Dr. Jonathan Kenigson, FRSA
The "Gaokao" system of mathematics is often criticized in the West for its difficulty. However, the result of this system is not just demonstrated mathematical achievement for more than one billion people but also an emphasis on deep mathematical thinking. In fact, the brilliance of the Chinese mathematical tradition is its emphasis on proficiency in classical problems of Arithmetic, Number Theory, Euclidean Geometry, and the Theory of Functions. High PISA scores for Chinese mathematics students are attributable to knowledge of the fundamentals of mathematical discourse before progression to more difficult and abstract concepts. The Ministry of Education of China understands that students cannot progress to abstract concepts without mastering basic ones. This position in classical Western mathematical education is known as the Grammar stage. In the Grammar stage, students must learn to complete a wide variety of mathematical problems rapidly and correctly. The next phase of learning is called the Logic stage. During this stage, students learn about the logical connections among mathematical concepts and begin to master demonstration and proof. This stage focuses principally upon clear and precise expression of formulae. Finally, students reach the maturity of the Rhetoric phase. During this phase, students learn about mathematical eloquence. They ruminate about how to make clear, concise, and aesthetically pleasing proofs. They continue to explore new connections among branches of mathematics. Exploring unrelated areas of mathematics and working to unite them is part of mathematical eloquence learned in this phase. Western mathematical instruction should follow this ancient and tested model in which students master fundamental operations of Arithmetic and Algebra without progressing to more abstract topics.
The British A system as practiced at Eton and similar schools, involves a methodologically related system of mathematical inquiry. In my estimation, UK students and those enrolled at classical schools in the USA should pursue studies similar to the "Gaokao" model. The suggested course of study is modified to emphasize Western mathematical sources and permit translation from Greek and Latin to English and other modern European Languages. The Grammar Stage, beginning in early elementary, should emphasize memorization (yes, memorization) of basic mathematical facts until their recall is swift, accurate, and reliable. At this stage, "critical thinking" should not be emphasized, but rather a rote memorization and repetition. An edifice of authentic discovery cannot be founded upon shifting sand. Students who do not know times tables, addition, percents, fractions, and decimals with lightning efficiency do not have any business venturing into the realms of more abstract mathematics. Doing so merely makes future mastery of necessary skills more difficult, as it diverts time from mastery of topics that are prerequisite to any more abstract thought. Constructivist mathematics misses an essential aspect of the discipline: Mathematical reasoning, like reasoning in most other fields, is an inductive enterprise. Truths are discovered not from formal theorems and rules but from the rapid and accurate application of a few fundamental ideas that are then applied in successively more abstract fashions. Generality is a consequence or fruit of repeated observation that certain patterns exist.
Certain precepts are matters of debate, and others are neither profitable nor wise to debate. Percents, decimals, and the fundamental operations or arithmetic should be taken to be held as tacitly true by all human societies inasmuch as they possess a divisible medium of exchange and a system of numeration sufficiently robust to conduct commerce, trade, warfare, or exploration. When prosecuted at the scale of societies, these operations all require minimal assent to the universal truth of arithmetic knowledge. All people inhabiting such societies merit the dignity of being educated in the application of the principles that so intimately concern their welfare, commerce, and economic development. Students should only progress to abstract mathematical thought the logos of mathematical practice after such skills have been thoroughly and completely mastered. It is the province of the Logic stage of middle and beginning-secondary school for students to adduce the rules and principles that govern the basic facts hitherto learned. To entreat, as reform-minded educators in the US, UK, and Canada do that reasons should be mastered before the most basic of facts is comparably ridiculous to asseverating that doctors should practice Endocrinology before learning what cells are or that linguists should write Shakespearean verse before having mastered orthography of the alphabet. Behind this delusion is the prevalent notion that the truths of mathematics and physics are constructed rather than discovered. Any consistent notion of mathematics or mathematical physics must contend with the apparently profound anti-relativism of mathematical knowledge. While cultural influences can be profitably considered in the discourse concerning the evolution of paradigms of mathematical knowledge, it is profoundly absurd to assert that there exists any empirical evidence for the relativism of mathematical facts.