Personally, I think that kids should learn simple mental arithmetic. It's not that I expect them to use it when they become engineers or cashiers, but it is useful for making quick approximations. It is useful to know when they have to pull out a calculator to check the price they are being charged at the store or to know when to check what they fed the computer (since they have the tools to figure out when an answer is wrong).
While I agree with being able to check the results with many modern calculators, my argument is that it is useful to know when the results have to be checked. For example: if you know that all of the points are less than 5 and you are adding 20 of them together, a result greater than 100 means you have to go back and check what was entered.
(I agree with you, btw).
wow
I'd rather call it "information". Knowledge is one level above that, knowledge is information integrated in a coherent whole. Nowadays it's very easy to find information about anything, but most people still know very little about most things. Confusing information with knowledge and knowing nothing at the same time is why misinformation spreads so easily.
People have access to limitless information but have little interest in actually looking for and using it.
(Granted, that was in physics where the graders usually use the final answer as a check and rarely deduct many marks if all of the work leading up to an incorrect numerical answer is correct.)
When I taught high school in the 00s, I had to impose rules about what sort of calculators were permitted to avoid cheating, especially as devices which could be used for communication like network-connected palm pilots were available.¹ I also wanted to have the kids avoid buying a $99 TI-89 when a much cheaper calculator was sufficient for the work they were doing.
⸻⸻⸻
1. I remember one cheating technique that my friends in high school AP Chem used where a group of us were sharing a calculator. You would type in the question number you needed an answer for and someone else would type in the letter of the answer (we generally had multiple choice tests).
I recall wondering out loud in hs geometry if one infinity could be bigger than another in typical early high school "big brain" thinking, and at the time my high school geometry teacher basically shrugged their shoulders.
Years later when I discovered cardinalities of infinite sets, which comes up pretty quickly when studying math in college, I was fairly disappointed that such a great potential teaching moment was passed over. I'm pretty sure early high school me would have been dramatically more engaged with math if a teacher has said "funny you should mention that, do you know about the difference between integers and real numbers..."
The tragedy of this is that most Americans, even the ones teaching math, believe mathematics is calculation, and even at the advanced levels will only rarely be exposed to real mathematical thinking.
He had a policy of no calculators/computer use, but he relaxed it for this assignment as he didn't want students to needlessly suffer.
All the kids that used software (Mathematica and similar ones) got it wrong. All of them. Each and every one made a typo somewhere.
Thing is, had any of these students done a very simple dimensional analysis, they would have known very quickly that their answer was incorrect. I'm sure they would have done that analysis had they attempted solving it by hand. However, people tend to forget sanity checks when working on a computer.
I wonder what things we consider "lazy" at the moment, will become the standard in 20 years