Can you explain this differently at all? I just checked with a 1 foot mirror, and if I stand 1 foot away from it, I can’t see my entire self in it. So it MUST depend on the size of the mirror, or the distance I am from it, doesn’t it?
I thought the date something happened WAS part of history? Seems weird to ask someone “When did the US Constitution take effect” and they say “I don’t know, I just know it did”
The first really correct thing you’ve contributed to the thread. And, of course, completely unrelated to the issue of learning “math facts” (to take one example).
You again go off base here. I’ll give you a news flash: almost NO student cares about learning much of ANYTHING that has importance to them that they learn in elementary, jr. high, or high school. If we let the students decide what they “wish” to learn, we’d end up with a society of uneducated boobs.
Again, quite incorrect. You’d be surprised how many adults remember all sorts of things from high school that they’ve never used. Used to quiz the parents of my Geometry students to see what they recalled; a lot of them remembered a fair amount of Geometry, albeit in a very unsorted, uncalibrated way.
This is not the fault of what they are learning, but of the teachers who are teaching it. It’s perfectly possible to end up understanding that math is NOT pointless, even when learning, oh, say, the times tables. I’ve seen it done, and it’s a wonderful thing. But one of the biggest issues with math is that most elementary school teachers don’t really understand math. And many of them didn’t like it themselves. So they hate teaching it, or fear teaching it, because it’s not “fun” to them. And they do a poor job of it, because no one shows them how to do it well, let alone explains to them the underlying math reasoning behind what they are teaching. If you started talking about inverse functions and arithmetical identities, they’d give you blank stares.
There are oodles of reasons why knowing math facts is an important skill, besides just preparing you for more complicated math later. Math facts help with number sense. That’s simply a truism. I offered an example of how this was true upthread, and you totally ignored it and went past it without comment. I’ll give another example:
I once was teaching a simple statistics lesson involving compounded probabilities. The students were doing a set of problems helping to cement the skill they were trying to learn. One problem had a university with a certain number of students (say, 40,000), of which 40% were female and 40% were smokers. The problem asked the students to determine what number of female smokers there were, based upon these percentages. As I went by one student’s desk, and reviewed her answers, I saw that she had responded with something like 20,000 for an answer (all numbers are made up, but they get to the gist of the situation). I suggested to her gently that she should rethink that result. She shot me a dirty look, grabbed the calculator, and proceeded to re-calculate the answer, using the exact same incorrect sequence of buttons as previously. She turned the calculator up to me with a look of triumph and said, “SEE!?”
I almost didn’t know where to start with her. Apart from her unfortunate belief that whatever the calculator told her must be right, there was the simple fact that, while she truly was trying to accomplish .4 x .4 x 40,000, she somehow couldn’t see that the answer couldn’t be correct, because she had no actual number sense. And that lack of skill was something that, in a junior math class, I was never going to be able to overcome easily (it didn’t help that the class in question was not my own, but rather that of my supervising teacher during the semester before my student teaching). And, I daresay, based upon my experience in the years after that, most modern students don’t have that number sense, precisely because they don’t really know their arithmetic facts any more. They learn them, but then lose them because they end up using calculators to do that for them thereafter.
Which is why I said way upthread that, in the modern world, while your opinion about math facts was (IMO) wrong, it was probably a moot point.
Math CAN and SHOULD be made interesting and fun for students, while at the same time getting them to learn the things that we as a society consider important for them to know. Yes, in the long run, much of what they learn will leak back out of their heads, as is true of ANY subject. But that doesn’t matter. Until we have the magic touchstone that says, when placed on any student, “Here is exactly what they need to know in the future”, we are obligated to present them a wide range of knowledge, so that they can pursue what they want when the time to choose comes. I, as I have stated upthread, tend to think this point of decision should come earlier than it does; thus I don’t see any particular reason to require knowledge of “Algebra I” to graduate from high school. But I adamantly oppose handicapping a student at the age of 8 in his/her pursuit of higher-order math later in life by failing to see to it that that student can multiply 8 x 7 and get 56 instantly, without having to figure it out each and every time. The same can be said for addition facts. And, frankly, the same can be suggested as true for some math facts we’ve stopped teaching (my sixth grade teacher required us to learn integer square roots up to 15 to the precision of the tenths unit).
Again, I am not being argumentative or “just asking questions” or anything else like that. I realize this is General Questions, and I am honestly seeking clarification of new ways of teaching, learning, understanding, and points of view on these things. I don’t want the great information that you guys are posting to stop because you think I am being argumentative or dismissive or things I get accused of in other forums. I say this only because I have kids, and learning of a better way to teach them stuff is extremely helpful in parenting I think. I’m not being a dick in these questions 
Well, we used computers (it was a 3-D problem, after all, and the missile moves fairly fast!). But in early days, it would have been done by hand, as needed.
Oh, how well I recall those days, using a machine to punch up programs on Hollerith cards, adding the ones that the machines created based on the data from the films, turning the whole stack in to the remote time-share center in your building, and then waiting until the result got spat back at you some hours later, so you could do more programming and more data reduction. Such joy, much fun.
I’m sensing sarcasm 
That’s cool. But this is the problem. You personally don’t need to know the ASA or SAS theorem because the computer took care of it for you.
I’m afraid my mantra to my kids of “Don’t do something if you don’t know why you are doing it” is coming back to bite me in the ass. They are smart, and would ask me the same questions I’m asking here (which is why I’m seeking the answers from other smart people )
Somebody has to program that computer…
I agree. But it seems like the tone in this thread seems to be “Explain the ASA or SAS to the students, show why it is needed and when it is used, maybe a couple of problems for them to work on, and then move on”
I don’t see how that style of teaching would enable someone to program a “missile tracking system” for lack of a better term.
That’s why I am asking.
For what it’s worth, you don’t need to memorize any multiplication tables to program a computer to multiply. You don’t need to drill yourself on manually carrying out algorithms quickly to program a computer to do them for you.
But also, separately, there’s lots of things somebody has to do that not everyone is forced to learn how to do. Someone has to program a computer to calculate sines and logarithms and all the rest of them too, and yet how many students are taught how to do that efficiently?
For what it’s worth, I fear some people may take me as suggesting that people shouldn’t understand arithmetic. People should absolutely understand arithmetic, which includes understanding how one could patiently figure out basic arithmetic calculations for themselves. If someone doesn’t realize how they could sit down and slowly carefully work out any arithmetic calculation of their choice, something has gone wrong. If someone doesn’t realize how they could sit down and slowly carefully work out the factorization of 54931, something has gone wrong.
But drilling them to actually quickly do 50 factorizations at a stretch for homework or tests or Kumon, slapping their wrists over each 8 mistranscribed into a 9 along the way or whatever, is a separate thing. Any idiot can follow mechanical rules slowly, patiently, carefully, with the rules in front of them; that’s what makes them mechanical. So let the electronic idiot do it.
Ah, my last post was just me trying to re-express what was already expressed here:
Yeah, but Geometry, Trig, and Calculus are not required to graduate high school. Nobody is forced into taking those classes.
Funny thing, i was helping my son with his homework, and I used Wolfram to calculate the roots of a polynomial to check his work
But how do you get someone to learn how to do something without practice? Practice that 50 factorizations would give you?
The thing I want them to learn isn’t the thing the drills focus on. It’s the thing which exists prior to or separate from the drills. I don’t care if they can run algorithms quickly.
I know how to calculate sines from scratch. I do, I really do. I have never once attempted to do so by hand; I have zero practice at it. That’s fine by me; I still feel competent saying I understand sines very well, and that were you to ask me to compute one to desired precision, and give me slow patient time to do so, I could pull it off just fine.
As an example that may apply to others, you probably never drilled yourself on solving mazes in school. It’s not a task that came up. But if I gave you a very large but simple maze and asked you to find a path from start to end by brute-force search, just plodding along till you hit dead-ends and then backing up and trying a different choice, that simple algorithm, you could probably do it just fine. It’s not something you ever drilled on, running that simple algorithm, but the concept behind it is one you understand regardless. You don’t have to drill at doing a bunch of mazes this way to understand how you would do them, slowly if not quickly; mechanical rules are mechanically followable, so long as you’re not tired or hungry or distracted. So it goes.
Okay, that’s fair. But in the lesson on factorization, what DO you want them to learn?
That’s cool. Did you learn that in school, or just on your own? (just wondering, because I never learned that in school, and never tried to learn on my own).
That’s cool too. But solving mazes is not required for trig or calc. Factoring is. How can students do factoring in trig or calc when they never learned (and practiced) how to do it?
Very good, that’s true: It does depend on the size of the mirror or the distance you are from it. Now, which one of those two is the one it depends on?
Whoops, I phrased the last line here in a way I don’t like. What I meant was, the mirror being halfway between your eye and your twin, how much length of the mirror lies between your eye and your twin’s body. If it were a window, how wide from top to bottom would it have to be?
It’s true that a 6 foot person doesn’t need a 6 foot mirror. But that doesn’t mean they can get away with a 6 inch mirror. Something inbetween suffices.
My impression was that in order to create such a tracking system, first and foremost you would need to understand those geometric/trigonometric theorems to come up with the concept in the first place, before getting into the question of numerical computation. You would certainly need to know explicitly how it is possible to calculate where the vertex of a triangle is given the opposite side and angles. So I do not understand your comment about the teaching style— seems like it would be excellent and indispensable preparation for such an application.
(If you meant that knowing the ASA theorem does not mean you know how to prepare a set of trigonometric tables or a computer maths library from scratch, you’d be right; that is a separate subject)
Geometry is required to graduate high school here in Louisiana. Has been since at least the mid 1970s, if not longer.