Revisiting Mathematics Class

Keith Devlin, a senior researcher at Stanford University and a British mathematician, recently wrote an article on mathematics education. This article discusses the mathematics education in high schools and universities from the perspective of the history of mathematics development, providing significant insights.

Today, I will interpret this article, examining which aspects of mathematics we have encountered through years of mathematics classes and how mathematics classes should be improved in the future.

High School Mathematics Does Not Cover the Last 300 Years of Mathematical Development

The mathematics that continues to this day has only one main developmental thread.

Mathematics historians believe that numbers and arithmetic are the origins of mathematics, starting from 10,000 years ago. Mathematics that appeared at that time was closely linked to the use of currency.

In the following centuries, the ancient Babylonians and Egyptians expanded geometry into mathematics. Mathematics at that time was purely practical, like a cookbook, telling you what to do step by step, and then you were guaranteed to get an answer.

From 500 BC to 300 BC, it was the period of ancient Greek mathematics. The ancient Greeks paid great attention to geometry, trying to solve all problems by converting them into geometric figures, most notably considering "numbers" as measurements of length. Because of the ancient Greeks' unique way of thinking, when they discovered values that could not correspond to real lengths, irrational numbers were discovered. For example, a right-angled triangle with two sides of length 1 has a hypotenuse equal to the square root of 2. The value of the square root of 2 cannot correspond to a real length and can only be infinitely approximated.

Although there are significant archaeological discoveries like the Archimedes Palimpsest found in the 20th century, proving he was very close to discovering calculus, it was not inherited, so from the overall progress of civilization, ancient Greek mathematics stopped with the discovery of irrational numbers.

The influence of ancient Greece on mathematics is not just about specific knowledge but about turning mathematics into a field of study. Before this, mathematics was entirely limited to applications like measurement, counting, and accounting.

For example, around 500 BC, Thales of Miletus defined the activity paradigm of the mathematical field as follows: All concrete numerical assertions can be proven in a formalized argument in a logical manner. This paradigm is what later became known as the "process of proving theorems," with the proof of a theorem underpinning an infinite number of specific numbers.

In ancient Greece, the peak of this paradigm was Euclid's "Elements." It used the method of proving theorems to expound on mathematical ideas, not sticking to specific numerical calculations.

In the 8th and 9th centuries, the next leap in mathematics was driven by the Arabs. The word "algebra" comes from the Arabic "al-jabr," meaning "something" in Arabic. Algebra evolved from the trade between Arabs and people around the world.

Besides these, although there were sporadic mathematical studies in other regions of the world, they did not continue and did not have a significant impact on today's mathematics.

The knowledge points that appeared in these early developments of mathematics, along with two advancements from the seventeenth century, namely calculus and the basics of probability theory, are essentially the content base of high school mathematics classes worldwide. That is to say, the advancements in mathematics in the last 300 years have not entered high school mathematics classes.

But today, all research objects within the field of mathematics are developments from the last two to three hundred years. This gives most people a wrong impression of the mathematical world, thinking it is not a thriving and developing field.

So, what changes have occurred in the mathematical world in the last two hundred years?

Firstly, there are many more branches of mathematics.

Mathematics 300 years ago could roughly be divided into four different fields - arithmetic, geometry, calculus, and algebra. A hundred years ago, with the emergence of Boolean algebra, various non-Euclidean geometries, and set theory, mathematics could be divided into 12 fields. Today, mathematics has sixty to seventy subfields, such as differential geometry, game theory, topology, measure theory, Lie groups, and Lie algebras, which are familiar to many.

These are the visible changes, but the more profound change can be summarized as a change in the research paradigm.

This change is as significant as when ancient Greek mathematics proposed that mathematics revolved around theorems, changing the previous research focused on specific numbers by ancient Babylonians and Indians. The most advanced concept to understand the discipline of mathematics is that mathematics is a science of patterns. When mathematics is reclassified from the perspective of patterns, mathematics appears differently -

Traditional content related to numbers, such as arithmetic and number theory, falls under the pattern of computation.

Traditional geometry falls under the pattern of shapes.

Traditional calculus belongs to the pattern of dealing with motion.

Traditional logic belongs to the pattern of reasoning.

Traditional topology belongs to the pattern of studying closure and position.

Traditional fractals belong to the pattern of self-similarity.

New Understanding of Mathematical Symbols

Mathematics employs a special system of symbols, each representing an abstract concept.

Mathematical symbols and musical notes share a highly similar status. A page of notes, though representing a piece of music, only becomes music itself when those notes are played. Similarly, mathematical symbols are not mathematics in themselves when they are merely on paper; mathematics reveals its true form only when someone capable of "interpreting" these symbols understands them in their mind. Since each symbol is abstract, when mathematics reveals its true form, it is also abstract. Yet, this abstract existence helps us comprehend patterns in the universe that are utterly invisible.

Galileo's statement in 1623 offers the best description of mathematical symbols: "Only those who understand the language in which nature is written can read the great book of nature, and this language is mathematics."

Knowing this, we can also reinterpret what physics is. Physics is the image of the universe seen through the lens of mathematics. When an airplane flies overhead, we see no support beneath it; only through mathematics can we "see" the forces and fields that keep the airplane aloft. The abstract symbols of mathematics turn the invisible in the natural world into something visible.

The Leap in University Mathematics

Entering university to study mathematics as a major significantly surpasses the knowledge domain of high school mathematics, introducing many important ideas completely detached from physical reality.

Simpler issues, such as "Which infinity is bigger than another infinity?" or "What exactly does one infinitesimal divided by another infinitesimal equal?"

More complex issues abound. For instance, the Russell paradox asks if "a set S consists of all sets that are not members of themselves," does the set S contain itself? This leads to a contradiction no matter how it's approached.

This paradox is colloquially equivalent to saying, "If 'no statement is absolutely true,' is that statement absolutely true?" or "If a barber swears to shave those, and only those, who do not shave themselves, should he shave himself?"

These paradoxes led to the third crisis in mathematics, which was eventually resolved through patching up, allowing the mathematical edifice to remain standing. The method of patching involved adding some extra restrictions when defining sets to prevent these paradoxes from arising. Are these extra restrictions naturally correct? They are assumed correct for now, allowing the subsequent mathematical structure to remain stable.

What extra restrictions should be set to keep the mathematical edifice from collapsing?

There is more than one method. For example, the most famous ZF axiomatization (Z for Zermelo and F for Fraenkel) requires the addition of nine axioms—Extensionality, Specification, Pairing, Union, Regularity, etc. Other methods, like the ZFC axioms, add the Axiom of Choice to the nine axioms, ensuring that previously proven mathematical theorems remain valid. Von Neumann proposed the NBG axiom system.

Choosing different axiom systems leads to different worlds.

For instance, the Banach–Tarski paradox suggests that, under the Axiom of Choice, a three-dimensional solid ball can be divided into a finite number of parts, which can then be rearranged by rotation and translation to form two complete balls of the same radius as the original. Simply put, as long as the Axiom of Choice holds, it can make one ball into two, two into four, and so on.

Rejecting the Axiom of Choice leads to even more bizarre conclusions, such as a space having two different dimensions.

And what does the Axiom of Choice say? In simple terms, even if there are infinitely many sets, each with infinitely many elements, there always exists a rule that can select exactly one element from each set. The nature of this rule is unknown, but its existence is assumed.

The proofs and thoughts of the above content are starkly different from high school mathematics. If you're feeling overwhelmed, you've just touched the edge of the developments in mathematics over the last 200 years.

Failed Reform

In the 1960s, the United States attempted a reform in mathematics education called New Math, aiming to introduce recent mathematical content from the last 200 years into textbooks, essentially shifting college-level mathematics content to the high school stage.

Though well-intentioned, the implementation deviated. The deviation was mainly caused by the notion: "Let students forget about complex calculation skills and focus only on new concepts." However, for mathematics, there's a harsh reality: only by mastering complex calculations to a certain level can one truly understand the nature of mathematical concepts later on. Before mastering calculations, any self-perceived understanding is riddled with errors. Thus, the mathematics education reform failed within a few years, and university-level mathematics content has never entered high school textbooks since.

Methods for Reforming Mathematics Classes

What's the use of learning mathematics?

The advancement of industrial society requires a large number of people proficient in mathematics, naturally dividing into two categories:

The first category can find solutions to given mathematical problems. They make up the majority.

The second category can translate problems encountered, for example, in manufacturing, into mathematical language, then use mathematical tools for precise analysis. This work is also known as creating mathematical models, then posing and solving problems based on those models. People in this category are rare but invaluable. They require a deeper understanding of mathematical concepts.

Today's mathematics textbooks, although still focusing on training students' computational skills, could allow schools to offer optional courses on mathematical concepts and the history of mathematics. This would be an excellent way for students who have already mastered computational skills to advance further.

That's the content for today. See you tomorrow.