When I was in school, calculators were new enough, that they really weren't something that the average student had. They were expensive, and quite limited. Hence, we learnt how to do arithmetic with pencil and paper.
In those days, the lesson plans were written to be approachable in a reasonable time frame to students to solve problems with pencil and paper easily.
Even in my physics class (3rd year) we never once used a calculator. But I did learn some things that hang with me to this very day.
Scale – in high school mathematics, it isn't obvious, but when you take science courses, you learn about this thing called "Scientific Notation". That is typically, mantissa and an exponent. The mantissa being a number like a.bbbbb where the number of "significant digits" is related to how precisely you know a value.
Estimation – In my high school physics class, we didn't use calculators, but we did learn how to use a slide rule. Don't laugh, this was extremely useful. You learn that you can quickly do large calculations by just operating on the mantissa's and then figuring out the magnitude. Thus 6.03e23 * 1.47e-7 is the equivalent of (6.021.47)x1023 – 7 or 8.85e16 (note, if you use a calculator to calculate 6.02×1.47 you will get a lot more numbers to the right of the decimal point, but I truncated it at two, as there was two digits of significant figures). Thus you can quickly get the scale of the answer (or estimate) by fiddling with the exponents. In this case, you can also get close to the mantissa by noting that 6.02 is almost 6, and 1.47 is a little less than 1.5. 1.56 is 9, so you know that it is a little less than 9, so guess 8.8, and then the exponent of 23 – 7, and you get 8.8e16, pretty damn close, and you can do it in your head.
There are a lot of other tricks that I use almost daily, often to claims if being magic by the people I interact with, but reality is just common sense combined with numeracy.
Today, students are armed with calculators, and the art of estimation, and an innate sense of scale is dying. I firmly believe that, plus the rise of misinformation packaged as fact on the internet is leading to the rise in the anti-vaccine trends.
Next up, I will take apart the common justification of not vaccinating.