Discovery

Conventional education is almost always focused on getting the approved "right" answer. But in the real world, there is almost never a single right answer. Multiple answers may be appropriate to the same question, and sometimes the standard answer turns out to be wrong.

It is commonly supposed that a theorem in mathematics, once proven, stays proven and remains a theorem forever, apart from occasional mistakes. This turns out not always to be the case, although exceptions are fairly rare. The best-known examples are supposed proofs that Euclidean geometry is the only one possible, that there can be no consistent arithmetic containing infinitesimals, and that there is only one structure satisfying the Peano axioms for the natural numbers. There have been extended periods of confusion in mathematics: plane geometry from Euclid to Gauss, Riemann, Bolyai, and Lobachevsky; calculus, from the time of Newton to the rigorous constructions of Dedekind, Weierstrass, and Cauchy; mathematical logic and set theory, from the time of the discovery of the great paradoxes to the creation of non-standard mathematics and model theory.

Another kind of problem has been much more common. The mathematical profession has often objected to the introduction of new notations (Arabic numerals, for example), and new objects (zero, negative numbers, complex numbers, infinite sets, infinitesimals) into mathematics, or to particular methods of proof (non-constructive proofs, use of the Axiom of Choice and its equivalents).

Thomas Kuhn proposed a theory of scientific discovery in which a single contrary fact meant doom for a theory. This turns out not to be true for science in general, and even for mathematics. This is fortunate for us, considering how many paradoxes of logic and set theory there were early in the 20th century, and how many physical theories failed to match up with experiment. A single verified, thoroughly tested fact may require the creation of a more general theory or a new branch of a science, but the old theory is usually kept on in the domain where it is adequate. Examples include celestial navigation (as though the Earth stands still), and the use of Newtonian mechanics and Clerk-Maxwell electromagnetism where none of Special Relativity, General Relativity, and Quantum Mechanics is required. Mendelian genetics is still taught as the introductory version, although our understanding has gone much further. Valence is still taught in introductory chemistry, even though chemical bonds are due to the structure of quantum mechanical electron orbitals.

In general, science advances fastest when practitioners get the wrong answers according to the prevailing theory, and math advances fastest when current models fail to support the demands of new constructions.

In the schools, discovery has long been suppressed, both as content and as activity. Children are capable of discovering laws of nature and of mathematics far beyond the level generally expected, and of learning how to validate them properly, whether through mathematical proof or through theoretical modeling and experiment in science. See, for example, the work of Caleb Gattegno, Jerome Bruner, Seymour Papert, and Alan Kay.