Regarding "How to Count"

I have been researching Japanese counters. Counters are a field that studies how we perceive objects; it is extremely interesting and a concept that doesn't exist in the West.

For example, a pencil and an umbrella are completely different objects, yet people often ask me why they are counted using the same counter (hon). Or why a home run is called 'one hon' even though it has no physical shape and the ball is round, or why people say they 'wrote one hon of a paper' when they go abroad (laughs).

I am currently supervising international students, and even those who are quite fluent in Japanese seem to find counters considerably difficult. I have been researching how to theorize what we as Japanese people have acquired naturally through living here, and how to explain it clearly to foreigners.

I am in a position involved in arithmetic and mathematics education, but the first thing I want to say is that no matter how anyone thinks about it, you must not add 6 pencils and 5 notebooks. This is incorrect as a problem. When I was at the Faculty of Education at Okayama University, I used to tell elementary school teachers that until I was blue in the face.

Some children feel that you shouldn't add 2 white tulips, 3 yellow tulips, and 5 red tulips. That way of thinking is actually more correct.

First, I'd like to start with the 'embodiment of counting.' The diagram in the paper below is from a cultural anthropology paper about how a certain tribe in Papua New Guinea counts things. Starting from the little finger of the left hand, they count 1, 2, 3, 4, 5, the palm is 6, the wrist is 7, the forearm is 8, the elbow is 9, the upper arm is 10, the shoulder is 11, the shoulder blade is 12, and so on, then moving to the right side, counting up to 37. In this case, they grasp the situation by mapping it to the 'body' rather than the 'concept' of numbers. This is the most primitive way of counting.

That's fascinating. It's a perception that goes from left to right up to 37. Do the people of Papua New Guinea have a bodily sensation where smaller numbers are on the left and larger numbers are on the right?

That might be the case. Once you exceed 37, the second round begins.

Does it start from 38?

The concept of 38 doesn't exist originally; it's the sensation of the 'little finger of the left hand' or 'ring finger of the left hand.' Once you reach the right hand, you start over from the left hand again. They rarely count numbers larger than 37, and if they do, they apparently refer to it as the second round.

So it's almost like a base-37 system.

That's right.

Base-10 and base-20 systems are well known, and it's often explained that their origins lie in humans having 10 fingers, and 20 if you include toes. I don't know how true that is, but that's the usual explanation.

Looking at this Papua New Guinea example, it's quite interesting that they go toward the head instead of the feet. Does the finger you start counting from differ by culture?

I think there are differences. Since most people are right-handed, they probably start counting from the left.

Because you point with your right hand. I've heard of cultures that count in base-12 using finger joints. Like 1, 2, 3, 4, 5, 6... there are three joints on each of the four fingers, and by indicating them with the thumb, you represent 12. There are also cultures where you fold a finger once you've counted a dozen. Some systems are self-contained within the fingers, while others use the entire upper body. It's fascinating.

I've heard that in Ancient Rome, people used the fingers of both hands to represent numbers up to about 10,000. Through that method, they were able to conduct trade even with people who didn't speak Latin—the so-called barbarians from the Roman perspective.

To put it in extreme terms, it's like 'holding one, two in my hands... oh, I can't hold any more, so it's a lot, I give up.' It's about counting with the body. Naturally, there is no abstract concept of number there, only the idea of one-to-one correspondence.

Ms. Iida, the fact that Japanese has so many counters is, in terms of thinking, a matter of one-to-one correspondence, isn't it?

We do count in groups sometimes, but in Japanese, while we might use both hands, I don't think we ever go around the head and end on an odd number. There is occasionally a debate about whether we count by folding fingers in or extending them. In the love poems of the Man'yoshu, people fold their fingers while wondering how many more days they must wait to see their beloved. So Japanese is a culture of folding fingers, whereas in America, they point by extending fingers. Is it the same in France?

France is the same as America. However, I often see the gesture of starting from the thumb when counting. Usually, by the time they get to the ring finger or little finger, it looks like they're struggling to lift them properly.

The fingers just won't move (laughs).

When counting, some people just raise their fingers, while others count 1, 2, 3 while flicking them one by one with the opposite hand. In Japan, you fold them in, right?

The reason I brought this up is because I believe 'one-to-one correspondence = counting.' Even the 'grouping' that Ms. Iida mentioned is treating the group as 'one' for correspondence.

In mathematics, we use the word 'mapping,' and there are ways of counting based on what you correspond with what, but the basis is one-to-one correspondence. However, the issue is what you are corresponding to. Whether it's folding or extending fingers, I think the fundamental position of counting is that everything corresponds one-to-one with a part of the body.

In other words, is it the idea that there must always be a concrete object?

Yes. That's why elementary school education is the most difficult. For example, when adding '3 + 5,' should we accept the stage of 'counting on'—counting them out one by one—or should we quickly tell them to move away from that and think of '3' and '5' as concepts?

This jump to abstract concepts is quite difficult in the early years of elementary school. That's why I think we should let them count thoroughly using their bodies. Since we are human, I believe that as intelligence develops, abstraction will happen naturally on its own.

Abstraction is very difficult. I have an elementary school child, and as long as they were counting by folding fingers, they weren't that bad at calculation. But when it switched to using counters (ohajiki) or treating a rod as '10,' the teacher would naturally say, 'This rod is 10, so 13 is one rod and three counters.' Suddenly the shape and the way of counting change, and they get lost.

As Mr. Sobukawa just said, that is a huge wall. Elementary school teachers push forward intently, but I feel that whether or not one can make that jump successfully makes a big difference in their future understanding of arithmetic.

Exactly. I often say that for the early years of elementary school and below, the three main subjects are arts and crafts, PE, and music.

By putting embodiment at the forefront and doing it enthusiastically, it eventually leads to the concept of numbers. So, I tell people that if they want their children to be good at math in the future, let them play outside as much as possible. Let them climb jungle gyms, and let them dance to music on TV.

That's true. In music, teachers talk about what kind of note it is and what the time signature is, and even before they learn multiples or divisors in math, the teacher says it's worth so many quarter notes, but the kids don't understand it at all.

I feel that unless we emphasize a sense of rhythm or the everyday feeling of 'this is about half,' no matter how much we teach conceptually, it won't stick.

Mr. Miyashiro, when you were in France, did you ever experience confusion regarding communication with numbers?

I was constantly confused. In Japanese, you can count all numbers using a base-10 system, but the way of counting in French is quite complex—some explain it as a mix of base-10, base-20, and for some, even base-60.

For example, numbers in the 70s are expressed as '60 + something,' so 71 is perceived as '60 + 11.' For native French speakers, the number '71' probably pops into their heads instantly when they hear '60 + 11,' but at least for me, it was hard to get used to.

Furthermore, when you get to the 80s, it becomes base-20, saying 'four 20s' for '80,' so my head would gradually get muddled.

Why has it become such a cumbersome way of speaking? Just like Japanese, French has influences from Latin, and the ideas of the Celtic people who were there before have also entered the language; historical circumstances resonate strongly in words. There are many intriguing points in the history of these names.

Naturally, astronomical elements enter into mathematics. So the reasons why base-60, base-20, or base-12 appear are mostly related to astronomy. Things like a year having 12 months—it's certainly interesting to trace the history. I also reviewed the French way of counting numbers this time, and it would be painful if I were told to do multiplication tables with that. After all, the multiplication table would be like '4 x 20 + 1'.

Elementary school math teachers have a hard time teaching it. It's not that base-10 expressions never existed. There were words corresponding to '70,' '80,' and '90,' but it seems they gradually took their current form due to the intentions of the Académie Française and dictionary compilers. However, even at the end of the 19th century, some dictionaries stated that the so-called base-10 expressions should be used.

You don't see them often, but there are 2-euro coins. I remember seeing '2' coins like guilders in other countries before the euro, and I thought, 'So this is the culture.' In Japan, the 2,000-yen bill never became popular, did it?

That's one of the reasons why Japanese people don't feel much familiarity with multiples of 2.

My sense is that the idea of '1, 2, and now it's full, so it's a block' is deeply ingrained in the hearts of Europeans. I've never seen one, but apparently America has 2-dollar bills as well.

In America, they use the quarter—the 25-cent coin—very often.

That's dividing into two, twice. The unit of 5 is, of course, the number of fingers, but it probably also means half of a base-10 block. I wonder if Japanese people think of it as dividing in half. Oh, come to think of it, gold coins in the Edo period were 4 'isshukin' for 1 'ibu,' and 4 'ibukin' for 1 'ryo' koban. Maybe that was dividing in half?

I think the units of bills and coins represent the concept of numbers most culturally. I believe Japanese people have a sense that odd numbers lined up, like 1-5-10, 7-5-3, or 5-7-5-7-7, are beautiful. In the West, when people read poetry, are they conscious of the number of sequences or sounds?

In France, the 12-syllable meter called the alexandrine is very famous. It's used not only in poetry but also in theatrical dialogue. The 17th-century classical plays include masterpieces that are still performed today, and this alexandrine is used in them. The pronunciation at the end of each line of dialogue is also designed to rhyme according to certain rules, so it has a pleasant rhythm when read aloud.

Regarding counters, Japan has many ways of counting. Is this a characteristic feature compared to other cultures? For example, how does it compare to China?

I think Japanese falls into the category of languages with a rich variety. Within the scope of my research, there are about 500 types of things called counters, though some are only used in literature or by certain experts. In daily life, about 120 to 130 are commonly used. It seems to be the sense that an adult native speaker should naturally be able to master this many.

Chinese also has 'measure words,' and apparently there are about 500 of those if you look them up. So I think it might be a common human cognitive sense that having up to a maximum of about 500 in a language allows for sufficient differentiation when counting various things.

There are also counters in Korean, Indonesian, and Tibetan dialects, so it's not a feature unique to Japanese. However, the breadth of the framework—where so many different things can be differentiated, yet a home run, an umbrella, and a pencil are all represented by the same 'hon'—is something I consider a unique Japanese sensibility.

Is counting a home run as 'one hon' unique to the Japanese?

Yes. What I find unique and interesting about the Japanese way of counting is how it often jumps to shape or function. For example, we categorize by shape like 'one mai' (flat objects), 'one men' (surfaces), or 'one ko' (small objects), or use 'one dai' widely for a car, a computer, or a machine.

Even though there seem to be more important things like color, softness, or whether it's edible, the fact that we boldly focus on function or shape and count things together that don't look like they belong in the same group is very daring and interesting.

In your book, Ms. Iida, there was a story about whether to call a robot dog 'one dai,' 'one tai,' or 'one hiki.' It was a story about how the situation changes depending on the function.

That's right. So it depends on how that person subjectively perceives it. Even if it's just one robot dog, for someone who cherishes it as a pet, it might be 'one hiki' (animal counter), or even 'one nin' (person counter).

I once asked Nobuyo Oyama about how to count Doraemon, and she said Doraemon is 'one nin, two nin.' At the manufacturing stage, he might be 'one dai' because he's a cat-shaped robot, but she said that because he is a special existence who helps Nobita, you must never count him as 'one dai'.

In mathematician-speak, I think it comes down to what you are corresponding with what. The way of making correspondences differs from person to person. It's the same object, but the way it's corresponded changes based on subjectivity.

That's why there are children who think that because 2 yellow tulips, 3 white tulips, and 5 red tulips are different things, they shouldn't be corresponded together—meaning they shouldn't be added. In a sense, this is correct.

Analyzing this philosophically, is it better to think that because the colors are different, the objects are also different?

How people correspond objects with numbers likely varies considerably depending on language and culture, and there is a kind of disconnect between perceiving concrete objects and perceiving numbers as concepts.

Indeed, perceiving that 3 cats and 3 blocks of tofu are the same '3' is what '3' is as a number, but that abstraction seems philosophically interesting.

There is a very famous quote by Poincaré: 'Mathematics is the art of giving the same name to different things,' and I think that is the human capacity for abstraction and universalization. Stepping beyond the perception of numbers tied to concrete things is a human ability. In that sense, the ability to handle numbers was highly valued in philosophy by people like Plato in ancient times, who even said that leaders of the state should learn arithmetic.

To give an example of where I stumbled in French, in the English I learned in middle and high school, I was taught that 'there are countable and uncountable things,' and I was taught something similar in French.

For example, they say things like coffee are uncountable. Therefore, you use the expression ' a certain amount of coffee.' It was explained that these differences between countable and uncountable are shown by definite, indefinite, and partitive articles attached to nouns.

But if you think about it, if you go to a cafe and say, 'Please give me a certain amount of coffee,' you don't know how much will come. Naturally, in actual language use, you can of course say one coffee, two coffees.

In old textbooks, some wrote that French has uncountable and countable nouns, but relatively recent ones do not. It's explained that it depends on whether the speaker perceives the object as something to count or not. Or, the use of articles changes depending on whether you want to convey it to the listener as something not to be counted or something to be counted, which makes sense. Indeed, that's how it works in both actual use and the structure of the language.

So, perhaps the consideration of whether to use 'one hiki' or 'one nin' in Japanese is also related to how the speaker perceives it and how they want to convey it to the listener.

I have an image that languages with definite articles have concepts with quite clear outlines, and based on linguistic classification, they don't particularly require counter words.

In languages without definite or indefinite articles and no grammatical gender, the outlines of nouns themselves become very blurred. When counting, it's like using a cookie cutter; I hypothesize that the framework of counting what has been 'cut out' serves the role of a counter word.

Looking at a world map, regions that practice rice cultivation have a strong tendency to have many counter words. On the other hand, in places where hunter-gatherer tribes once lived, there are plural forms and distinctions between definite and indefinite articles. I believe that lifestyles significantly influence the sense of nouns through counter words.

In elementary school mathematics education, there is the concept of 'number' as in 1 or 2 items, and another concept of 'quantity.' I believe that units are essentially the way we measure quantity.

Ms. Iida's view in her book—that units are things that allow for substitution—is certainly one valid perspective. However, one could also say that units are what convert how we perceive quantity into the form of numbers. With hunting, you can just say you caught one rabbit or two rabbits, so the concept of 'number of items' suffices. But with grain, that doesn't work, so the concept of quantity becomes essential.

Actually, this is a key point in math education. Talking about numbers that can be counted as 1 or 2 is fine, but the problem is fractions. The concept of quantity enters into fractions. One-third is a whole divided into three parts. Fractions are difficult because the concept of quantity gets mixed in.

That's why children don't understand why 1/3 + 1/2 is not 2/5. If they think of them just as numbers, it feels like that should work, but it's not like that; because it's a concept of quantity, it has a different definition.

Objectivity is also important for grasping quantity accurately, isn't it? When trading something, what I might call 'a lot' could be 'a little' to the other person. I think it was crucial for people in distant locations to standardize units of quantity when conducting trade.

My daughter also struggles a lot with fractions. Doesn't that require a whole new set of skills, like finding common denominators or reducing fractions? When you do that, you end up returning to concrete examples like 'how many buckets of water,' so the mathematical ability that was finally abstracted has to be made concrete again to be understood. That back-and-forth feels frustrating.

Exactly, that's what makes elementary education so difficult. Compared to that, giving a university math lecture is much easier. Teaching what you just described to a first-grader is incredibly hard.

Does that mean the movement between the concrete and the abstract is important?

Yes, that's true at any level. In a certain university class—which is already abstract enough—I might go two levels deeper into abstraction from a certain point. However, the students can't keep up with that abstraction.

So, as a concrete example, I come down to a level they can normally handle, then go back up two levels of abstraction, and repeat this process until they can tolerate that higher level of abstraction as a sensation. After doing it repeatedly, they get used to it and accept it as inevitable.

Mathematics relatively always has a correct answer, doesn't it? Is the most important thing to acquire the ability to think logically toward that correct answer?

I think thinking logically is enough. Too many people try to acquire only 'warp techniques' that skip the process without thinking logically. With just warp techniques, the act of memorizing itself becomes painful.

This is something I've thought about for decades, and it leads to the question: what does it mean to 'understand'?

First, I don't accept 'logical understanding.' Logical understanding is usually just a string of words. Instead, 'understanding' is when you can feel it intuitively. I tell people to think until they reach that point. At first, you have no choice but to accept it logically and understand the structure. But ultimately, I want them to arrive at an intuitive understanding.

It's important how deeply it connects with the experience and knowledge you already have. I think you intuitively understand it when you feel it has become one with you.

Numbers themselves are invisible to our eyes, aren't they? We can see two cats, but we can't see the concept of the number '2' itself. Does it mean that if you can't grasp such things intuitively, you won't understand the world of mathematics?

Exactly. It's about whether you can come to feel it as something certain within your own senses through repeated practice. When you can perceive '2' as something certain, then you can say you understand '3' or that you understand the number of items.

People who can do it without thinking about it every time can be said to have recognized quite abstract things as concepts.

Nowadays, the abacus (soroban) is no longer taught in elementary schools. Has the loss of the abacus changed the calculation ability or the concept of numbers for Japanese people in any way?

I wouldn't go so far as to say concepts or abilities have changed, but I think there will be an impact. However, compared to when Ms. Iida and I were children, it's only been a few decades, so I don't think it's enough to destroy society as a whole. And since we have calculators, it doesn't really surface.

However, in high school chemistry, students use calculators, and some might accidentally be off by two digits and write that 2 kilograms of salt dissolve in 1 liter of water. No matter how you think about it, that's impossible, yet they write it without a second thought.

In that sense, I think there's a possibility that the concept of quantity will be forgotten. That's exactly why I want to value subjects that nurture physical and sensory experiences in the lower grades.

I'm not that good at the abacus myself, but I'm a person of the Showa era who calculates three-digit addition and subtraction by flicking an imaginary abacus in my head. But when I explain it to my daughter using abacus logic, she doesn't understand at all. I felt this generational gap might be significant. I wondered if the concept of calculation changes depending on the tools used.

You're absolutely right. I tell people everywhere not to teach numbers too much to children before they enter elementary school. There's no point in teaching numbers when the concept isn't well understood.

But even without understanding the concept, just by memorizing the sequence of numbers, one could—to an extreme—end up being able to do integral calculus in high school.

As long as they remember that '2' follows '1,' '3' follows '2,' and '4' follows '3,' they can get a passing mark on that extension. I think that's dangerous. That's why I believe it's more correct to say that the concept of numbers is something mysterious. I think numbers are strange things. But nobody listens to me (laughs).

True, people simplistically think that being good at math means being smart. While you can handle things with superficial techniques before the concepts are solidified, I feel that, as you say, it will eventually lead to stumbling or misunderstandings.

I think it's the same in English, but in French, when expressing a small amount or length, we sometimes say 'one finger' or 'two fingers.' It's the same as saying 'a finger' of whiskey. Are there similar expressions in Japanese that are linked to the body?

I think they exist as units. Like 'shaku' or 'ata,' which represent the span of a hand. 'Ata' is an ancient unit of length corresponding to the distance between the thumb and middle finger when spread. For the length of chopsticks that fit one's hand, it is said that 'a length of one and a half ata is best.'

France tried to change the system of weights and measures all at once during the French Revolution. The meter is the most famous example, but they also boldly tried to unify various things artificially, even if they conflicted with everyday senses. I think the calendar (the Revolutionary Calendar) was one of them.

The Revolutionary Calendar made a day 10 hours, a week 10 days, and a month 30 days, and changed the names of each month. A calendar should be linked to the image and physical sensation of weeks and months. For several reasons, perhaps because it didn't fit people's lives, the Revolutionary Calendar eventually fell out of use.

The length of time might also be a matter of physicality, but the French example shows that there are units that don't work if they deviate from everyday senses and daily life.

Was there confusion in Japan when the Shaku-kan system changed?

In the Meiji era, they were used interchangeably. Depending on the item, they would use the Shaku-kan system as is, or change to the yard-pound system, and there was a mix of British and American counting methods, so I think there was quite a bit of confusion.

For example, the counter 'kin' for bread comes from one pound and had a proper objective weight, but that also became unclear, and it was decided to count any loaf of bread as 'one kin' regardless of size. There are examples of things becoming loose like that.

In that case, 'one kin' is not a unit because it's not one pound, right?

Nowadays, many companies sell bread, and everyone calls it 'one kin' even though the weights are different, so it just means 'a loaf of bread.'

In France, was there a constant struggle between the idea of using the decimal system and the idea that base-20 or base-60 systems were better?

Even within France, there were regions that used decimal-style ways to say numbers in the 60s, 70s, 80s, and 90s. It seems that was the case in parts of the south and east even into the 20th century. French-speaking Belgium and Switzerland also use decimal-style terms. However, in Belgium, they seem to use a vigesimal (base-20) style for the 80s, like in France.

Currently, there is an international student from the Democratic Republic of the Congo at SFC. The official language of the DRC is French. However, that student said that for the numbers '70' and '90,' they use decimal-style terms, unlike in France.

Why do they use decimal-style terms? Even though they use the same French language, it's because the DRC was originally a Belgian colony. Numbers seem universal, but I realized then that how numbers are called is quite deeply connected to political and social issues.

When I was in the US, many people didn't get it when I said my height was 1 meter 50 centimeters. As I thought, feet are ingrained in their daily physical sensation, and they don't understand meters. Aren't units built on the experience of numbers and time that we use every day?

When I went to buy shoes, even if I said 'Please give me size 23 centimeter shoes,' they were like 'What?', and I had to try everything on like Cinderella, which was inconvenient (laughs).

When expressing the volume of liquid, I think in Japan it's written in cooking recipes as 'cc,' but in French recipes, the unit 'centiliter' (cl) is often used. One centiliter is 10 cc.

So when I was cooking while looking at French recipes, I thought the taste was a bit weak. It was very strange, but I realized that since they measure in centiliters, if I cook with the sensation of cc, it ends up being one-tenth (laughs).

Europe uses centiliters, don't they? Most liquor bottles also have centiliters written on them.

We learn 'deciliter' in math, but which country uses that?

It's not used anywhere. You could say that unit exists for the sake of Japanese math education. In practice, one deciliter is one small cup. That makes it easy to experience physically.

So it exists specifically for that purpose.

Just like centiliters, it's naturally possible as an original concept. It's used because it's perfect for math examples.

There's often a math independent research project to look up units around us, where you pull things out of the fridge and find that one Yakult is 70 cc and milk is 1 liter or 1000 milliliters. But since I've never seen anything that's 1 deciliter, I thought it was a fictional unit (laughs).

Prefixes like 'deci' for one-tenth or 'deca' for ten times are auxiliary units that are hardly ever used. I don't really remember 'decimeter' either.

What's even worse is that a square centimeter is not one-hundredth of a square meter. It's one-ten-thousandth of a square meter because it's one centimeter on each side, so it becomes confusing.

That's why you can't possibly remember it just by memorizing numbers; you have to understand it intuitively. So in the end, what we return to is, of course, quantity, but I believe we have no choice but to grasp it as a sensation of counting using the body—as an intuition.

Counting methods have a different kind of interest when viewed from the perspective of mathematics. I learned a lot today.