Upvoted because I think this is a really good point, which is almost totally missed in the surrounding discussion.

For example, it's interesting to see that a lot of the experiments were directly attempting to measure C: The researcher tries to persuade the child to believe something about A, and then measures their performance. But then that research gets translated in the lay press as demonstrating something about A!

I feel that if emr's post were put as a header to Scott's, the amount of confusion in the rebuttals would be reduced considerably.

Incidentally, I've observed a similarly common difficulty understanding the distinction between derivative orders of a quantity, eg. distinguishing between something "being large" vs. "growing fast", etc. This seems less common among people trained in calculus, but even then, often people confuse these. I see it all the time in the press, and I wonder if there is a similar level-hopping neural circuit at work.

For example, there are three or four orders of differentiation that exist in common discussion of climate change, eg:

• A: Scientists recommend that atmospheric CO2 be kept below 350 ppm.
• B: Canada emits only about half a gigaton of CO2 per year, whereas China emits nearly twenty times that much.
• BB: Canada emits 15.7 tons of CO2 annually per capita, among the highest in the world, whereas China emits less than half of that amount per capita.
• C: China's emissions are among the fastest-growing in the world, up by nearly 500 million tonnes over last year. Canada decreased its emissions by 10 million tonnes over the same period.
• D: The growth in Canadian oil-industry emissions could slow if low prices force the industry to reduce expansion plans.

Et cetera...

Ostensibly what actually matters is A, which is dependent on the fourth integral of what is being discussed in D! People end up having a very hard time keeping these levels distinct, and much confusion and miscommunication ensues.

I wonder -- do you think students of calculus will be better at understanding the levels of indirection in either case?