This is a short one and, necessarily, is WitCH-like. It is an elaboration in the new Curriculum that smelled wrong to us. We checked enough to confirm there was sufficient wrongness for the elaboration to be added to the Awfulnesses list, but we haven’t sorted it out further. The comments may be interesting (or non-existent).

*comparing the Hindu-Arabic numeral system to other numeral systems; for example, investigating the Japanese numeral system, 一、十、百、千、万* (AC9M3N01)

1) What’s the point of comparing?

2) The Hindu-Arabic equivalents to the Japanese symbols given are 1, 10, 100, 1000, 10000. (No zero …?) How is this the “Japanese numeral system”. And again, what’s the point of comparing?

And if you’re gonna go down that path, surely Roman numerals would be explicitly mentioned before Japanese.

Maybe we should also compare the “Hindu-Arabic numeral system to” the colours IR, Red, Orange, … UV. Or the known isotopes of helium. Or the planets of the solar system (Pluto will always be a planet to me). Or the members of Bruce Springsteen and the E Street Band. Or the lives of a cat. Or …

My (30 years ago, roughly) memories of the Japanese way of writing numbers is that it was quite logical in itself but devoid of place value (more or less).

Is there a point to learning about Egyptian number systems (which seem to be in every Year 7 textbook I can find)?

Not convinced.

I’m also pretty sure the Japanese have a symbol for zero, it is not a place-holding symbol though, so is only used when one wants to write the number zero itself.

JF, you joke but if a cat has 9 lives, then we maybe have a base ten system…

Let’s hope that a cat has at least eight lives more than the acara curriculum.

(Isn’t the Japanese system base ten?)

It is, but instead of writing 21 they write which uses three characters (2, 10, 1).

Hence, place value becomes less obvious to someone used to the Hindu-Arabic system.

Then it’s just the thing we’d be wanting to include when teaching this stuff. Just to muddy the water and keep students, especially the weakest ones, on their toes. Good job, acara. Someone send in the ninjas.

More precisely, but without hieroglyphs, in ancient Egypt they would write

(10)(10)(1).

In their system there is no need for placed value; 21 could be equally well expressed as (10)(1)(10) or (1)(10)(10).

I’d be more convinced if the example was for example, using French (where 80 is read as “four 20’s” with a couple of other oddities), or perhaps counting in a different base (e.g. base 12).

Even then, nothing overly worthy beyond one or two lessons’ worth in my opinion.

Given the morbid fetish our education ‘authorities’ have for assimilating pseudocode, algorithmic thinking, use of technologies etc. into the mathematics curricula, I’m surprised there’s no explicit suggestion to compare with the hexadecimal system.

Even one lesson is pushing it and even then I question the value (pun partially intended).

That said, the “addition algorithms” I see in recent textbooks at Year 7 level are a bit of a worry for the title alone, so maybe delaying those lessons is a good idea.

Some exposure to other systems of numeral has potential benefits.

First, it enhances appreciation of our system of numerals which we take for granted.

Second, this can link mathematics with other subjects. For example, if the students are studying ancient Egypt in history, learning about mathematics in ancient Egypt links history and mathematics. It is unlikely that the history teacher will explore this aspect of ancient History. Perhaps students could be set an assignment that would count for both history and mathematics.

Third, students might be asked to create their own system of numerals as suggested in the attached.

2020-MillsSacrez

Please accept my apologies for my typos.

The quote from Hilbert may be found under item 1 below, without two typos from the paper:

https://de.wikipedia.org/wiki/Cantors_Paradies

Thanks, Terry. I know you’re a glass half full guy (even when the glass is 0.99 empty). But I still pose the question: Why?

一 ) There are many things we take for granted. Why choose our system of numerals for special treatment? We take for granted even the process of simple transposition when solving an equation …

二 ) You’re talking about ancient mathematics. I agree that there’s value in teaching the history and philosophy of mathematics. But acara is explicitly giving an example “comparing the Hindu-Arabic numeral system to [modern] systems”. What’s the point?

三 ) When there’s so much for a student to learn and understand, why spend time doing this? What will students learn from it? Or is it less about learning and more about having fun? As the Sensei says – Don’t think! Do!

Why choose our system of numerals for special treatment? If one reflects on very elementary mathematical ideas in different cultures, technical difficulties do not get in the way. Finger counting is a good example. I recall that finger counting is different in China and Japan, and this is how a spy from one of these countries was caught out in the other.

I can’t explain why ACARA suggests this comparison. My point is that considering other systems of numeral has potential benefits. Why did the Romans write L for 50? Perhaps ACARA suggests Japanese because this idea came into the head of the person while writing that part of the curriculum without a great deal of thought, or perhaps the author knew something about Japanese mathematics. Anyway, I read it only as a suggestion. I agree with the implication of Red Five: the treatment of Egyptian mathematics in lower secondary mathematics is superficial and not enlightening.

There is indeed a great deal to learn. But so much time is wasted in teaching mathematics at school. Students are exposed to the same ideas year after year. Surely there is a more efficient way to teach students about mathematics.

BTW, here is a paper from the Mediterranean Youth Mathematical Championship in 2013 with lovely historical references.

2013-mymc

Interesting comments. I agree, it is far from clear what ACARA imagines is the purpose of this “comparing”. But that’s garden variety nonsense for ACARA.

None of the comments have gotten to why I posted on this specific elaboration.

The elaboration is for students at Year 3 level. I have often found that such elaborations might be better at a more advanced level. Let me say at least Year 7, perhaps Years 9 or 10. It’s possible that some students at Year 3 level are still struggling with the Hindu-Arabic system. The beauty of our system is a subtle but important feature that can be best appreciated when one has a firm grip on basic arithmetic. Perhaps this is a case of “elementary mathematics from an advanced standpoint” – sounds like a good title for a book.

“The elaboration is for students at Year 3 level. ”

As I said earlier, the suggestion is just the thing we want to include when teaching this stuff. Just to muddy the water and keep students, especially the weakest ones, on their toes. Good job, acara.

“Perhaps ACARA suggests Japanese because this idea came into the head of the person while writing that part of the curriculum without a great deal of thought, or perhaps the author knew something about Japanese mathematics.”

The former, I’m sure. It raises the question: On what basis – apart from its name – is acara qualified to write a national curriculum? It reminds me of Dr Enos Pork MD (ENT, brain surgery by appointment). The name maketh the man it seems.

ACARA’s qualifications are not really the point here as they are not a person applying to write the curriculum, they are a group of people who have been given the task of writing said curriculum.

I would suggest to my lawyer (in full client-lawyer confidentiality, of course) that the stone best looked under is that of those assigning tasks to organizations.

Circles within circles, as they (not ACARA) say.

OK – a question for those still playing the original game: ACARA, who as we know likes to choose their words (and numbers…) carefully (sarcasm, apologies) has chosen to say Japanese NUMERAL system and not NUMBER system.

That seems interesting for a couple of reasons but mainly one:

Unlike the Hindu-Arabic system, the Japanese number system has (what I would describe as) one NUMERAL for 10 which is also a number and one NUMERAL for 100.

There are advantages and disadvantages to this system (Cue: Terry…)

So when is a NUMBER system different to a NUMERAL system?

Good question. There is a subtlety in the answer, a subtlety I think that is beyond the year 3 level. I wonder if acara understands the subtlety.

https://en.wikipedia.org/wiki/Numeral_system

Sure – but WHY did ACARA choose to say NUMERAL system?

It is the thinking behind the decisions that I’m really interested in; although I understand many commenters here believe that is often a null-set.

The generous answer is that acara (rightly) thinks that a numeral system “is a systematic method of using symbols to represent numbers.” (as Marty says).

In which case I would have thought that a much better suggestion- particularly at the grade 3 level!! – would be the unary numeral system:

https://en.wikipedia.org/wiki/Unary_numeral_system#:~:text=The%20unary%20numeral%20system%20is,the%20absence%20of%20a%20symbol.

Every single student would be familiar with representing numbers in this way. Asking what 11 means a formal discussion on numeral systems would be very effective and appropriate for Grade 3. There is all sorts of valuable stuff that can be taught.

This assumes the aforementioned generous answer. Realistically, I think a much less generous answer is much more likely, because I think the example suggested by acara is way too complicated for Grade 3.

Thanks, RF. I don’t exactly understand your specific concern, but I think you’re questioning is getting to the important aspects.

A numeral is a symbol(s) for representing a number. A numeral system is a systematic method of using symbols to represent numbers. So, on its face, it is plausible that the characters above are bones of a Japanese numeral system.

There are questions, however:

1) Is it reasonable to refer to the characters in the above elaboration as “numerals”?

2) In any case, what is the precise meaning of these characters?

3) Whatever name one wants to give to it, are these characters the basis of a system for representing numbers?

4) What is the origin of these characters, and of whatever system to which they might be related?

Thanks Marty – as someone who has taught Year 7 more than a few times, I have what I hope is a decent understanding of the difference between numbers and numerals – but I still think the distinction is a bit vague to many teachers.

In all the examples given in these comments, I think the Japanese numbers, as written, are examples of both a number system and a numeral system.

But… in the absence of any clue as to which base they are working in…

So, in answer to your four questions:

1. Yes. They are symbols to represent numbers or numbers in place.

2. They are counting numbers/numerals and when combined using the structure of Japanese numbers, they have contextual meaning.

3. Yes, but one has to be clear on the structure of the system for writing numbers above 10.

4. The Japanese characters, historically, came from Chinese because the Japanese did not have a written language until relatively recently. The numbers 1, 2, 3 are what you could call pictograms or ideograms. Beyond 3, my history knowledge is sadly lacking.

Whether or not that is what ACARA was hoping to convey… I have no idea.

Thanks, RF. Let’s begin with 4.

Given the Chinese origin of the characters, is it accurate or reasonable to refer to them as (part of) a “Japanese” numeral system (or whatever it is)?

That is a VERY good point.

We refer to the Hindu-Arabic number system even though we use English names for the numerals, so there is internal consistency in NOT referring to “the” Japanese number system.

Then again… do the Japanese refer to their system of numerals as “Japanese”? I do not know.

It’s irrelevant whether the Japanese refer to the characters as Japanese. If the character system is fundamentally Chinese in origin, then of course the characters should be identified as such. Just, as you point out, we refer to 0, 1, 2 etc as being Hindu-Arabic, rather than European or whatever.

I believe Wikipedia sheds some light on the matter:

https://en.wikipedia.org/wiki/Japanese_numerals

Marty, you raised a very good point/question with (1) below:

“If I write “three” then those are (alphabetical) symbols used to refer to the number we commonly write as 3. We refer to “3” as a numeral, but “three” as a name. So, are the “Japanese” characters in the elaboration more like numerals or more like names?”

This is a very good question, one that is particularly reasonable to ask for any language (such as Japanese or Chinese) that uses logosyllabic characters (glyphs) rather than an alphabet.

It’s a great pity that acara did not better research its thought bubbles.

Thanks, JF. So, so far, we’ve got that the characters are not properly “Japanese” and are not properly “numerals”. On to “system”.

Hi, RF. On to 1.

If I write “three” then those are (alphabetical) symbols used to refer to the number we commonly write as 3. We refer to “3” as a numeral, but “three” as a name.

So, are the “Japanese” characters in the elaboration more like numerals or more like names?

Again, fair point.

(Your passing mention of “system” is currently occupying a lot of my thinking, so this answer is not a well-thought-out one)

I’m not sure that the distinction between numerals and names is that clear in this case.

From what I have learned about Japanese writing (thanks JF for the references), Chinese characters are sometimes used in place of the Japanese letters for common words/names. So we can write 1 with two Japanese characters or a single Japanese/Chinese character. I would suggest that the single horizontal line therefore meets the requirement of being a numeral.

But I have a few difficulties accepting this as a fair test/criterion. More thinking required.

From a practical standpoint, I will be ignoring ACARA on this dot-point, in case that matters.

Hi, Rf. I agree that the numeral-name distinction is unclear for such character languages. and I agree, “一” is pretty numeral-ish. But then we get to Q2 and Q3.

What about the other characters: 十、百、千、万? What is the precise meaning(s) of each character?

How, precisely, are (more “were”, I believe) these characters used?

Is your suggestion that writing (2)(100)(3)(10)(6) using Japanese characters is more akin to writing the word two hundred and thirty six than writing the basic numeral 236?

(Not a judgement, just a question to see if I understand what I think you are getting at).

If my interpretation of your question is correct, then the “Japanese numeral system” is none of Japanese, Numeral nor System!

(Out of interest… how do you add the Chinese/Japanese characters in comments?)

Yes. I’m very unfamiliar with the use of such characters. But that’s how it’s been explained to me. The characters have more the meaning of “tens” and “hundreds” than “ten” and “hundred”.

1. They are words, not numerals

2. 一 one、十 ten、百 a hundred、千 a thousand、万 ten thousand

3. They are the equivalent to writing “two thousand and twenty three” in English

4. Their origin is Chinese. Japan adopted Chinese characters in the 4th-5th Centuries – back when Celtic and Latin were spoken in the British Isles, so calling this “relatively recently” is probably not accurate.

Calculations were done using rods (and rod notation), then the abacus, and now Hindu-Arabic numerals. Chinese characters were used, along with other notation, for Wasan (traditional Japanese maths, done with an abacus), while it was a thing.

Although they said “numeral system”, they probably meant “counting system”, i.e. the Sino-Japanese counting system – as that’s worth learning about.

Japan has more than one counting system, a Japanese system only used for numbers 1 to 10, and a Sino-Japanese system for other counting.

The Sino-Japanese system is very logical, and so it is easier to learn than irregular counting systems such as English or French (and is slightly more simple that the Chinese system). After the numbers 1 to 10, the teens are all said in the format ten + units, e.g. ten one (11), ten two (12), ten three (13) (no elevens, twelves or quatorzes). Numbers from twenty onwards are said in the format multiple of ten, ten, units, e.g. two ten three (23), five ten six (56) (no confusable thirties and thirteens, forties and fourteens, fifties and fifteens). And this pattern continues on for hundreds, thousands, and ten thousands .

Learning about other counting systems may not enhance calculation skills, but mathematics education should be about more than that.

Thanks, Tim. That is very interesting and very helpful.

I think most people here would agree with you, that there is educational value in considering other counting systems, and other numeral systems. I don’t think this changes the fact that the elaboration above is garbled, culturally inaccurate and undirected. In particular, listing characters for “one and “ten” and so on is hardly indicating anything of a different “counting system”.

Although the Sino-Japanese naming system is more logical, with “two ten three” instead of “twenty-three” and so on, is that such a big deal? Is it pointing to a fundamentally different sense of recording or computing with numbers, in the manner of e.g Roman or Babylonian systems?

I’m still happy with the idea of including plenty of exposure to different systems of naming and counting and computing, including the system poorly presented in the elaboration above. I think there is plenty to be learned there, mostly cultural rather than computational, but not only cultural. As Simon just commented on an another post, the false identification of numbers with names and symbols is very tempting, and presenting other systems can definitely help address this.

For all who have potential pop culture shortcomings/gaps and therefore don’t quite understand Marty’s title:

(It is an unfulfilled dream of mine to be a one-hit wonder and live off the royalties forever after. But alas, I think that boat has sailed).

Yep, that was the reference.

Smith, D.E. & Mikami, Y. (1913). A history of Japanese mathematics. Open Court Publishing Co.

Click to access historyofjapanes00smituoft.pdf

Thanks again, Terry. Very interesting.