I’ve seen this game of Telephone played out in my cell biology class, but to a greater degree.
Square units? Far too easy. Our first lab in cell biology is about observing cells and then doing a bunch of unit conversions — but we’re working with volumes and cubic units. For instance, we look at a sample of microorganisms taken from a local lake, and they calculate the density and size of cells, and have to estimate how many cells are in the lake and the volume they take up. I hate to tell you this, but our lake either contains one gigantic cell that fills the entire thing, or is a thick soup containing 10googolplex cells…mmmm, sounds tasty and nutritious.
I have found students I have taught dislike units.
Units are hard…and they shouldn’t be.
also, uncertainty analysis and significant figures.
Something similar seems to have happened with the reported number of Europeans (mostly women IIRC) killed during the Burning Times. Historian Ronald Hutton admits the most reliable figure is about half-a-million, but somehow it got up to 9 million in some people’s retellings.
[ObRef]
https://xkcd.com/882/
This is why you should convert everything to fundamental units and use exponential notation.
If you normalise your numbers not so there is exactly one non-zero digit before the point, but so the exponents are triple, then the exponent will line up with a prefix. (We’ll mostly ignore deka-, hecto-, deci-, centi- and sesqui for now; just say they have found their niches.) So instead of 5.89e-7, write 589e-9; and this way you can see, very clearly, it’s 589 nano-somethings. (On the Casio calculators up with which I grew, there was an ENG button which did this; 5.89e-7 would become 589e-9 or 0.589e-6, depending whether you used it with or without the INVerse button.)
Of course, you then have to remember that if the unit is squared, the prefix also gets squared with it. So a square millimetre is a micro(square metre) — or you can just not bother, because a number expressed in exponential notation is still perfectly valid.
My generation probably was the last to be taught to use log tables and slide rules “because calculators are unreliable, and if the battery fails at the wrong moment you will have to know how to do without”. Examination questions would be deliberately contrived to minimise the advantage conferred by a calculator, which has led to the slight side effect that we can’t look at, say, a 5 and a 13 in the wild without expecting there to be a 12 lurking somewhere nearby; but we did get damned good at understanding the underlying concept of exponential notation.