The schools minister Nick Gibb, who is responsible for the Grammar, Punctuation and Spelling (GPS) tests that UK school children sit at ages 7 and 11, was asked a grammar question in a BBC Radio 4 news programme. The question had previously appeared on one of the GPS test papers. Here's a snippet of the interview:
Interviewer Let me give you this sentence: I went to the cinema after I’d eaten my dinner. Is the word after there being used as a subordinating conjunction or as a preposition? Minister Well, it’s a proposition. After - it's... Interviewer I don’t think it is. Minister After is a preposition, it can be used in some contexts as a, as a, word that coordinates a subclause, but this isn’t about me. Interviewer No, I think in this sentence it’s being used a subordinating conjunction!
The press had a wonderful time pouring scorn on the minister for not being able to answer a question that very young children need to be able to answer by the time they leave primary school. There has subsequently been some debate over the question of whether the minister may actually have answered the question correctly. Oliver Kamm wrote a piece in the Times with this heading:

Kamm writes that Gibb was correct in saying that after is a preposition, and there "was no need for the minister to be defensive of his grasp of grammar." So was Gibb's answer correct or not? Well, that depends on the grammar you consult. The two most influential standard grammars of modern English, namely A comprehensive grammar of the English language (CGEL; by Quirk, Greenbaum, Leech, and Svartvik) and the Cambridge grammar of the English language (CamGEL; by Huddleston and Pullum et al.) have different analyses of the sentence that was presented to the minister. In CGEL a distinction is made between the preposition after and the subordinating conjunction after, such that we have the two analyses below:
I went to the cinema [PP [P after] [NP dinner]]
I went to the cinema [clause [subordinating conjunction after] I'd eaten my dinner]
[In this notation the brackets indicate the grammatical units, called constituents. P = preposition; PP = prepositional phrase; NP = noun phrase.] So in the CGEL framework after, as a preposition, takes a noun phrase as its complement, but as a subordinating conjunction it introduces a subordinate clause. By contrast, in CamGEL (which was influenced by the Danish grammarian Otto Jespersen, as Kamm points out) we have this analysis:
I went to the cinema [PP [P after] dinner]
I went to the cinema [PP [P after] [clause I'd eaten my dinner]]
In the CamGEL framework after is always a preposition which takes either a noun phrase or a clause as its complement. Both after dinner and after I'd eaten my dinner are prepositional phrases. In my Oxford modern English grammar I adopt the CamGEL analysis. It would be fair to say that the CGEL analysis is the dominant one and in this grammar, along with after, we also find words like although, as, because, before, if, since, (even) though, while, when, while, etc. within the word class of subordinating conjunctions. Because it is more widely adopted, and because the analysis is conceptually easier to grasp, the CGEL approach was adopted in the National Curriculum (NC) after a process of consultation which my former UCL colleague Dick Hudson conducted to find out what fellow linguists thought about the best way to present particular grammatical analyses for use in schools. Now, because the National Curriculum adopts the CGEL analysis, we can say that Gibb was in fact wrong in saying that after is a preposition in the sentence I went to the cinema after I'd eaten my dinner. Some commentators have been saying that we should not be teaching children about subordinating conjunctions because grammarians themselves can't agree about how they should be defined. This is misguided. The study of grammar is no different from other fields of enquiry in being replete with disgreements about how we should look at particular phenomena in the world around us. Think of the Expanding Earth Hypothesis in science, or disagreements among art historians about the significance of Renaissance art, or the different ways in which we can interpret the character of Iago in Shakespeare's Othello. Such disagreements make these fields fascinating, and discussions of them propel our knowledge of the world forward. But what does all this entail for the teaching of grammatical concepts such as 'subordinating conjunction' in primary schools? One approach is to say that such notions are too hard for 11-year olds. This is what most commentators in the press and social media seem to have been claiming. Some of these comments were along these lines: "I have an English degree from Oxford and do not know what a subordinating conjunction is." The conclusion that is then drawn is that nobody else needs to know either! The alternative is that we do not shy away from teaching this notion, since it really is not all that difficult to grasp, at least no more difficult than learning about, say, the division rules at KS2. However, if we do decide to teach grammatical notions it's best to use a particular model of grammar to teach the basic concepts, and to refine the concepts that are learnt in primary school at a later stage, e.g. at secondary or university level. Pedagogically it makes sense to use models, even if they are not agreed on by all specialists in a particular field. In science something similar happens, for example when the acceleration of free-falling objects is taught: we then pretend that there is no air resistance. In the classroom making use of idealisations and models is a very useful and effective strategy. PS If you want to find out more about Otto Jespersen, have a look here. PPS The basic concepts of grammar in the NC are all explained on the free Englicious site.