We should aim to train our children not to be good calculators, but to be good thinkers.
Not so simple. Being a good calculator is sometimes a good prerequisite for other kinds of thinking. Or, at the very least it makes some tasks easier. I have a lot of students that come through my laboratory. As part of their training it is necessary to do some algebra and arithmetic fairly regularly. I typically bar calculators, because I see them as a mental crutch. Being good at calculations is a good proxy for being able to integrate complex thoughts all at once and do something useful with the information. If one can do a multi-digit calculation without the aid of a piece of paper or a calculator, I think it shows that one is able to keep several things organized in one's head simultaneously (as place value calculations are accomplished most easily by doing several simple calculations which are then combined linearly). Maybe I'm nuts and/or old fashioned, but I find arithmetic to have a lot of value (so long as we consider it a means and not an ends).
I completely agree and I would further posit that the phrase NotPhil quoted contains the sort of buzzwords -- what the hell is a good thinker -- that are ruining American education. Can't do mental math? Oh well, you're still a good thinker. Who cares if everyone you later compete for jobs with has more mental acuity than you.
I find it interesting that one would use this as a gauge because there are so many "tricks" one can use to do simple calculations without having to put the mathematics in the "brain register". This book for example goes through many mental tricks to quickly do calculations without the aid of calculators. Because you have personal experience with this, I assume that you have at least anecdotal evidence that you are correct, so can you give me an example of a calculation that you might consider a good "test"?
I know I'm not who you replied to, but -- I had the dubious honor of teaching some 10-12 year olds a little math last week. (They didn't understand adding/subtracting negative numbers, couldn't do any long division, functions were an unknown quantity; but that's all beside the point.) I was trying to explain how, if you have say 12 x 30 and you can't do that in your head immediately, you can mentally break it down into 10 x 30 and 2 x 30, add the components, and voila. And then you can add harder and harder numbers depending on your capacity. I'm not sure if that's a good "test"; it may even be too simple to have been in your book of tricks even though it is sort of a trick. But they couldn't do it. I was a smart kid so I tend to be a bit tough (not to their faces, of course) on children that age as far as what basics they know* -- am I unreasonable for thinking they should know how to do that, and for putting a good bit of blame on their teachers? Maybe we should push less "everyone's a winner let's do puzzles" and more "everyone can do basic mental math, so you need to learn it." *the week before, I found out that many of them couldn't distinguish between states and state capitals i.e. "what's the capital of Missouri"; "Kansas?"
Actually flagamuffin, this is exactly the thing they teach in the book! By the way, just to be clear I think that doing mental math is absolutely one of the best skills one can acquire. I was just wondering if b_b gave them a sort of aptitude test of if it was a general rule. If a student walked in and he gave them what's 111 x 23 or if it was more on a general "Oh I found this by calculating 111 x 23 in my head". Not sure if I am explaining myself clearly here. I do not think there is anything wrong with pushing kids. People might conflate "pushing" with "punishing". This is not the case. In teaching I have noticed that often times there is a heavy mental barrier for most kids. By pushing them you get them past that barrier and into where they need to be. Honestly, I just love thinking about the different ways to teach this stuff. Maybe I'll go through some of them with a self post one day, it's a bit off topic for this post.
I am not officially a teacher, I did not go to school for it. My wife is the teacher in the classroom and I help her out during the day. Because her class is split (1st and 2nd grade). We will usually cut the class into groups and I will re-enforce something that she was teaching or review concepts with the kids. However if I were to wax pedagogical (I am not sure if that's the right adjective) just for a bit, I am a teacher, the students are teachers, everyone is a teacher! My primary philosophy is that everyone knows something that is worth learning.
My primary philosophy is that everyone knows something that is worth learning.
I agree with your philosophy. I was just saying the other night at a party, after asking a lot of questions of someone, that I'd almost always rather be the one learning in the conversation. I ask a lot of questions and it's made all the difference. Of course, I'm happy to "teach" when the opportunity arises but for the most part people are so interesting, it's a shame to not learn more about/from them.
Asking questions and teaching interactively is a actually a great way to learn something interactively too. If you are with someone else, and you teach them something, often times you'll find that they will give you a perspective on the subject you hadn't thought of before. So you've gained in at least 2 ways by asking questions, first you have helped yourself by learning more and second you have helped the "teacher" gain a new perspective on the problem.
Ah I see I misunderstood your post then on my first look. I can certainly see how one's ability to do mental arithmetic would help them with issues of stoichiometry. A "big picture" person who has practiced their math will mentally "see" ratios in their head while someone who'd rely on a calculator might just see "Oh I need to find the numbers to plugin". So if I were the second type of student, I would do more practice with basic arithmetic.