I think I'm closer to understanding you now. For a professional, they need to know (that is, reproduce without looking up) about 70k ± 20k facts in their field

I think I'm closer to understanding you now. For a professional, they need to know (that is, reproduce without looking up) about 70k ± 20k facts in their field to be effective (https://apps.dtic.mil/dtic/tr/fulltext/u2/a219064.pdf https://apps.dtic.mil/dtic/tr/fulltext/u2/a219064.pdf). Otherwise they're just too slow. Over time though, those facts change. For some concepts, especially ones early on like arithmetic, sometimes memorization and mechanistic drilling is a prerequisite for solidifying the more general concepts at play which is what will really boost your skills (understanding what is actually happening when doing arithmetic). And sometimes you just need to do the simpler activity over and over before 'getting it' (like Feynman's counting with beans story he gave in order to justify his ability to explain fundamentals of QED without years of undergrad work first). I agree that if you just show how to use the tool and never explain what's going on (and have no way to vet understanding that any explanation was sufficient) then it can cheat learning and further understanding... The most apt comparison is that of always asking your friend what the answer is, never sparing a single neuron for trying to figure it out yourself. Calculators can be like that to some people. Maybe the key is the subtle difference between "let me ask the calculator (or hey, Alexa)" and "let me use the calculator/Alexa". The second implies a belief of being able to do without, of seeing an alternative potential path towards the goal even if it takes more time. And as one of the first bodies of math children will be exposed to with expectation to master, there's room in arithmetic for manual tedium to help solidify concepts that will be present in the future like copying things down in different places, value substitutions, learning and following (or programming) an algorithm, experiencing directly the tight feedback loop of seeing a problem or sub-problem and recalling a memorized answer instantly, parsing out numerical information from text or graphs, and so forth -- but eventually too, using a calculator or some device for almost all actual arithmetic. Even professional mathematicians will often outsource their brain for simple arithmetic, and anyone trying to multiply two 3 digit numbers in their head is likely just wasting the effort of trillions of firing synapses... Anyway there's certainly room and need for some by-hand or in-head stuff, but I still maintain there's also a lot of wasted time (especially as subjects get more and more advanced) taken up by such activities and we'd do well to cast critical eyes on both traditional and newer computer-aware curricula. And then as a pipe dream, not holding everyone to the same pace.