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@hooser's aside about being a mathematician who's benefited from "get more women into math" initiatives despite not necessarily believing in them, reminded me I wanted to ask:
What can I do to help out my mathy daughter?
Her dad and I are both everything-but-math types (like Scott); she OTOH is so far shaping up to be your stereotypical math-and-music kid. She's very young, maybe she'll stop being mathy or w/e, but if she stays mathy...I just wonder how to help her in the future. (Right now we homeschool.)
From my own experience and studies I recall reading, seems like affinity groups have been the most effective way to help minorities-in-a-field achieve to their potential. IOW, put people who have an interest in the field and are not in the majority-in-the-field in contact with one another. (As opposed to "try to change the field to be more in line with the targeted identity group's average group preferences" or some such.) I did see an online group for mathy girls 5th grade and up, maybe that'll be good for when she's older...
So, any math-and-music types here, what would you have wanted someone to tell your parents?
As a parent:
For "mathy":
The best thing for my son was Khan Academy, which basically covers everything you'd want in grade school, up to a little basic undergraduate-engineering-major-level math. When Covid hit and all the other kids' brains started atrophying, he instead was thrilled to discover he could now go as fast as he wanted (or occasionally as slow as he needed to?) He started squeezing it in to the sad little "online school" schedule that had been hectically thrown together, then he eventually got permission from a teacher to let his little sister do his "real" math homework so he could spend more time studying way ahead, and he got far ahead before testing out of a few physical classes and joining others. (or "auditing" others; they have to call his Calculus class "independent study in math" due to some age/grade restriction, but fortunately there's no restriction on the AP tests).
When he got a bit older he really started getting into math competitions, so instead of just racing ahead he's spent a lot of time getting better at the sorts of questions that a typical kid his age knows enough math to understand but still can't necessarily solve. These have also been helpful at avoiding ego inflation, putting him up against the sharpest math students in the city or the country rather than just a few classmates and a standardized curriculum.
We've done a lot of talking together about the basics of things like set theory, boolean algebra, group theory, linear algebra. There's a lot of math that's understandable to very small kids but that doesn't get covered in a standard curriculum. This is reasonable of the standard curriculum, since most non-scientist non-engineers will never need to know e.g. what a power set is, and even most scientists and engineers can get away with believing theorems without picking up the groundwork to prove them themselves ... but if you've got a kid who's interested in math, then she may be interested in math enough not to worry too hard about which of the things she's learning have what future applications. Being able to learn multiple things at a time can also be helpful if one doesn't "click"; even professional mathematicians often just dive into one subfield they really enjoy, or end up mostly on one side of a divide like the "algebraist-vs-analyst" rift.
For "daughter":
I'm actually not sure? I might have screwed this one up badly somehow! My oldest daughter has a great talent for and a great dislike for math. She's taking Calculus at 15 and doing a couple math competitions, but solely to spruce up her college applications and get ahead on engineering major requirements. Watch from 4:55 through 6:10 of NewsRadio "Houses of the Holy", but imagine a girl doing math instead of a man doing magic tricks.
My youngest doesn't really dislike math, but she doesn't have any of the interest her brother does. She's joined him in a math club, but whereas he's there for the "math" part she's there for the "club" part. Your daughter might not have the stereotypical people-vs-things gender difference, but for any child having a community to work with and (to a limited extent) compete with can be strongly motivating.
As a former mathy child:
I really wish I'd grown up with more resources like Khan Academy, but also Wikipedia and especially math YouTube. The adults around me all tried to help me hit my full potential, to great effect, but there are interesting subfields that I didn't even know existed until I stumbled across references to them somewhere else.
I wish I'd somehow challenged myself more in high school. I took a few night courses at the local university after exhausting the available high school math classes, but still had a sudden shock when I got into a selective university and could no longer just ace every math test without serious study first.
I ended up in applied math, after nearly just going full engineering, and in hindsight this was more of a good idea than a mere diversion. The state of the art in pure math is so far ahead of us lowly applied mathematicians, and I think it's good for my morale to come up with ideas that I can then immediately implement and use, rather than ideas that for all I know might be foundational to 22nd century physics but that might more likely then be nothing more than footnotes in papers nobody reads. And despite what I said above about proper math brains not caring about future applications, it's still easier for me to remember new ideas I learn if I can immediately see a few ways to apply them to something connected to reality rather than if they just feel like a neat self-contained game.
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I highly recommend Math Circles, if there is one in your area. Typically, a Math Circle is a group of kids of about similar age who meet once a week for like a semester to explore an interesting math idea. Such circles get organized and led by mathematicians (professional or amateur), and they can start quite young. For example, "Math from Three to Seven" (or great review of the book) is basically a diary of a guy running a Math Circle for his young kids and their friends.
At their best, the kind of activity the kids do in a Math Circle actually models an authentic mathematical exploration. Even when not at their best, it gets your daughter together with other kids who are interested in math, and connects you with at least one math adult who is interested in math outreach for kids and will therefore probably know of other local opportunities for STEM extracurriculars.
People mean different things when they say "math". For many not in the field it means doing drills or word problems, which at best are skill challenges posed by others for educational purpose, and do not--cannot--reflect authentic problems that require a mathematical approach. The authentic problems are messy; they are vague; you have to choose what to measure and how, and what to define and how, and sometimes all your options are but poor approximations, and sometimes you can't even begin to tackle the problem as is until you have considered many much simpler similar problems that may (you hope) give you ideas on how to approach the big messy one. That's what "math" is to a mathematician, and those are great problem-solving skills to practice no matter where your life takes you.
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In my experience, you need to get your daughter to be ok with getting things wrong. It is uncomfortable to be wrong and it seems a lot of girls shy away from it more than boys do, but if you're going to keep going in STEM-y stuff you're going to be wrong. You're going to be confused. You're going to not get it when other people do. She needs to be ok with that and if possible even embrace it.
We celebrated our daughter trying and failing and trying again. And again. Music actually helped with that because she hit the limits of her natural talent at a younger age than she did with math. So we could remind her how she wasn't able to sight read a particular piece perfectly, how it required thoughtful, diligent practice. But then she got it, and yadda yadda. We may have done this a bit too much because she perversely seeks out things she really has to work at and almost neglects areas of natural talent, so maybe keep an eye on that.
Also, we found co-ed STEM activities were likely to be at a higher level than single sex ones (more participants? Less focused on achieving consensus? More competitive?) but our daughter had to be comfortable with failure and trash talk to enjoy them, especially at the middle and high school level. Keep an eye on girl focused activities though because they can be a great introduction or place to get comfortable.
Oh, wow, I can't believe I didn't think about this when talking about math competitions.
Coaching MathCounts, I think this is probably the biggest benefit I see for the kids vs a typical math class. In a class, you're given all the material before you're given the test, and the test problems are all pretty similar, and getting 100% is a reasonable expectation. In MathCounts, well, I just got the results of our chapter-level competition, and among the hundreds of top math students in a big techie city this year there was only a single kid who (barely) broke 90%. The experience of seeing problems you have no idea how to solve, and not panicking, and going on to solve the ones you can along with perhaps figuring out creative ways to solve some of the ones you initially couldn't, is huge.
On the other hand, as a correction to the correction, I should point out that there are some very unrealistic things about math competitions too. The level of speed that's beneficial for even MathCounts sprint rounds is purely a contest thing rather than a simulation of any real-world work, and especially the "you ought to be able to answer the question before most people are done reading it" level of competition among the kids who make it to a big MathCounts countdown round is basically just a fun game show for the audience.
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