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I was talking with a child who, I was told, has learned some basic multiplication that he can do in his head, so I asked him Сколько будет восемь на семь?

I'm not a native speaker, but I've given lectures in Russian on higher math and I say just на in place of умножить на and разделить на; the context makes it obvious which one is meant. With the child also it was obvious (to me) that I meant multiplication, since division, square roots, etc. are not yet part of the child's lexicon.

Anyway, the parent corrected me by repeating the task but said восемь умножить на семь. That made me realize that perhaps when children learn multiplication this operation is always given in the form умножить на instead of на.

When does the transition usually take place to abbreviate multiplication (and division) to на when the choice is clear from context?

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  • I think that when people (including children) learn something new it is necessary for them that task is explained using all the words they familiar with and in that particular order as teacher does. So, for a child that does not know that many operations can have this 'на' part, it is not obvious that 'на' can be used with 'умножить' omitted. The child knows what's "умножить-на", but does not what just "на" means (or child is not sure that умножить is an action you expect). – Artemix Apr 24 '15 at 15:37
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I think most people will realize this form if listener knows that you talking about math. Also, a little bit more precise, you can say "восемью семь" or "трижды семь" for 3 * 7.

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  • A 7-year old should realize it? That's the person I was directing the question to, so the parent's correction made me suspect that simplifying to на was perhaps not "child-speak" for math yet. This was the point of my question. I'm wondering not what most people will think but when do children start to use на instead of умножить на? – KCd Apr 22 '15 at 6:17
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    @KCd I don't think there's a clear answer to be given here, because the established way to verbalise multiplication was, until recently, the one brought up by Anton Maximov: дважды, трижды, четырежды and past that, the instrumental — пятью, шестью, etc. As far as I'm aware it's on the decline now, replaced by [умножить] на, which adds the inconvenience of ambiguity with division, but I think it's too early to talk about any formalised rules to get around that. – Nikolay Ershov Apr 22 '15 at 7:26
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    @KCd After reading this thread I remembered one story with my own son (native Russian speaker) when he was 7 and knew how to add/subtract numbers but didn't know how to multiply them properly yet. He heard phrases like 'семью восемь равно 56' and thought that multiplication sign is always called 'ю' ! This was really funny, he said 'один ю два равно два'! So multiplication can be difficult even for native speakers :) – demonplus Jun 18 '16 at 8:41
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Imagine the teacher stands in front of the class.

Addressing the audience the teacher will never say Сколько будет восемь на семь? but always Сколько будет восемь умножить на семь?

If a teacher examines complex expression 100 + 8 × 7 - (5 + 12) standing at the blackboard and points at (5 + 12) he will say сколько будет пять плюс двенадцать?, but highly unlikely Чему равна сумма пяти и двенадцати? or Чему равно пять прибавить двенадцать?. Further pointing at 8 × 7 he most likely will say а восемь на семь? and a bit less likely Сколько будет восемь на семь?

The teacher say that because the symbols of a multiplication operation ×,*,· dosn't have his own name, while the symbol + on the contrary, has the name плюс. Let us remark here that Умножить is related to the name of operation, not to the name of conventional sign ×, which is assigned to this operation. So, he just replaces nameless with something neutral, namely, на. The same is true for division.

The parent are not going to correct you, but perhaps trying to be extremely pedantic in the role of the teacher who standing in front of the class ))

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Умножить на is formal and "official" name of operation, на is colloquial abbreviation. Also, на could be abbreviation of division (разделить на or делить на), that's why it can be unclear, when you say just восемь на семь. But with context, there is no problem:

Таблицу умножения выучил? Сколько будет семь на восемь?

Here is obviously multiplication. And here:

Сколько будет десять разделить на пять? А шестнадцать на четыре?

is obviously division, but usually на means multiplication by default, if no context provided.
In your case problem was more educational then translational. Parents of the child could think that he didn't understand question (for example, because of he didn't answered for a long time). When children had just learned multiplication, it can be a bit hard for them to understand question fast when question is unclear or ambiguous.
There is a lot of confusing abbreviation in Russian. For example:

  • people often say теория относительности (just "relativity", which is incorrect) instead of общая теория относительности (general relativity) or специальная теория относительности (special relativity)
  • there is a term in photography глубина резко изображаемого пространства (Depth of field), but ususally people use term глубина резкости, which is a bit incorrect, because it has another meaning in optics.
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  • The "correction" of my question to the child, by the parent, happened immediately after I asked it. So a time lag in the child understanding the question was not an issue. I suspected the parent made the correction because children that young do not yet know about на meaning умножить на, which is why I posted my question in the first place asking when children start to learn about such an abbreviation. – KCd Apr 24 '15 at 0:29
  • It was just my guess about possible reason. In any case, you were right, and correction was addressed to a child, not to you. – Dmitriy Apr 24 '15 at 20:32
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AFAIR the teachers in school insisted on saying "умножить на" but everyone just says "на" as it's more convenient.

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Yes, на can mean both division and multiplication depending of context. But if there is no any context, I would interpret it as multiplication by default.

There are a lot of defaults in math, for example operation priorities with standard functions:
sin 2 x = sin(2*x)
sin x cos x = (sin(x)) * (cos(x))
You are not only saying in in a same way (without any operation), but you are also writing it in a same way - without brackets.

which was about the time when children start to know that на is a common abbreviation in speech for умножить на (or разделить на)

I think they know as soon as they start to multiply numbers, at least numbers more than ten. As for smaller numbers we usually use "дважды два", "пятью три" instead of "два на два", "пять на три".

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  • I know на means multiplication or division. This answer is not addressing the question I asked, which was about the time when children start to know that на is a common abbreviation in speech for умножить на (or разделить на). – KCd Apr 24 '15 at 0:26
  • @KCd, I've updated my answer. – Qwertiy Apr 24 '15 at 6:56

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