Tea is a ubiquitous beverage well enjoyed the world over. Its method of brewing is almost common knowledge: we have the leaves, and we have the hot water; the leaves distinguish flavour, while the water offers its volume. This process might take a variety of forms depending on culture, but at its origins, traditional tea making is viewed as an art form.
A brief history on tea
The traditional Japanese ceremony of tea making was introduced along with the teachings of Zen Buddhism in the 12th century. As a mere island nation neighbouring the great political power of the East at the time, of course Japan would adopt much of its culture, whether it be politics or literature, from the powerhouse that was China. What the tea ceremony came to be called in Japanese is sadō, written in Japanese kanji 茶道: 茶 (sa) meaning “tea”, and 道 (dō) meaning “path” or “way”. Now as anyone familiar with the linguistics of the Japanese language would know, the Japanese kanji are adopted from Chinese, and in this case, the characters are pronounced with their Chinese reading or on’yomi, literally “sound reading”. Indeed, these same two characters 茶 道 (chádào) when used in Chinese refer to the same practice of traditional tea making.
Now what we have here is an amazing phenomenon of the linguistic orthography of ideas transcending the barriers of language. While pronounced differently and used in two largely different languages of two separate continents, the same two characters 茶 道 are able to be written and read, and they convey the same conceptual idea to people who may not share a common tongue. What has been created over the course of history is a careful brew of linguistic tea. In the figurative pot of cultural exchange between the two countries, each linguistic background has introduced their own version of tea:
The phonology, syntactics, grammar, lexicon are the tea leaves. They belong to each language uniquely, and supply flavour and meaning.
The orthography, writing system, is the hot water. It is the common element of all teas and distributes the flavour.
With each country adding their own linguistic elements, these tea leaves infused within the pot as they soaked in the universal hot water. The Chinese writing system is the common supporting volume that provides to the beverage, while the native leaves are a brew of each language individually. But nonetheless, tea is enjoyed by all.
Arising out of all this is a remarkable consequence in the Japanese writing system. There is not one, not two, but three different scripts used within Japanese. Many may be familiar with the Chinese characters, called the kanji. So too are there the hiragana, the “smooth borrowed characters”, and the katakana, the “partial borrowed characters”. Both hiragana and katakana are derived originally from Chinese characters themselves, and are used as a syllabary,
meaning each character represents one syllable of sound. Chinese characters, on the other hand, do not have a one-to-one correspondence with sounds. They are instead often used for their meaning; they convey ideas.
A linguistic brew
Here is one very peculiar example illustrating this linguistic point. Below is an excerpt of Latin poetry from before Christ, in the time of antiquity.
467 stat sua cuique dies, breve et irreparabile tempus omnibus est vitae; sed famam extendere factis,
469 hoc virtutis opus.
— The Aeneid, Book 10 lines 467-469, Virgil
The English translation of which is:
For each, one’s own day is given, for all the time of life is lost and ephemeral. But to prolong life’s glory by great deeds, such is the feat of virtue.
— Yours truly
The first sentence out of this excerpt from Virgil is “stat sua cuique dies”, which literally translates to “For each man, his day stands”. The verb used in this case is “stat”, meaning “to stand”: it indicates a passage of time, whereby each person is given the limited number of days, of which they must make use to their fullest potential.
A rendition of Virgil’s line in Japanese might be:
人々には、自分の日が経つ。
Hitobito niwa, jibun no hi ga tatsu.
For each person, their own day passes.
— Yours truly
The verb in question here is of course 経 つ (tatsu) “to pass (of time)”. It is written using the
kanji 経 , and the hiragana つ (tsu). The kanji 経 provides the meaning of the “elapsing or passage of time”, while the hiragana つ acts as what is called the okurigana, literally “accompanying letters”, that indicate grammatical information such as tense, called inflection in linguistics. Written together, the kanji gains the pronunciation 経 つ (tatsu), and this is referred to as the kun’yomi, literally “meaning reading”.
Now, you may be thinking:
Okay, there’s a word in Japanese that describes the passage of time.
How does this have any relation to Virgil, or standing, or tea for that matter?
Let’s consider what has happened here. We talked of on’yomi, “sound reading” at the very beginning. But here it is contrasted with kun’yomi, “meaning reading”. We are brought back to this peculiar point: Chinese characters are not only used for sound, they are also used for
meaning. Think of the English verb “to smile”. If we were to write this in an equivalent to how Japanese uses kanji, we might have the following examples using emoji:
- I am 😊ling.
I am smiling.
- He 😊led at me brilliantly.
He smiled at me brilliantly.
There is a concept in linguistics called cognates, which refers to different words within a language that have the same origin. As it so happens, there is a cognate to 経 つ (tatsu) in Japanese, pronounced exactly the same, but written 立 つ (tatsu). Their homophony means that they can essentially be considered the same word in normal speech, distinguished only by their written forms; they are two faces of the same conceptual coin. Now what might come as surprising is that 立つ(tatsu) exactly means “to stand”; the kanji it uses is 立, which has the meaning of “standing”.
Bringing ourselves back to antique poetry, this is exactly the same verb as used in Virgil’s “stat sua cuique dies”. We witness the convergence of two languages: one ancient, from the European continent that existed more than two millennia ago; one still alive in modern usage, from an island nation to the east of Asia. Yet despite sharing no linguistic relation, the common idea of the passage of time becomes communicated through the same verb “to stand”. In Japanese, this verb tatsu has been made distinct into two different verbs in their written form: 経つ (tatsu) and 立つ (tatsu). But nonetheless, we brew the same tea when we pour the kanji of hot water onto the phonetic tea leaves.
The problem with mathematical pedagogy
There seems to be the widely held notion that mathematics is a hard STEM subject. Particularly, a very large number of students seem to struggle to learn mathematics throughout their school life, leaving with incomplete, partial and perhaps even inadequate understanding of what mathematics seeks to convey. The notion I will propose here is not novel, and it has certainly been touched upon by many throughout history, yet the impact of the message on popular culture seems little, and the belief that mathematics is difficult remains yet to be dispelled.
When I look at a standard mathematics textbook used at the secondary level, its layout is colourful yet uninspiring, and its content equally dry. It is filled with a barrage of symbols that purport to teach students all the formal notions of mathematics, and how we might manipulate them to calculate various values, but it lacks the effort to convey what ideas the symbols capture, and therefore the artistic lure that catches a student’s interest and hooks them on to learn.
The symbols used to represent mathematics are no different from any language. They convey ideas that one might have conceived in their minds such that others might read these symbols and come to the same ideas. But the symbols are never the ideas themselves, and it is a tea without the tea leaves when we attempt to write out symbols that have no meaning behind them.
As was once said by G. H. Hardy in his book A Mathematician’s Apology,
“A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.”
The symbols we are taught in school maths are only a communication of these ideas.
The Binomial theorem is not a summation of arbitrary terms in relation to Pascal’s triangle that happens to be the powers of a term (a+b), it is a remark at how the act of multiplication of terms exactly corresponds to the selection of a number of objects from an arbitrary collection.
The quadratic formula is not just a set formula for plugging in numbers, it is a note of symmetry and tells us exactly how such a symmetry manifests.
A function f(x) is not just another definition where we substitute in numbers for x, it is the notion of relating two different objects together by some meaningful pattern.
The derivative dy/dx is not just a way of saying we’re calculating the gradient of a curve, it encapsulates the relation between two quantities and how they vary with each other.
These are the ideas underlying the various topics presented throughout highschool, but without a proper understanding of what we wish to convey, the symbols fall short of being meaningful, and only go towards creating a barrier to entry.
Paul Lockhart makes such a comparison in his essay A Mathematician’s Lament:
“If your art teacher were to tell you that painting is all about filling in numbered regions, you would know that something was wrong.”
— Paul Lockhart, A Mathematician’s Lament
To get at the heart of mathematics, one must understand the notions they are playing with, have a grasp of the ideas, patterns, abstractions, concepts that may be useful to express a sequence of logical arguments. A language and its symbols only go to facilitate the communication of such ideas, but they are not a substitute, and they cannot be used to their full potential when there are no ideas to express. While universally understood, the symbols become the water of a tea without leaves if we never imbue them with meaning.
If we wish to teach mathematics to students, we must teach them how to deal with notions rather than how to deal with symbols. A mathematical concept taught at the highschool level is seldom too contrived to be beyond a student’s reach of understanding. Should a student understand what they wish to convey, as with language, like writing a story, the mathematical symbols will flow out naturally like any author who has struck upon a spurt of creativity.
If we wish to brew a tea that we can share at the table, we must add the tea leaves, or else what are we left with, but a pot of hot water.
