Hi Chris,

I happened to chance upon your post, and I thought I would share what I know.

A: "The maths lesson"    B:"The skating rink"

These are noun phrases, and every word revolves, functionally speaking, around a main noun (noun head). 

In your noun phrase A:

The words "The" and "maths" come before the noun head "lesson" -- they are pre-modifiers. Sometimes, pre-modifiers can simply be an adjective (Example: Charming (adjective) lady (noun head). Sometimes, the pre-modifier can be another noun which is used before the noun head. (Example: Math (noun) lesson (noun)). The first noun now functions as an adjective of the noun head.

Similarly, "skating" becomes an adjective of the noun head "rink". 

To end off with an example: 

The best safety device which turned out to be a hoax.

Pre-modifiers:

"The" -- determiner   /   "best" - adjective   /   "safety" - adjective of the noun head "device"

Noun head (main noun):

"device"

Post-modifiers: rest of the noun phrase that still modifies the noun head

"which turned out to be a hoax" -- relative clause that provides more information about the device. 

Hope this helps.
Cheers,

Lance

I agree with Michael chambers and Mr Lance.

Of course, one noun modifies the other, that is clear, but this function doesn't make it an adjective. 

Intermediate students need to learn about compound nouns, and then keep learning.

So if your students are being taught simply that in "maths/math lesson", "maths/math" is an adjective, then this is unhelpful, in my view.  They could think that "maths/math" is equivalent to "mathematical", or even not bother learning "mathematical" and "mathematically". 

At elementary level, when teaching adjectives, students will get confused if the material includes compound nouns, in my opinion.

Skating is also a noun rather than an adjective. Television is a noun, rather than an adjective. The related adjective is Televised e.g. televised football match (with football match being the compound noun). Similarly "People carrier" is a car carrying several people, but I would not say "people" is an adjective.

There are many discussions about this subject on the internet, but in my mind I see any word or combinations of words that act as a noun as being a noun. I see any words which then further describe the combination of words as being an adjective e.g. [interesting] (adjective) [television program] (noun) .

In the end I am not sure it matters that much in English, but in other languages e.g. Italian or French it matters because an adjective changes its ending depending on whether the noun is masculine, feminine and singular/plural.