Good Essentialism: "Men and women have tiny differences in their brains that show which sex they are."
Bad Essentialism: "Men and women have major differences in their genitals, reproductive system and secondary sex characteristics"
roll eyes
So anyway, this has been long exploded. As someone else pointed out there are tendencies rather than essential differences but if you go down that route some people go "Sex is a spectrum" and it's not. Any more than picking height or muscle mass as your criteria to measure makes sex a spectrum. You wouldn't say "men tend to be taller therefore sex is a spectrum".
A more productive route sometimes is to point out that the differences in brains are (and keep emphasising that said differences are very minor compared to far larger biological differences that TRAs reject as defining sex), developmental not essential. For example, men's brains tend to be slightly bigger. Is that because of some essential difference in the brains? No, it's because men have larger skulls on average so there's more space to grow. There are some slightly different relative average sizes of some areas of the brain. Maybe that is because of some essential difference in the brains? No - it's again developmental. Male and female brains exist in a bath of hormones and as they grew the slightly different hormone levels lead to marginally different outcomes. Is this guy saying that trans people have opposite sex hormone levels? No - clearly that's not the case.
Finally, some people would argue that of course male and female brains are different because they have different DNA so you could check the chromosomes and see that they are different. Well yes - by the same criteria you'd have male and female eyeballs, male and female livers and male and female skin. It's a nonsense argument.
In my experience the best routes of challenging this nonsense are:
- There are some mildly different averages which are developmental due to different hormone levels, not an essential characteristic. And trans identifying people don't have opposite sex hormones which is observationally easy to prove.
- It has never been shown that trans people have opposite sex brains even aside from differences not being clear. This is made up.
- For such an extraordinary claim, it's on the scientist making it to demonstrate evidence of such a thing and nobody has ever been able to do so.
And returning to the issue of some biological differences mysteriously determining whether or not you're trans and other biological differences arbitrarily not counting (dick or no dick), the expected argument would be "but a brain determines how you actually feel and who you are". Nah, high testosterone throughout your life has more effect on brain development and we can make a pretty good guess from the name that testosterone is made in the... testes. But I'd start with the first couple of points.
Anyway, you asked for actual references so here is one:
The 'Pink and Blue Brain' Myth - Transgender Trend
I thought it might be useful to have someone's account of how they approach this myth.
I would also contact your school and have very strong words about them teaching unsubstantiated pseudo-science.