I was asked to write a response to this article; I did.
If teaching is an art, then it is no more rigid and concrete than it is abstract and fluid. Therefore, Polya’s outlined tenants of teaching and learning are no more than his own teaching philosophy. Polya even supports this by saying, “Yet it is far from me to say that you must accept these principles,” because he knows how subjective teaching is, “…it does not matter much what your philosophy is or is not. It matters more whether you have a philosophy or not.” I do have a philosophy, which I have submitted, and one can clearly see that Polya and I agree on many stances. However, there is no need to convolute teaching with such subjective abstractions. We should try to reduce the role of a teacher to a single idea that can apply to all teachers, similar to the golden rule.
Polya and I both believe a teacher should do more than regurgitate information. A teacher must develop students. But, in what way? In regards to mathematical educators, Polya believes teachers should develop their students by teaching purposeful thinking, thought processes, and applications of concepts. I firmly believe these attributes to be necessary for all subjects. Some forgiveness is needed because I have taken Polya’s words out of context—this article was published in a mathematics journal. Nevertheless, it begs the definition of a teacher.
What is the purpose of reading and discussing To Kill a Mocking Bird or mastering hyperbolic geodesics? The importance of these pursuits is the development of critical thinking. The language arts focus on critical thinking for subjective matters, whereas the STEM subjects focus on the objective topics. The two pursuits are equally important because they develop our inductive and deductive logic—critical thinking—which is a necessary requirement for independent thinking. Without critical thinking one cannot understand. Richard Feynman exemplifies the difference between knowing and understanding with this quote from his father, “You can know the name of a bird in all the languages of the world, but when you're finished, you'll know absolutely nothing whatever about the bird... So let's look at the bird and see what it's doing—that's what counts.” Without understanding one cannot be an independent thinker, else all conclusions are simply borrowed, which can be found in the full context of Feynman’s anecdote. When we understand something we are then able to form conclusions from our own perspective, the best way to learn anything is to discover it by yourself.
Polya says the “principles of learning” can be regarded as the “principles of teaching,” and I agree. Polya provides the principle of active learning, the principle of best motivation, and the principle of consecutive phases. The same are given to teaching. I agree that one cannot understand without these principles, but I contend that these principles alone do not define learning. Furthermore, Polya provides some pedagogical approaches to incentivize students. Again, I argue that the pedagogy is more or less his own teaching philosophy. I think a more appropriate claim is to say that the act of learning and the act of teaching are synonymous. I also argue that Polya has erroneously presented the role of a teacher.
Imagine the way a wild animal learns. Surely these animals do not congregate around a chalkboard and have engaging conversations where the students guess at solutions and teachers
sell motivation. Wild animals don’t even have teachers per se. In other words, a teacher is not necessary for learning, and I believe students to be oblivious to this fact. A teacher cannot learn for a student. A teacher can only expedite the learning process. And, now we have a hint at the synonymy of learning and teaching.
One is only capable of teaching after having mastered a topic. A teacher by definition has previously undergone the learning process. In the classroom a teacher revisits the learning process again, but this intending to build maximum conceptual connections in a shortened time. From another perspective, the teacher intends to present the material in such a way that the amount of effort required is significantly reduced when compared to individual learning. A learner without a teacher builds conceptual understanding one concept at a time. Moreover, learners teach themselves and teachers relearn material.
Polya’s pedagogy can now easily be seen as his own approach to motivate a student to learn a particular subject when the material seems peripheral. It goes without saying that humans are more likely to pursue learning when benefits of that knowledge are known. We are lazy creatures, no different from the rest of nature. So, it is only natural that a student refrain from learning if there is no benefit. It should seem obvious now that figuring out the benefit of learning a subject is part of the process. In fact, this is often how new branches of fields are established, for example, applying advanced physics and psychology to economics to birth econometrics and behavioral economics. A teacher did not tell the student of this value—it was discovered!
I can further philosophize about the role of a teacher and suggest that they should constantly challenge, support, and motivate. However, this adds no more value that one golden rule for teaching. I mean, how does one define something with infinitely many characteristics? It is like defining “table” with colors, textures, shapes, and dimensions. This is preposterous because a table is more than that. Sure, a table can have all of those qualities, but it can be none of those things as well. Must a table have four legs? Must a table be made of wood? A table serves a purpose. Anything that serves the purpose of a table could be classified as a table. Similarly, anything that supports the role of teaching can be considered a principle of teaching.
In conclusion, I conjecture that a teacher is anyone or anything that expedites one’s learning process; learning is the act of understanding.