# What is the mathematical value of children learning and being tested on Roman numerals?

My 11 year old child recently took an important numeracy test. One of the questions required her to know that M = 1000 in Roman numerals. This made me very angry: I could not see how this relatively obscure piece of trivia was a useful way of measuring a child's numeracy.

I complained about it on social media, and the complaint gained a great deal of traction. My opinions on this - and other aspects of testing - are now being sought by the national media. Being someone who believes in informed opinion where possible, especially at that level, I wanted to make sure I was correct in assuming this was a somewhat superfluous part of the curriculum.

Some defenses I've seen offered are:

• The linguistics of "M" have lead to it being used as part of the mili- and mega- prefixes in metric measurement. But units of measurement and conversion between them are already taught (and tested) separately.

• It helps children understand that there are number systems other than decimal. But surely very basic algebra (which is already taught and tested) also demonstrates this? And it would also seem possible to understand (and test) different number systems without this precise piece of information?

• It is useful general/cultural knowledge. Well fine: I'm not going to dispute that. But that doesn't make it a useful aspect of measuring a child's numeracy.

Are my assumptions on this correct? Or have I overlooked some other value in testing this knowledge?

• This seems like a tempest in a teacup to me. For better or for worse, there are all kinds of culturally relevant pieces of information that stem from previous eras that are nonetheless important for an educated person to know. For an 11-year-old to know a few Roman numerals does not seem unreasonable, given their continued limited use today and significant use in the past for dates and other numeration. – kcrisman May 12 '17 at 11:49
• This seems more like a rant than a real question. There are many sub-questions in the post. For this site it would be better if it were cleaned up to focus on one single question. – Daniel R. Collins May 12 '17 at 13:34
• Regarding your comment directed at @celeriko: there is a qualitative difference there between roman and decimal, binary, hex, etc. The latter are all positional, the roman is not. Fair enough, the comment said only non-decimal, yet as mentioned in the first comment this one has the feature of not being positional, and as such is a lot more different than decimal, binary, hex, octal etc which are 'essentially' all the same. – quid May 12 '17 at 16:23
• – Dmiters May 12 '17 at 18:07
• Don't you mean educational value ? It has nothing to do with math. It's been taught, therefore it should be tested for. Why are Roman numerals still used? "It's mainly just inertia." – Mazura May 13 '17 at 1:11

## 12 Answers

Learning about different numeral system develops the critical thinking and understanding. Arabic numerals are so ubiquitous that most of us take them for granted. Seeing other numeral systems and analyzing them creates a lot of opportunities for thinking about why did we pick one over the other and that there was a pick at some time, not a given (somebody had to come up with the design of characters, somebody had to create the numerical system and somebody decided that Arabic is better).

Part of this understanding comes from doing operations and experiencing first hand how cumbersome it can be in other number systems. It also enforces the idea that digits are just symbols to express ideas (i. e. numbers). Actually, the whole purpose of a numerical system is communication - it provides a way of communicating abstract concepts (numbers) and perform operations on them. Thus, they can see that 3 and Ⅲ are two different ways of expressing the same thing.

I find it interesting that the numerical symbols are so embedded in our brains that it becomes very hard to switch them or even be aware of it. This also helps when learning new numerical bases that might use different symbols by keeping their mind open.

And also, knowing Roman numerals has a practical application. Every once in a while, I come across a year or a quite big number written in Roman numerals.

• This is important. Notation affect perception of the concepts. The placement of number symbols ( also in spoken language, which curiously differ a lot, some say "3 and twenty" and some others "twenty and 3" and more obscure ones surely exist) often bust a dependence between how to calculate a number and how to represent it. $IV$ means not only 4, but "subtract 1 from 5" just as you can claim that "23" in decimal means "multiply 2 by 10 and add 3". – mathreadler May 14 '17 at 7:09
• @mathreadler. Tricker ones definitely exist. Tom Scott argues that 58 is a very confusing number. – TRiG May 14 '17 at 22:43
• @TRiG: Yep, knew you'd go for that one, but it's too late in this case. ;) – mathreadler May 15 '17 at 3:27
• The copyright dates of BBC TV programmes are an example of Roman numerals in action. And, if you want to know when a programme was made, you'd better be fluent because you only have a few seconds to parse MCMLXXXVI. Though kids these days have it easy and are more likely to have to read MMXV. – David Richerby May 15 '17 at 21:05
• @DavidRicherby or the date listed in the opening credits of movies (at least back in the MCMXL's ) – Carl Witthoft May 16 '17 at 13:19

# Common knowledge

Roman numerals are part of the common knowledge in Western cultures for representing numbers. That's why it's valuable for children growing up in these societies to learn them. It's the same as the reason we teach children Arabic numerals.

If you don't know how to read Roman numerals, you'll be effectively illiterate—or "innumerate"—whenever you encounter one. Roman numerals function like "italics for numbers": a distinctive, alternative way of writing the same thing. For example, they're used for page numbers in long prefaces to distinguish them from the main page numbers of the book, for items in hierarchies (such as essay outlines) to distinguish different levels, for the roots of musical chords where Arabic numerals denote intervals, for groups in the periodic table, etc. They denote ordinal numbers of monarchs (e.g. Henry VIII), sporting events (Super Bowl LI), yearly formal gatherings, world wars, version numbers, movie sequels, clotting factors in the fibrin cascade, etc. And they represent years or hours in very formal contexts such as stone inscriptions, sundials, and clocks meant to last a long time. When people write Roman numerals, they expect that any member of our society with even a minimal education will understand them. They're even used in slang: a "C-note" is a hundred-dollar bill.

That's what I mean by "common knowledge": knowledge or conventions that are shared by most people in the culture and that are known by most people to be shared. When something is common knowledge, we can depend on other people knowing it, other people knowing that we know it, etc. The classic example of common knowledge is language: without that, we couldn't communicate much at all. Roman numerals are just part of our common language of numbers.

If you're thinking that numeracy, since it relates to mathematics, should be independent of culture, that's an error. By that reasoning, Arabic numerals would also be excluded. From the standpoint of pure mathematics, the fact that the symbol 6 stands for the same abstract quantity that we refer to by the word "six" is an obscure piece of trivia. "Numeracy" is a vague concept, but I understand it to emphasize practical matters, and that includes knowing your culture's conventions for representing numbers—even a less-often-used convention that exists alongside the most common one.

# Tangential mathematics and history

Some people argue that Roman numerals should be taught because they illustrate things like multiple ways to represent a number (a kind of flexibility that you need for algebra and anything beyond), or the value of positional notation, or the value of a symbol for zero (since you really come to appreciate something by trying to do without it), or that the C and M stand for the Latin roots of our prefixes centi- and milli-. Roman numerals provide an opportunity to teach all those things, and children may learn them as a happy side-effect even if the teacher doesn't point them out, but the related topics aren't the reason for teaching Roman numerals. Kids need to learn Roman numerals so they can participate in the common knowledge of the adult world—the same reason that nearly everything in elementary school through high school is taught. Notice that if learning the related topics were sufficient reason to teach Roman numerals, then Roman numerals should be taught even in Mongolia.

By the way, the numerals used by the ancient Romans did not exactly follow the conventions that today we call "Roman numerals". They were less standardized and the forms changed over time. For example, one convention represented 500 as IƆ; 18 might be represented as IIXX. More here—but those sorts of things are relatively obscure trivia, not common knowledge. While the Roman numerals taught in elementary school today have deep roots in our past, they are part of our present-day culture.

• "If you're thinking that numeracy, since it relates to mathematics, should be independent of culture, that's an error." Nice. – kcrisman May 13 '17 at 12:08
• The 'value' is in knowing "common knowledge of the adult world—the same reason for teaching nearly everything" (should be the second sentence in the first paragraph). The above comment should be a preamble. – Mazura May 13 '17 at 18:46
• @Mazura Thanks. I just reworded to explicitly mention 'value' in the opening paragraph. (I had to puzzle a while about exactly where to place the emphasis.) – Ben Kovitz May 18 '17 at 8:43

As I said in the question this has attracted quite a bit of attention. And after a lot of nonsense, I did finally get an answer from a British maths teacher.

Although measurements and conversions are taught, schools are also supposed to pass on the direct equivalences between these and Roman numerals. i.e. to teach that C and M are the basis for centi-, mili- and that each case where it's used is an operation on 100 and 1000 respectively.

While each piece of knowledge by itself may not be particularly useful, the key thing is that learning the connection between them aids understanding and recall. It becomes a contiguous piece of information where each bit supports the other.

I suspect a lot of schools are perhaps not making this connection explicit, or that it's not being properly understood by the children. But it's good to see the reasoning here.

• +1 for your third paragraph, which I think is very nice and should be made more explicit in teacher training. – Dave L Renfro May 12 '17 at 15:05
• +1 Rarely have a I seen a better self-answer to a question on this site. – Daniel R. Collins May 12 '17 at 17:08
• It’s also worth noting that, if education is poorly thought out, children may not even realize that facts are connected to each other in real life. By drawing these connections, the curriculum not only teaches about connections that help reinforce these particular ideas, but also teaches that everything is connected and understanding and knowledge hinges upon discovering and understanding their connections to everything else. A curriculum that fails to do this will have students thinking that they are just supposed to memorize a whole bunch of random facts, missing the greater whole. – KRyan May 12 '17 at 20:23
• I totally never realized it until now, the relationship between C/M and centi- and mili-. I just liked Roman numerals because they were fun; of course, while C/M are useful for that relationship, something like D=500 or even the lower numbers (I, V, X, L) are less so in this regard. – Roddy of the Frozen Peas May 12 '17 at 21:54
• @Dorus Indeed "C and M are the basis for centi- and milli-" sounds dubious—but almost true. Centi- and milli- come from the French words centième and millième, meaning hundredth and thousandth. The Romans used various other symbols for hundred and thousand for a while, which morphed gradually into C and M, apparently "attracted" by the Latin words centum and mille, meaning hundred and thousand—the roots of the French words. Often "mutually supporting information" is information distorted to make just-so stories. – Ben Kovitz May 14 '17 at 1:52

One other little defense of roman numerals: a lot of students see the "place value" notation we have for numbers as inevitable, since they've used it from before they can remember. But of course it took hundreds of years to invent. Try multiplying CXI by X, for example -- imagining that you're a Roman who can't just translate these into Arabic numerals, multiply, and translate back. It quickly helps you realize what a powerful technology we have hidden in the notation.

• (+1) Incidentally, multiplying CXI (or CIX) by X works quite nicely - there are certain tricks to deal with multiplication in Roman numerals, which serve both as a reminder that there are other systems with different algorithms to the ones we are used to using, but also have some lessons to teach us about e.g. the distributive property of multiplication. – Silverfish May 12 '17 at 22:59
• Multiplying by X is easy - C times X is M, X times X is C, I times X is X, and you get MCX. If you want to show the value of positional notation, try multiplying CLXXII by VIII, or worse, dividing. – Mark May 12 '17 at 23:00
• In that context, it's also worth taking a cross-glance on what astonishing feats of engineering the romans were able to pull off, in particular in architecture. Today these disciplines rely heavily on numerical calculations, which are rather unthinkable if all you have is those awkward numerals. Good to remind ourselves every once in a while that the “modern” a-point-is-a-vector-is-just-a-bunch-of-numbers attitude may often not be the most effective approach. I reckon educated romans were rather more skilled in geometry than most people today. – leftaroundabout May 13 '17 at 13:09
• +1 To me this is the important part of Roman numerals. It's not just a "see you can do things a little different in base 8" or "see you can use letters as numbers in base 16." With Roman numerals it's a "here is a completely different way of handling numbers that looks nothing like what you are used to." – Cort Ammon May 15 '17 at 20:46

You forget, that school is not only about knowledge, but also about thinking.

Learning Roman numerals and how they work teaches child, that the same thing may be expressed in different ways. And that we may define new system with a new set of rules and now we have to use these different rules to work with this system - hell, the whole maths is just about it! Roman numerals are perfect way to slowly introduce child to this specific way of thinking.

I have a master's degree in mathematics - I studied on Wrocław's University in Poland. Currently I'm working for 10 years as a programmer and I don't use much knowledge I gained while studying. But even now, knowing that, I definitely would still chose mathematics as my studies, because I utilize way of thinking that studies taught me.

I'm having an impression, that nowadays people tend to change the educational system to teach only practical things. And such system produces robots, who can go straight, but when problems gets more complicated, they can't think outside the box. In our company we have trouble with hiring good programmers, because a lot of candidates are just like that.

Maybe, instead of getting angry and going to teacher and telling, that he's teaching your child stupid things, it would be wiser to widen horizon of your child even more and show him Mayan numeric system, which is 20-based or Indian one, which uses separators in different way than western systems? Maybe it is a good occasion, to tell your child about binary or hexadecimal system and how it is used to transmit data today? Or sit down and invent your own numeric system and try to use it? Did you think about it?

• +1, and I wish I could upvote this more times. – Martin Argerami May 13 '17 at 17:52
• While this may be the ideal (that education teaches thinking), in my experience schools don't really work this way. Using it as justification for teaching content requires substantial evidence that, to my knowledge, does not exist. Furthermore, it may work for you, as an advanced learner, while not working for 11 year-olds. This may seem overly harsh, but your idea is a prevalent one, and is rarely given with any real backing. – Jacob Maibach May 14 '17 at 17:22
• I joined this community just so I could upvote this answer. – Nelz May 17 '17 at 13:13

Aside from your own answer, I also am a firm believer in providing our children with a good educational background.

Knowledge in itself is value, even if you cannot always use it for any practical reason, and even if it stands pretty much on its own. In the case of the roman numerals, you cannot do much with them, they do not lead on to other insights, and they have nothing leading up to them. They are remarkably on their own.

And yet.

Your child will encounter such numerals plenty, in his future. Kings, year-of-print in books, dice, street signs, clocks, etc.. At least this is the case in my country. And while I certainly do not remember every last of their letters, and would probably struggle a bit to decode or encode larger number spontaneously, I am still pretty happy that I know, in principle, what they're about, how the Romans thought about it, and how I could, in principle, work out how to decode one, if I really had the dire need to do so.

The same goes for everything else. Does a child really need to know what a minuet is, whether stalactites grow up or down, how many planets our solar system has or how a worm digests its nutrients? You could do away with an awful lot of our (higher level) education with that thinking. I know people who made it all the way through to pension age without ever doing a single mathematical calculation except basic +, - and * (seldomly /) in the low digit range. I'm not sure even if they can really do * and / or if they cheated with x*2 = x+x and x/2 = "try numbers until their double is x". And still we teach our kids all kinds of maths for many years.

So in summary, I would probably not care that much if my child did not understand the roman numerals, or forgot about them soon enough. But I am certainly not getting angry by the fact that he is taught the topic. I would be pretty sad if I learned that they were not taught anymore, though.

Maybe it's not purely a maths thing, but maths comes into other subjects, and the crossover is both ways. As a former teacher in U.K. I can't believe that the national curriculum contains only 'M' and 'C'. Actually I can! Usual half-cock.

From a maths standpoint, as already discussed here, it's good that children understand other number systems - decimal being one of very many. I've taught base 5 to 11 yr olds, and it helps them understand place value. Then moved on to base two - binary - which is in everyday use. The French have a strange way of counting, so maybe that ought to be addressed as well - normal till 60, then 70 is 60+10, but 80 is called 4x20s, with 90 = 4x20+10. Just another interesting bit of maths, that could spark the attention of bored little Willy at the back of the class...

At 11, I could read the burial dates in graveyards, and the BBC still dates films etc in R.N.

Incidentally, algebra normally gets used in base 10, so no big deal there.

To answer, there are many many things that get tested at school, at 11 - and younger and older - and the vast majority of them can be, and are questioned by parents and pupils, and sometimes by the teachers themselves.Certainly by me! So, if that's the only issue you have with the testing regime, you're very lucky!

To finish - why doesn't someone tell crossword compilers that 99 is not IC?

• Hey Tim! I've never seen a crossword break the Roman numeral rules so egregiously. If I did, I would stop doing crosswords in that publication. That said, crossword writers do get things wrong quite often, sadly. I can only hope their deadlines allow no time for research. LXLIX would be an interesting puzzle answer though. – Todd Wilcox May 14 '17 at 6:28
• @ToddWilcox - yes, I meant to give 'ic' such as the ending of 'toxic'. XCIX doesn't quite work! Still pondering on your clue... – Tim May 14 '17 at 7:01
• IC: Roman numerals weren't very standard though: this straight dope article gives some examples, but lacks citations; wikipedia cites a few examples. In other words the setter was being difficult (or archaic), not wrong. – Chris H May 15 '17 at 9:28
• Found a better source: A History of Mathematical Notations, Florian Cajori 1928 starting at P46 of the PDF (also @ToddWilcox) – Chris H May 15 '17 at 9:29

In my country, many buildings (city halls and other official buildings) have their construction date, sculpted upon the building, in Roman numerals. Imagine my standing next to a 12-year old child, before such a building, I ask him/her when this building has been built and (s)he has no clue about the meaning of the MCMxxx letters, I'd be very disappointed in this child's education.

As stated before, it's up to schools to pass general knowledge to their students, and the mathematics lectures should be no exception on that.

The Roman numerals are used. It is useful to be able to read them, isn't it?

• They are used because of aesthetics. vsz noted Star Wars, episode IV versus Star Wars, episode 4. Old buildings in Europe have them as designation when they were built.
• They are used to emphasize different (level of) enumeration. Chapter I, section 1, for example. Or you can use Roman numerals to count Contents or Attachments and Arabic numerals to count the "true" pages.

It is also good to train in levels of abstraction. You have five particular apples on the desk, then we move to abstract number five described with particular symbol 5 or V. It can make another level of mathematical abstraction a bit easier, the step where a number five is replaced by a variable.

I think the difference between systems of writing down the numbers - I, II, III, IV, V, VI, VII, VII, IX, X, ... - compared to the common decimal and background binary, octal and hexadecimal systems - 0 1 2 3 10 11 12 13 20, ... (base 4) - is The Lesson to be taught. And the "M for 1000" is the mandatory step.

The rule: If the numerals are in ascending order the smaller one is substracted, if they are in descending order they are sumarized forces us to use nontrivial "decoder" to handle Roman numbers. It is a good training to have agile mind.

Side note: In Czech, there is dummy sentence "Lidé Co Dávají Málo" (people who give a few) which resembles in the Roman numerals for 50 and above (I V X L C D M). I think it is easy to create simillar dummy sentence for any language.

Successfully completing one's schooling requires clearing many hurdles, some of which have are very useful in later schooling and career, and others that may not add much value. Roman numerals mostly falls into the latter category. Many books still use i, ii, iii for forwards and older films may list the copyright as MCMXLIX. You can survive without this knowledge (or google it when needed), but it is useful cultural knowledge. In any case, this particular hurdle should not be too hard to clear.

The testing described in the question sounds more like a cultural than numeracy test. In mathematics, Roman numerals are one example of a different ancient numeral system. Did this same test ask about Mayan or Babylonian numeral systems? As a base 60 system, the Babylonian system is very interesting from a mathematical point of view. :)

Note that I have a BA in Mathematics and have been a software engineer for many years.

In this high-tech world, your child will eventually find it useful to understand Binary, Hexadecimal and other number systems that are not Base Ten. (By the way, "decimals" are included in our base-ten system.) As an understanding of exponents is needed to learn these systems, it makes sense to begin learning a new number system (Roman numerals) that does not require any complex mathematics.

Roman numbers should be taught to show the historical failure of math in the ancient west - both Greece and Rome. It is unfortunate that most of the math used today be it numbers, counting, concept of zero, infinite series, basic algorithms almost all come from India. Yet if you google mathematicians some imaginary pictures of Aristotle and Euclid come up, like they invented something. BTW modern numbers are NOT arabic numbers - they are the Indian place value system.