Timetables separate the physical from the intellectual. There is maths, and then there is PE, and the two are understood to be different kinds of thing. It is an administratively convenient division and a poor description of how development works.
Each new movement changes the world
The clearest way to see the entanglement is to follow what happens when an infant acquires a new capacity.
Sitting independently frees the hands. An infant who no longer needs their arms for support can hold, turn and examine objects, which changes what can be learned about objects.
Crawling changes the relationship with space. Distance becomes something that can be crossed. It also changes what adults do: mobile infants provoke far more instruction, warning and following, so the social environment shifts too.
Walking changes it again, and more than crawling did. Walking infants carry things, which means objects travel between places and people. They cover more ground, see more, and receive more language directed at them.
None of that requires any special link between motor and cognitive systems. It follows from the fact that a body is how a mind gets at the world.
The hand and what it does for attention
Manual skill deserves particular mention because of how much it changes the input.
An infant who can bring an object to the midline, turn it and look at it while turning is generating a structured sequence of views. That is a far better data stream for learning about three dimensional shape than passively watching an object be moved.
Work using head mounted cameras on infants has shown that the visual scene a young child actually receives is dominated by their own hands and by objects they are holding. Their view is not the adult's view of the room; it is a close, cluttered, hand centred scene. What they can do with their hands substantially determines what they see.
Handwriting, and why it keeps coming up
Handwriting is the point where motor skill meets academic work most directly, and it illustrates the workspace argument again.
Forming letters is initially effortful. An effortful hand consumes capacity that would otherwise be available for composing a sentence, which is why children's written output is often far poorer than their spoken output at the same age. As letter formation becomes automatic, capacity is released.
There is also a live question about whether writing by hand supports letter learning more than typing, on the grounds that producing a letter involves a motor programme that reinforces the visual form. The evidence is suggestive rather than conclusive, and it is frequently reported with more certainty than it has earned.
Interpreting the associations carefully
Studies find associations between motor coordination measures and cognitive or academic measures. What should be concluded from that is less obvious than it looks.
The access explanation is strong: movement changes experience, and experience drives learning. There are also shared factors. General health, sleep, opportunity to be active, and the amount of time spent in environments that afford movement all influence both sides. And measurement of both is imperfect, so associations can be inflated or attenuated by the instruments.
What cannot be concluded is that improving coordination improves attainment. That would require intervention evidence, and the intervention evidence is not strong. Movement programmes with academic aims have produced disappointing results relative to their popularity, which is a familiar pattern and one we describe in correlation and cause.
Why physical activity is worth defending anyway
This is worth saying clearly, because the previous section could be misread.
The case for children moving does not depend on it raising test scores. Physical activity has well established benefits for physical health, and guidance from the NHS and the chief medical officers reflects that. It is also, for a great many children, the most enjoyable part of the week.
The habit of justifying everything by its effect on attainment is a bad one. It implies that anything without an attainment effect is expendable, which is how physical activity, music and outdoor time end up being cut in the run up to assessments. The argument for movement is that bodies are for moving and children like it, which ought to be sufficient.
The reframing worth keeping
Not that movement makes children clever. That would be the same overreach in a more sympathetic direction.
The reframing is that a developing child is not a mind being carried around by a body. What a child can do determines what they encounter, and what they encounter is what they learn from. On that view, motor development is not a parallel track. It is the mechanism by which the world is delivered.
For related ground, see unstructured play is not idleness, particularly on risky play and calibration, and attention in small children.