New Cur 18: To be two, or not to be 2

This one may be of little interest to others, but it’s been bugging us.

A peculiar puzzle of writing mathematics is deciding when to use names and when to use numerals: should we write “two” or “2”? There is no one (1?) answer, and there are conflicting principles. Along with other rules, English style guides instruct that names should be used for small numbers, and numerals for large (with varying interpretations of “small” and “large”).

Reasonable enough, but there also appears to be a semi-conflicting mathematical principle to follow (unless we made it up): if the scenario explicitly or implicitly contains arithmetic then numerals seem preferable. So, writing “6 is an even number” feels correct, and the more explicitly arithmetic the context, the stronger the case for numerals. By contrast, if we are simply noting the count of something then, at least if the count is small, names seems better: “Marty committed five cardinal sins”.

Of course the distinction is not always so clear or constant. We might note there are fifty-two cards in a deck, for example, but with the immediate intention of dividing up the deck, or of calculating some probabilities. Moreover, the context and the audience should always be considered. For an exercise or test question, one should typically use numerals, particularly if the students are young or weak. The proper concern, always, is for clarity and for style, in that order.

The rule is there are no rules, but the principle is there are principles. Or at least there should be. Which brings us to the Australian Curriculum.

The Curriculum is very numeral-heavy. That includes plenty of judgment calls, where we would have gone for names but numerals also seem reasonable. But it also includes plenty of instances where there is no arithmetic in sight, where the use of numerals feels plain wrong. For example,

investigating sufficient conditions to establish that 2 triangles are congruent (AC9M8SP01)

It is even more jarring in instances where there is arithmetic nearby, but the numeral is not relevant to this arithmetic:

building a rectangular prism out of unit cubes and showing that the measure of volume is the same as would be found by multiplying the 3 edge lengths or by multiplying the area of the base by the height/length (AC9M7M02)

recognising the features of circles and their relationships to one another; for example, labelling the parts of a circle including centre, radius, diameter, circumference and using one of radius, diameter or circumference to determine the measure of the other 2; understanding that the diameter of a circle is twice the radius, or that the radius is the circumference divided by 2π (AC9M7M03)

Then there’s the possibility of switching mid-stream:

using a tree diagram to represent a three-stage event and assigning probabilities to these events; for example, selecting 3 cards from a deck, assigning the probability of drawing an ace, then a king, then a queen of the same suit, with and without replacing the cards after every draw (AC9M9P01)

The Curriculum contains dozens of other such usages, many just as clunky.

Of course there are many smellier fish to fry: any numeral-name screw-ups pale in comparison to the Curriculum’s other failings. But such misuse of numerals is incredibly aggravating, at least for us. It is inelegant and, more importantly, it makes the reading awkward, almost demanding a double-take.

It amounts to one more instance of ACARA’s inability to manage even the small aspects with a minimal competence.

33 Replies to “New Cur 18: To be two, or not to be 2”

  1. I agree. You’re quite right to be incredibly aggravated. It drives us spare when we see this sort of acara writing (we often see it in the context of probability questions). Yes, it’s only a small thing. As is the stinger on a wasp. And all these small things add up to death (of the acara curriculum, I wish) by a thousand cuts. It’s thoughtless and incompetent – a common theme with this curriculum.

    “investigating sufficient conditions to establish that 2 triangles are congruent (AC9M8SP01)”

    It should definitely be two not 2.

    “the measure of volume is the same as would be found by multiplying the 3 edge lengths”.

    Definitely should be three not 3.

    “diameter or circumference to determine the measure of the other 2; understanding that the diameter of a circle is twice the radius, or that the radius is the circumference divided by 2π (AC9M7M03)”

    The use of 2 here is definitely wrong. It should be two.

    “selecting 3 cards from a deck, assigning the probability of drawing an ace, then a king, then a queen of the same suit, with and without replacing the cards after every draw (AC9M9P01)”

    The use of 3 here is wrong and also ambiguous – There are 3 cards (the 3 of clubs, 3 of diamonds etc) in a deck of cards. I doubt acara means selecting those cards. It definitely should be three NOT 3.

    I agree with your conclusions, stated in the last two (not 2) paragraphs. I notice that the vcaa exams generally do it better (*). Examples from the 2022 Maths Methods Exam 2:

    “Find the average population during the first 300 weeks for the combined population.”

    “Find the average rate of change between the first two times when the population of rabbits is at a
    maximum.”

    “Let X be the binomial random variable representing the number of times that the coin lands on heads.
    Mika flips the coin five times.”

    “Bella flips her coin 25 times in order to estimate p.”

    * Rare ‘praise’ but it’s all relative. The acara writing is even more aggravating for this reason because of the change in writing style students are potentially exposed to once they enter VCE mathematics.

  2. Oof, yes, ACARA’s writing does bug me!

    Through my contract work with publishers I’ve also corrected a few incorrect instances of those popping up in textbook drafts. Thank goodness for not having smellier/bigger fish to fry as far as my work is concerned! I don’t remember which editor I inquired with, but the bigger companies (e.g. Cambridge) definitely have style guides that advise on that sort of thing.

    1. Thanks, Matt. That’s interesting. I don’t think the line by line writing in textbooks is very good, but it does appear to conform to some minimally functional style. Which is a million miles better than ACARA.

  3. They also write “base-10 number system” not “base-ten number system”. Every number system is base-10. That’s the whole point: in base-two, 10 represents the base, two. in base-three, 10 represents the base, three. I even put feedback in about this for the draft but it wasn’t corrected.

    1. Thanks, Alex. The extent to which ACARA doesn’t give a stuff for anyone else’s opinion is truly remarkable. They are the most secretive, arrogant and incompetent organisation imaginable. They are criminally stupid.

      On your specific point, I have a year 5 “base 10” elaboration included in my Awfulnesses list. Were there others?

      1. Year 3 description also has a reference to it. I thought there was more but I can’t seem to find them now.

    2. Alex, I’d be very interested to know \displaystyle how you worded your feedback to acara. I doubt that acara understood what you were saying (particularly since it doesn’t understand any distinction between 2 and two etc). I think you made the big mistake of assuming that acara is mathematically literate.

      For those who read these blogs but might be wondering what you mean, allow me to further clarify:
      In base-two, the decimal number 2 is written 10 (\displaystyle \underline{1} \times 2^1 + \underline{0} \times 2^0).
      In base-three the decimal number 3 is written 10 (\displaystyle \underline{1} \times 3^1 + \underline{0} \times 3^0).
      In base-four the decimal number 4 is written 10 (\displaystyle \underline{1} \times 4^1 + \underline{0} \times 4^0).

      And, obviously, in base-ten the decimal number 10 is written 10 (\displaystyle \underline{1} \times 10^1 + \underline{0} \times 10^0).

      So every number system is indeed base-10.

      It is a damning indictment of the mathematical competency of acara that your feedback on this issue was ignored. It is also an excellent testimonial for why the acara curriculum cannot be trusted.

      1. I think I see what’s happened with my feedback, they just cut the references I pointed our rather than fixing them 🤦🏻‍♂️

        Feedback: AC9M2N02_E3, AC9M2N02_E5, AC9M3N01 need to use “base ten” and “base five” as the numerals 10 and 5 refer to different values in different base systems (e.g. 10 base 5 is five)

        AC9M2N02_E3, AC9M2N02_E5, are both removed
        AC9M3N01 “represent read, write, rename and order natural numbers to at least 10 000 using naming and writing conventions for larger numbers and relate these representations to place value in the base 10 number system” changed to “recognise, represent and order natural numbers using naming and writing conventions for numerals beyond 10000”

        1. Thanks very much, Alex. Very interesting, and it semi-solves another mystery, which I’ll New Cur soon.

          It’s not clear what role your feedback had in those excisions/rewording. ACARA was anyway under pressure to reduce the insane number of elaborations. It’s almost impossible to untangle or to guess ACARA’s thinking (if that’s the correct word). But as far as I can tell, the approved curriculum never shines a reflective light on the ten-ness of our system. The Year 3 level description has been reworded, but both refer to “base 10”, are equivalently bad, and the base concept doesn’t seem to be teased out by anything in the Year 3 syllabus. The Year 2 elaborations you flagged were considering more directly the concept of base, so perhaps your feedback did have an unintended, and bad, consequence.

          For the Year 5 elaboration you flagged, I don’t see that the reference to “base” helped much, so the rewording is ok by me (except “numerals” should be “numbers”). They reworded a Year 5 elaboration to refer to “base-10 blocks” (AC9M5N020), which is a typical own goal, but is not considering base in the same direct manner.

  4. The quote about tree diagrams made me wonder: Should we assume that students know the composition of a standard deck of playing cards?

    1. Between students not often playing with an English 52-deck card, there’s plenty of card games that use different size decks with different types of cards.

      1. You’re right, TM and Alex.
        acara makes a big deal of its inclusiveness. And yet it assumes that ‘deck’ will be understood by \displaystyle everyone as meaning an English 52-deck card.

        (Might there be indigenous students who do not know even know what a deck, any deck, of cards is …?)

  5. I recall that Littlewood wrote arguing that the use of numerical digits should be acceptable (preferable?) everywhere. Maybe it was in {\it A Mathematician’s Miscellany} ?

    1. Find the probability that when 2 standard dice are rolled 3 times the total will be greater than 7.
      I don’t think so.

    2. The literary convention that numbers less than 10 should be given in words is often highly unsuitable in mathematics (though delicate distinctions are possible). The excessive use of the word forms is regrettably spreading at the present time. I lately came across (the lowest depth, from a very naive writer) functions never taking the values nought or one ‘. I myself favour using figures practically always (and am acting up to the principle in the book).

      1. “there also appears to be a semi-conflicting mathematical principle to follow (unless we made it up): if the scenario explicitly or implicitly contains arithmetic then numerals seem preferable. So, writing “6 is an even number” feels correct, and the more explicitly arithmetic the context, the stronger the case for numerals. By contrast, if we are simply noting the count of something then, at least if the count is small, names seems better”

        The above principle would dictate “functions never taking the values 0 or 1”.

      2. I get the temptation to use as few lines as possible but this doesn’t read well to me either. If you are going to break with convention at least be internally consistent. This apparent lack of care doesn’t inspire a whole lot of confidence.

        I recently vented, in passing, about the quality of writing in the recent VCE specialist mathematics to my head of maths. I was shocked by how defensive he was. Turns out he is aquatinted with some of the educators who write this stuff. An anecdote I know but there seem to be this strange fear of criticism out there. Might explain the how.

        (My undergrad honours supervisor wouldn’t have let me get away with writing like this.)

        1. Thanks, Jay. Did your head attempt to offer any substantive defense of the writing on VCAA exams? Of course there is none, but if he attempted one, I’d love to hear it.

            1. Those sorts of people are enablers of VCAA’s exam writing incompetence.

              (He probably harbors ambitions to join the exam writing panel and hopes to use his acquaintanceships to get recommended).

  6. This may seem trivial.
    But it is the simplest instance of the regular confrontation between ‘language’ (where everything starts and finishes), and mathematics/notation.
    In linguistic contexts one needs some simple guide (to avoid having to write out “one hundred and one” every time – unless, for example, they are Dalmations).
    I (and old-fashioned editors) try to work to:
    – if it is less or equal to ten, use the word;
    – if it is more than ten, use the numerals.

    The exception is where the meaning refers to something (small) that refers to, or makes more sense as, numerals, or that is genuinely mathematical. So
    “conditions to establish that 2 triangles are congruent”,
    “multiplying the 3 edge lengths”, and
    “to determine the measure of the other 2”
    seem ugly/wrong.

    In contrast,
    “using a tree diagram to represent a three-stage event [TG: linguistic, and <10, so fine] and assigning
    probabilities to these events; for example, selecting 3 cards [TG: clearly a number, so "3" is appropriate,
    as would be "three"] from a deck"
    does not jar in any way – except for the possible omission of "conditional".

    1. Thanks, Tony. I agree with all that except for your conclusion on the last quote.

      I agree that “three-stage event” is fine. I sort of agree that “3 cards” is fine, although I’d typically go with “three cards”. But I think the “three” and the “3” in the same sentence, with the 3 an example of the three, is extremely jarring.

  7. I sometimes correct students who say “oh” when they mean “zero”. But, to be consistent, should I say “double zero 7”?

        1. In the original script Bond says it’s only an alphanumeric. But it was decided the retort was not pithy enough.
          (And changing to number required only a trivial suspension of disbelief in the broader context).
          Happy New Year to all.

    1. When I was in school that digit was called “nought”. Whatever happened to that fine old English word? Can we revive it?

      1. Yeah, they switched from “nought” to zero” while I was in primary school. I hated and hate “nought”. The vowel sound is terrible.

      2. Your plans will come to nought, Tom. Which reminds me of the best way to greet an extra-terrestrial:

        Gnorts, Mr Alien.

        (You need some backward logic to see why).

        PS – I can’t imagine using a 24 hour clock and saying that the time was nought six hundred hours.

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