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 things, yet people often ask me why they are counted with the same word (hon). Or why a home run is called 'ippon' (one long object) even though it has no shape and the ball is round, or why people say they 'wrote one paper' (ronbun ippon) 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 life 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 think 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 pinky 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 interesting. It's a perception that goes from left to right up to 37. Do the people of Papua New Guinea have a physical sensation where small numbers are on the left and large numbers are on the right?

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

Does it start from 38?

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

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

That's right.

Base-10 and base-20 systems are well known. The explanation usually given is that base-10 comes from humans having 10 fingers, and base-20 comes from having 20 if you include toes. I don't know how true that is, but that's the common 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 count one 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 Roman times, they used the fingers of both hands to represent numbers up to about 10,000. That method allowed them 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: 'Hold one in my hand, two... oh, I can't hold any more, so it's "many," I give up.' It's about counting with the body. Naturally, there is no abstract concept of number there; it's just a matter of one-to-one correspondence.

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

We do count things 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, they fold their fingers while wondering how many more days until they can see their beloved. So Japanese is a culture of folding fingers, whereas in America, they extend fingers to count. 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 pinky, it looks like they have trouble raising 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. The discussion about whether to fold or extend fingers is all based on the fundamental position that counting is a one-to-one correspondence with body parts.

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,' do we accept the stage of 'counting on'—adding them by counting one by one—or do 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 lower grades of elementary school. That's why I think we should let them count thoroughly using their bodies. Since they are human, I believe they will naturally abstract things as their intelligence develops.

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 was replaced by counters (ohajiki) or when a stick was used to represent '10,' the teacher would naturally say, 'This stick is 10, so 13 is one stick and three counters.' Suddenly the shape and the way of counting changed, and they became lost.

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

I completely agree. I often say that from the lower grades of elementary school down, 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, if you want your child to be able to do math in the future, I tell people to let them play outside as much as possible. Let them climb the jungle gym, or let them dance to music on TV.

That's true. In music, teachers talk about what kind of note it is and how many beats it has, or how many quarter notes it equals, even before the children have learned multiples or divisors in math, so they don't understand it at all.

I feel that unless we teach by emphasizing 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, did you have any confusing experiences with numerical communication when you were in France?

I was confused all the time. 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, 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 head instantly when they hear '60 + 11,' but at least for me, it was hard to get used to.

Furthermore, when it gets to the 80s, it becomes base-20, where 80 is said as 'four 20s,' so my head starts to get muddled.

Why has it become such a cumbersome way of speaking? Just like Japanese, French is heavily influenced by Latin, and the ideas of the earlier Celtic people have also entered the language; history resonates strongly in language. There are many intriguing points in the history of these names.

Naturally, astronomical elements enter into mathematics. So the reasons for base-60, base-20, or base-12 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 for this occasion, and if I were told to do multiplication tables with that, it would be painful. After all, 81 is '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 base-10 way should be used.

You don't see them often, but there are 2-euro coins. Before the euro, I remember seeing '2' coins like guilders in other countries, 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.

In my sense, the idea that '1' and '2' make it full, so it's a single 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 a half-division. Oh, come to think of it, gold coins in the Edo period were 4 'ishu-kin' for 1 'ibu,' and 4 'ibu-kin' for 1 'ryo' koban. Was that a half-division?

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 reading poetry, are people 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 they use this alexandrine. 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 belongs to the category of languages rich in counters. As far as I have investigated, 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 as well if you look into it. So I think it might be a common human cognitive sense that having up to a maximum of about 500 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, I think, a sensibility unique to the Japanese.

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

Yes. What I find unique and interesting about the Japanese way of counting is how it jumps onto shape and function. For example, we categorize by shape like 'ichimai' (one flat object), 'ichimen' (one surface), 'ikko' (one small object), or we use 'ichidai' (one machine) broadly for a car, a computer, or a piece of machinery.

Even though there seem to be more important things like color, softness, or whether it's edible, the fact that we deliberately focus on function or shape and count things together that don't look like companions at all is, I think, very bold and interesting.

In your book, Ms. Iida, there was a story about whether a robot dog is called 'ichidai' (one machine), 'ichitai' (one body), or 'ippiki' (one small animal). 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 'ippiki,' or even 'hitori' (one person).

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

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

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

Analyzing this philosophically, is it better to think that different colors mean different objects?

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

Indeed, perceiving 3 cats and 3 blocks of tofu as the same '3' is what '3' as a number is, 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 ability for abstraction and universalization. Stepping beyond the perception of numbers tied to concrete things is a human capability. In that sense, the ability to handle numbers was highly valued in philosophy by, for example, Plato in ancient times, who 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 in a similar way for 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, whether they want to convey it to the listener as something not to count or something to count, and the use of articles changes accordingly—which makes sense. Indeed, that's how it works in actual use and as a linguistic mechanism.

So, perhaps the consideration of whether it's 'ippiki' (one animal) or 'hitori' (one person) in Japanese also has an aspect 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 the nouns themselves become very blurred. When counting, it's like using a cookie cutter; I hypothesize that the framework of the counter word acts as that cutter, and counting the pieces cut out is how the system works.

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 the sense of nouns is significantly influenced by lifestyle, which in turn affects counter words.

In elementary school mathematics education, there is the concept of number as a count—one item, two items—and another concept, which is quantity. I think units are essentially the way we measure quantity.

Ms. Iida, your view in your book that units are things that allow for substitution is certainly a good perspective, but one could also say that units are what convert how we perceive quantity into the form of numbers. If it's hunting, you can just say you caught one rabbit or two rabbits, so the concept of count suffices. But with grain, that doesn't work, so the concept of quantity becomes important.

Actually, this is a key point in math education. Talking about numbers that can be counted as one or two 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 in a messy way.

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; 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' might be a small amount to the other person. I think it was crucial for people in distant locations to standardize units of quantity when engaging in trade.

My daughter also struggles a lot with fractions. Doesn't it require a whole new set of skills, like finding common denominators or simplifying? When you do that, you end up going back to concrete examples like 'how many buckets of water,' so the mathematical ability that was finally abstracted becomes impossible to understand unless it's made concrete again. 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.

So you're saying that moving back and forth 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 take it two levels further into abstraction. However, everyone fails to keep up with that abstraction.

So, out of necessity, I descend to a concrete example that everyone should normally be able to do, then go back up two levels of abstraction, and repeat this process until they can tolerate the two-level abstraction as a sensation. After doing it over and over, 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 sufficient. Too many people try to acquire only 'warp techniques' that skip the process without thinking logically. With only warp techniques, the act of memorizing itself becomes painful.

This is something I've thought about for decades, but it comes down 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 intuitively feel that you've understood. I tell people to think to that point. At first, you have no choice but to accept it logically and understand that 'this is the structure.' But ultimately, I want them to reach the point of intuitive understanding.

It's important how deeply it connects with the experience and knowledge you already have. I think you understand intuitively when you feel that 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 as you do it many times. When you can think of '2' as something certain, that's when you can say you've understood '3' or understood the count.

People who can do it without thinking about it every time can be said to have recognized something quite abstract as a concept.

Nowadays, they've stopped doing the abacus in elementary schools, haven't they? Has the calculation ability or the concept of numbers for Japanese people changed because they no longer use the abacus?

I wouldn't go so far as to say the concept or ability has changed, but I think an influence will emerge. However, compared to when you and I were children, it's only been a few decades, so I don't think it will be enough to destroy society as a whole. And since we have calculators, it doesn't really come to the surface.

However, in high school chemistry, they use calculators, and some people accidentally make a two-digit error and write that 2 kilograms of salt dissolve in 1 liter of water. No matter how you think about it, that's impossible, but 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 things in the lower grades.

I'm not that good at the abacus either, 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.

That's exactly right. I tell people in various places 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 if they don't understand the concept, just by memorizing the sequence of numbers, in the extreme, they end up being able to do integral calculus like a high schooler.

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

True, people simplistically think that being good at math means you're smart. While you can handle things with superficial tricks before the concepts are solidified, as you say, I feel that eventually, you'll stumble or misunderstandings will arise.

I think it's the same in English, but in French, there are ways to express small quantities or lengths by saying '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 own hand, people say 'a length of one and a half ata is good.'

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 did things that conflicted with everyday senses in an attempt to unify various things artificially. I believe 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 even changed the names of each month. A calendar should be linked to the image and physical sensation of weeks and months. Whatever the 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 catch on if they deviate from everyday senses and daily life.

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

In the Meiji era, they were used interchangeably. Depending on the item, they might keep using the Shaku-kan system, change to the yard-pound system, or have a mix of British and American counting methods; I think there was quite a bit of confusion.

For example, the 'kin' used for counting 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.' There are examples of things becoming loose like that.

In that case, since one 'kin' is not one pound, it's not really a unit, is it?

Nowadays, bread is released by various companies, 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 phrasing. 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 phrasing, unlike in France.

Why do they use decimal-style phrasing? 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 as an everyday physical sensation, and they don't understand meters. Aren't units built on the experience of numbers and time that we use daily?

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

When expressing the volume of liquids, 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 a French recipe, 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? Liquor bottles usually have centiliters written on them.

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

Nowhere uses it. 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, as an original concept, it's naturally possible. It's used because it's perfect for math examples.

Often in math independent research projects, students investigate units around them, pulling things out of the fridge to find that one Yakult is 70 cc and milk is 1 liter or 1000 milliliters. Since I've never seen anything that's 1 deciliter, I thought it might be a fictional unit (laughs).

Deci for one-tenth and deca for ten times are auxiliary units that are almost never used. I don't have much memory of decimeters either.

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

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

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