Probably more to come, but had to respond to this:
Lol, wow, I would have said the opposite! That the way the school system is set up, you're responsible for a finite set of assigned material, and the authority figure knows exactly what that material is. A wrong answer means you haven't learned something you've been taught, and you need to fix that, because the authorities have decided you need to know this.
I wonder if this is a classics vs. tech learning divide? Because I would say, if you don't know how to do the problem, that's not a wrong answer! (*) Often, a wrong answer means either that you have been careless or (if you're making the mistakes a lot) that you haven't got the sufficient automaticity to be able to do the problem without making careless mistakes, PLUS which you might not understand the material well enough to realize that your answer is off and that you should recheck it. (I think I've told you this? or maybe it was someone else? that my physics teacher would give partial points if your answer was orders of magnitude off and you said, "hey, this answer looks weird and I'm pretty sure it's not right.")
The thing is, if you are a nurse who is calculating drug dosages, or an engineer who is calculating how to build a bridge, I absolutely DO NOT WANT you to get a wrong answer! But okay, everyone gets wrong answers sometimes, so at least I want you to know your subject well enough to realize that prescribing 1 mg instead of 1 ug is WAY TOO MUCH!
In my job, while people's lives don't generally depend on my calculations, yeah, the customer relies on me checking my work and not producing something that's wrong. Of course I've made mistakes before, and I try to be transparent about that to the customer, but -- yeah, it's actually reasonably important to be able to check one's work and not make careless mistakes to the extent possible, or catch your mistakes when you do. (That's obviously only part of being a good engineer, but it is a part! I have worked with junior engineers before who could write superfast code... but you always had to check their work because it would be sloppy and often give totally the wrong answer (e.g., when you do a coordinate transform from one coordinate system to another there ACTUALLY IS a right answer, not that I'm bitter about this or anything!)... and this is NOT a good thing.)
Now, I'm not saying that this is something E needs to worry about! She doesn't at all -- she's the most thorough person I've ever met. (Let's put it this way: when she got 49/50 on the local math competition and said she'd answered all 50, both her teacher and I figured it was the test that was wrong, not her -- and we were right.) I do however think that in general for math education (I can see how it would be different for other subjects) at the level she's at, there's actually a reason why it's set up that way and there's a reason why, after being told for many years that it's okay to be wrong, at some point students start getting told that they need to start getting the right answer, at least on the second try.
(*) Which she also hates, but I also think it's also very important for her to be able to tackle problems that she can't immediately solve. But that's another issue. I would like to say more about it, but let me just say right now that I never got much practice in sitting with problems I didn't know how to do and struggling with them for a while and figuring them out, and I think my life would have been a lot better if I'd gotten more practice with that. And also that she does get a lot of practice along the lines of the steps you outline, for various reasons. Maybe more later on this if I have more time.
Re: Another book rec
Lol, wow, I would have said the opposite! That the way the school system is set up, you're responsible for a finite set of assigned material, and the authority figure knows exactly what that material is. A wrong answer means you haven't learned something you've been taught, and you need to fix that, because the authorities have decided you need to know this.
I wonder if this is a classics vs. tech learning divide? Because I would say, if you don't know how to do the problem, that's not a wrong answer! (*) Often, a wrong answer means either that you have been careless or (if you're making the mistakes a lot) that you haven't got the sufficient automaticity to be able to do the problem without making careless mistakes, PLUS which you might not understand the material well enough to realize that your answer is off and that you should recheck it. (I think I've told you this? or maybe it was someone else? that my physics teacher would give partial points if your answer was orders of magnitude off and you said, "hey, this answer looks weird and I'm pretty sure it's not right.")
The thing is, if you are a nurse who is calculating drug dosages, or an engineer who is calculating how to build a bridge, I absolutely DO NOT WANT you to get a wrong answer! But okay, everyone gets wrong answers sometimes, so at least I want you to know your subject well enough to realize that prescribing 1 mg instead of 1 ug is WAY TOO MUCH!
In my job, while people's lives don't generally depend on my calculations, yeah, the customer relies on me checking my work and not producing something that's wrong. Of course I've made mistakes before, and I try to be transparent about that to the customer, but -- yeah, it's actually reasonably important to be able to check one's work and not make careless mistakes to the extent possible, or catch your mistakes when you do. (That's obviously only part of being a good engineer, but it is a part! I have worked with junior engineers before who could write superfast code... but you always had to check their work because it would be sloppy and often give totally the wrong answer (e.g., when you do a coordinate transform from one coordinate system to another there ACTUALLY IS a right answer, not that I'm bitter about this or anything!)... and this is NOT a good thing.)
Now, I'm not saying that this is something E needs to worry about! She doesn't at all -- she's the most thorough person I've ever met. (Let's put it this way: when she got 49/50 on the local math competition and said she'd answered all 50, both her teacher and I figured it was the test that was wrong, not her -- and we were right.) I do however think that in general for math education (I can see how it would be different for other subjects) at the level she's at, there's actually a reason why it's set up that way and there's a reason why, after being told for many years that it's okay to be wrong, at some point students start getting told that they need to start getting the right answer, at least on the second try.
(*) Which she also hates, but I also think it's also very important for her to be able to tackle problems that she can't immediately solve. But that's another issue. I would like to say more about it, but let me just say right now that I never got much practice in sitting with problems I didn't know how to do and struggling with them for a while and figuring them out, and I think my life would have been a lot better if I'd gotten more practice with that. And also that she does get a lot of practice along the lines of the steps you outline, for various reasons. Maybe more later on this if I have more time.