It is well recognized that learning number names is not an easy task for native English speaking children. For example, the number name for "11" should be learned quite independently from the name of previous numbers, rather than as "ten-one" like in Chinese and a number of other languages. The latter is not only easier to remember but also goes better with the idea of place value. Indeed as Iranian, I could explain the situation just using examples from different region of Iran. Children has to learn "national" language (Farsi) when they start school at the age of 7. The structure of number names in Farsi is more or less like English (even a bit worse). Thus, learning number names is not an easy task for Iranian children in general, and even harder for Iranian Turk children whom the language they grow up with (Turkish) has a different (and better) structure for number names (e.g., "11" is "ten-one").

Let us move beyond early number learning. Years later, one of the algebraic problems of the national mathematics textbook is the following:

Express in words: $(a+b)^2$

Hard for children in general, harder for Iranian Turks. In Farsi, it reads the same as for the reader of this post, outside operation first, inside operation second: "square of the sum of two numbers". For Iranian Turks, it reads the other way around, inside operation first, outside operation second, something like: "the sum of two numbers to the power of 2". Not wrong, but not what is expected from them.

Do you know any other examples in which the structure of native language of students is in conflict with the structure of language of mathematics that they are expected to learn?

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    $\begingroup$ (Incidentally, in colloquial English I am not sure that this would be described as you mention, i.e., square of the sum of two numbers. Somewhat common, perhaps, would be to say: a plus b, quantity squared.) $\endgroup$ – Benjamin Dickman May 4 '15 at 21:59
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    $\begingroup$ @BenjaminDickman Interestingly, I learnt mathematics English before colloquial English-if I have ever learnt the latter :) $\endgroup$ – Amir Asghari May 4 '15 at 22:05
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    $\begingroup$ Even as a child, I noticed the anomaly of the numbers 11-19, glad to this discussion come up here. $\endgroup$ – JTP - Apologise to Monica May 4 '15 at 23:57
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    $\begingroup$ I find this question very interesting, but was wondering if a claim such as "learning number names in one language is harder than in another" is a claim that can actually be substantiated with some sort of observations/measurement. For example, can one observe that Farsi native speakers outperform Turkish speakers in math in school or later life? $\endgroup$ – Michael Bächtold Mar 17 '18 at 10:23

Some examples:

  • "One plus five squared" could be read as $(1+5)^2$ or $1+5^2$. This is a well-known ambiguity in natural language, for example in the following sentence

    I saw a man on a hill with a telescope.

    we don't know if it was a man who had the telescope, or the speaker, or perhaps the telescope was just standing on some hill.

    See also this list.

  • Double negation in some languages doesn't always turn sentences into positives, while in classical logic in math we have $\neg\neg x \iff x$.

  • The word order may have more or less meaning and it may result in wrong interpretation. For example when you ask "Take 10 and subtract the difference between 3 and 5" you may get both $10-(3-5) = 12$ and $10-(5-3) = 8$. This effect is even stronger, if the context suggest that negative value doesn't make sense in a non-mathematical way (e.g. number of apples).
  • Some words have special meaning, e.g.

    No cat has eight tails. A cat has one tail more than no cat. Therefore, a cat has nine tails.

  • Mathematical expressions have highly recursive structure, while spoken natural language isn't prepared to handle more than a few levels. For example

    The rat the cat the dog bit chased escaped.

    is an understandable sentence while dissected as follows:

    The rat $\Big($the cat $\big($the dog bit$\big)$ chased$\Big)$ escaped.

    The rat escaped. Which rat? The rat the cat chased. Which cat? The cat the dog bit.

  • When the term comes from a different language, it may become confusing. A mild example would be "limes", a bit more confusing could be $P$ and $V$ in Semaphores.

  • Not exactly about language, but I will include nevertheless: context matters (and different language is a different context).

    Let $f(x,y) = x^2+y^2$, what would be $f(\phi, r)$?

    Mathematician: $f(\phi, r) = \phi^2 + r^2$.

    Physicist: $f(\phi, r) = r^2$.

I hope this helps $\ddot\smile$

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    $\begingroup$ Your second point reminds me of one of my favorite jokes / observations. A linguistics professor is giving a lecture about the variation of double negatives in different languages. To add some spice to his lecture, he concludes with, "But, of course, in no language does a double positive become a negative." One of the student pipes up in the back and says, "Yeah, right." $\endgroup$ – Michael Joyce May 7 '15 at 15:57
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    $\begingroup$ "I saw a man on a hill with a telescope." Perhaps the telescope has serrations and the man on the hill is being gruesomely sawed by the telescope. :-) $\endgroup$ – Joseph O'Rourke May 9 '15 at 20:35

German/Deutsch: dreihundert fünfundzwanzig:


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  • $\begingroup$ Could you explain if it is something about this particular example or it is an example of a more general structure. $\endgroup$ – Amir Asghari May 5 '15 at 10:06
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    $\begingroup$ English numbers are not easy, but at least the words match the digits left-to-right. I find unraveling the German ordering a challenge (three hundred five and twenty), especially when told verbally the price of an item in a shop. $\endgroup$ – Joseph O'Rourke May 5 '15 at 10:58
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    $\begingroup$ @AmirAsghari This is an example of a general structure in German. For numbers between 21 and 99, where the decimal representation is ab and the English name is something like "a-ty b", the German is something like "b und a-zig", with b named before a. This weirdness percolates up to bigger numbers. Decimal ab000, in English "a-ty b thousand", is "b und a-zig tausend" in German. $\endgroup$ – Andreas Blass May 5 '15 at 12:02
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    $\begingroup$ Yes, so 54,321 is four-and-fifty thousand, three hundred one-and-twenty. $\endgroup$ – Joseph O'Rourke May 5 '15 at 12:06
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    $\begingroup$ Dutch/Nederlands has a similar pattern: "b en a-tig" and "b en a-tig duizend". This order is fossilized in English in the names of 13 to 19. For example, "sixteen" from six and ten. There is also the nursery rhyme "Six a Song of Sixpence" with its line "Four and twenty blackbirds baked in a pie". $\endgroup$ – J W May 6 '15 at 5:50

Given the preponderance of repetition in mathematics, reuse of symbols for various meanings in differing contexts, interesting fashions of modifying words to indicate additional properties or lack there of, I don't think the result is something that should instill pride. When one wants to abbreviate gcd(a,b) as (a,b), as well as use (a,b) for the result of other binary operations or relations, one stretches the reader's ability to include context to interpret a particular glyph. Is the class of near-rings inclusive of the class of rings, or a subclass, or neither? Should I take a dot product, a cross product, a cartesian product, an ultraproduct, or a semidirect product? Am I producing when I do this?

Gerhard "Time For A Mathematical Esperanto" Paseman, 2015.05.05

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  • $\begingroup$ Or consider the multiple meanings of "simple" in mathematics... $\endgroup$ – Joseph O'Rourke May 5 '15 at 17:15
  • $\begingroup$ There is a vast background literature behind mathematics as a language. For example, the book written by David Pimm, "Speaking mathematically, communication in mathematics classrooms", is one among many which discusses similar issues like the one you mentioned in depth. That is why I was very careful when posting the question avoiding many different language-related problems, focusing on a very particular one. $\endgroup$ – Amir Asghari May 5 '15 at 18:14

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