disconnected from each other that the students have the foggiest idea of how or why such a thing might have come up in the first place. A complete prescription for permanently disabling young minds. We could all be having so much more fun.
The trouble is that science, like painting or poetry, is hard creative work. That makes it very difficult to teach. I have read, gathered and inasmuch as my experiences with the projects I have done, it is a slow, contemplative process. It takes time to produce a work of art, and it takes a skilled teacher to recognize one. Yes, it's easier to post a set of rules than to guide aspiring young minds. Should we take the path which seems easier or which is joyous throughout and gives the freedom to understand nature?
It is far too painless to be a passive recipient of some publisher’s “materials” and to follow the shampoo-bottle instruction “lecture, test, repeat” than to think deeply and thoughtfully about the meaning of one’s subject and how best to convey that meaning directly and honestly to one’s students. We are encouraged to forego the difficult task of making decisions based on our individual wisdom and conscience, and to “get along with the program.” Of course it is far easier to test someone’s knowledge of a pointless definition than to inspire them to create something beautiful and to find their own meaning.
To do science is to engage in an act of discovery and conjecture, intuition and inspiration; to be in a state of confusion — not just because it makes no sense, but because one gave it sense and one still does not understand what his/her creation is up to; to have a breakthrough idea; to be frustrated as an artist; to be awed and overwhelmed by an almost painful beauty; to be alive, damn it.
Applied science is about problems and solving them, and problems must be made the focus of a student's scientific life. Creatively frustrating as it may be, students and their teachers should be engaged in the process of - having ideas, not having ideas, discovering patterns, making conjectures, constructing examples and counterexamples, devising arguments, and critiquing each other’s work. Of course this is not meant to say they should have no other interest. Specific techniques and methods will arise naturally out of this process, as they did historically: not isolated from, but organically connected to, and as an outgrowth of, their problem-background.
A proof (be it theoretical or experimental), that is, a scientific or mathematical argument, is a work of fiction, a poem. A beautiful proof should explain clearly, deeply, and elegantly. A well-written, well-crafted argument should feel like a splash of cool water, and be a beacon of light— it should refresh the spirit and illuminate the mind. And it should be charming.
What exactly do I suggest is a perfectly simple idea, maybe much more work. However, we reap what we sow.
How do we try to resolve the issue at hand? By choosing an engaging set of natural problems suitable to their tastes, personalities, and level of experience. By giving them time to make discoveries and formulate conjectures. By helping them to refine their arguments and creating an atmosphere of healthy and vibrant scientific criticism. By being flexible and open to sudden changes in direction to which their curiosity may lead. In short, by having an honest intellectual relationship with our students and our subject. Students are not aliens. They respond to beauty and pattern, and are naturally curious like anyone else. Just talk to them! And more importantly, listen to them if they have problems!