The Silver Chair begins in a fictional school called Experiment House. The school is obviously based on ‘modern’ or ‘progressive’ schools along the lines of the real life Summerhill.
In Experiment House, children are allowed to do as they like. In the book, this inevitably leads to institutionalized bullying, which isn’t punished because the Head sees them as interesting psychological cases.
Incidentally, Lewis also seems to see boys and girls attending the same school, and a female Head, as signs of a school gone mad.
However, my main problem with the thesis that ‘modern’ schooling = bullying is that Lewis himself went to the most traditional kinds of schools, and suffered bullying there.
This post points out that Lewis called his old school, Wynard, ‘Belsen’, after the Nazi concentration camp. The headmaster of this school was actually eventually criminally charged and sent to an insane asylum.
He then moved to another school which used the British system of fagging: that is, older boys being able to give orders to younger boys. This system was associated with abuse, which was generally ignored or even approved of. In some cases (though apparently not Lewis’) it went as far as rape.
For a relatively recent example of the culture of abuse in the most ‘traditional’ schools, see this article.
So Lewis must have known that thesis he presented in The Silver Chair, that bullying was associated with then-modern methods of education, was the opposite of the truth.
He seems to have lied for ideological reasons: that is, because he associated ‘modern schooling’ with skepticism about Christianity.