Addressing a gathering of girls’ state-school heads today, the now-famous Baroness Hale of Richmond was wearing one of her signature animal-themed brooches: a diamante dragonfly. When presiding over the Supreme Court last month, it was her out-sized, glittering spider brooch, as well as her acid legal delivery, that caught the public imagination.

Her powerful statement jewellery is, perhaps, suggestive of the determination and resolve that have enabled her to ascend to the top of a male-dominated legal profession: she is the first female president of the Supreme Court and was the first woman appointed to the Law Commission in 1984.

Last month, of course, she shot to fame delivering the historic verdict on prime minister Boris Johnson’s prorogation of Parliament, declaring it “unlawful.”

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Speaking at the Association of State Girls’ Schools’ (ASGS) annual leadership conference in Westminster - she is one of the organisation’s patrons -  she had one message for headteachers at all-girls’ schools: “Let’s hear it for the girly swots!”

The phrase “girly swot” was at first intended to be derogatory - Mr Johnson used it to insult former prime minister David Cameron in a leaked Cabinet paper. Yet it seems that it is being reclaimed as a badge of pride by intelligent, high-profile women such as Lady Hale.

This morning, she argued that single-sex schools allow girls the space to develop intellectual confidence, and revel in being “swots”.

Born Brenda Hale, Lady Hale’s parents were both headteachers, and she herself went to a state girls’ school in Yorkshire. She said leaders at girls’ schools should use the changing nature of the legal profession, with its increasing diversity and gender balance, as a way of boosting their pupils’ confidence and aspirations.

“When I last talked to you, we had just doubled the number of women on the Supreme Court from one to two - a very good observation of how unreliable statistics are because doubling sounds so good, then you realise from what a small base. And then last year we went up to three - a quarter of the Supreme Court,” she said.

But she added that, across the legal profession, “things are getting better”.

“The story of what’s happened to law and legal education, women and diversity in law, is a good model for all the other things you want to get your young women into - the STEM subjects and technology, computing.”

“I’m sure those are all the messages you want to give to your young women - and you can say, ‘Well, look how things have changed over Lady Hale’s lifetime and they can continue to change over your lifetime, and for heaven’s sake, don’t let anybody put you off.’”

In a line that echoed a phrase in Margaret Atwood’s feminist novel The Handmaid’s Tale, she quipped: “I always say, ‘Don’t let the bastards grind you down.’ When young women ask me for advice, that’s number one!”

Lady Hale said all-girls’ education prevented louder boys from dominating lesson time, although she noted that during her own school days, the aspirations for pupils could be somewhat limited, with nursing and teaching seen as the preferred career options.

“My father was headmaster of a similarly sized boys’ grammar school in North Yorkshire and he was firmly of the view that single-sex education should be compulsory for girls but forbidden for boys,” Lady Hale said.

“Now, these are two inconsistent positions,” she said, to laughter from the audience, “But you can work out why. Because…I found this in Cambridge - I was a girly swot and there were quite a few young men who were, similarly, girly swots: they wanted to get on with their work and their lives.”

“But sometimes supervisions were invaded by the other sort of male student, who wasn’t particularly interested in doing much in the way of work, and who concentrated on trying to put the supervisor off with silly questions, and just generally not do a lot of work.”

“Now, you’ll all be familiar with that as a pattern. So, one of the reasons why it’s a good thing to be in a girls’ school is you don’t get so much of that,” she said, adding, “Of course, there are some girls like that as well, but you don’t get so much.”

Lady Hale said she was “honoured” to be joining the conference and wished the association and “all your wonderful girls” good luck.

“And thank you for wearing brooches!” she said, to rapturous applause.