You have probably seen the claim: children who learn an instrument do better at school, particularly in maths. It is repeated often enough to feel settled. It is worth looking at more carefully, because the honest answer is more interesting than the headline.
What the research actually shows
There is a consistent correlation. Students who learn instruments do tend, on average, to achieve more highly academically. That much is well established across a lot of studies.
The difficulty is what causes what. Families who arrange and pay for years of instrumental lessons differ from those who do not in all sorts of ways, including income, parental education and how much structure there is around homework. Children who stick with an instrument for eight years also tend to be children who are reasonably good at sticking with difficult things, which is not a small advantage in exams.
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When researchers try to isolate the effect by assigning music training at random, the results are much more modest than the correlation suggests. There is something there, but it is not the dramatic transfer effect the popular version implies. Learning the violin does not straightforwardly make you better at algebra.
Where the more plausible benefits sit
Two areas hold up better than general intelligence claims.
The first is sustained practice. Learning an instrument involves working at something difficult, in small increments, without visible progress for weeks at a time, and coming back to it anyway. That is precisely the skill that separates students who cope with A level workloads from those who do not. Whether it transfers automatically is debatable. Whether it is worth having is not.
The second is performance under pressure. A student who has played a solo to a hall of parents has a useful reference point when they walk into an exam room. They have already discovered that their hands shake and that they can carry on anyway.
The revision question
There is one place where the evidence is clearer and points the other way. Music with lyrics, played while studying, generally makes verbal work harder rather than easier, because the language centres you need for reading are busy processing the song. Instrumental music at low volume is less disruptive, and silence is often best for demanding material.
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So if a student is committed to background music, that is manageable. It is just not doing what they think it is doing.
The better reason
The case for music does not need to rest on grades. It is worth doing because it is worth doing: it is one of the few things a teenager can get demonstrably better at through their own effort, and it gives them something that is entirely theirs at a point when a lot of their life is being assessed by other people.
Sixth form is where this often gets squeezed, as students narrow down to three or four subjects and drop everything else. Colleges that keep space for music and other co-curricular activity alongside a demanding academic programme, as at Brampton College in Hendon, tend to produce students who are better company and, not coincidentally, better at handling pressure.
If you are deciding
Do not fund instrumental lessons as an academic investment; the return is uncertain. Fund them because your child wants to play, and because the discipline involved is genuinely valuable. If the grades benefit too, treat it as a bonus rather than the business case.