It's natural to lump these together as if they are a unit, but I think it's important to be explicit about the fact (which of course you know) that they are all too easily separable: it is incredibly easy, given sufficiently powerful tools, to get correct answers with little to no understanding. True understanding (not just "oh, I think I get it") that doesn't help to get answers is not nearly as common, but of course it exists, too.
Yes I totally agree with your sentiment that it is vitally important to separate the correct answer from the method by which it is achieved. Honestly understanding mathematics is more important at some level than getting the right answer to a particular problem. That's the whole point, once you understand the grammar and vocabulary of mathematical thinking you can attempt to solve problems that are totally unrelated to things you know and still sometimes get useful results. The overwhelming question of the ages is how do you teach with that goal in mind and take account of all of the different learning pathologies while doing it in a universally accessible and measurable way. I think the problem is intractable as it is currently being approached.
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It's natural to lump these together as if they are a unit, but I think it's important to be explicit about the fact (which of course you know) that they are all too easily separable: it is incredibly easy, given sufficiently powerful tools, to get correct answers with little to no understanding. True understanding (not just "oh, I think I get it") that doesn't help to get answers is not nearly as common, but of course it exists, too.
Yes I totally agree with your sentiment that it is vitally important to separate the correct answer from the method by which it is achieved. Honestly understanding mathematics is more important at some level than getting the right answer to a particular problem. That's the whole point, once you understand the grammar and vocabulary of mathematical thinking you can attempt to solve problems that are totally unrelated to things you know and still sometimes get useful results. The overwhelming question of the ages is how do you teach with that goal in mind and take account of all of the different learning pathologies while doing it in a universally accessible and measurable way. I think the problem is intractable as it is currently being approached.