I had a conversation with Grace Chen and Christopher Danielson, among others, about how languages can affect students’ early learning of mathematics. For example, in Mandarin the phrasing for twelve is equivalent to “one ten two” in English.
Paul Goldenberg sent me this and I got his permission to publish it here. Thanks Paul!
I don’t know of any language that uses an essentially spelling pronunciation of the numerals (e.g., 23 as “two three”) although that is essentially what common (out-of-school) practice does with decimals. Only U.S. elementary schools insist that 3.1416 be pronounced “three and one-thousand sixteen ten thousandths.” (It is, or at least was in 2006, required of school teachers in Austin, Texas.)
But I did say that there were many variants on fully or nearly place-value pronunciations. French ranks near 0 (with near regularity in low numbers not beginning until 16 and succumbing to IE’s general love of toes later on; 91 is four twenties eleven), Danish is slowly reforming from a 5 to a 6, English a 6 or 7. But, for example, Kinyarwanda does 11 (etc.) as “ten and one” and has only minor anomalies in the naming of numbers (20, as in French, is not built the way larger multiples of 10 are built, though it is far more recognizably two-ish than the French is).
The use of numbers for counting, and for counting things, differs in some languages where agreement with noun gender (or other class) is required, but people are so used to accounting for the affixes that they’re almost invisible (inaudible?) anyway.
But the regularity in structure seems to be more important than regularity in application. For example, Kinyarwanda’s regularity in structure (and, I suppose, the fact that tiny kids handle money in the hundreds and thousands) gets kids really good really early. Their first graders are better at counting by ones and tens (the latter even if we start at 4 or 7 and proceed to 14 or 17, etc.) than many kids here. And that’s despite the fact that number-noun agreement requires some eight (I think I’m getting that right) different grammatical forms. (Bleh!)
What I do know firsthand is how differently children respond to print than to spoken (mental) calculations set in some (not necessarily “concrete”) context. I don’t think it has anything to do with abstract vs. concrete, but rather about some existing semantics. For example, adding 9 in the context of adding 10 is a no-brainer for kids, but it takes them much longer to look at 23 + 9, especially if it’s set up vertically, and think of doing anything other than what they were taught to do—recalling a specific fact and going through specific notational moves—and they can easily blow steps that have no semantic sense and only syntax (rules and order) to guide them.