Not only research mathematicians but basically anyone who’s supposed to research previously unencountered problems.
Isn’t that rather “problems that can’t be solved using currently existing mathematics”? If it’s just a previously unencountered problem, but can be solved using the tools from an existing branch of math, then what you actually need is experience from working with those tools so that you can recognize it as a problem that can be tackled with those tools. As well as having had plenty of instruction in actually breaking down big problems into smaller pieces.
And even those research mathematicians will primarily need a good and thorough understanding of the more basic mathematics that they’re building on. The ability to tackle complex unencountered problems that you have no idea of how to solve is definitely important, but I would still prioritize giving them a maximally strong understanding of the existing mathematics first.
But I wasn’t thinking that much in the context of university education, more in the context of primary/secondary school. Math offers plenty of general-purpose contexts that may greatly enhance one’s ability to think in precise terms: to the extent that we can make the whole general population learn and enjoy those concepts, it might help raise the sanity waterline.
I agree that calculus probably isn’t very useful for that purpose, though. A thorough understanding of basic statistics and probability would seem much more important.
There an interesting paper about how doing science is basically about coping with feeling stupid.
No matter whether you do research in math or whether you do research in biology, you have to come to terms with tackling problems that aren’t easily solved.
One of the huge problems with Reddit style New Atheists is that they don’t like to feel stupid. They want their science education to be in easily digestible form.
As well as having had plenty of instruction in actually breaking down big problems into smaller pieces.
I agree, that’s an important skill and probably undertaught.
The ability to tackle complex unencountered problems that you have no idea of how to solve is definitely important, but I would still prioritize giving them a maximally strong understanding of the existing mathematics first.
Nobody understands all math. For practical purposes it’s often more important to know which mathematical tools exist and having an ability to learn to use those tools.
I don’t need to be able to solve integrals. It’s enough to know that integrals exists and that Wolfram Alpha will solve them for me.
And even those research mathematicians will primarily need a good and thorough understanding of the more basic mathematics that they’re building on.
I’m not saying that one shouldn’t spend any time on easy exercises. Spending a third of the time on problems that are really hard might be a ratio that’s okay.
A thorough understanding of basic statistics and probability would seem much more important.
Statistics are important, but it’s not clear that math statistics classes help. Students that take them often think that real world problems follow a normal distribution.
Isn’t that rather “problems that can’t be solved using currently existing mathematics”? If it’s just a previously unencountered problem, but can be solved using the tools from an existing branch of math, then what you actually need is experience from working with those tools so that you can recognize it as a problem that can be tackled with those tools. As well as having had plenty of instruction in actually breaking down big problems into smaller pieces.
And even those research mathematicians will primarily need a good and thorough understanding of the more basic mathematics that they’re building on. The ability to tackle complex unencountered problems that you have no idea of how to solve is definitely important, but I would still prioritize giving them a maximally strong understanding of the existing mathematics first.
But I wasn’t thinking that much in the context of university education, more in the context of primary/secondary school. Math offers plenty of general-purpose contexts that may greatly enhance one’s ability to think in precise terms: to the extent that we can make the whole general population learn and enjoy those concepts, it might help raise the sanity waterline.
I agree that calculus probably isn’t very useful for that purpose, though. A thorough understanding of basic statistics and probability would seem much more important.
There an interesting paper about how doing science is basically about coping with feeling stupid.
No matter whether you do research in math or whether you do research in biology, you have to come to terms with tackling problems that aren’t easily solved.
One of the huge problems with Reddit style New Atheists is that they don’t like to feel stupid. They want their science education to be in easily digestible form.
I agree, that’s an important skill and probably undertaught.
Nobody understands all math. For practical purposes it’s often more important to know which mathematical tools exist and having an ability to learn to use those tools.
I don’t need to be able to solve integrals. It’s enough to know that integrals exists and that Wolfram Alpha will solve them for me.
I’m not saying that one shouldn’t spend any time on easy exercises. Spending a third of the time on problems that are really hard might be a ratio that’s okay.
Statistics are important, but it’s not clear that math statistics classes help. Students that take them often think that real world problems follow a normal distribution.