It is not exactly rocket science to understand that people get better at doing something with training (although it is almost brain surgery to understand why this happens). Not everyone can learn to play chess like Garry Kasparov, but almost everyone can learn to play it quite well, with enough training.
The same is true for abstract reasoning, solving arithmetic or geometric problems, understanding your mother tongue, and other factors that influence IQ. If you spent ten years learning to see relations in sequences of numbers, you would be able to recognize many more of these much faster than someone who never did so, just like experienced chess players are able to recognize many more tactical and strategic ideas than beginners, no matter how much talent they have.
All the skills that constitute IQ can be further developed with training. We are not all equal in terms of how fast or how far we can develop these skills, but we can all develop them to a certain degree. And that’s where education comes into play.
Pupils and students of all ages develop (among others) skills that help them score better on IQ tests. Smarter children develop such skills fast, slower children may need a lot of assistance before they are able to grasp certain concepts (and may never be able to grasp some), but the point is:
IQ tests may have been understood as a measurement of innate intelligence a hundred years ago when they came into the world, but this is no longer the case, as we have strong empirical evidence that quality and quantity of schooling influences IQ scores.
One of the strongest supporting arguments is a study conducted in Norway. Norway switched from 7 to 9 years of compulsory schooling in the 1960s, which allowed researchers to measure the effect of additional schooling on the IQ of adolescents.
Amazingly, they discovered that each additional year of schooling led to an increase of up to 5 IQ points at the age of 19.
A sad example of this phenomenon occurred in the United States. Quoting from Rising Scores on Intelligence Tests by Ulric Neisser:
Six IQ points per year is a significant number. Most importantly, it tells us that quality and quantity of education a certain ethnic group receives is probably the most important factor when it comes to differences in average IQ scores (but, of course, there may also be genetic differences). Studies comparing the average IQ of different ethnicities are not a proof of genetically predetermined racial inequality; that would require studies controlling for environmental factors, which is hard to do, and there is some evidence to the contrary.