Translation of "He has 8 more quarters than dimes." into symbols?

The sentence "He has 8 more quarters than dimes." is often mistranslated by students into the equation 8 + q = d. This is an extremely common error in the usage of English to express daily counting and word problems. My question is: how can I explain in grammatical terms where the error in reasoning has occurred? What are the strictly grammatical reasons why this confusion occurs?

Students are taught early on to translate the phrase "8 more than a number" as 8 + n. What is the grammatical difference between the original sentence and that phrase which causes the mistranslation?

Could it be the presence of the word "quarters" between "more" and "than", and if so, what would be the grammatical function of these words in the original sentence, "He has 8 more quarters than [he has] dimes," as compared to the phrase "8 more than a number"?

I'd go quite the other direction than you. The error is not in usage of English to express the concept, or a fault in the symbol presentation required by the equation, but in the incommensurability of the conceptual metaphors used by the two different systems, natural English and symbolic arithmetic.

He has 8 more quarters than dimes.

is translated into

q = d + 8

This makes sense with the number of quarters being q, dimes d.

but is easily seen to have the thinko of 8 + q = d

The difficulty with symbolic translation between two conceptual systems is that it is easy to make a faulty correspondence between the symbols, rather than working with the concepts themselves. 'more' seems like '+', but its objects are not in the same syntax as '+'.

'Z more X than Y looks like 'Z + X = Y', in the order of the operands, but that's just not the same syntax. 'Z more X than Y', to translate, becomes

X is a set bigger than Y, by Z.

This has the closer correspondence in syntax of '+'. The syntax of 'more ... than' is different from the syntax of +.

If you're like me, you learned English first, and so learning the slightly different Math syntax took some getting used to.

This is a major difficulty in teaching of mathematics because the syntax of English is slightly different from Math. Many brain teasers and puzzles are essentially language puzzles and questions posing culture rather than purely mathematical. "Bob has 8 apples. Jane took all but 5 from Bob. How many does Bob have?"

I think the important idea to stress is that it's more than, not just more. So, in this case, it's more than dimes. The fact that quarters is in there in the middle is just to explain what he has that numbers 8 more than dimes.

One trick I use to explain things is to drop out unnecessary parts and figure it out, then add things back in to see how they should affect the sentence. (e.g., if someone says "Charlie told Frank and I", if you drop Frank out, you'd get "Charlie told I" which is more clearly wrong. Adding Frank back in doesn't change me to I.)

So, if they drop the quarters (for the moment), they can put more and than together to see what needs to be added -- 8 and dimes. Then, when they ask what that equals, you add back in the quarters -- The quarters number 8 more than dimes.