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Is it good idea to encourage students to look up more about particular topics that interest them? The idea is that I don't think they will understand how to read math literature.

for example, if a student in a higher level algebra course asks "can every polynomial be solved?" Is it beneficial to point them to the Abel-Ruffini theorem? They can clearly understand that the result is no, at least not written in radicals, without proof. In order for them to actually understand the depth of the result requires a whole course on abstract algebra which is way beyond their level of comprehension.

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    $\begingroup$ I think it is a good idea to encourage students. Of course there are topics that will be poor choices of study, and there are also poor approaches we could take. But if structured well, with guidance, reflection, and check-ins, I think it can be a powerful learning experience. Some of my students have done "research" on unsolved problems, not necessarily attempting to solve them, but understanding why they're still unsolved. Other students have done work on genetic algorithms and path-finding. Some computer skills really broaden the scope of what they can work on. $\endgroup$
    – Carser
    Oct 21, 2021 at 12:33
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    $\begingroup$ Your question seems to present a false dichotomy, in which a student is not told anything or has to read higher level math literature. What about the teacher recommending appropriate semi-popular books that the teacher or the school library might have (although I would imagine most any student interested in math would already have long since been familiar with the math selections in their school's library) or papers the teacher might know of (continued) $\endgroup$ Oct 21, 2021 at 14:10
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    $\begingroup$ (in the U.S., this would likely from the teacher's old copies of Mathematics Teacher)? $\endgroup$ Oct 21, 2021 at 14:11
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    $\begingroup$ The problem we're running into is that there's a rising trend among education professionals to prescribe "student research" as a silver bullet. E.g.: In a sociology class, one can interview people in a local community who have never been interviewed before, and that can be legitimate new information. The idea of unique, novel research is much harder to pull off in a STEM context. With math likely the hardest of all. $\endgroup$ Oct 22, 2021 at 1:56
  • $\begingroup$ Bit of a tangent, but if you are giving students research problems, you might have more luck with finding alternative proofs of known results, rather than completely new results. Quadratic reciprocity, for example, is easy to state (if a little harder to actually prove) and has dozens of substantially different proofs. $\endgroup$
    – anomaly
    Oct 29, 2021 at 16:40

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In general, no. First of all, definitely don't send them to original (old or cutting edge) literature. It's just too hard. But really sending them to advanced textbooks is too tricky also. Kids in Algebra 2 need to learn that stuff first, not Artin.

Second, when I say "in general", probably the majority of your kids don't have A/A+ level capability in what they are doing now. Throwing more on them is not kind, in that case.

I do think there is a way to do enrichment. But it's maybe a bit more of a tease. "If you ever take abstract algebra, they'll show you why quintic general solution doesn't exist." I say this as someone who had it explained to me thus. AND who still has only this level of understanding. (I've never taken abstract algebra, so that's my level of understanding of it.)

For the sharper, more curious kids, you can maybe direct them to contests. It won't just be the problems then, but some interaction with other kids in an after class setting. Also Numberphile or Mathologer videos are good stuff. Or the Andrew Wiles video (ripped copy has been up on Vimeo for years). There are some good books also, in particular Flatland.

Avoid any push for doing original research. It's just not feasible for high school kids in math. If you have to, just do some statistics things or descriptives (like the thread we had on pictures/sculptures of shapes). I.e. take a "big tent" view of what passes for math. Your administrators won't know the difference and will probably think you are complying. (Maybe even you are, if you think a little sideways.)

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